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Metamath Proof Explorer |
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| 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... |
| pm2.18i 129 | Inference associated with ... |
| notnotr 130 | Double negation eliminatio... |
| notnotri 131 | Inference associated with ... |
| notnotriALT 132 | Alternate proof of ~ notno... |
| notnotrd 133 | Deduction associated with ... |
| con2d 134 | A contraposition deduction... |
| con2 135 | Contraposition. Theorem *... |
| mt2d 136 | Modus tollens deduction. ... |
| mt2i 137 | Modus tollens inference. ... |
| nsyl3 138 | A negated syllogism infere... |
| con2i 139 | A contraposition inference... |
| nsyl 140 | A negated syllogism infere... |
| nsyl2 141 | A negated syllogism infere... |
| notnot 142 | Double negation introducti... |
| notnoti 143 | Inference associated with ... |
| notnotd 144 | Deduction associated with ... |
| con1d 145 | A contraposition deduction... |
| con1 146 | Contraposition. Theorem *... |
| con1i 147 | A contraposition inference... |
| mt3d 148 | Modus tollens deduction. ... |
| mt3i 149 | Modus tollens inference. ... |
| pm2.24i 150 | Inference associated with ... |
| pm2.24d 151 | Deduction form of ~ pm2.24... |
| con3d 152 | A contraposition deduction... |
| con3 153 | Contraposition. Theorem *... |
| con3i 154 | A contraposition inference... |
| con3rr3 155 | Rotate through consequent ... |
| nsyld 156 | A negated syllogism deduct... |
| nsyli 157 | A negated syllogism infere... |
| nsyl4 158 | A negated syllogism infere... |
| nsyl5 159 | A negated syllogism infere... |
| pm3.2im 160 | Theorem *3.2 of [Whitehead... |
| jc 161 | Deduction joining the cons... |
| jcn 162 | Theorem joining the conseq... |
| jcnd 163 | Deduction joining the cons... |
| impi 164 | An importation inference. ... |
| expi 165 | An exportation inference. ... |
| simprim 166 | Simplification. Similar t... |
| simplim 167 | Simplification. Similar t... |
| pm2.5g 168 | General instance of Theore... |
| pm2.5 169 | Theorem *2.5 of [Whitehead... |
| conax1 170 | Contrapositive of ~ ax-1 .... |
| conax1k 171 | Weakening of ~ conax1 . G... |
| pm2.51 172 | Theorem *2.51 of [Whitehea... |
| pm2.52 173 | Theorem *2.52 of [Whitehea... |
| pm2.521g 174 | A general instance of Theo... |
| pm2.521g2 175 | A general instance of Theo... |
| pm2.521 176 | Theorem *2.521 of [Whitehe... |
| expt 177 | Exportation theorem ~ pm3.... |
| impt 178 | Importation theorem ~ pm3.... |
| pm2.61d 179 | Deduction eliminating an a... |
| pm2.61d1 180 | Inference eliminating an a... |
| pm2.61d2 181 | Inference eliminating an a... |
| pm2.61i 182 | Inference eliminating an a... |
| pm2.61ii 183 | Inference eliminating two ... |
| pm2.61nii 184 | Inference eliminating two ... |
| pm2.61iii 185 | Inference eliminating thre... |
| ja 186 | Inference joining the ante... |
| jad 187 | Deduction form of ~ ja . ... |
| pm2.01 188 | Weak Clavius law. If a fo... |
| pm2.01d 189 | Deduction based on reducti... |
| pm2.6 190 | Theorem *2.6 of [Whitehead... |
| pm2.61 191 | Theorem *2.61 of [Whitehea... |
| pm2.65 192 | Theorem *2.65 of [Whitehea... |
| pm2.65i 193 | Inference for proof by con... |
| pm2.21dd 194 | A contradiction implies an... |
| pm2.65d 195 | Deduction for proof by con... |
| mto 196 | The rule of modus tollens.... |
| mtod 197 | Modus tollens deduction. ... |
| mtoi 198 | Modus tollens inference. ... |
| mt2 199 | A rule similar to modus to... |
| mt3 200 | A rule similar to modus to... |
| peirce 201 | Peirce's axiom. A non-int... |
| looinv 202 | The Inversion Axiom of the... |
| bijust0 203 | A self-implication (see ~ ... |
| bijust 204 | Theorem used to justify th... |
| impbi 207 | Property of the biconditio... |
| impbii 208 | Infer an equivalence from ... |
| impbidd 209 | Deduce an equivalence from... |
| impbid21d 210 | Deduce an equivalence from... |
| impbid 211 | Deduce an equivalence from... |
| dfbi1 212 | Relate the biconditional c... |
| dfbi1ALT 213 | Alternate proof of ~ dfbi1... |
| biimp 214 | Property of the biconditio... |
| biimpi 215 | Infer an implication from ... |
| sylbi 216 | A mixed syllogism inferenc... |
| sylib 217 | A mixed syllogism inferenc... |
| sylbb 218 | A mixed syllogism inferenc... |
| biimpr 219 | Property of the biconditio... |
| bicom1 220 | Commutative law for the bi... |
| bicom 221 | Commutative law for the bi... |
| bicomd 222 | Commute two sides of a bic... |
| bicomi 223 | Inference from commutative... |
| impbid1 224 | Infer an equivalence from ... |
| impbid2 225 | Infer an equivalence from ... |
| impcon4bid 226 | A variation on ~ impbid wi... |
| biimpri 227 | Infer a converse implicati... |
| biimpd 228 | Deduce an implication from... |
| mpbi 229 | An inference from a bicond... |
| mpbir 230 | An inference from a bicond... |
| mpbid 231 | A deduction from a bicondi... |
| mpbii 232 | An inference from a nested... |
| sylibr 233 | A mixed syllogism inferenc... |
| sylbir 234 | A mixed syllogism inferenc... |
| sylbbr 235 | A mixed syllogism inferenc... |
| sylbb1 236 | A mixed syllogism inferenc... |
| sylbb2 237 | A mixed syllogism inferenc... |
| sylibd 238 | A syllogism deduction. (C... |
| sylbid 239 | A syllogism deduction. (C... |
| mpbidi 240 | A deduction from a bicondi... |
| syl5bi 241 | A mixed syllogism inferenc... |
| syl5bir 242 | A mixed syllogism inferenc... |
| syl5ib 243 | A mixed syllogism inferenc... |
| syl5ibcom 244 | A mixed syllogism inferenc... |
| syl5ibr 245 | A mixed syllogism inferenc... |
| syl5ibrcom 246 | A mixed syllogism inferenc... |
| biimprd 247 | Deduce a converse implicat... |
| biimpcd 248 | Deduce a commuted implicat... |
| biimprcd 249 | Deduce a converse commuted... |
| syl6ib 250 | A mixed syllogism inferenc... |
| syl6ibr 251 | A mixed syllogism inferenc... |
| syl6bi 252 | A mixed syllogism inferenc... |
| syl6bir 253 | A mixed syllogism inferenc... |
| syl7bi 254 | A mixed syllogism inferenc... |
| syl8ib 255 | A syllogism rule of infere... |
| mpbird 256 | A deduction from a bicondi... |
| mpbiri 257 | An inference from a nested... |
| sylibrd 258 | A syllogism deduction. (C... |
| sylbird 259 | A syllogism deduction. (C... |
| biid 260 | Principle of identity for ... |
| biidd 261 | Principle of identity with... |
| pm5.1im 262 | Two propositions are equiv... |
| 2th 263 | Two truths are equivalent.... |
| 2thd 264 | Two truths are equivalent.... |
| monothetic 265 | Two self-implications (see... |
| ibi 266 | Inference that converts a ... |
| ibir 267 | Inference that converts a ... |
| ibd 268 | Deduction that converts a ... |
| pm5.74 269 | Distribution of implicatio... |
| pm5.74i 270 | Distribution of implicatio... |
| pm5.74ri 271 | Distribution of implicatio... |
| pm5.74d 272 | Distribution of implicatio... |
| pm5.74rd 273 | Distribution of implicatio... |
| bitri 274 | An inference from transiti... |
| bitr2i 275 | An inference from transiti... |
| bitr3i 276 | An inference from transiti... |
| bitr4i 277 | An inference from transiti... |
| bitrd 278 | Deduction form of ~ bitri ... |
| bitr2d 279 | Deduction form of ~ bitr2i... |
| bitr3d 280 | Deduction form of ~ bitr3i... |
| bitr4d 281 | Deduction form of ~ bitr4i... |
| syl5bb 282 | A syllogism inference from... |
| bitr2id 283 | A syllogism inference from... |
| bitr3id 284 | A syllogism inference from... |
| bitr3di 285 | A syllogism inference from... |
| bitrdi 286 | A syllogism inference from... |
| bitr2di 287 | A syllogism inference from... |
| bitr4di 288 | A syllogism inference from... |
| bitr4id 289 | A syllogism inference from... |
| 3imtr3i 290 | A mixed syllogism inferenc... |
| 3imtr4i 291 | A mixed syllogism inferenc... |
| 3imtr3d 292 | More general version of ~ ... |
| 3imtr4d 293 | More general version of ~ ... |
| 3imtr3g 294 | More general version of ~ ... |
| 3imtr4g 295 | More general version of ~ ... |
| 3bitri 296 | A chained inference from t... |
| 3bitrri 297 | A chained inference from t... |
| 3bitr2i 298 | A chained inference from t... |
| 3bitr2ri 299 | A chained inference from t... |
| 3bitr3i 300 | A chained inference from t... |
| 3bitr3ri 301 | A chained inference from t... |
| 3bitr4i 302 | A chained inference from t... |
| 3bitr4ri 303 | A chained inference from t... |
| 3bitrd 304 | Deduction from transitivit... |
| 3bitrrd 305 | Deduction from transitivit... |
| 3bitr2d 306 | Deduction from transitivit... |
| 3bitr2rd 307 | Deduction from transitivit... |
| 3bitr3d 308 | Deduction from transitivit... |
| 3bitr3rd 309 | Deduction from transitivit... |
| 3bitr4d 310 | Deduction from transitivit... |
| 3bitr4rd 311 | Deduction from transitivit... |
| 3bitr3g 312 | More general version of ~ ... |
| 3bitr4g 313 | More general version of ~ ... |
| notnotb 314 | Double negation. Theorem ... |
| con34b 315 | A biconditional form of co... |
| con4bid 316 | A contraposition deduction... |
| notbid 317 | Deduction negating both si... |
| notbi 318 | Contraposition. Theorem *... |
| notbii 319 | Negate both sides of a log... |
| con4bii 320 | A contraposition inference... |
| mtbi 321 | An inference from a bicond... |
| mtbir 322 | An inference from a bicond... |
| mtbid 323 | A deduction from a bicondi... |
| mtbird 324 | A deduction from a bicondi... |
| mtbii 325 | An inference from a bicond... |
| mtbiri 326 | An inference from a bicond... |
| sylnib 327 | A mixed syllogism inferenc... |
| sylnibr 328 | A mixed syllogism inferenc... |
| sylnbi 329 | A mixed syllogism inferenc... |
| sylnbir 330 | A mixed syllogism inferenc... |
| xchnxbi 331 | Replacement of a subexpres... |
| xchnxbir 332 | Replacement of a subexpres... |
| xchbinx 333 | Replacement of a subexpres... |
| xchbinxr 334 | Replacement of a subexpres... |
| imbi2i 335 | Introduce an antecedent to... |
| jcndOLD 336 | Obsolete version of ~ jcnd... |
| bibi2i 337 | Inference adding a bicondi... |
| bibi1i 338 | Inference adding a bicondi... |
| bibi12i 339 | The equivalence of two equ... |
| imbi2d 340 | Deduction adding an antece... |
| imbi1d 341 | Deduction adding a consequ... |
| bibi2d 342 | Deduction adding a bicondi... |
| bibi1d 343 | Deduction adding a bicondi... |
| imbi12d 344 | Deduction joining two equi... |
| bibi12d 345 | Deduction joining two equi... |
| imbi12 346 | Closed form of ~ imbi12i .... |
| imbi1 347 | Theorem *4.84 of [Whitehea... |
| imbi2 348 | Theorem *4.85 of [Whitehea... |
| imbi1i 349 | Introduce a consequent to ... |
| imbi12i 350 | Join two logical equivalen... |
| bibi1 351 | Theorem *4.86 of [Whitehea... |
| bitr3 352 | Closed nested implication ... |
| con2bi 353 | Contraposition. Theorem *... |
| con2bid 354 | A contraposition deduction... |
| con1bid 355 | A contraposition deduction... |
| con1bii 356 | A contraposition inference... |
| con2bii 357 | A contraposition inference... |
| con1b 358 | Contraposition. Bidirecti... |
| con2b 359 | Contraposition. Bidirecti... |
| biimt 360 | A wff is equivalent to its... |
| pm5.5 361 | Theorem *5.5 of [Whitehead... |
| a1bi 362 | Inference introducing a th... |
| mt2bi 363 | A false consequent falsifi... |
| mtt 364 | Modus-tollens-like theorem... |
| imnot 365 | If a proposition is false,... |
| pm5.501 366 | Theorem *5.501 of [Whitehe... |
| ibib 367 | Implication in terms of im... |
| ibibr 368 | Implication in terms of im... |
| tbt 369 | A wff is equivalent to its... |
| nbn2 370 | The negation of a wff is e... |
| bibif 371 | Transfer negation via an e... |
| nbn 372 | The negation of a wff is e... |
| nbn3 373 | Transfer falsehood via equ... |
| pm5.21im 374 | Two propositions are equiv... |
| 2false 375 | Two falsehoods are equival... |
| 2falsed 376 | Two falsehoods are equival... |
| 2falsedOLD 377 | Obsolete version of ~ 2fal... |
| pm5.21ni 378 | Two propositions implying ... |
| pm5.21nii 379 | Eliminate an antecedent im... |
| pm5.21ndd 380 | Eliminate an antecedent im... |
| bija 381 | Combine antecedents into a... |
| pm5.18 382 | Theorem *5.18 of [Whitehea... |
| xor3 383 | Two ways to express "exclu... |
| nbbn 384 | Move negation outside of b... |
| biass 385 | Associative law for the bi... |
| biluk 386 | Lukasiewicz's shortest axi... |
| pm5.19 387 | Theorem *5.19 of [Whitehea... |
| bi2.04 388 | Logical equivalence of com... |
| pm5.4 389 | Antecedent absorption impl... |
| imdi 390 | Distributive law for impli... |
| pm5.41 391 | Theorem *5.41 of [Whitehea... |
| pm4.8 392 | Theorem *4.8 of [Whitehead... |
| pm4.81 393 | A formula is equivalent to... |
| imim21b 394 | Simplify an implication be... |
| pm4.63 397 | Theorem *4.63 of [Whitehea... |
| pm4.67 398 | Theorem *4.67 of [Whitehea... |
| imnan 399 | Express an implication in ... |
| imnani 400 | Infer an implication from ... |
| iman 401 | Implication in terms of co... |
| pm3.24 402 | Law of noncontradiction. ... |
| annim 403 | Express a conjunction in t... |
| pm4.61 404 | Theorem *4.61 of [Whitehea... |
| pm4.65 405 | Theorem *4.65 of [Whitehea... |
| imp 406 | Importation inference. (C... |
| impcom 407 | Importation inference with... |
| con3dimp 408 | Variant of ~ con3d with im... |
| mpnanrd 409 | Eliminate the right side o... |
| impd 410 | Importation deduction. (C... |
| impcomd 411 | Importation deduction with... |
| ex 412 | Exportation inference. (T... |
| expcom 413 | Exportation inference with... |
| expdcom 414 | Commuted form of ~ expd . ... |
| expd 415 | Exportation deduction. (C... |
| expcomd 416 | Deduction form of ~ expcom... |
| imp31 417 | An importation inference. ... |
| imp32 418 | An importation inference. ... |
| exp31 419 | An exportation inference. ... |
| exp32 420 | An exportation inference. ... |
| imp4b 421 | An importation inference. ... |
| imp4a 422 | An importation inference. ... |
| imp4c 423 | An importation inference. ... |
| imp4d 424 | An importation inference. ... |
| imp41 425 | An importation inference. ... |
| imp42 426 | An importation inference. ... |
| imp43 427 | An importation inference. ... |
| imp44 428 | An importation inference. ... |
| imp45 429 | An importation inference. ... |
| exp4b 430 | An exportation inference. ... |
| exp4a 431 | An exportation inference. ... |
| exp4c 432 | An exportation inference. ... |
| exp4d 433 | An exportation inference. ... |
| exp41 434 | An exportation inference. ... |
| exp42 435 | An exportation inference. ... |
| exp43 436 | An exportation inference. ... |
| exp44 437 | An exportation inference. ... |
| exp45 438 | An exportation inference. ... |
| imp5d 439 | An importation inference. ... |
| imp5a 440 | An importation inference. ... |
| imp5g 441 | An importation inference. ... |
| imp55 442 | An importation inference. ... |
| imp511 443 | An importation inference. ... |
| exp5c 444 | An exportation inference. ... |
| exp5j 445 | An exportation inference. ... |
| exp5l 446 | An exportation inference. ... |
| exp53 447 | An exportation inference. ... |
| pm3.3 448 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 449 | Theorem *3.31 (Imp) of [Wh... |
| impexp 450 | Import-export theorem. Pa... |
| impancom 451 | Mixed importation/commutat... |
| expdimp 452 | A deduction version of exp... |
| expimpd 453 | Exportation followed by a ... |
| impr 454 | Import a wff into a right ... |
| impl 455 | Export a wff from a left c... |
| expr 456 | Export a wff from a right ... |
| expl 457 | Export a wff from a left c... |
| ancoms 458 | Inference commuting conjun... |
| pm3.22 459 | Theorem *3.22 of [Whitehea... |
| ancom 460 | Commutative law for conjun... |
| ancomd 461 | Commutation of conjuncts i... |
| biancomi 462 | Commuting conjunction in a... |
| biancomd 463 | Commuting conjunction in a... |
| ancomst 464 | Closed form of ~ ancoms . ... |
| ancomsd 465 | Deduction commuting conjun... |
| anasss 466 | Associative law for conjun... |
| anassrs 467 | Associative law for conjun... |
| anass 468 | Associative law for conjun... |
| pm3.2 469 | Join antecedents with conj... |
| pm3.2i 470 | Infer conjunction of premi... |
| pm3.21 471 | Join antecedents with conj... |
| pm3.43i 472 | Nested conjunction of ante... |
| pm3.43 473 | Theorem *3.43 (Comp) of [W... |
| dfbi2 474 | A theorem similar to the s... |
| dfbi 475 | Definition ~ df-bi rewritt... |
| biimpa 476 | Importation inference from... |
| biimpar 477 | Importation inference from... |
| biimpac 478 | Importation inference from... |
| biimparc 479 | Importation inference from... |
| adantr 480 | Inference adding a conjunc... |
| adantl 481 | Inference adding a conjunc... |
| simpl 482 | Elimination of a conjunct.... |
| simpli 483 | Inference eliminating a co... |
| simpr 484 | Elimination of a conjunct.... |
| simpri 485 | Inference eliminating a co... |
| intnan 486 | Introduction of conjunct i... |
| intnanr 487 | Introduction of conjunct i... |
| intnand 488 | Introduction of conjunct i... |
| intnanrd 489 | Introduction of conjunct i... |
| adantld 490 | Deduction adding a conjunc... |
| adantrd 491 | Deduction adding a conjunc... |
| pm3.41 492 | Theorem *3.41 of [Whitehea... |
| pm3.42 493 | Theorem *3.42 of [Whitehea... |
| simpld 494 | Deduction eliminating a co... |
| simprd 495 | Deduction eliminating a co... |
| simprbi 496 | Deduction eliminating a co... |
| simplbi 497 | Deduction eliminating a co... |
| simprbda 498 | Deduction eliminating a co... |
| simplbda 499 | Deduction eliminating a co... |
| simplbi2 500 | Deduction eliminating a co... |
| simplbi2comt 501 | Closed form of ~ simplbi2c... |
| simplbi2com 502 | A deduction eliminating a ... |
| simpl2im 503 | Implication from an elimin... |
| simplbiim 504 | Implication from an elimin... |
| impel 505 | An inference for implicati... |
| mpan9 506 | Modus ponens conjoining di... |
| sylan9 507 | Nested syllogism inference... |
| sylan9r 508 | Nested syllogism inference... |
| sylan9bb 509 | Nested syllogism inference... |
| sylan9bbr 510 | Nested syllogism inference... |
| jca 511 | Deduce conjunction of the ... |
| jcad 512 | Deduction conjoining the c... |
| jca2 513 | Inference conjoining the c... |
| jca31 514 | Join three consequents. (... |
| jca32 515 | Join three consequents. (... |
| jcai 516 | Deduction replacing implic... |
| jcab 517 | Distributive law for impli... |
| pm4.76 518 | Theorem *4.76 of [Whitehea... |
| jctil 519 | Inference conjoining a the... |
| jctir 520 | Inference conjoining a the... |
| jccir 521 | Inference conjoining a con... |
| jccil 522 | Inference conjoining a con... |
| jctl 523 | Inference conjoining a the... |
| jctr 524 | Inference conjoining a the... |
| jctild 525 | Deduction conjoining a the... |
| jctird 526 | Deduction conjoining a the... |
| iba 527 | Introduction of antecedent... |
| ibar 528 | Introduction of antecedent... |
| biantru 529 | A wff is equivalent to its... |
| biantrur 530 | A wff is equivalent to its... |
| biantrud 531 | A wff is equivalent to its... |
| biantrurd 532 | A wff is equivalent to its... |
| bianfi 533 | A wff conjoined with false... |
| bianfd 534 | A wff conjoined with false... |
| baib 535 | Move conjunction outside o... |
| baibr 536 | Move conjunction outside o... |
| rbaibr 537 | Move conjunction outside o... |
| rbaib 538 | Move conjunction outside o... |
| baibd 539 | Move conjunction outside o... |
| rbaibd 540 | Move conjunction outside o... |
| bianabs 541 | Absorb a hypothesis into t... |
| pm5.44 542 | Theorem *5.44 of [Whitehea... |
| pm5.42 543 | Theorem *5.42 of [Whitehea... |
| ancl 544 | Conjoin antecedent to left... |
| anclb 545 | Conjoin antecedent to left... |
| ancr 546 | Conjoin antecedent to righ... |
| ancrb 547 | Conjoin antecedent to righ... |
| ancli 548 | Deduction conjoining antec... |
| ancri 549 | Deduction conjoining antec... |
| ancld 550 | Deduction conjoining antec... |
| ancrd 551 | Deduction conjoining antec... |
| impac 552 | Importation with conjuncti... |
| anc2l 553 | Conjoin antecedent to left... |
| anc2r 554 | Conjoin antecedent to righ... |
| anc2li 555 | Deduction conjoining antec... |
| anc2ri 556 | Deduction conjoining antec... |
| pm4.71 557 | Implication in terms of bi... |
| pm4.71r 558 | Implication in terms of bi... |
| pm4.71i 559 | Inference converting an im... |
| pm4.71ri 560 | Inference converting an im... |
| pm4.71d 561 | Deduction converting an im... |
| pm4.71rd 562 | Deduction converting an im... |
| pm4.24 563 | Theorem *4.24 of [Whitehea... |
| anidm 564 | Idempotent law for conjunc... |
| anidmdbi 565 | Conjunction idempotence wi... |
| anidms 566 | Inference from idempotent ... |
| imdistan 567 | Distribution of implicatio... |
| imdistani 568 | Distribution of implicatio... |
| imdistanri 569 | Distribution of implicatio... |
| imdistand 570 | Distribution of implicatio... |
| imdistanda 571 | Distribution of implicatio... |
| pm5.3 572 | Theorem *5.3 of [Whitehead... |
| pm5.32 573 | Distribution of implicatio... |
| pm5.32i 574 | Distribution of implicatio... |
| pm5.32ri 575 | Distribution of implicatio... |
| pm5.32d 576 | Distribution of implicatio... |
| pm5.32rd 577 | Distribution of implicatio... |
| pm5.32da 578 | Distribution of implicatio... |
| sylan 579 | A syllogism inference. (C... |
| sylanb 580 | A syllogism inference. (C... |
| sylanbr 581 | A syllogism inference. (C... |
| sylanbrc 582 | Syllogism inference. (Con... |
| syl2anc 583 | Syllogism inference combin... |
| syl2anc2 584 | Double syllogism inference... |
| sylancl 585 | Syllogism inference combin... |
| sylancr 586 | Syllogism inference combin... |
| sylancom 587 | Syllogism inference with c... |
| sylanblc 588 | Syllogism inference combin... |
| sylanblrc 589 | Syllogism inference combin... |
| syldan 590 | A syllogism deduction with... |
| sylbida 591 | A syllogism deduction. (C... |
| sylan2 592 | A syllogism inference. (C... |
| sylan2b 593 | A syllogism inference. (C... |
| sylan2br 594 | A syllogism inference. (C... |
| syl2an 595 | A double syllogism inferen... |
| syl2anr 596 | A double syllogism inferen... |
| syl2anb 597 | A double syllogism inferen... |
| syl2anbr 598 | A double syllogism inferen... |
| sylancb 599 | A syllogism inference comb... |
| sylancbr 600 | A syllogism inference comb... |
| syldanl 601 | A syllogism deduction with... |
| syland 602 | A syllogism deduction. (C... |
| sylani 603 | A syllogism inference. (C... |
| sylan2d 604 | A syllogism deduction. (C... |
| sylan2i 605 | A syllogism inference. (C... |
| syl2ani 606 | A syllogism inference. (C... |
| syl2and 607 | A syllogism deduction. (C... |
| anim12d 608 | Conjoin antecedents and co... |
| anim12d1 609 | Variant of ~ anim12d where... |
| anim1d 610 | Add a conjunct to right of... |
| anim2d 611 | Add a conjunct to left of ... |
| anim12i 612 | Conjoin antecedents and co... |
| anim12ci 613 | Variant of ~ anim12i with ... |
| anim1i 614 | Introduce conjunct to both... |
| anim1ci 615 | Introduce conjunct to both... |
| anim2i 616 | Introduce conjunct to both... |
| anim12ii 617 | Conjoin antecedents and co... |
| anim12dan 618 | Conjoin antecedents and co... |
| im2anan9 619 | Deduction joining nested i... |
| im2anan9r 620 | Deduction joining nested i... |
| pm3.45 621 | Theorem *3.45 (Fact) of [W... |
| anbi2i 622 | Introduce a left conjunct ... |
| anbi1i 623 | Introduce a right conjunct... |
| anbi2ci 624 | Variant of ~ anbi2i with c... |
| anbi1ci 625 | Variant of ~ anbi1i with c... |
| anbi12i 626 | Conjoin both sides of two ... |
| anbi12ci 627 | Variant of ~ anbi12i with ... |
| anbi2d 628 | Deduction adding a left co... |
| anbi1d 629 | Deduction adding a right c... |
| anbi12d 630 | Deduction joining two equi... |
| anbi1 631 | Introduce a right conjunct... |
| anbi2 632 | Introduce a left conjunct ... |
| anbi1cd 633 | Introduce a proposition as... |
| 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 ... |
| biadanid 819 | Deduction associated with ... |
| pm5.1 820 | Two propositions are equiv... |
| pm5.21 821 | Two propositions are equiv... |
| pm5.35 822 | Theorem *5.35 of [Whitehea... |
| abai 823 | Introduce one conjunct as ... |
| pm4.45im 824 | Conjunction with implicati... |
| impimprbi 825 | An implication and its rev... |
| nan 826 | Theorem to move a conjunct... |
| pm5.31 827 | Theorem *5.31 of [Whitehea... |
| pm5.31r 828 | Variant of ~ pm5.31 . (Co... |
| pm4.15 829 | Theorem *4.15 of [Whitehea... |
| pm5.36 830 | Theorem *5.36 of [Whitehea... |
| annotanannot 831 | A conjunction with a negat... |
| pm5.33 832 | Theorem *5.33 of [Whitehea... |
| syl12anc 833 | Syllogism combined with co... |
| syl21anc 834 | Syllogism combined with co... |
| syl22anc 835 | Syllogism combined with co... |
| syl1111anc 836 | Four-hypothesis eliminatio... |
| syldbl2 837 | Stacked hypotheseis implie... |
| mpsyl4anc 838 | An elimination deduction. ... |
| pm4.87 839 | Theorem *4.87 of [Whitehea... |
| bimsc1 840 | Removal of conjunct from o... |
| a2and 841 | Deduction distributing a c... |
| animpimp2impd 842 | Deduction deriving nested ... |
| pm4.64 845 | Theorem *4.64 of [Whitehea... |
| pm4.66 846 | Theorem *4.66 of [Whitehea... |
| pm2.53 847 | Theorem *2.53 of [Whitehea... |
| pm2.54 848 | Theorem *2.54 of [Whitehea... |
| imor 849 | Implication in terms of di... |
| imori 850 | Infer disjunction from imp... |
| imorri 851 | Infer implication from dis... |
| pm4.62 852 | Theorem *4.62 of [Whitehea... |
| jaoi 853 | Inference disjoining the a... |
| jao1i 854 | Add a disjunct in the ante... |
| jaod 855 | Deduction disjoining the a... |
| mpjaod 856 | Eliminate a disjunction in... |
| ori 857 | Infer implication from dis... |
| orri 858 | Infer disjunction from imp... |
| orrd 859 | Deduce disjunction from im... |
| ord 860 | Deduce implication from di... |
| orci 861 | Deduction introducing a di... |
| olci 862 | Deduction introducing a di... |
| orc 863 | Introduction of a disjunct... |
| olc 864 | Introduction of a disjunct... |
| pm1.4 865 | Axiom *1.4 of [WhiteheadRu... |
| orcom 866 | Commutative law for disjun... |
| orcomd 867 | Commutation of disjuncts i... |
| 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... |
| olcnd 873 | A lemma for Conjunctive No... |
| unitreslOLD 874 | Obsolete version of ~ olcn... |
| orcnd 875 | A lemma for Conjunctive No... |
| mtord 876 | A modus tollens deduction ... |
| pm3.2ni 877 | Infer negated disjunction ... |
| pm2.45 878 | Theorem *2.45 of [Whitehea... |
| pm2.46 879 | Theorem *2.46 of [Whitehea... |
| pm2.47 880 | Theorem *2.47 of [Whitehea... |
| pm2.48 881 | Theorem *2.48 of [Whitehea... |
| pm2.49 882 | Theorem *2.49 of [Whitehea... |
| norbi 883 | If neither of two proposit... |
| nbior 884 | If two propositions are no... |
| orel1 885 | Elimination of disjunction... |
| pm2.25 886 | Theorem *2.25 of [Whitehea... |
| orel2 887 | Elimination of disjunction... |
| pm2.67-2 888 | Slight generalization of T... |
| pm2.67 889 | Theorem *2.67 of [Whitehea... |
| curryax 890 | A non-intuitionistic posit... |
| exmid 891 | Law of excluded middle, al... |
| exmidd 892 | Law of excluded middle in ... |
| pm2.1 893 | Theorem *2.1 of [Whitehead... |
| pm2.13 894 | Theorem *2.13 of [Whitehea... |
| pm2.621 895 | Theorem *2.621 of [Whitehe... |
| pm2.62 896 | Theorem *2.62 of [Whitehea... |
| pm2.68 897 | Theorem *2.68 of [Whitehea... |
| dfor2 898 | Logical 'or' expressed in ... |
| pm2.07 899 | Theorem *2.07 of [Whitehea... |
| pm1.2 900 | Axiom *1.2 of [WhiteheadRu... |
| oridm 901 | Idempotent law for disjunc... |
| pm4.25 902 | Theorem *4.25 of [Whitehea... |
| pm2.4 903 | Theorem *2.4 of [Whitehead... |
| pm2.41 904 | Theorem *2.41 of [Whitehea... |
| orim12i 905 | Disjoin antecedents and co... |
| orim1i 906 | Introduce disjunct to both... |
| orim2i 907 | Introduce disjunct to both... |
| orim12dALT 908 | Alternate proof of ~ orim1... |
| orbi2i 909 | Inference adding a left di... |
| orbi1i 910 | Inference adding a right d... |
| orbi12i 911 | Infer the disjunction of t... |
| orbi2d 912 | Deduction adding a left di... |
| orbi1d 913 | Deduction adding a right d... |
| orbi1 914 | Theorem *4.37 of [Whitehea... |
| orbi12d 915 | Deduction joining two equi... |
| pm1.5 916 | Axiom *1.5 (Assoc) of [Whi... |
| or12 917 | Swap two disjuncts. (Cont... |
| orass 918 | Associative law for disjun... |
| pm2.31 919 | Theorem *2.31 of [Whitehea... |
| pm2.32 920 | Theorem *2.32 of [Whitehea... |
| pm2.3 921 | Theorem *2.3 of [Whitehead... |
| or32 922 | A rearrangement of disjunc... |
| or4 923 | Rearrangement of 4 disjunc... |
| or42 924 | Rearrangement of 4 disjunc... |
| orordi 925 | Distribution of disjunctio... |
| orordir 926 | Distribution of disjunctio... |
| orimdi 927 | Disjunction distributes ov... |
| pm2.76 928 | Theorem *2.76 of [Whitehea... |
| pm2.85 929 | Theorem *2.85 of [Whitehea... |
| pm2.75 930 | Theorem *2.75 of [Whitehea... |
| pm4.78 931 | Implication distributes ov... |
| biort 932 | A disjunction with a true ... |
| biorf 933 | A wff is equivalent to its... |
| biortn 934 | A wff is equivalent to its... |
| biorfi 935 | A wff is equivalent to its... |
| pm2.26 936 | Theorem *2.26 of [Whitehea... |
| pm2.63 937 | Theorem *2.63 of [Whitehea... |
| pm2.64 938 | Theorem *2.64 of [Whitehea... |
| pm2.42 939 | Theorem *2.42 of [Whitehea... |
| pm5.11g 940 | A general instance of Theo... |
| pm5.11 941 | Theorem *5.11 of [Whitehea... |
| pm5.12 942 | Theorem *5.12 of [Whitehea... |
| pm5.14 943 | Theorem *5.14 of [Whitehea... |
| pm5.13 944 | Theorem *5.13 of [Whitehea... |
| pm5.55 945 | Theorem *5.55 of [Whitehea... |
| pm4.72 946 | Implication in terms of bi... |
| imimorb 947 | Simplify an implication be... |
| oibabs 948 | Absorption of disjunction ... |
| orbidi 949 | Disjunction distributes ov... |
| pm5.7 950 | Disjunction distributes ov... |
| jaao 951 | Inference conjoining and d... |
| jaoa 952 | Inference disjoining and c... |
| jaoian 953 | Inference disjoining the a... |
| jaodan 954 | Deduction disjoining the a... |
| mpjaodan 955 | Eliminate a disjunction in... |
| pm3.44 956 | Theorem *3.44 of [Whitehea... |
| jao 957 | Disjunction of antecedents... |
| jaob 958 | Disjunction of antecedents... |
| pm4.77 959 | Theorem *4.77 of [Whitehea... |
| pm3.48 960 | Theorem *3.48 of [Whitehea... |
| orim12d 961 | Disjoin antecedents and co... |
| orim1d 962 | Disjoin antecedents and co... |
| orim2d 963 | Disjoin antecedents and co... |
| orim2 964 | Axiom *1.6 (Sum) of [White... |
| pm2.38 965 | Theorem *2.38 of [Whitehea... |
| pm2.36 966 | Theorem *2.36 of [Whitehea... |
| pm2.37 967 | Theorem *2.37 of [Whitehea... |
| pm2.81 968 | Theorem *2.81 of [Whitehea... |
| pm2.8 969 | Theorem *2.8 of [Whitehead... |
| pm2.73 970 | Theorem *2.73 of [Whitehea... |
| pm2.74 971 | Theorem *2.74 of [Whitehea... |
| pm2.82 972 | Theorem *2.82 of [Whitehea... |
| pm4.39 973 | Theorem *4.39 of [Whitehea... |
| animorl 974 | Conjunction implies disjun... |
| animorr 975 | Conjunction implies disjun... |
| animorlr 976 | Conjunction implies disjun... |
| animorrl 977 | Conjunction implies disjun... |
| ianor 978 | Negated conjunction in ter... |
| anor 979 | Conjunction in terms of di... |
| ioran 980 | Negated disjunction in ter... |
| pm4.52 981 | Theorem *4.52 of [Whitehea... |
| pm4.53 982 | Theorem *4.53 of [Whitehea... |
| pm4.54 983 | Theorem *4.54 of [Whitehea... |
| pm4.55 984 | Theorem *4.55 of [Whitehea... |
| pm4.56 985 | Theorem *4.56 of [Whitehea... |
| oran 986 | Disjunction in terms of co... |
| pm4.57 987 | Theorem *4.57 of [Whitehea... |
| pm3.1 988 | Theorem *3.1 of [Whitehead... |
| pm3.11 989 | Theorem *3.11 of [Whitehea... |
| pm3.12 990 | Theorem *3.12 of [Whitehea... |
| pm3.13 991 | Theorem *3.13 of [Whitehea... |
| pm3.14 992 | Theorem *3.14 of [Whitehea... |
| pm4.44 993 | Theorem *4.44 of [Whitehea... |
| pm4.45 994 | Theorem *4.45 of [Whitehea... |
| orabs 995 | Absorption of redundant in... |
| oranabs 996 | Absorb a disjunct into a c... |
| pm5.61 997 | Theorem *5.61 of [Whitehea... |
| pm5.6 998 | Conjunction in antecedent ... |
| orcanai 999 | Change disjunction in cons... |
| pm4.79 1000 | Theorem *4.79 of [Whitehea... |
| pm5.53 1001 | Theorem *5.53 of [Whitehea... |
| ordi 1002 | Distributive law for disju... |
| ordir 1003 | Distributive law for disju... |
| andi 1004 | Distributive law for conju... |
| andir 1005 | Distributive law for conju... |
| orddi 1006 | Double distributive law fo... |
| anddi 1007 | Double distributive law fo... |
| pm5.17 1008 | Theorem *5.17 of [Whitehea... |
| pm5.15 1009 | Theorem *5.15 of [Whitehea... |
| pm5.16 1010 | Theorem *5.16 of [Whitehea... |
| xor 1011 | Two ways to express exclus... |
| nbi2 1012 | Two ways to express "exclu... |
| xordi 1013 | Conjunction distributes ov... |
| pm5.54 1014 | Theorem *5.54 of [Whitehea... |
| pm5.62 1015 | Theorem *5.62 of [Whitehea... |
| pm5.63 1016 | Theorem *5.63 of [Whitehea... |
| niabn 1017 | Miscellaneous inference re... |
| ninba 1018 | Miscellaneous inference re... |
| pm4.43 1019 | Theorem *4.43 of [Whitehea... |
| pm4.82 1020 | Theorem *4.82 of [Whitehea... |
| pm4.83 1021 | Theorem *4.83 of [Whitehea... |
| pclem6 1022 | Negation inferred from emb... |
| bigolden 1023 | Dijkstra-Scholten's Golden... |
| pm5.71 1024 | Theorem *5.71 of [Whitehea... |
| pm5.75 1025 | Theorem *5.75 of [Whitehea... |
| ecase2d 1026 | Deduction for elimination ... |
| ecase2dOLD 1027 | Obsolete version of ~ ecas... |
| ecase3 1028 | Inference for elimination ... |
| ecase 1029 | Inference for elimination ... |
| ecase3d 1030 | Deduction for elimination ... |
| ecased 1031 | Deduction for elimination ... |
| ecase3ad 1032 | Deduction for elimination ... |
| ecase3adOLD 1033 | Obsolete version of ~ ecas... |
| ccase 1034 | Inference for combining ca... |
| ccased 1035 | Deduction for combining ca... |
| ccase2 1036 | Inference for combining ca... |
| 4cases 1037 | Inference eliminating two ... |
| 4casesdan 1038 | Deduction eliminating two ... |
| cases 1039 | Case disjunction according... |
| dedlem0a 1040 | Lemma for an alternate ver... |
| dedlem0b 1041 | Lemma for an alternate ver... |
| dedlema 1042 | Lemma for weak deduction t... |
| dedlemb 1043 | Lemma for weak deduction t... |
| cases2 1044 | Case disjunction according... |
| cases2ALT 1045 | Alternate proof of ~ cases... |
| dfbi3 1046 | An alternate definition of... |
| pm5.24 1047 | Theorem *5.24 of [Whitehea... |
| 4exmid 1048 | The disjunction of the fou... |
| consensus 1049 | The consensus theorem. Th... |
| pm4.42 1050 | Theorem *4.42 of [Whitehea... |
| prlem1 1051 | A specialized lemma for se... |
| prlem2 1052 | A specialized lemma for se... |
| oplem1 1053 | A specialized lemma for se... |
| dn1 1054 | A single axiom for Boolean... |
| bianir 1055 | A closed form of ~ mpbir ,... |
| jaoi2 1056 | Inference removing a negat... |
| jaoi3 1057 | Inference separating a dis... |
| ornld 1058 | Selecting one statement fr... |
| dfifp2 1061 | Alternate definition of th... |
| dfifp3 1062 | Alternate definition of th... |
| dfifp4 1063 | Alternate definition of th... |
| dfifp5 1064 | Alternate definition of th... |
| dfifp6 1065 | Alternate definition of th... |
| dfifp7 1066 | Alternate definition of th... |
| ifpdfbi 1067 | Define the biconditional a... |
| anifp 1068 | The conditional operator i... |
| ifpor 1069 | The conditional operator i... |
| ifpn 1070 | Conditional operator for t... |
| ifpnOLD 1071 | Obsolete version of ~ ifpn... |
| ifptru 1072 | Value of the conditional o... |
| ifpfal 1073 | Value of the conditional o... |
| ifpid 1074 | Value of the conditional o... |
| casesifp 1075 | Version of ~ cases express... |
| ifpbi123d 1076 | Equivalence deduction for ... |
| ifpbi123dOLD 1077 | Obsolete version of ~ ifpb... |
| ifpbi23d 1078 | Equivalence deduction for ... |
| ifpimpda 1079 | Separation of the values o... |
| 1fpid3 1080 | The value of the condition... |
| elimh 1081 | Hypothesis builder for the... |
| dedt 1082 | The weak deduction theorem... |
| con3ALT 1083 | Proof of ~ con3 from its a... |
| 3orass 1088 | Associative law for triple... |
| 3orel1 1089 | Partial elimination of a t... |
| 3orrot 1090 | Rotation law for triple di... |
| 3orcoma 1091 | Commutation law for triple... |
| 3orcomb 1092 | Commutation law for triple... |
| 3anass 1093 | Associative law for triple... |
| 3anan12 1094 | Convert triple conjunction... |
| 3anan32 1095 | Convert triple conjunction... |
| 3ancoma 1096 | Commutation law for triple... |
| 3ancomb 1097 | Commutation law for triple... |
| 3anrot 1098 | Rotation law for triple co... |
| 3anrev 1099 | Reversal law for triple co... |
| anandi3 1100 | Distribution of triple con... |
| anandi3r 1101 | Distribution of triple con... |
| 3anidm 1102 | Idempotent law for conjunc... |
| 3an4anass 1103 | Associative law for four c... |
| 3ioran 1104 | Negated triple disjunction... |
| 3ianor 1105 | Negated triple conjunction... |
| 3anor 1106 | Triple conjunction express... |
| 3oran 1107 | Triple disjunction in term... |
| 3impa 1108 | Importation from double to... |
| 3imp 1109 | Importation inference. (C... |
| 3imp31 1110 | The importation inference ... |
| 3imp231 1111 | Importation inference. (C... |
| 3imp21 1112 | The importation inference ... |
| 3impb 1113 | Importation from double to... |
| 3impib 1114 | Importation to triple conj... |
| 3impia 1115 | Importation to triple conj... |
| 3expa 1116 | Exportation from triple to... |
| 3exp 1117 | Exportation inference. (C... |
| 3expb 1118 | Exportation from triple to... |
| 3expia 1119 | Exportation from triple co... |
| 3expib 1120 | Exportation from triple co... |
| 3com12 1121 | Commutation in antecedent.... |
| 3com13 1122 | Commutation in antecedent.... |
| 3comr 1123 | Commutation in antecedent.... |
| 3com23 1124 | Commutation in antecedent.... |
| 3coml 1125 | Commutation in antecedent.... |
| 3jca 1126 | Join consequents with conj... |
| 3jcad 1127 | Deduction conjoining the c... |
| 3adant1 1128 | Deduction adding a conjunc... |
| 3adant2 1129 | Deduction adding a conjunc... |
| 3adant3 1130 | Deduction adding a conjunc... |
| 3ad2ant1 1131 | Deduction adding conjuncts... |
| 3ad2ant2 1132 | Deduction adding conjuncts... |
| 3ad2ant3 1133 | Deduction adding conjuncts... |
| simp1 1134 | Simplification of triple c... |
| simp2 1135 | Simplification of triple c... |
| simp3 1136 | Simplification of triple c... |
| simp1i 1137 | Infer a conjunct from a tr... |
| simp2i 1138 | Infer a conjunct from a tr... |
| simp3i 1139 | Infer a conjunct from a tr... |
| simp1d 1140 | Deduce a conjunct from a t... |
| simp2d 1141 | Deduce a conjunct from a t... |
| simp3d 1142 | Deduce a conjunct from a t... |
| simp1bi 1143 | Deduce a conjunct from a t... |
| simp2bi 1144 | Deduce a conjunct from a t... |
| simp3bi 1145 | Deduce a conjunct from a t... |
| 3simpa 1146 | Simplification of triple c... |
| 3simpb 1147 | Simplification of triple c... |
| 3simpc 1148 | Simplification of triple c... |
| 3anim123i 1149 | Join antecedents and conse... |
| 3anim1i 1150 | Add two conjuncts to antec... |
| 3anim2i 1151 | Add two conjuncts to antec... |
| 3anim3i 1152 | Add two conjuncts to antec... |
| 3anbi123i 1153 | Join 3 biconditionals with... |
| 3orbi123i 1154 | Join 3 biconditionals with... |
| 3anbi1i 1155 | Inference adding two conju... |
| 3anbi2i 1156 | Inference adding two conju... |
| 3anbi3i 1157 | Inference adding two conju... |
| syl3an 1158 | A triple syllogism inferen... |
| syl3anb 1159 | A triple syllogism inferen... |
| syl3anbr 1160 | A triple syllogism inferen... |
| syl3an1 1161 | A syllogism inference. (C... |
| syl3an2 1162 | A syllogism inference. (C... |
| syl3an3 1163 | A syllogism inference. (C... |
| 3adantl1 1164 | Deduction adding a conjunc... |
| 3adantl2 1165 | Deduction adding a conjunc... |
| 3adantl3 1166 | Deduction adding a conjunc... |
| 3adantr1 1167 | Deduction adding a conjunc... |
| 3adantr2 1168 | Deduction adding a conjunc... |
| 3adantr3 1169 | Deduction adding a conjunc... |
| ad4ant123 1170 | Deduction adding conjuncts... |
| ad4ant124 1171 | Deduction adding conjuncts... |
| ad4ant134 1172 | Deduction adding conjuncts... |
| ad4ant234 1173 | Deduction adding conjuncts... |
| 3adant1l 1174 | Deduction adding a conjunc... |
| 3adant1r 1175 | Deduction adding a conjunc... |
| 3adant2l 1176 | Deduction adding a conjunc... |
| 3adant2r 1177 | Deduction adding a conjunc... |
| 3adant3l 1178 | Deduction adding a conjunc... |
| 3adant3r 1179 | Deduction adding a conjunc... |
| 3adant3r1 1180 | Deduction adding a conjunc... |
| 3adant3r2 1181 | Deduction adding a conjunc... |
| 3adant3r3 1182 | Deduction adding a conjunc... |
| 3ad2antl1 1183 | Deduction adding conjuncts... |
| 3ad2antl2 1184 | Deduction adding conjuncts... |
| 3ad2antl3 1185 | Deduction adding conjuncts... |
| 3ad2antr1 1186 | Deduction adding conjuncts... |
| 3ad2antr2 1187 | Deduction adding conjuncts... |
| 3ad2antr3 1188 | Deduction adding conjuncts... |
| simpl1 1189 | Simplification of conjunct... |
| simpl2 1190 | Simplification of conjunct... |
| simpl3 1191 | Simplification of conjunct... |
| simpr1 1192 | Simplification of conjunct... |
| simpr2 1193 | Simplification of conjunct... |
| simpr3 1194 | Simplification of conjunct... |
| simp1l 1195 | Simplification of triple c... |
| simp1r 1196 | Simplification of triple c... |
| simp2l 1197 | Simplification of triple c... |
| simp2r 1198 | Simplification of triple c... |
| simp3l 1199 | Simplification of triple c... |
| simp3r 1200 | Simplification of triple c... |
| simp11 1201 | Simplification of doubly t... |
| simp12 1202 | Simplification of doubly t... |
| simp13 1203 | Simplification of doubly t... |
| simp21 1204 | Simplification of doubly t... |
| simp22 1205 | Simplification of doubly t... |
| simp23 1206 | Simplification of doubly t... |
| simp31 1207 | Simplification of doubly t... |
| simp32 1208 | Simplification of doubly t... |
| simp33 1209 | Simplification of doubly t... |
| simpll1 1210 | Simplification of conjunct... |
| simpll2 1211 | Simplification of conjunct... |
| simpll3 1212 | Simplification of conjunct... |
| simplr1 1213 | Simplification of conjunct... |
| simplr2 1214 | Simplification of conjunct... |
| simplr3 1215 | Simplification of conjunct... |
| simprl1 1216 | Simplification of conjunct... |
| simprl2 1217 | Simplification of conjunct... |
| simprl3 1218 | Simplification of conjunct... |
| simprr1 1219 | Simplification of conjunct... |
| simprr2 1220 | Simplification of conjunct... |
| simprr3 1221 | Simplification of conjunct... |
| simpl1l 1222 | Simplification of conjunct... |
| simpl1r 1223 | Simplification of conjunct... |
| simpl2l 1224 | Simplification of conjunct... |
| simpl2r 1225 | Simplification of conjunct... |
| simpl3l 1226 | Simplification of conjunct... |
| simpl3r 1227 | Simplification of conjunct... |
| simpr1l 1228 | Simplification of conjunct... |
| simpr1r 1229 | Simplification of conjunct... |
| simpr2l 1230 | Simplification of conjunct... |
| simpr2r 1231 | Simplification of conjunct... |
| simpr3l 1232 | Simplification of conjunct... |
| simpr3r 1233 | Simplification of conjunct... |
| simp1ll 1234 | Simplification of conjunct... |
| simp1lr 1235 | Simplification of conjunct... |
| simp1rl 1236 | Simplification of conjunct... |
| simp1rr 1237 | Simplification of conjunct... |
| simp2ll 1238 | Simplification of conjunct... |
| simp2lr 1239 | Simplification of conjunct... |
| simp2rl 1240 | Simplification of conjunct... |
| simp2rr 1241 | Simplification of conjunct... |
| simp3ll 1242 | Simplification of conjunct... |
| simp3lr 1243 | Simplification of conjunct... |
| simp3rl 1244 | Simplification of conjunct... |
| simp3rr 1245 | Simplification of conjunct... |
| simpl11 1246 | Simplification of conjunct... |
| simpl12 1247 | Simplification of conjunct... |
| simpl13 1248 | Simplification of conjunct... |
| simpl21 1249 | Simplification of conjunct... |
| simpl22 1250 | Simplification of conjunct... |
| simpl23 1251 | Simplification of conjunct... |
| simpl31 1252 | Simplification of conjunct... |
| simpl32 1253 | Simplification of conjunct... |
| simpl33 1254 | Simplification of conjunct... |
| simpr11 1255 | Simplification of conjunct... |
| simpr12 1256 | Simplification of conjunct... |
| simpr13 1257 | Simplification of conjunct... |
| simpr21 1258 | Simplification of conjunct... |
| simpr22 1259 | Simplification of conjunct... |
| simpr23 1260 | Simplification of conjunct... |
| simpr31 1261 | Simplification of conjunct... |
| simpr32 1262 | Simplification of conjunct... |
| simpr33 1263 | Simplification of conjunct... |
| simp1l1 1264 | Simplification of conjunct... |
| simp1l2 1265 | Simplification of conjunct... |
| simp1l3 1266 | Simplification of conjunct... |
| simp1r1 1267 | Simplification of conjunct... |
| simp1r2 1268 | Simplification of conjunct... |
| simp1r3 1269 | Simplification of conjunct... |
| simp2l1 1270 | Simplification of conjunct... |
| simp2l2 1271 | Simplification of conjunct... |
| simp2l3 1272 | Simplification of conjunct... |
| simp2r1 1273 | Simplification of conjunct... |
| simp2r2 1274 | Simplification of conjunct... |
| simp2r3 1275 | Simplification of conjunct... |
| simp3l1 1276 | Simplification of conjunct... |
| simp3l2 1277 | Simplification of conjunct... |
| simp3l3 1278 | Simplification of conjunct... |
| simp3r1 1279 | Simplification of conjunct... |
| simp3r2 1280 | Simplification of conjunct... |
| simp3r3 1281 | Simplification of conjunct... |
| simp11l 1282 | Simplification of conjunct... |
| simp11r 1283 | Simplification of conjunct... |
| simp12l 1284 | Simplification of conjunct... |
| simp12r 1285 | Simplification of conjunct... |
| simp13l 1286 | Simplification of conjunct... |
| simp13r 1287 | Simplification of conjunct... |
| simp21l 1288 | Simplification of conjunct... |
| simp21r 1289 | Simplification of conjunct... |
| simp22l 1290 | Simplification of conjunct... |
| simp22r 1291 | Simplification of conjunct... |
| simp23l 1292 | Simplification of conjunct... |
| simp23r 1293 | Simplification of conjunct... |
| simp31l 1294 | Simplification of conjunct... |
| simp31r 1295 | Simplification of conjunct... |
| simp32l 1296 | Simplification of conjunct... |
| simp32r 1297 | Simplification of conjunct... |
| simp33l 1298 | Simplification of conjunct... |
| simp33r 1299 | Simplification of conjunct... |
| simp111 1300 | Simplification of conjunct... |
| simp112 1301 | Simplification of conjunct... |
| simp113 1302 | Simplification of conjunct... |
| simp121 1303 | Simplification of conjunct... |
| simp122 1304 | Simplification of conjunct... |
| simp123 1305 | Simplification of conjunct... |
| simp131 1306 | Simplification of conjunct... |
| simp132 1307 | Simplification of conjunct... |
| simp133 1308 | Simplification of conjunct... |
| simp211 1309 | Simplification of conjunct... |
| simp212 1310 | Simplification of conjunct... |
| simp213 1311 | Simplification of conjunct... |
| simp221 1312 | Simplification of conjunct... |
| simp222 1313 | Simplification of conjunct... |
| simp223 1314 | Simplification of conjunct... |
| simp231 1315 | Simplification of conjunct... |
| simp232 1316 | Simplification of conjunct... |
| simp233 1317 | Simplification of conjunct... |
| simp311 1318 | Simplification of conjunct... |
| simp312 1319 | Simplification of conjunct... |
| simp313 1320 | Simplification of conjunct... |
| simp321 1321 | Simplification of conjunct... |
| simp322 1322 | Simplification of conjunct... |
| simp323 1323 | Simplification of conjunct... |
| simp331 1324 | Simplification of conjunct... |
| simp332 1325 | Simplification of conjunct... |
| simp333 1326 | Simplification of conjunct... |
| 3anibar 1327 | Remove a hypothesis from t... |
| 3mix1 1328 | Introduction in triple dis... |
| 3mix2 1329 | Introduction in triple dis... |
| 3mix3 1330 | Introduction in triple dis... |
| 3mix1i 1331 | Introduction in triple dis... |
| 3mix2i 1332 | Introduction in triple dis... |
| 3mix3i 1333 | Introduction in triple dis... |
| 3mix1d 1334 | Deduction introducing trip... |
| 3mix2d 1335 | Deduction introducing trip... |
| 3mix3d 1336 | Deduction introducing trip... |
| 3pm3.2i 1337 | Infer conjunction of premi... |
| pm3.2an3 1338 | Version of ~ pm3.2 for a t... |
| mpbir3an 1339 | Detach a conjunction of tr... |
| mpbir3and 1340 | Detach a conjunction of tr... |
| syl3anbrc 1341 | Syllogism inference. (Con... |
| syl21anbrc 1342 | Syllogism inference. (Con... |
| 3imp3i2an 1343 | An elimination deduction. ... |
| ex3 1344 | Apply ~ ex to a hypothesis... |
| 3imp1 1345 | Importation to left triple... |
| 3impd 1346 | Importation deduction for ... |
| 3imp2 1347 | Importation to right tripl... |
| 3impdi 1348 | Importation inference (und... |
| 3impdir 1349 | Importation inference (und... |
| 3exp1 1350 | Exportation from left trip... |
| 3expd 1351 | Exportation deduction for ... |
| 3exp2 1352 | Exportation from right tri... |
| exp5o 1353 | A triple exportation infer... |
| exp516 1354 | A triple exportation infer... |
| exp520 1355 | A triple exportation infer... |
| 3impexp 1356 | Version of ~ impexp for a ... |
| 3an1rs 1357 | Swap conjuncts. (Contribu... |
| 3anassrs 1358 | Associative law for conjun... |
| ad5ant245 1359 | Deduction adding conjuncts... |
| ad5ant234 1360 | Deduction adding conjuncts... |
| ad5ant235 1361 | Deduction adding conjuncts... |
| ad5ant123 1362 | Deduction adding conjuncts... |
| ad5ant124 1363 | Deduction adding conjuncts... |
| ad5ant125 1364 | Deduction adding conjuncts... |
| ad5ant134 1365 | Deduction adding conjuncts... |
| ad5ant135 1366 | Deduction adding conjuncts... |
| ad5ant145 1367 | Deduction adding conjuncts... |
| ad5ant2345 1368 | Deduction adding conjuncts... |
| syl3anc 1369 | Syllogism combined with co... |
| syl13anc 1370 | Syllogism combined with co... |
| syl31anc 1371 | Syllogism combined with co... |
| syl112anc 1372 | Syllogism combined with co... |
| syl121anc 1373 | Syllogism combined with co... |
| syl211anc 1374 | Syllogism combined with co... |
| syl23anc 1375 | Syllogism combined with co... |
| syl32anc 1376 | Syllogism combined with co... |
| syl122anc 1377 | Syllogism combined with co... |
| syl212anc 1378 | Syllogism combined with co... |
| syl221anc 1379 | Syllogism combined with co... |
| syl113anc 1380 | Syllogism combined with co... |
| syl131anc 1381 | Syllogism combined with co... |
| syl311anc 1382 | Syllogism combined with co... |
| syl33anc 1383 | Syllogism combined with co... |
| syl222anc 1384 | Syllogism combined with co... |
| syl123anc 1385 | Syllogism combined with co... |
| syl132anc 1386 | Syllogism combined with co... |
| syl213anc 1387 | Syllogism combined with co... |
| syl231anc 1388 | Syllogism combined with co... |
| syl312anc 1389 | Syllogism combined with co... |
| syl321anc 1390 | Syllogism combined with co... |
| syl133anc 1391 | Syllogism combined with co... |
| syl313anc 1392 | Syllogism combined with co... |
| syl331anc 1393 | Syllogism combined with co... |
| syl223anc 1394 | Syllogism combined with co... |
| syl232anc 1395 | Syllogism combined with co... |
| syl322anc 1396 | Syllogism combined with co... |
| syl233anc 1397 | Syllogism combined with co... |
| syl323anc 1398 | Syllogism combined with co... |
| syl332anc 1399 | Syllogism combined with co... |
| syl333anc 1400 | A syllogism inference comb... |
| syl3an1b 1401 | A syllogism inference. (C... |
| syl3an2b 1402 | A syllogism inference. (C... |
| syl3an3b 1403 | A syllogism inference. (C... |
| syl3an1br 1404 | A syllogism inference. (C... |
| syl3an2br 1405 | A syllogism inference. (C... |
| syl3an3br 1406 | A syllogism inference. (C... |
| syld3an3 1407 | A syllogism inference. (C... |
| syld3an1 1408 | A syllogism inference. (C... |
| syld3an2 1409 | A syllogism inference. (C... |
| syl3anl1 1410 | A syllogism inference. (C... |
| syl3anl2 1411 | A syllogism inference. (C... |
| syl3anl3 1412 | A syllogism inference. (C... |
| syl3anl 1413 | A triple syllogism inferen... |
| syl3anr1 1414 | A syllogism inference. (C... |
| syl3anr2 1415 | A syllogism inference. (C... |
| syl3anr3 1416 | A syllogism inference. (C... |
| 3anidm12 1417 | Inference from idempotent ... |
| 3anidm13 1418 | Inference from idempotent ... |
| 3anidm23 1419 | Inference from idempotent ... |
| syl2an3an 1420 | ~ syl3an with antecedents ... |
| syl2an23an 1421 | Deduction related to ~ syl... |
| 3ori 1422 | Infer implication from tri... |
| 3jao 1423 | Disjunction of three antec... |
| 3jaob 1424 | Disjunction of three antec... |
| 3jaoi 1425 | Disjunction of three antec... |
| 3jaod 1426 | Disjunction of three antec... |
| 3jaoian 1427 | Disjunction of three antec... |
| 3jaodan 1428 | Disjunction of three antec... |
| mpjao3dan 1429 | Eliminate a three-way disj... |
| mpjao3danOLD 1430 | Obsolete version of ~ mpja... |
| 3jaao 1431 | Inference conjoining and d... |
| syl3an9b 1432 | Nested syllogism inference... |
| 3orbi123d 1433 | Deduction joining 3 equiva... |
| 3anbi123d 1434 | Deduction joining 3 equiva... |
| 3anbi12d 1435 | Deduction conjoining and a... |
| 3anbi13d 1436 | Deduction conjoining and a... |
| 3anbi23d 1437 | Deduction conjoining and a... |
| 3anbi1d 1438 | Deduction adding conjuncts... |
| 3anbi2d 1439 | Deduction adding conjuncts... |
| 3anbi3d 1440 | Deduction adding conjuncts... |
| 3anim123d 1441 | Deduction joining 3 implic... |
| 3orim123d 1442 | Deduction joining 3 implic... |
| an6 1443 | Rearrangement of 6 conjunc... |
| 3an6 1444 | Analogue of ~ an4 for trip... |
| 3or6 1445 | Analogue of ~ or4 for trip... |
| mp3an1 1446 | An inference based on modu... |
| mp3an2 1447 | An inference based on modu... |
| mp3an3 1448 | An inference based on modu... |
| mp3an12 1449 | An inference based on modu... |
| mp3an13 1450 | An inference based on modu... |
| mp3an23 1451 | An inference based on modu... |
| mp3an1i 1452 | An inference based on modu... |
| mp3anl1 1453 | An inference based on modu... |
| mp3anl2 1454 | An inference based on modu... |
| mp3anl3 1455 | An inference based on modu... |
| mp3anr1 1456 | An inference based on modu... |
| mp3anr2 1457 | An inference based on modu... |
| mp3anr3 1458 | An inference based on modu... |
| mp3an 1459 | An inference based on modu... |
| mpd3an3 1460 | An inference based on modu... |
| mpd3an23 1461 | An inference based on modu... |
| mp3and 1462 | A deduction based on modus... |
| mp3an12i 1463 | ~ mp3an with antecedents i... |
| mp3an2i 1464 | ~ mp3an with antecedents i... |
| mp3an3an 1465 | ~ mp3an with antecedents i... |
| mp3an2ani 1466 | An elimination deduction. ... |
| biimp3a 1467 | Infer implication from a l... |
| biimp3ar 1468 | Infer implication from a l... |
| 3anandis 1469 | Inference that undistribut... |
| 3anandirs 1470 | Inference that undistribut... |
| ecase23d 1471 | Deduction for elimination ... |
| 3ecase 1472 | Inference for elimination ... |
| 3bior1fd 1473 | A disjunction is equivalen... |
| 3bior1fand 1474 | A disjunction is equivalen... |
| 3bior2fd 1475 | A wff is equivalent to its... |
| 3biant1d 1476 | A conjunction is equivalen... |
| intn3an1d 1477 | Introduction of a triple c... |
| intn3an2d 1478 | Introduction of a triple c... |
| intn3an3d 1479 | Introduction of a triple c... |
| an3andi 1480 | Distribution of conjunctio... |
| an33rean 1481 | Rearrange a 9-fold conjunc... |
| an33reanOLD 1482 | Obsolete version of ~ an33... |
| nanan 1485 | Conjunction in terms of al... |
| dfnan2 1486 | Alternative denial in term... |
| nanor 1487 | Alternative denial in term... |
| nancom 1488 | Alternative denial is comm... |
| nannan 1489 | Nested alternative denials... |
| nanim 1490 | Implication in terms of al... |
| nannot 1491 | Negation in terms of alter... |
| nanbi 1492 | Biconditional in terms of ... |
| nanbi1 1493 | Introduce a right anti-con... |
| nanbi2 1494 | Introduce a left anti-conj... |
| nanbi12 1495 | Join two logical equivalen... |
| nanbi1i 1496 | Introduce a right anti-con... |
| nanbi2i 1497 | Introduce a left anti-conj... |
| nanbi12i 1498 | Join two logical equivalen... |
| nanbi1d 1499 | Introduce a right anti-con... |
| nanbi2d 1500 | Introduce a left anti-conj... |
| nanbi12d 1501 | Join two logical equivalen... |
| nanass 1502 | A characterization of when... |
| xnor 1505 | Two ways to write XNOR (ex... |
| xorcom 1506 | The connector ` \/_ ` is c... |
| xorcomOLD 1507 | Obsolete version of ~ xorc... |
| xorass 1508 | The connector ` \/_ ` is a... |
| excxor 1509 | This tautology shows that ... |
| xor2 1510 | Two ways to express "exclu... |
| xoror 1511 | Exclusive disjunction impl... |
| xornan 1512 | Exclusive disjunction impl... |
| xornan2 1513 | XOR implies NAND (written ... |
| xorneg2 1514 | The connector ` \/_ ` is n... |
| xorneg1 1515 | The connector ` \/_ ` is n... |
| xorneg 1516 | The connector ` \/_ ` is u... |
| xorbi12i 1517 | Equality property for excl... |
| xorbi12iOLD 1518 | Obsolete version of ~ xorb... |
| xorbi12d 1519 | Equality property for excl... |
| anxordi 1520 | Conjunction distributes ov... |
| xorexmid 1521 | Exclusive-or variant of th... |
| norcom 1524 | The connector ` -\/ ` is c... |
| norcomOLD 1525 | Obsolete version of ~ norc... |
| nornot 1526 | ` -. ` is expressible via ... |
| nornotOLD 1527 | Obsolete version of ~ norn... |
| noran 1528 | ` /\ ` is expressible via ... |
| noranOLD 1529 | Obsolete version of ~ nora... |
| noror 1530 | ` \/ ` is expressible via ... |
| nororOLD 1531 | Obsolete version of ~ noro... |
| norasslem1 1532 | This lemma shows the equiv... |
| norasslem2 1533 | This lemma specializes ~ b... |
| norasslem3 1534 | This lemma specializes ~ b... |
| norass 1535 | A characterization of when... |
| norassOLD 1536 | Obsolete version of ~ nora... |
| trujust 1541 | Soundness justification th... |
| tru 1543 | The truth value ` T. ` is ... |
| dftru2 1544 | An alternate definition of... |
| trut 1545 | A proposition is equivalen... |
| mptru 1546 | Eliminate ` T. ` as an ant... |
| tbtru 1547 | A proposition is equivalen... |
| bitru 1548 | A theorem is equivalent to... |
| trud 1549 | Anything implies ` T. ` . ... |
| truan 1550 | True can be removed from a... |
| fal 1553 | The truth value ` F. ` is ... |
| nbfal 1554 | The negation of a proposit... |
| bifal 1555 | A contradiction is equival... |
| falim 1556 | The truth value ` F. ` imp... |
| falimd 1557 | The truth value ` F. ` imp... |
| dfnot 1558 | Given falsum ` F. ` , we c... |
| inegd 1559 | Negation introduction rule... |
| efald 1560 | Deduction based on reducti... |
| pm2.21fal 1561 | If a wff and its negation ... |
| truimtru 1562 | A ` -> ` identity. (Contr... |
| truimfal 1563 | A ` -> ` identity. (Contr... |
| falimtru 1564 | A ` -> ` identity. (Contr... |
| falimfal 1565 | A ` -> ` identity. (Contr... |
| nottru 1566 | A ` -. ` identity. (Contr... |
| notfal 1567 | A ` -. ` identity. (Contr... |
| trubitru 1568 | A ` <-> ` identity. (Cont... |
| falbitru 1569 | A ` <-> ` identity. (Cont... |
| trubifal 1570 | A ` <-> ` identity. (Cont... |
| falbifal 1571 | A ` <-> ` identity. (Cont... |
| truantru 1572 | A ` /\ ` identity. (Contr... |
| truanfal 1573 | A ` /\ ` identity. (Contr... |
| falantru 1574 | A ` /\ ` identity. (Contr... |
| falanfal 1575 | A ` /\ ` identity. (Contr... |
| truortru 1576 | A ` \/ ` identity. (Contr... |
| truorfal 1577 | A ` \/ ` identity. (Contr... |
| falortru 1578 | A ` \/ ` identity. (Contr... |
| falorfal 1579 | A ` \/ ` identity. (Contr... |
| trunantru 1580 | A ` -/\ ` identity. (Cont... |
| trunanfal 1581 | A ` -/\ ` identity. (Cont... |
| falnantru 1582 | A ` -/\ ` identity. (Cont... |
| falnanfal 1583 | A ` -/\ ` identity. (Cont... |
| truxortru 1584 | A ` \/_ ` identity. (Cont... |
| truxorfal 1585 | A ` \/_ ` identity. (Cont... |
| falxortru 1586 | A ` \/_ ` identity. (Cont... |
| falxorfal 1587 | A ` \/_ ` identity. (Cont... |
| trunortru 1588 | A ` -\/ ` identity. (Cont... |
| trunortruOLD 1589 | Obsolete version of ~ trun... |
| trunorfal 1590 | A ` -\/ ` identity. (Cont... |
| trunorfalOLD 1591 | Obsolete version of ~ trun... |
| falnortru 1592 | A ` -\/ ` identity. (Cont... |
| falnorfal 1593 | A ` -\/ ` identity. (Cont... |
| falnorfalOLD 1594 | Obsolete version of ~ faln... |
| hadbi123d 1597 | Equality theorem for the a... |
| hadbi123i 1598 | Equality theorem for the a... |
| hadass 1599 | Associative law for the ad... |
| hadbi 1600 | The adder sum is the same ... |
| hadcoma 1601 | Commutative law for the ad... |
| hadcomaOLD 1602 | Obsolete version of ~ hadc... |
| hadcomb 1603 | Commutative law for the ad... |
| hadrot 1604 | Rotation law for the adder... |
| hadnot 1605 | The adder sum distributes ... |
| had1 1606 | If the first input is true... |
| had0 1607 | If the first input is fals... |
| hadifp 1608 | The value of the adder sum... |
| cador 1611 | The adder carry in disjunc... |
| cadan 1612 | The adder carry in conjunc... |
| cadbi123d 1613 | Equality theorem for the a... |
| cadbi123i 1614 | Equality theorem for the a... |
| cadcoma 1615 | Commutative law for the ad... |
| cadcomb 1616 | Commutative law for the ad... |
| cadrot 1617 | Rotation law for the adder... |
| cadnot 1618 | The adder carry distribute... |
| cad11 1619 | If (at least) two inputs a... |
| cad1 1620 | If one input is true, then... |
| cad0 1621 | If one input is false, the... |
| cad0OLD 1622 | Obsolete version of ~ cad0... |
| cadifp 1623 | The value of the carry is,... |
| cadtru 1624 | The adder carry is true as... |
| minimp 1625 | A single axiom for minimal... |
| minimp-syllsimp 1626 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1627 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1628 | Derivation of a commuted f... |
| minimp-ax2 1629 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1630 | Derivation of ~ pm2.43 (al... |
| impsingle 1631 | The shortest single axiom ... |
| impsingle-step4 1632 | Derivation of impsingle-st... |
| impsingle-step8 1633 | Derivation of impsingle-st... |
| impsingle-ax1 1634 | Derivation of impsingle-ax... |
| impsingle-step15 1635 | Derivation of impsingle-st... |
| impsingle-step18 1636 | Derivation of impsingle-st... |
| impsingle-step19 1637 | Derivation of impsingle-st... |
| impsingle-step20 1638 | Derivation of impsingle-st... |
| impsingle-step21 1639 | Derivation of impsingle-st... |
| impsingle-step22 1640 | Derivation of impsingle-st... |
| impsingle-step25 1641 | Derivation of impsingle-st... |
| impsingle-imim1 1642 | Derivation of impsingle-im... |
| impsingle-peirce 1643 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1644 | Derivation of ~ ax-2 from ... |
| meredith 1645 | Carew Meredith's sole axio... |
| merlem1 1646 | Step 3 of Meredith's proof... |
| merlem2 1647 | Step 4 of Meredith's proof... |
| merlem3 1648 | Step 7 of Meredith's proof... |
| merlem4 1649 | Step 8 of Meredith's proof... |
| merlem5 1650 | Step 11 of Meredith's proo... |
| merlem6 1651 | Step 12 of Meredith's proo... |
| merlem7 1652 | Between steps 14 and 15 of... |
| merlem8 1653 | Step 15 of Meredith's proo... |
| merlem9 1654 | Step 18 of Meredith's proo... |
| merlem10 1655 | Step 19 of Meredith's proo... |
| merlem11 1656 | Step 20 of Meredith's proo... |
| merlem12 1657 | Step 28 of Meredith's proo... |
| merlem13 1658 | Step 35 of Meredith's proo... |
| luk-1 1659 | 1 of 3 axioms for proposit... |
| luk-2 1660 | 2 of 3 axioms for proposit... |
| luk-3 1661 | 3 of 3 axioms for proposit... |
| luklem1 1662 | Used to rederive standard ... |
| luklem2 1663 | Used to rederive standard ... |
| luklem3 1664 | Used to rederive standard ... |
| luklem4 1665 | Used to rederive standard ... |
| luklem5 1666 | Used to rederive standard ... |
| luklem6 1667 | Used to rederive standard ... |
| luklem7 1668 | Used to rederive standard ... |
| luklem8 1669 | Used to rederive standard ... |
| ax1 1670 | Standard propositional axi... |
| ax2 1671 | Standard propositional axi... |
| ax3 1672 | Standard propositional axi... |
| nic-dfim 1673 | This theorem "defines" imp... |
| nic-dfneg 1674 | This theorem "defines" neg... |
| nic-mp 1675 | Derive Nicod's rule of mod... |
| nic-mpALT 1676 | A direct proof of ~ nic-mp... |
| nic-ax 1677 | Nicod's axiom derived from... |
| nic-axALT 1678 | A direct proof of ~ nic-ax... |
| nic-imp 1679 | Inference for ~ nic-mp usi... |
| nic-idlem1 1680 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1681 | Lemma for ~ nic-id . Infe... |
| nic-id 1682 | Theorem ~ id expressed wit... |
| nic-swap 1683 | The connector ` -/\ ` is s... |
| nic-isw1 1684 | Inference version of ~ nic... |
| nic-isw2 1685 | Inference for swapping nes... |
| nic-iimp1 1686 | Inference version of ~ nic... |
| nic-iimp2 1687 | Inference version of ~ nic... |
| nic-idel 1688 | Inference to remove the tr... |
| nic-ich 1689 | Chained inference. (Contr... |
| nic-idbl 1690 | Double the terms. Since d... |
| nic-bijust 1691 | Biconditional justificatio... |
| nic-bi1 1692 | Inference to extract one s... |
| nic-bi2 1693 | Inference to extract the o... |
| nic-stdmp 1694 | Derive the standard modus ... |
| nic-luk1 1695 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1696 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1697 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1698 | This alternative axiom for... |
| lukshefth1 1699 | Lemma for ~ renicax . (Co... |
| lukshefth2 1700 | Lemma for ~ renicax . (Co... |
| renicax 1701 | A rederivation of ~ nic-ax... |
| tbw-bijust 1702 | Justification for ~ tbw-ne... |
| tbw-negdf 1703 | The definition of negation... |
| tbw-ax1 1704 | The first of four axioms i... |
| tbw-ax2 1705 | The second of four axioms ... |
| tbw-ax3 1706 | The third of four axioms i... |
| tbw-ax4 1707 | The fourth of four axioms ... |
| tbwsyl 1708 | Used to rederive the Lukas... |
| tbwlem1 1709 | Used to rederive the Lukas... |
| tbwlem2 1710 | Used to rederive the Lukas... |
| tbwlem3 1711 | Used to rederive the Lukas... |
| tbwlem4 1712 | Used to rederive the Lukas... |
| tbwlem5 1713 | Used to rederive the Lukas... |
| re1luk1 1714 | ~ luk-1 derived from the T... |
| re1luk2 1715 | ~ luk-2 derived from the T... |
| re1luk3 1716 | ~ luk-3 derived from the T... |
| merco1 1717 | A single axiom for proposi... |
| merco1lem1 1718 | Used to rederive the Tarsk... |
| retbwax4 1719 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1720 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1721 | Used to rederive the Tarsk... |
| merco1lem3 1722 | Used to rederive the Tarsk... |
| merco1lem4 1723 | Used to rederive the Tarsk... |
| merco1lem5 1724 | Used to rederive the Tarsk... |
| merco1lem6 1725 | Used to rederive the Tarsk... |
| merco1lem7 1726 | Used to rederive the Tarsk... |
| retbwax3 1727 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1728 | Used to rederive the Tarsk... |
| merco1lem9 1729 | Used to rederive the Tarsk... |
| merco1lem10 1730 | Used to rederive the Tarsk... |
| merco1lem11 1731 | Used to rederive the Tarsk... |
| merco1lem12 1732 | Used to rederive the Tarsk... |
| merco1lem13 1733 | Used to rederive the Tarsk... |
| merco1lem14 1734 | Used to rederive the Tarsk... |
| merco1lem15 1735 | Used to rederive the Tarsk... |
| merco1lem16 1736 | Used to rederive the Tarsk... |
| merco1lem17 1737 | Used to rederive the Tarsk... |
| merco1lem18 1738 | Used to rederive the Tarsk... |
| retbwax1 1739 | ~ tbw-ax1 rederived from ~... |
| merco2 1740 | A single axiom for proposi... |
| mercolem1 1741 | Used to rederive the Tarsk... |
| mercolem2 1742 | Used to rederive the Tarsk... |
| mercolem3 1743 | Used to rederive the Tarsk... |
| mercolem4 1744 | Used to rederive the Tarsk... |
| mercolem5 1745 | Used to rederive the Tarsk... |
| mercolem6 1746 | Used to rederive the Tarsk... |
| mercolem7 1747 | Used to rederive the Tarsk... |
| mercolem8 1748 | Used to rederive the Tarsk... |
| re1tbw1 1749 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1750 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1751 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1752 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1753 | Justification for ~ rb-imd... |
| rb-imdf 1754 | The definition of implicat... |
| anmp 1755 | Modus ponens for ` { \/ , ... |
| rb-ax1 1756 | The first of four axioms i... |
| rb-ax2 1757 | The second of four axioms ... |
| rb-ax3 1758 | The third of four axioms i... |
| rb-ax4 1759 | The fourth of four axioms ... |
| rbsyl 1760 | Used to rederive the Lukas... |
| rblem1 1761 | Used to rederive the Lukas... |
| rblem2 1762 | Used to rederive the Lukas... |
| rblem3 1763 | Used to rederive the Lukas... |
| rblem4 1764 | Used to rederive the Lukas... |
| rblem5 1765 | Used to rederive the Lukas... |
| rblem6 1766 | Used to rederive the Lukas... |
| rblem7 1767 | Used to rederive the Lukas... |
| re1axmp 1768 | ~ ax-mp derived from Russe... |
| re2luk1 1769 | ~ luk-1 derived from Russe... |
| re2luk2 1770 | ~ luk-2 derived from Russe... |
| re2luk3 1771 | ~ luk-3 derived from Russe... |
| mptnan 1772 | Modus ponendo tollens 1, o... |
| mptxor 1773 | Modus ponendo tollens 2, o... |
| mtpor 1774 | Modus tollendo ponens (inc... |
| mtpxor 1775 | Modus tollendo ponens (ori... |
| stoic1a 1776 | Stoic logic Thema 1 (part ... |
| stoic1b 1777 | Stoic logic Thema 1 (part ... |
| stoic2a 1778 | Stoic logic Thema 2 versio... |
| stoic2b 1779 | Stoic logic Thema 2 versio... |
| stoic3 1780 | Stoic logic Thema 3. Stat... |
| stoic4a 1781 | Stoic logic Thema 4 versio... |
| stoic4b 1782 | Stoic logic Thema 4 versio... |
| alnex 1785 | Universal quantification o... |
| eximal 1786 | An equivalence between an ... |
| nf2 1789 | Alternate definition of no... |
| nf3 1790 | Alternate definition of no... |
| nf4 1791 | Alternate definition of no... |
| nfi 1792 | Deduce that ` x ` is not f... |
| nfri 1793 | Consequence of the definit... |
| nfd 1794 | Deduce that ` x ` is not f... |
| nfrd 1795 | Consequence of the definit... |
| nftht 1796 | Closed form of ~ nfth . (... |
| nfntht 1797 | Closed form of ~ nfnth . ... |
| nfntht2 1798 | Closed form of ~ nfnth . ... |
| gen2 1800 | Generalization applied twi... |
| mpg 1801 | Modus ponens combined with... |
| mpgbi 1802 | Modus ponens on biconditio... |
| mpgbir 1803 | Modus ponens on biconditio... |
| nex 1804 | Generalization rule for ne... |
| nfth 1805 | No variable is (effectivel... |
| nfnth 1806 | No variable is (effectivel... |
| hbth 1807 | No variable is (effectivel... |
| nftru 1808 | The true constant has no f... |
| nffal 1809 | The false constant has no ... |
| sptruw 1810 | Version of ~ sp when ` ph ... |
| altru 1811 | For all sets, ` T. ` is tr... |
| alfal 1812 | For all sets, ` -. F. ` is... |
| alim 1814 | Restatement of Axiom ~ ax-... |
| alimi 1815 | Inference quantifying both... |
| 2alimi 1816 | Inference doubly quantifyi... |
| ala1 1817 | Add an antecedent in a uni... |
| al2im 1818 | Closed form of ~ al2imi . ... |
| al2imi 1819 | Inference quantifying ante... |
| alanimi 1820 | Variant of ~ al2imi with c... |
| alimdh 1821 | Deduction form of Theorem ... |
| albi 1822 | Theorem 19.15 of [Margaris... |
| albii 1823 | Inference adding universal... |
| 2albii 1824 | Inference adding two unive... |
| sylgt 1825 | Closed form of ~ sylg . (... |
| sylg 1826 | A syllogism combined with ... |
| alrimih 1827 | Inference form of Theorem ... |
| hbxfrbi 1828 | A utility lemma to transfe... |
| alex 1829 | Universal quantifier in te... |
| exnal 1830 | Existential quantification... |
| 2nalexn 1831 | Part of theorem *11.5 in [... |
| 2exnaln 1832 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1833 | Theorem *11.25 in [Whitehe... |
| alimex 1834 | An equivalence between an ... |
| aleximi 1835 | A variant of ~ al2imi : in... |
| alexbii 1836 | Biconditional form of ~ al... |
| exim 1837 | Theorem 19.22 of [Margaris... |
| eximi 1838 | Inference adding existenti... |
| 2eximi 1839 | Inference adding two exist... |
| eximii 1840 | Inference associated with ... |
| exa1 1841 | Add an antecedent in an ex... |
| 19.38 1842 | Theorem 19.38 of [Margaris... |
| 19.38a 1843 | Under a nonfreeness hypoth... |
| 19.38b 1844 | Under a nonfreeness hypoth... |
| imnang 1845 | Quantified implication in ... |
| alinexa 1846 | A transformation of quanti... |
| exnalimn 1847 | Existential quantification... |
| alexn 1848 | A relationship between two... |
| 2exnexn 1849 | Theorem *11.51 in [Whitehe... |
| exbi 1850 | Theorem 19.18 of [Margaris... |
| exbii 1851 | Inference adding existenti... |
| 2exbii 1852 | Inference adding two exist... |
| 3exbii 1853 | Inference adding three exi... |
| nfbiit 1854 | Equivalence theorem for th... |
| nfbii 1855 | Equality theorem for the n... |
| nfxfr 1856 | A utility lemma to transfe... |
| nfxfrd 1857 | A utility lemma to transfe... |
| nfnbi 1858 | A variable is nonfree in a... |
| nfnbiOLD 1859 | Obsolete version of ~ nfnb... |
| nfnt 1860 | If a variable is nonfree i... |
| nfn 1861 | Inference associated with ... |
| nfnd 1862 | Deduction associated with ... |
| exanali 1863 | A transformation of quanti... |
| 2exanali 1864 | Theorem *11.521 in [Whiteh... |
| exancom 1865 | Commutation of conjunction... |
| exan 1866 | Place a conjunct in the sc... |
| alrimdh 1867 | Deduction form of Theorem ... |
| eximdh 1868 | Deduction from Theorem 19.... |
| nexdh 1869 | Deduction for generalizati... |
| albidh 1870 | Formula-building rule for ... |
| exbidh 1871 | Formula-building rule for ... |
| exsimpl 1872 | Simplification of an exist... |
| exsimpr 1873 | Simplification of an exist... |
| 19.26 1874 | Theorem 19.26 of [Margaris... |
| 19.26-2 1875 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1876 | Theorem ~ 19.26 with tripl... |
| 19.29 1877 | Theorem 19.29 of [Margaris... |
| 19.29r 1878 | Variation of ~ 19.29 . (C... |
| 19.29r2 1879 | Variation of ~ 19.29r with... |
| 19.29x 1880 | Variation of ~ 19.29 with ... |
| 19.35 1881 | Theorem 19.35 of [Margaris... |
| 19.35i 1882 | Inference associated with ... |
| 19.35ri 1883 | Inference associated with ... |
| 19.25 1884 | Theorem 19.25 of [Margaris... |
| 19.30 1885 | Theorem 19.30 of [Margaris... |
| 19.43 1886 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1887 | Obsolete proof of ~ 19.43 ... |
| 19.33 1888 | Theorem 19.33 of [Margaris... |
| 19.33b 1889 | The antecedent provides a ... |
| 19.40 1890 | Theorem 19.40 of [Margaris... |
| 19.40-2 1891 | Theorem *11.42 in [Whitehe... |
| 19.40b 1892 | The antecedent provides a ... |
| albiim 1893 | Split a biconditional and ... |
| 2albiim 1894 | Split a biconditional and ... |
| exintrbi 1895 | Add/remove a conjunct in t... |
| exintr 1896 | Introduce a conjunct in th... |
| alsyl 1897 | Universally quantified and... |
| nfimd 1898 | If in a context ` x ` is n... |
| nfimt 1899 | Closed form of ~ nfim and ... |
| nfim 1900 | If ` x ` is not free in ` ... |
| nfand 1901 | If in a context ` x ` is n... |
| nf3and 1902 | Deduction form of bound-va... |
| nfan 1903 | If ` x ` is not free in ` ... |
| nfnan 1904 | If ` x ` is not free in ` ... |
| nf3an 1905 | If ` x ` is not free in ` ... |
| nfbid 1906 | If in a context ` x ` is n... |
| nfbi 1907 | If ` x ` is not free in ` ... |
| nfor 1908 | If ` x ` is not free in ` ... |
| nf3or 1909 | If ` x ` is not free in ` ... |
| empty 1910 | Two characterizations of t... |
| emptyex 1911 | On the empty domain, any e... |
| emptyal 1912 | On the empty domain, any u... |
| emptynf 1913 | On the empty domain, any v... |
| ax5d 1915 | Version of ~ ax-5 with ant... |
| ax5e 1916 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1917 | If a formula holds for som... |
| nfv 1918 | If ` x ` is not present in... |
| nfvd 1919 | ~ nfv with antecedent. Us... |
| alimdv 1920 | Deduction form of Theorem ... |
| eximdv 1921 | Deduction form of Theorem ... |
| 2alimdv 1922 | Deduction form of Theorem ... |
| 2eximdv 1923 | Deduction form of Theorem ... |
| albidv 1924 | Formula-building rule for ... |
| exbidv 1925 | Formula-building rule for ... |
| nfbidv 1926 | An equality theorem for no... |
| 2albidv 1927 | Formula-building rule for ... |
| 2exbidv 1928 | Formula-building rule for ... |
| 3exbidv 1929 | Formula-building rule for ... |
| 4exbidv 1930 | Formula-building rule for ... |
| alrimiv 1931 | Inference form of Theorem ... |
| alrimivv 1932 | Inference form of Theorem ... |
| alrimdv 1933 | Deduction form of Theorem ... |
| exlimiv 1934 | Inference form of Theorem ... |
| exlimiiv 1935 | Inference (Rule C) associa... |
| exlimivv 1936 | Inference form of Theorem ... |
| exlimdv 1937 | Deduction form of Theorem ... |
| exlimdvv 1938 | Deduction form of Theorem ... |
| exlimddv 1939 | Existential elimination ru... |
| nexdv 1940 | Deduction for generalizati... |
| 2ax5 1941 | Quantification of two vari... |
| stdpc5v 1942 | Version of ~ stdpc5 with a... |
| 19.21v 1943 | Version of ~ 19.21 with a ... |
| 19.32v 1944 | Version of ~ 19.32 with a ... |
| 19.31v 1945 | Version of ~ 19.31 with a ... |
| 19.23v 1946 | Version of ~ 19.23 with a ... |
| 19.23vv 1947 | Theorem ~ 19.23v extended ... |
| pm11.53v 1948 | Version of ~ pm11.53 with ... |
| 19.36imv 1949 | One direction of ~ 19.36v ... |
| 19.36imvOLD 1950 | Obsolete version of ~ 19.3... |
| 19.36iv 1951 | Inference associated with ... |
| 19.37imv 1952 | One direction of ~ 19.37v ... |
| 19.37iv 1953 | Inference associated with ... |
| 19.41v 1954 | Version of ~ 19.41 with a ... |
| 19.41vv 1955 | Version of ~ 19.41 with tw... |
| 19.41vvv 1956 | Version of ~ 19.41 with th... |
| 19.41vvvv 1957 | Version of ~ 19.41 with fo... |
| 19.42v 1958 | Version of ~ 19.42 with a ... |
| exdistr 1959 | Distribution of existentia... |
| exdistrv 1960 | Distribute a pair of exist... |
| 4exdistrv 1961 | Distribute two pairs of ex... |
| 19.42vv 1962 | Version of ~ 19.42 with tw... |
| exdistr2 1963 | Distribution of existentia... |
| 19.42vvv 1964 | Version of ~ 19.42 with th... |
| 3exdistr 1965 | Distribution of existentia... |
| 4exdistr 1966 | Distribution of existentia... |
| weq 1967 | Extend wff definition to i... |
| speimfw 1968 | Specialization, with addit... |
| speimfwALT 1969 | Alternate proof of ~ speim... |
| spimfw 1970 | Specialization, with addit... |
| ax12i 1971 | Inference that has ~ ax-12... |
| ax6v 1973 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1974 | At least one individual ex... |
| spimw 1975 | Specialization. Lemma 8 o... |
| spimew 1976 | Existential introduction, ... |
| speiv 1977 | Inference from existential... |
| speivw 1978 | Version of ~ spei with a d... |
| exgen 1979 | Rule of existential genera... |
| extru 1980 | There exists a variable su... |
| 19.2 1981 | Theorem 19.2 of [Margaris]... |
| 19.2d 1982 | Deduction associated with ... |
| 19.8w 1983 | Weak version of ~ 19.8a an... |
| spnfw 1984 | Weak version of ~ sp . Us... |
| spvw 1985 | Version of ~ sp when ` x `... |
| 19.3v 1986 | Version of ~ 19.3 with a d... |
| 19.8v 1987 | Version of ~ 19.8a with a ... |
| 19.9v 1988 | Version of ~ 19.9 with a d... |
| 19.39 1989 | Theorem 19.39 of [Margaris... |
| 19.24 1990 | Theorem 19.24 of [Margaris... |
| 19.34 1991 | Theorem 19.34 of [Margaris... |
| 19.36v 1992 | Version of ~ 19.36 with a ... |
| 19.12vvv 1993 | Version of ~ 19.12vv with ... |
| 19.27v 1994 | Version of ~ 19.27 with a ... |
| 19.28v 1995 | Version of ~ 19.28 with a ... |
| 19.37v 1996 | Version of ~ 19.37 with a ... |
| 19.44v 1997 | Version of ~ 19.44 with a ... |
| 19.45v 1998 | Version of ~ 19.45 with a ... |
| spimevw 1999 | Existential introduction, ... |
| spimvw 2000 | A weak form of specializat... |
| spvv 2001 | Specialization, using impl... |
| spfalw 2002 | Version of ~ sp when ` ph ... |
| chvarvv 2003 | Implicit substitution of `... |
| equs4v 2004 | Version of ~ equs4 with a ... |
| alequexv 2005 | Version of ~ equs4v with i... |
| exsbim 2006 | One direction of the equiv... |
| equsv 2007 | If a formula does not cont... |
| equsalvw 2008 | Version of ~ equsalv with ... |
| equsexvw 2009 | Version of ~ equsexv with ... |
| cbvaliw 2010 | Change bound variable. Us... |
| cbvalivw 2011 | Change bound variable. Us... |
| ax7v 2013 | Weakened version of ~ ax-7... |
| ax7v1 2014 | First of two weakened vers... |
| ax7v2 2015 | Second of two weakened ver... |
| equid 2016 | Identity law for equality.... |
| nfequid 2017 | Bound-variable hypothesis ... |
| equcomiv 2018 | Weaker form of ~ equcomi w... |
| ax6evr 2019 | A commuted form of ~ ax6ev... |
| ax7 2020 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2021 | Commutative law for equali... |
| equcom 2022 | Commutative law for equali... |
| equcomd 2023 | Deduction form of ~ equcom... |
| equcoms 2024 | An inference commuting equ... |
| equtr 2025 | A transitive law for equal... |
| equtrr 2026 | A transitive law for equal... |
| equeuclr 2027 | Commuted version of ~ eque... |
| equeucl 2028 | Equality is a left-Euclide... |
| equequ1 2029 | An equivalence law for equ... |
| equequ2 2030 | An equivalence law for equ... |
| equtr2 2031 | Equality is a left-Euclide... |
| stdpc6 2032 | One of the two equality ax... |
| equvinv 2033 | A variable introduction la... |
| equvinva 2034 | A modified version of the ... |
| equvelv 2035 | A biconditional form of ~ ... |
| ax13b 2036 | An equivalence between two... |
| spfw 2037 | Weak version of ~ sp . Us... |
| spw 2038 | Weak version of the specia... |
| cbvalw 2039 | Change bound variable. Us... |
| cbvalvw 2040 | Change bound variable. Us... |
| cbvexvw 2041 | Change bound variable. Us... |
| cbvaldvaw 2042 | Rule used to change the bo... |
| cbvexdvaw 2043 | Rule used to change the bo... |
| cbval2vw 2044 | Rule used to change bound ... |
| cbvex2vw 2045 | Rule used to change bound ... |
| cbvex4vw 2046 | Rule used to change bound ... |
| alcomiw 2047 | Weak version of ~ alcom . ... |
| alcomiwOLD 2048 | Obsolete version of ~ alco... |
| hbn1fw 2049 | Weak version of ~ ax-10 fr... |
| hbn1w 2050 | Weak version of ~ hbn1 . ... |
| hba1w 2051 | Weak version of ~ hba1 . ... |
| hbe1w 2052 | Weak version of ~ hbe1 . ... |
| hbalw 2053 | Weak version of ~ hbal . ... |
| 19.8aw 2054 | If a formula is true, then... |
| exexw 2055 | Existential quantification... |
| spaev 2056 | A special instance of ~ sp... |
| cbvaev 2057 | Change bound variable in a... |
| aevlem0 2058 | Lemma for ~ aevlem . Inst... |
| aevlem 2059 | Lemma for ~ aev and ~ axc1... |
| aeveq 2060 | The antecedent ` A. x x = ... |
| aev 2061 | A "distinctor elimination"... |
| aev2 2062 | A version of ~ aev with tw... |
| hbaev 2063 | All variables are effectiv... |
| naev 2064 | If some set variables can ... |
| naev2 2065 | Generalization of ~ hbnaev... |
| hbnaev 2066 | Any variable is free in ` ... |
| sbjust 2067 | Justification theorem for ... |
| sbt 2070 | A substitution into a theo... |
| sbtru 2071 | The result of substituting... |
| stdpc4 2072 | The specialization axiom o... |
| sbtALT 2073 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2074 | A double specialization us... |
| sbi1 2075 | Distribute substitution ov... |
| spsbim 2076 | Distribute substitution ov... |
| spsbbi 2077 | Biconditional property for... |
| sbimi 2078 | Distribute substitution ov... |
| sb2imi 2079 | Distribute substitution ov... |
| sbbii 2080 | Infer substitution into bo... |
| 2sbbii 2081 | Infer double substitution ... |
| sbimdv 2082 | Deduction substituting bot... |
| sbbidv 2083 | Deduction substituting bot... |
| sban 2084 | Conjunction inside and out... |
| sb3an 2085 | Threefold conjunction insi... |
| spsbe 2086 | Existential generalization... |
| sbequ 2087 | Equality property for subs... |
| sbequi 2088 | An equality theorem for su... |
| sb6 2089 | Alternate definition of su... |
| 2sb6 2090 | Equivalence for double sub... |
| sb1v 2091 | One direction of ~ sb5 , p... |
| sbv 2092 | Substitution for a variabl... |
| sbcom4 2093 | Commutativity law for subs... |
| pm11.07 2094 | Axiom *11.07 in [Whitehead... |
| sbrimvlem 2095 | Common proof template for ... |
| sbrimvw 2096 | Substitution in an implica... |
| sbievw 2097 | Conversion of implicit sub... |
| sbiedvw 2098 | Conversion of implicit sub... |
| 2sbievw 2099 | Conversion of double impli... |
| sbcom3vv 2100 | Substituting ` y ` for ` x... |
| sbievw2 2101 | ~ sbievw applied twice, av... |
| sbco2vv 2102 | A composition law for subs... |
| equsb3 2103 | Substitution in an equalit... |
| equsb3r 2104 | Substitution applied to th... |
| equsb1v 2105 | Substitution applied to an... |
| nsb 2106 | Any substitution in an alw... |
| sbn1 2107 | One direction of ~ sbn , u... |
| wel 2109 | Extend wff definition to i... |
| ax8v 2111 | Weakened version of ~ ax-8... |
| ax8v1 2112 | First of two weakened vers... |
| ax8v2 2113 | Second of two weakened ver... |
| ax8 2114 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2115 | An identity law for the no... |
| elsb1 2116 | Substitution for the first... |
| cleljust 2117 | When the class variables i... |
| ax9v 2119 | Weakened version of ~ ax-9... |
| ax9v1 2120 | First of two weakened vers... |
| ax9v2 2121 | Second of two weakened ver... |
| ax9 2122 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2123 | An identity law for the no... |
| elequ2g 2124 | A form of ~ elequ2 with a ... |
| elsb2 2125 | Substitution for the secon... |
| ax6dgen 2126 | Tarski's system uses the w... |
| ax10w 2127 | Weak version of ~ ax-10 fr... |
| ax11w 2128 | Weak version of ~ ax-11 fr... |
| ax11dgen 2129 | Degenerate instance of ~ a... |
| ax12wlem 2130 | Lemma for weak version of ... |
| ax12w 2131 | Weak version of ~ ax-12 fr... |
| ax12dgen 2132 | Degenerate instance of ~ a... |
| ax12wdemo 2133 | Example of an application ... |
| ax13w 2134 | Weak version (principal in... |
| ax13dgen1 2135 | Degenerate instance of ~ a... |
| ax13dgen2 2136 | Degenerate instance of ~ a... |
| ax13dgen3 2137 | Degenerate instance of ~ a... |
| ax13dgen4 2138 | Degenerate instance of ~ a... |
| hbn1 2140 | Alias for ~ ax-10 to be us... |
| hbe1 2141 | The setvar ` x ` is not fr... |
| hbe1a 2142 | Dual statement of ~ hbe1 .... |
| nf5-1 2143 | One direction of ~ nf5 can... |
| nf5i 2144 | Deduce that ` x ` is not f... |
| nf5dh 2145 | Deduce that ` x ` is not f... |
| nf5dv 2146 | Apply the definition of no... |
| nfnaew 2147 | All variables are effectiv... |
| nfnaewOLD 2148 | Obsolete version of ~ nfna... |
| nfe1 2149 | The setvar ` x ` is not fr... |
| nfa1 2150 | The setvar ` x ` is not fr... |
| nfna1 2151 | A convenience theorem part... |
| nfia1 2152 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2153 | The setvar ` x ` is not fr... |
| modal5 2154 | The analogue in our predic... |
| nfs1v 2155 | The setvar ` x ` is not fr... |
| alcoms 2157 | Swap quantifiers in an ant... |
| alcom 2158 | Theorem 19.5 of [Margaris]... |
| alrot3 2159 | Theorem *11.21 in [Whitehe... |
| alrot4 2160 | Rotate four universal quan... |
| sbal 2161 | Move universal quantifier ... |
| sbalv 2162 | Quantify with new variable... |
| sbcom2 2163 | Commutativity law for subs... |
| excom 2164 | Theorem 19.11 of [Margaris... |
| excomim 2165 | One direction of Theorem 1... |
| excom13 2166 | Swap 1st and 3rd existenti... |
| exrot3 2167 | Rotate existential quantif... |
| exrot4 2168 | Rotate existential quantif... |
| hbal 2169 | If ` x ` is not free in ` ... |
| hbald 2170 | Deduction form of bound-va... |
| hbsbw 2171 | If ` z ` is not free in ` ... |
| nfa2 2172 | Lemma 24 of [Monk2] p. 114... |
| ax12v 2174 | This is essentially Axiom ... |
| ax12v2 2175 | It is possible to remove a... |
| 19.8a 2176 | If a wff is true, it is tr... |
| 19.8ad 2177 | If a wff is true, it is tr... |
| sp 2178 | Specialization. A univers... |
| spi 2179 | Inference rule of universa... |
| sps 2180 | Generalization of antecede... |
| 2sp 2181 | A double specialization (s... |
| spsd 2182 | Deduction generalizing ant... |
| 19.2g 2183 | Theorem 19.2 of [Margaris]... |
| 19.21bi 2184 | Inference form of ~ 19.21 ... |
| 19.21bbi 2185 | Inference removing two uni... |
| 19.23bi 2186 | Inference form of Theorem ... |
| nexr 2187 | Inference associated with ... |
| qexmid 2188 | Quantified excluded middle... |
| nf5r 2189 | Consequence of the definit... |
| nf5rOLD 2190 | Obsolete version of ~ nfrd... |
| nf5ri 2191 | Consequence of the definit... |
| nf5rd 2192 | Consequence of the definit... |
| spimedv 2193 | Deduction version of ~ spi... |
| spimefv 2194 | Version of ~ spime with a ... |
| nfim1 2195 | A closed form of ~ nfim . ... |
| nfan1 2196 | A closed form of ~ nfan . ... |
| 19.3t 2197 | Closed form of ~ 19.3 and ... |
| 19.3 2198 | A wff may be quantified wi... |
| 19.9d 2199 | A deduction version of one... |
| 19.9t 2200 | Closed form of ~ 19.9 and ... |
| 19.9 2201 | A wff may be existentially... |
| 19.21t 2202 | Closed form of Theorem 19.... |
| 19.21 2203 | Theorem 19.21 of [Margaris... |
| stdpc5 2204 | An axiom scheme of standar... |
| 19.21-2 2205 | Version of ~ 19.21 with tw... |
| 19.23t 2206 | Closed form of Theorem 19.... |
| 19.23 2207 | Theorem 19.23 of [Margaris... |
| alimd 2208 | Deduction form of Theorem ... |
| alrimi 2209 | Inference form of Theorem ... |
| alrimdd 2210 | Deduction form of Theorem ... |
| alrimd 2211 | Deduction form of Theorem ... |
| eximd 2212 | Deduction form of Theorem ... |
| exlimi 2213 | Inference associated with ... |
| exlimd 2214 | Deduction form of Theorem ... |
| exlimimdd 2215 | Existential elimination ru... |
| exlimdd 2216 | Existential elimination ru... |
| nexd 2217 | Deduction for generalizati... |
| albid 2218 | Formula-building rule for ... |
| exbid 2219 | Formula-building rule for ... |
| nfbidf 2220 | An equality theorem for ef... |
| 19.16 2221 | Theorem 19.16 of [Margaris... |
| 19.17 2222 | Theorem 19.17 of [Margaris... |
| 19.27 2223 | Theorem 19.27 of [Margaris... |
| 19.28 2224 | Theorem 19.28 of [Margaris... |
| 19.19 2225 | Theorem 19.19 of [Margaris... |
| 19.36 2226 | Theorem 19.36 of [Margaris... |
| 19.36i 2227 | Inference associated with ... |
| 19.37 2228 | Theorem 19.37 of [Margaris... |
| 19.32 2229 | Theorem 19.32 of [Margaris... |
| 19.31 2230 | Theorem 19.31 of [Margaris... |
| 19.41 2231 | Theorem 19.41 of [Margaris... |
| 19.42 2232 | Theorem 19.42 of [Margaris... |
| 19.44 2233 | Theorem 19.44 of [Margaris... |
| 19.45 2234 | Theorem 19.45 of [Margaris... |
| spimfv 2235 | Specialization, using impl... |
| chvarfv 2236 | Implicit substitution of `... |
| cbv3v2 2237 | Version of ~ cbv3 with two... |
| sbalex 2238 | Equivalence of two ways to... |
| sb4av 2239 | Version of ~ sb4a with a d... |
| sbimd 2240 | Deduction substituting bot... |
| sbbid 2241 | Deduction substituting bot... |
| 2sbbid 2242 | Deduction doubly substitut... |
| sbequ1 2243 | An equality theorem for su... |
| sbequ2 2244 | An equality theorem for su... |
| sbequ2OLD 2245 | Obsolete version of ~ sbeq... |
| stdpc7 2246 | One of the two equality ax... |
| sbequ12 2247 | An equality theorem for su... |
| sbequ12r 2248 | An equality theorem for su... |
| sbelx 2249 | Elimination of substitutio... |
| sbequ12a 2250 | An equality theorem for su... |
| sbid 2251 | An identity theorem for su... |
| sbcov 2252 | A composition law for subs... |
| sb6a 2253 | Equivalence for substituti... |
| sbid2vw 2254 | Reverting substitution yie... |
| axc16g 2255 | Generalization of ~ axc16 ... |
| axc16 2256 | Proof of older axiom ~ ax-... |
| axc16gb 2257 | Biconditional strengthenin... |
| axc16nf 2258 | If ~ dtru is false, then t... |
| axc11v 2259 | Version of ~ axc11 with a ... |
| axc11rv 2260 | Version of ~ axc11r with a... |
| drsb2 2261 | Formula-building lemma for... |
| equsalv 2262 | An equivalence related to ... |
| equsexv 2263 | An equivalence related to ... |
| equsexvOLD 2264 | Obsolete version of ~ equs... |
| sbft 2265 | Substitution has no effect... |
| sbf 2266 | Substitution for a variabl... |
| sbf2 2267 | Substitution has no effect... |
| sbh 2268 | Substitution for a variabl... |
| hbs1 2269 | The setvar ` x ` is not fr... |
| nfs1f 2270 | If ` x ` is not free in ` ... |
| sb5 2271 | Alternate definition of su... |
| sb5OLD 2272 | Obsolete version of ~ sb5 ... |
| sb56OLD 2273 | Obsolete version of ~ sbal... |
| equs5av 2274 | A property related to subs... |
| 2sb5 2275 | Equivalence for double sub... |
| sbco4lem 2276 | Lemma for ~ sbco4 . It re... |
| sbco4lemOLD 2277 | Obsolete version of ~ sbco... |
| sbco4 2278 | Two ways of exchanging two... |
| dfsb7 2279 | An alternate definition of... |
| sbn 2280 | Negation inside and outsid... |
| sbex 2281 | Move existential quantifie... |
| nf5 2282 | Alternate definition of ~ ... |
| nf6 2283 | An alternate definition of... |
| nf5d 2284 | Deduce that ` x ` is not f... |
| nf5di 2285 | Since the converse holds b... |
| 19.9h 2286 | A wff may be existentially... |
| 19.21h 2287 | Theorem 19.21 of [Margaris... |
| 19.23h 2288 | Theorem 19.23 of [Margaris... |
| exlimih 2289 | Inference associated with ... |
| exlimdh 2290 | Deduction form of Theorem ... |
| equsalhw 2291 | Version of ~ equsalh with ... |
| equsexhv 2292 | An equivalence related to ... |
| hba1 2293 | The setvar ` x ` is not fr... |
| hbnt 2294 | Closed theorem version of ... |
| hbn 2295 | If ` x ` is not free in ` ... |
| hbnd 2296 | Deduction form of bound-va... |
| hbim1 2297 | A closed form of ~ hbim . ... |
| hbimd 2298 | Deduction form of bound-va... |
| hbim 2299 | If ` x ` is not free in ` ... |
| hban 2300 | If ` x ` is not free in ` ... |
| hb3an 2301 | If ` x ` is not free in ` ... |
| sbi2 2302 | Introduction of implicatio... |
| sbim 2303 | Implication inside and out... |
| 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... |
| sblbis 2309 | Introduce left bicondition... |
| sbrbis 2310 | Introduce right biconditio... |
| sbrbif 2311 | Introduce right biconditio... |
| sbiev 2312 | Conversion of implicit sub... |
| sbiedw 2313 | Conversion of implicit sub... |
| sbiedwOLD 2314 | Obsolete version of ~ sbie... |
| axc7 2315 | Show that the original axi... |
| axc7e 2316 | Abbreviated version of ~ a... |
| modal-b 2317 | The analogue in our predic... |
| 19.9ht 2318 | A closed version of ~ 19.9... |
| axc4 2319 | Show that the original axi... |
| axc4i 2320 | Inference version of ~ axc... |
| nfal 2321 | If ` x ` is not free in ` ... |
| nfex 2322 | If ` x ` is not free in ` ... |
| hbex 2323 | If ` x ` is not free in ` ... |
| nfnf 2324 | If ` x ` is not free in ` ... |
| 19.12 2325 | Theorem 19.12 of [Margaris... |
| nfald 2326 | Deduction form of ~ nfal .... |
| nfexd 2327 | If ` x ` is not free in ` ... |
| nfsbv 2328 | If ` z ` is not free in ` ... |
| nfsbvOLD 2329 | Obsolete version of ~ nfsb... |
| hbsbwOLD 2330 | Obsolete version of ~ hbsb... |
| sbco2v 2331 | A composition law for subs... |
| aaan 2332 | Distribute universal quant... |
| eeor 2333 | Distribute existential qua... |
| cbv3v 2334 | Rule used to change bound ... |
| cbv1v 2335 | Rule used to change bound ... |
| cbv2w 2336 | Rule used to change bound ... |
| cbvaldw 2337 | Deduction used to change b... |
| cbvexdw 2338 | Deduction used to change b... |
| cbv3hv 2339 | Rule used to change bound ... |
| cbvalv1 2340 | Rule used to change bound ... |
| cbvexv1 2341 | Rule used to change bound ... |
| cbval2v 2342 | Rule used to change bound ... |
| cbval2vOLD 2343 | Obsolete version of ~ cbva... |
| cbvex2v 2344 | Rule used to change bound ... |
| dvelimhw 2345 | Proof of ~ dvelimh without... |
| pm11.53 2346 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2347 | Special case of ~ 19.12 wh... |
| eean 2348 | Distribute existential qua... |
| eeanv 2349 | Distribute a pair of exist... |
| eeeanv 2350 | Distribute three existenti... |
| ee4anv 2351 | Distribute two pairs of ex... |
| sb8v 2352 | Substitution of variable i... |
| sb8ev 2353 | Substitution of variable i... |
| 2sb8ev 2354 | An equivalent expression f... |
| sb6rfv 2355 | Reversed substitution. Ve... |
| sbnf2 2356 | Two ways of expressing " `... |
| exsb 2357 | An equivalent expression f... |
| 2exsb 2358 | An equivalent expression f... |
| sbbib 2359 | Reversal of substitution. ... |
| sbbibvv 2360 | Reversal of substitution. ... |
| sbievg 2361 | Substitution applied to ex... |
| cleljustALT 2362 | Alternate proof of ~ clelj... |
| cleljustALT2 2363 | Alternate proof of ~ clelj... |
| equs5aALT 2364 | Alternate proof of ~ equs5... |
| equs5eALT 2365 | Alternate proof of ~ equs5... |
| axc11r 2366 | Same as ~ axc11 but with r... |
| dral1v 2367 | Formula-building lemma for... |
| dral1vOLD 2368 | Obsolete version of ~ dral... |
| drex1v 2369 | Formula-building lemma for... |
| drnf1v 2370 | Formula-building lemma for... |
| drnf1vOLD 2371 | Obsolete version of ~ drnf... |
| ax13v 2373 | A weaker version of ~ ax-1... |
| ax13lem1 2374 | A version of ~ ax13v with ... |
| ax13 2375 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2376 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2377 | An equation between setvar... |
| dveeq2 2378 | Quantifier introduction wh... |
| nfeqf1 2379 | An equation between setvar... |
| dveeq1 2380 | Quantifier introduction wh... |
| nfeqf 2381 | A variable is effectively ... |
| axc9 2382 | Derive set.mm's original ~... |
| ax6e 2383 | At least one individual ex... |
| ax6 2384 | Theorem showing that ~ ax-... |
| axc10 2385 | Show that the original axi... |
| spimt 2386 | Closed theorem form of ~ s... |
| spim 2387 | Specialization, using impl... |
| spimed 2388 | Deduction version of ~ spi... |
| spime 2389 | Existential introduction, ... |
| spimv 2390 | A version of ~ spim with a... |
| spimvALT 2391 | Alternate proof of ~ spimv... |
| spimev 2392 | Distinct-variable version ... |
| spv 2393 | Specialization, using impl... |
| spei 2394 | Inference from existential... |
| chvar 2395 | Implicit substitution of `... |
| chvarv 2396 | Implicit substitution of `... |
| cbv3 2397 | Rule used to change bound ... |
| cbval 2398 | Rule used to change bound ... |
| cbvex 2399 | Rule used to change bound ... |
| cbvalv 2400 | Rule used to change bound ... |
| cbvexv 2401 | Rule used to change bound ... |
| cbv1 2402 | Rule used to change bound ... |
| cbv2 2403 | Rule used to change bound ... |
| cbv3h 2404 | Rule used to change bound ... |
| cbv1h 2405 | Rule used to change bound ... |
| cbv2h 2406 | Rule used to change bound ... |
| cbvald 2407 | Deduction used to change b... |
| cbvexd 2408 | Deduction used to change b... |
| cbvaldva 2409 | Rule used to change the bo... |
| cbvexdva 2410 | Rule used to change the bo... |
| cbval2 2411 | Rule used to change bound ... |
| cbvex2 2412 | Rule used to change bound ... |
| cbval2vv 2413 | Rule used to change bound ... |
| cbvex2vv 2414 | Rule used to change bound ... |
| cbvex4v 2415 | Rule used to change bound ... |
| equs4 2416 | Lemma used in proofs of im... |
| equsal 2417 | An equivalence related to ... |
| equsex 2418 | An equivalence related to ... |
| equsexALT 2419 | Alternate proof of ~ equse... |
| equsalh 2420 | An equivalence related to ... |
| equsexh 2421 | An equivalence related to ... |
| axc15 2422 | Derivation of set.mm's ori... |
| ax12 2423 | Rederivation of Axiom ~ ax... |
| ax12b 2424 | A bidirectional version of... |
| ax13ALT 2425 | Alternate proof of ~ ax13 ... |
| axc11n 2426 | Derive set.mm's original ~... |
| aecom 2427 | Commutation law for identi... |
| aecoms 2428 | A commutation rule for ide... |
| naecoms 2429 | A commutation rule for dis... |
| axc11 2430 | Show that ~ ax-c11 can be ... |
| hbae 2431 | All variables are effectiv... |
| hbnae 2432 | All variables are effectiv... |
| nfae 2433 | All variables are effectiv... |
| nfnae 2434 | All variables are effectiv... |
| hbnaes 2435 | Rule that applies ~ hbnae ... |
| axc16i 2436 | Inference with ~ axc16 as ... |
| axc16nfALT 2437 | Alternate proof of ~ axc16... |
| dral2 2438 | Formula-building lemma for... |
| dral1 2439 | Formula-building lemma for... |
| dral1ALT 2440 | Alternate proof of ~ dral1... |
| drex1 2441 | Formula-building lemma for... |
| drex2 2442 | Formula-building lemma for... |
| drnf1 2443 | Formula-building lemma for... |
| drnf2 2444 | Formula-building lemma for... |
| nfald2 2445 | Variation on ~ nfald which... |
| nfexd2 2446 | Variation on ~ nfexd which... |
| exdistrf 2447 | Distribution of existentia... |
| dvelimf 2448 | Version of ~ dvelimv witho... |
| dvelimdf 2449 | Deduction form of ~ dvelim... |
| dvelimh 2450 | Version of ~ dvelim withou... |
| dvelim 2451 | This theorem can be used t... |
| dvelimv 2452 | Similar to ~ dvelim with f... |
| dvelimnf 2453 | Version of ~ dvelim using ... |
| dveeq2ALT 2454 | Alternate proof of ~ dveeq... |
| equvini 2455 | A variable introduction la... |
| equvel 2456 | A variable elimination law... |
| equs5a 2457 | A property related to subs... |
| equs5e 2458 | A property related to subs... |
| equs45f 2459 | Two ways of expressing sub... |
| equs5 2460 | Lemma used in proofs of su... |
| dveel1 2461 | Quantifier introduction wh... |
| dveel2 2462 | Quantifier introduction wh... |
| axc14 2463 | Axiom ~ ax-c14 is redundan... |
| sb6x 2464 | Equivalence involving subs... |
| sbequ5 2465 | Substitution does not chan... |
| sbequ6 2466 | Substitution does not chan... |
| sb5rf 2467 | Reversed substitution. Us... |
| sb6rf 2468 | Reversed substitution. Fo... |
| ax12vALT 2469 | Alternate proof of ~ ax12v... |
| 2ax6elem 2470 | We can always find values ... |
| 2ax6e 2471 | We can always find values ... |
| 2sb5rf 2472 | Reversed double substituti... |
| 2sb6rf 2473 | Reversed double substituti... |
| sbel2x 2474 | Elimination of double subs... |
| sb4b 2475 | Simplified definition of s... |
| sb4bOLD 2476 | Obsolete version of ~ sb4b... |
| sb3b 2477 | Simplified definition of s... |
| sb3 2478 | One direction of a simplif... |
| sb1 2479 | One direction of a simplif... |
| sb2 2480 | One direction of a simplif... |
| sb3OLD 2481 | Obsolete version of ~ sb3 ... |
| sb1OLD 2482 | Obsolete version of ~ sb1 ... |
| sb3bOLD 2483 | Obsolete version of ~ sb3b... |
| sb4a 2484 | A version of one implicati... |
| dfsb1 2485 | Alternate definition of su... |
| hbsb2 2486 | Bound-variable hypothesis ... |
| nfsb2 2487 | Bound-variable hypothesis ... |
| hbsb2a 2488 | Special case of a bound-va... |
| sb4e 2489 | One direction of a simplif... |
| hbsb2e 2490 | Special case of a bound-va... |
| hbsb3 2491 | If ` y ` is not free in ` ... |
| nfs1 2492 | If ` y ` is not free in ` ... |
| axc16ALT 2493 | Alternate proof of ~ axc16... |
| axc16gALT 2494 | Alternate proof of ~ axc16... |
| equsb1 2495 | Substitution applied to an... |
| equsb2 2496 | Substitution applied to an... |
| dfsb2 2497 | An alternate definition of... |
| dfsb3 2498 | An alternate definition of... |
| drsb1 2499 | Formula-building lemma for... |
| sb2ae 2500 | In the case of two success... |
| sb6f 2501 | Equivalence for substituti... |
| sb5f 2502 | Equivalence for substituti... |
| nfsb4t 2503 | A variable not free in a p... |
| nfsb4 2504 | A variable not free in a p... |
| sbequ8 2505 | Elimination of equality fr... |
| sbie 2506 | Conversion of implicit sub... |
| sbied 2507 | Conversion of implicit sub... |
| sbiedv 2508 | Conversion of implicit sub... |
| 2sbiev 2509 | Conversion of double impli... |
| sbcom3 2510 | Substituting ` y ` for ` x... |
| sbco 2511 | A composition law for subs... |
| sbid2 2512 | An identity law for substi... |
| sbid2v 2513 | An identity law for substi... |
| sbidm 2514 | An idempotent law for subs... |
| sbco2 2515 | A composition law for subs... |
| sbco2d 2516 | A composition law for subs... |
| sbco3 2517 | A composition law for subs... |
| sbcom 2518 | A commutativity law for su... |
| sbtrt 2519 | Partially closed form of ~... |
| sbtr 2520 | A partial converse to ~ sb... |
| sb8 2521 | Substitution of variable i... |
| sb8e 2522 | Substitution of variable i... |
| sb9 2523 | Commutation of quantificat... |
| sb9i 2524 | Commutation of quantificat... |
| sbhb 2525 | Two ways of expressing " `... |
| nfsbd 2526 | Deduction version of ~ nfs... |
| nfsb 2527 | If ` z ` is not free in ` ... |
| nfsbOLD 2528 | Obsolete version of ~ nfsb... |
| hbsb 2529 | If ` z ` is not free in ` ... |
| sb7f 2530 | This version of ~ dfsb7 do... |
| sb7h 2531 | This version of ~ dfsb7 do... |
| sb10f 2532 | Hao Wang's identity axiom ... |
| sbal1 2533 | Check out ~ sbal for a ver... |
| sbal2 2534 | Move quantifier in and out... |
| 2sb8e 2535 | An equivalent expression f... |
| dfmoeu 2536 | An elementary proof of ~ m... |
| dfeumo 2537 | An elementary proof showin... |
| mojust 2539 | Soundness justification th... |
| nexmo 2541 | Nonexistence implies uniqu... |
| exmo 2542 | Any proposition holds for ... |
| moabs 2543 | Absorption of existence co... |
| moim 2544 | The at-most-one quantifier... |
| moimi 2545 | The at-most-one quantifier... |
| moimdv 2546 | The at-most-one quantifier... |
| mobi 2547 | Equivalence theorem for th... |
| mobii 2548 | Formula-building rule for ... |
| mobidv 2549 | Formula-building rule for ... |
| mobid 2550 | Formula-building rule for ... |
| moa1 2551 | If an implication holds fo... |
| moan 2552 | "At most one" is still the... |
| moani 2553 | "At most one" is still tru... |
| moor 2554 | "At most one" is still the... |
| mooran1 2555 | "At most one" imports disj... |
| mooran2 2556 | "At most one" exports disj... |
| nfmo1 2557 | Bound-variable hypothesis ... |
| nfmod2 2558 | Bound-variable hypothesis ... |
| nfmodv 2559 | Bound-variable hypothesis ... |
| nfmov 2560 | Bound-variable hypothesis ... |
| nfmod 2561 | Bound-variable hypothesis ... |
| nfmo 2562 | Bound-variable hypothesis ... |
| mof 2563 | Version of ~ df-mo with di... |
| mo3 2564 | Alternate definition of th... |
| mo 2565 | Equivalent definitions of ... |
| mo4 2566 | At-most-one quantifier exp... |
| mo4f 2567 | At-most-one quantifier exp... |
| eu3v 2570 | An alternate way to expres... |
| eujust 2571 | Soundness justification th... |
| eujustALT 2572 | Alternate proof of ~ eujus... |
| eu6lem 2573 | Lemma of ~ eu6im . A diss... |
| eu6 2574 | Alternate definition of th... |
| eu6im 2575 | One direction of ~ eu6 nee... |
| euf 2576 | Version of ~ eu6 with disj... |
| euex 2577 | Existential uniqueness imp... |
| eumo 2578 | Existential uniqueness imp... |
| eumoi 2579 | Uniqueness inferred from e... |
| exmoeub 2580 | Existence implies that uni... |
| exmoeu 2581 | Existence is equivalent to... |
| moeuex 2582 | Uniqueness implies that ex... |
| moeu 2583 | Uniqueness is equivalent t... |
| eubi 2584 | Equivalence theorem for th... |
| eubii 2585 | Introduce unique existenti... |
| eubidv 2586 | Formula-building rule for ... |
| eubid 2587 | Formula-building rule for ... |
| nfeu1 2588 | Bound-variable hypothesis ... |
| nfeu1ALT 2589 | Alternate proof of ~ nfeu1... |
| nfeud2 2590 | Bound-variable hypothesis ... |
| nfeudw 2591 | Bound-variable hypothesis ... |
| nfeud 2592 | Bound-variable hypothesis ... |
| nfeuw 2593 | Bound-variable hypothesis ... |
| nfeu 2594 | Bound-variable hypothesis ... |
| dfeu 2595 | Rederive ~ df-eu from the ... |
| dfmo 2596 | Rederive ~ df-mo from the ... |
| euequ 2597 | There exists a unique set ... |
| sb8eulem 2598 | Lemma. Factor out the com... |
| sb8euv 2599 | Variable substitution in u... |
| sb8eu 2600 | Variable substitution in u... |
| sb8mo 2601 | Variable substitution for ... |
| cbvmovw 2602 | Change bound variable. Us... |
| cbvmow 2603 | Rule used to change bound ... |
| cbvmowOLD 2604 | Obsolete version of ~ cbvm... |
| cbvmo 2605 | Rule used to change bound ... |
| cbveuvw 2606 | Change bound variable. Us... |
| cbveuw 2607 | Version of ~ cbveu with a ... |
| cbveuwOLD 2608 | Obsolete version of ~ cbve... |
| cbveu 2609 | Rule used to change bound ... |
| cbveuALT 2610 | Alternative proof of ~ cbv... |
| eu2 2611 | An alternate way of defini... |
| eu1 2612 | An alternate way to expres... |
| euor 2613 | Introduce a disjunct into ... |
| euorv 2614 | Introduce a disjunct into ... |
| euor2 2615 | Introduce or eliminate a d... |
| sbmo 2616 | Substitution into an at-mo... |
| eu4 2617 | Uniqueness using implicit ... |
| euimmo 2618 | Existential uniqueness imp... |
| euim 2619 | Add unique existential qua... |
| moanimlem 2620 | Factor out the common proo... |
| moanimv 2621 | Introduction of a conjunct... |
| moanim 2622 | Introduction of a conjunct... |
| euan 2623 | Introduction of a conjunct... |
| moanmo 2624 | Nested at-most-one quantif... |
| moaneu 2625 | Nested at-most-one and uni... |
| euanv 2626 | Introduction of a conjunct... |
| mopick 2627 | "At most one" picks a vari... |
| moexexlem 2628 | Factor out the proof skele... |
| 2moexv 2629 | Double quantification with... |
| moexexvw 2630 | "At most one" double quant... |
| 2moswapv 2631 | A condition allowing to sw... |
| 2euswapv 2632 | A condition allowing to sw... |
| 2euexv 2633 | Double quantification with... |
| 2exeuv 2634 | Double existential uniquen... |
| eupick 2635 | Existential uniqueness "pi... |
| eupicka 2636 | Version of ~ eupick with c... |
| eupickb 2637 | Existential uniqueness "pi... |
| eupickbi 2638 | Theorem *14.26 in [Whitehe... |
| mopick2 2639 | "At most one" can show the... |
| moexex 2640 | "At most one" double quant... |
| moexexv 2641 | "At most one" double quant... |
| 2moex 2642 | Double quantification with... |
| 2euex 2643 | Double quantification with... |
| 2eumo 2644 | Nested unique existential ... |
| 2eu2ex 2645 | Double existential uniquen... |
| 2moswap 2646 | A condition allowing to sw... |
| 2euswap 2647 | A condition allowing to sw... |
| 2exeu 2648 | Double existential uniquen... |
| 2mo2 2649 | Two ways of expressing "th... |
| 2mo 2650 | Two ways of expressing "th... |
| 2mos 2651 | Double "there exists at mo... |
| 2eu1 2652 | Double existential uniquen... |
| 2eu1v 2653 | Double existential uniquen... |
| 2eu2 2654 | Double existential uniquen... |
| 2eu3 2655 | Double existential uniquen... |
| 2eu4 2656 | This theorem provides us w... |
| 2eu5 2657 | An alternate definition of... |
| 2eu6 2658 | Two equivalent expressions... |
| 2eu7 2659 | Two equivalent expressions... |
| 2eu8 2660 | Two equivalent expressions... |
| euae 2661 | Two ways to express "exact... |
| exists1 2662 | Two ways to express "exact... |
| exists2 2663 | A condition implying that ... |
| barbara 2664 | "Barbara", one of the fund... |
| celarent 2665 | "Celarent", one of the syl... |
| darii 2666 | "Darii", one of the syllog... |
| dariiALT 2667 | Alternate proof of ~ darii... |
| ferio 2668 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2669 | Lemma for ~ barbari and th... |
| barbari 2670 | "Barbari", one of the syll... |
| barbariALT 2671 | Alternate proof of ~ barba... |
| celaront 2672 | "Celaront", one of the syl... |
| cesare 2673 | "Cesare", one of the syllo... |
| camestres 2674 | "Camestres", one of the sy... |
| festino 2675 | "Festino", one of the syll... |
| festinoALT 2676 | Alternate proof of ~ festi... |
| baroco 2677 | "Baroco", one of the syllo... |
| barocoALT 2678 | Alternate proof of ~ festi... |
| cesaro 2679 | "Cesaro", one of the syllo... |
| camestros 2680 | "Camestros", one of the sy... |
| datisi 2681 | "Datisi", one of the syllo... |
| disamis 2682 | "Disamis", one of the syll... |
| ferison 2683 | "Ferison", one of the syll... |
| bocardo 2684 | "Bocardo", one of the syll... |
| darapti 2685 | "Darapti", one of the syll... |
| daraptiALT 2686 | Alternate proof of ~ darap... |
| felapton 2687 | "Felapton", one of the syl... |
| calemes 2688 | "Calemes", one of the syll... |
| dimatis 2689 | "Dimatis", one of the syll... |
| fresison 2690 | "Fresison", one of the syl... |
| calemos 2691 | "Calemos", one of the syll... |
| fesapo 2692 | "Fesapo", one of the syllo... |
| bamalip 2693 | "Bamalip", one of the syll... |
| axia1 2694 | Left 'and' elimination (in... |
| axia2 2695 | Right 'and' elimination (i... |
| axia3 2696 | 'And' introduction (intuit... |
| axin1 2697 | 'Not' introduction (intuit... |
| axin2 2698 | 'Not' elimination (intuiti... |
| axio 2699 | Definition of 'or' (intuit... |
| axi4 2700 | Specialization (intuitioni... |
| axi5r 2701 | Converse of ~ axc4 (intuit... |
| axial 2702 | The setvar ` x ` is not fr... |
| axie1 2703 | The setvar ` x ` is not fr... |
| axie2 2704 | A key property of existent... |
| axi9 2705 | Axiom of existence (intuit... |
| axi10 2706 | Axiom of Quantifier Substi... |
| axi12 2707 | Axiom of Quantifier Introd... |
| axbnd 2708 | Axiom of Bundling (intuiti... |
| axexte 2710 | The axiom of extensionalit... |
| axextg 2711 | A generalization of the ax... |
| axextb 2712 | A bidirectional version of... |
| axextmo 2713 | There exists at most one s... |
| nulmo 2714 | There exists at most one e... |
| eleq1ab 2717 | Extension (in the sense of... |
| cleljustab 2718 | Extension of ~ cleljust fr... |
| abid 2719 | Simplification of class ab... |
| vexwt 2720 | A standard theorem of pred... |
| vexw 2721 | If ` ph ` is a theorem, th... |
| vextru 2722 | Every setvar is a member o... |
| nfsab1 2723 | Bound-variable hypothesis ... |
| hbab1 2724 | Bound-variable hypothesis ... |
| hbab1OLD 2725 | Obsolete version of ~ hbab... |
| hbab 2726 | Bound-variable hypothesis ... |
| hbabg 2727 | Bound-variable hypothesis ... |
| nfsab 2728 | Bound-variable hypothesis ... |
| nfsabg 2729 | Bound-variable hypothesis ... |
| dfcleq 2731 | The defining characterizat... |
| cvjust 2732 | Every set is a class. Pro... |
| ax9ALT 2733 | Proof of ~ ax-9 from Tarsk... |
| eleq2w2 2734 | A weaker version of ~ eleq... |
| eqriv 2735 | Infer equality of classes ... |
| eqrdv 2736 | Deduce equality of classes... |
| eqrdav 2737 | Deduce equality of classes... |
| eqid 2738 | Law of identity (reflexivi... |
| eqidd 2739 | Class identity law with an... |
| eqeq1d 2740 | Deduction from equality to... |
| eqeq1dALT 2741 | Alternate proof of ~ eqeq1... |
| eqeq1 2742 | Equality implies equivalen... |
| eqeq1i 2743 | Inference from equality to... |
| eqcomd 2744 | Deduction from commutative... |
| eqcom 2745 | Commutative law for class ... |
| eqcoms 2746 | Inference applying commuta... |
| eqcomi 2747 | Inference from commutative... |
| neqcomd 2748 | Commute an inequality. (C... |
| eqeq2d 2749 | Deduction from equality to... |
| eqeq2 2750 | Equality implies equivalen... |
| eqeq2i 2751 | Inference from equality to... |
| eqeqan12d 2752 | A useful inference for sub... |
| eqeqan12rd 2753 | A useful inference for sub... |
| eqeq12d 2754 | A useful inference for sub... |
| eqeq12 2755 | Equality relationship amon... |
| eqeq12i 2756 | A useful inference for sub... |
| eqeq12OLD 2757 | Obsolete version of ~ eqeq... |
| eqeq12dOLD 2758 | Obsolete version of ~ eqeq... |
| eqeqan12dOLD 2759 | Obsolete version of ~ eqeq... |
| eqeqan12dALT 2760 | Alternate proof of ~ eqeqa... |
| eqtr 2761 | Transitive law for class e... |
| eqtr2 2762 | A transitive law for class... |
| eqtr2OLD 2763 | Obsolete version of eqtr2 ... |
| eqtr3 2764 | A transitive law for class... |
| eqtr3OLD 2765 | Obsolete version of ~ eqtr... |
| eqtri 2766 | An equality transitivity i... |
| eqtr2i 2767 | An equality transitivity i... |
| eqtr3i 2768 | An equality transitivity i... |
| eqtr4i 2769 | An equality transitivity i... |
| 3eqtri 2770 | An inference from three ch... |
| 3eqtrri 2771 | An inference from three ch... |
| 3eqtr2i 2772 | An inference from three ch... |
| 3eqtr2ri 2773 | An inference from three ch... |
| 3eqtr3i 2774 | An inference from three ch... |
| 3eqtr3ri 2775 | An inference from three ch... |
| 3eqtr4i 2776 | An inference from three ch... |
| 3eqtr4ri 2777 | An inference from three ch... |
| eqtrd 2778 | An equality transitivity d... |
| eqtr2d 2779 | An equality transitivity d... |
| eqtr3d 2780 | An equality transitivity e... |
| eqtr4d 2781 | An equality transitivity e... |
| 3eqtrd 2782 | A deduction from three cha... |
| 3eqtrrd 2783 | A deduction from three cha... |
| 3eqtr2d 2784 | A deduction from three cha... |
| 3eqtr2rd 2785 | A deduction from three cha... |
| 3eqtr3d 2786 | A deduction from three cha... |
| 3eqtr3rd 2787 | A deduction from three cha... |
| 3eqtr4d 2788 | A deduction from three cha... |
| 3eqtr4rd 2789 | A deduction from three cha... |
| eqtrid 2790 | An equality transitivity d... |
| syl5eq 2791 | Renamed to ~ eqtrid . Kep... |
| eqtr2id 2792 | An equality transitivity d... |
| eqtr3id 2793 | An equality transitivity d... |
| eqtr3di 2794 | An equality transitivity d... |
| eqtrdi 2795 | An equality transitivity d... |
| eqtr2di 2796 | An equality transitivity d... |
| eqtr4di 2797 | An equality transitivity d... |
| eqtr4id 2798 | An equality transitivity d... |
| sylan9eq 2799 | An equality transitivity d... |
| sylan9req 2800 | An equality transitivity d... |
| sylan9eqr 2801 | An equality transitivity d... |
| 3eqtr3g 2802 | A chained equality inferen... |
| 3eqtr3a 2803 | A chained equality inferen... |
| 3eqtr4g 2804 | A chained equality inferen... |
| 3eqtr4a 2805 | A chained equality inferen... |
| eq2tri 2806 | A compound transitive infe... |
| abbi1 2807 | Equivalent formulas yield ... |
| abbidv 2808 | Equivalent wff's yield equ... |
| abbii 2809 | Equivalent wff's yield equ... |
| abbid 2810 | Equivalent wff's yield equ... |
| abbi 2811 | Equivalent formulas define... |
| cbvabv 2812 | Rule used to change bound ... |
| cbvabw 2813 | Rule used to change bound ... |
| cbvabwOLD 2814 | Obsolete version of ~ cbva... |
| cbvab 2815 | Rule used to change bound ... |
| abeq2w 2816 | Version of ~ abeq2 using i... |
| dfclel 2818 | Characterization of the el... |
| elissetv 2819 | An element of a class exis... |
| elisset 2820 | An element of a class exis... |
| eleq1w 2821 | Weaker version of ~ eleq1 ... |
| eleq2w 2822 | Weaker version of ~ eleq2 ... |
| eleq1d 2823 | Deduction from equality to... |
| eleq2d 2824 | Deduction from equality to... |
| eleq2dALT 2825 | Alternate proof of ~ eleq2... |
| eleq1 2826 | Equality implies equivalen... |
| eleq2 2827 | Equality implies equivalen... |
| eleq12 2828 | Equality implies equivalen... |
| eleq1i 2829 | Inference from equality to... |
| eleq2i 2830 | Inference from equality to... |
| eleq12i 2831 | Inference from equality to... |
| eqneltri 2832 | If a class is not an eleme... |
| eleq12d 2833 | Deduction from equality to... |
| eleq1a 2834 | A transitive-type law rela... |
| eqeltri 2835 | Substitution of equal clas... |
| eqeltrri 2836 | Substitution of equal clas... |
| eleqtri 2837 | Substitution of equal clas... |
| eleqtrri 2838 | Substitution of equal clas... |
| eqeltrd 2839 | Substitution of equal clas... |
| eqeltrrd 2840 | Deduction that substitutes... |
| eleqtrd 2841 | Deduction that substitutes... |
| eleqtrrd 2842 | Deduction that substitutes... |
| eqeltrid 2843 | A membership and equality ... |
| eqeltrrid 2844 | A membership and equality ... |
| eleqtrid 2845 | A membership and equality ... |
| eleqtrrid 2846 | A membership and equality ... |
| eqeltrdi 2847 | A membership and equality ... |
| eqeltrrdi 2848 | A membership and equality ... |
| eleqtrdi 2849 | A membership and equality ... |
| eleqtrrdi 2850 | A membership and equality ... |
| 3eltr3i 2851 | Substitution of equal clas... |
| 3eltr4i 2852 | Substitution of equal clas... |
| 3eltr3d 2853 | Substitution of equal clas... |
| 3eltr4d 2854 | Substitution of equal clas... |
| 3eltr3g 2855 | Substitution of equal clas... |
| 3eltr4g 2856 | Substitution of equal clas... |
| eleq2s 2857 | Substitution of equal clas... |
| eqneltrd 2858 | If a class is not an eleme... |
| eqneltrrd 2859 | If a class is not an eleme... |
| neleqtrd 2860 | If a class is not an eleme... |
| neleqtrrd 2861 | If a class is not an eleme... |
| cleqh 2862 | Establish equality between... |
| nelneq 2863 | A way of showing two class... |
| nelneq2 2864 | A way of showing two class... |
| eqsb1 2865 | Substitution for the left-... |
| clelsb1 2866 | Substitution for the first... |
| clelsb2 2867 | Substitution for the secon... |
| hbxfreq 2868 | A utility lemma to transfe... |
| hblem 2869 | Change the free variable o... |
| hblemg 2870 | Change the free variable o... |
| abeq2 2871 | Equality of a class variab... |
| abeq1 2872 | Equality of a class variab... |
| abeq2d 2873 | Equality of a class variab... |
| abeq2i 2874 | Equality of a class variab... |
| abeq1i 2875 | Equality of a class variab... |
| abbi2dv 2876 | Deduction from a wff to a ... |
| abbi1dv 2877 | Deduction from a wff to a ... |
| abbi2i 2878 | Equality of a class variab... |
| abbiOLD 2879 | Obsolete proof of ~ abbi a... |
| abid1 2880 | Every class is equal to a ... |
| abid2 2881 | A simplification of class ... |
| clelab 2882 | Membership of a class vari... |
| clelabOLD 2883 | Obsolete version of ~ clel... |
| clabel 2884 | Membership of a class abst... |
| sbab 2885 | The right-hand side of the... |
| nfcjust 2887 | Justification theorem for ... |
| nfci 2889 | Deduce that a class ` A ` ... |
| nfcii 2890 | Deduce that a class ` A ` ... |
| nfcr 2891 | Consequence of the not-fre... |
| nfcrALT 2892 | Alternate version of ~ nfc... |
| nfcri 2893 | Consequence of the not-fre... |
| nfcd 2894 | Deduce that a class ` A ` ... |
| nfcrd 2895 | Consequence of the not-fre... |
| nfcriOLD 2896 | Obsolete version of ~ nfcr... |
| nfcriOLDOLD 2897 | Obsolete version of ~ nfcr... |
| nfcrii 2898 | Consequence of the not-fre... |
| nfcriiOLD 2899 | Obsolete version of ~ nfcr... |
| nfcriOLDOLDOLD 2900 | Obsolete version of ~ nfcr... |
| nfceqdf 2901 | An equality theorem for ef... |
| nfceqdfOLD 2902 | Obsolete version of ~ nfce... |
| nfceqi 2903 | Equality theorem for class... |
| nfcxfr 2904 | A utility lemma to transfe... |
| nfcxfrd 2905 | A utility lemma to transfe... |
| nfcv 2906 | If ` x ` is disjoint from ... |
| nfcvd 2907 | If ` x ` is disjoint from ... |
| nfab1 2908 | Bound-variable hypothesis ... |
| nfnfc1 2909 | The setvar ` x ` is bound ... |
| clelsb1fw 2910 | Substitution for the first... |
| clelsb1f 2911 | Substitution for the first... |
| nfab 2912 | Bound-variable hypothesis ... |
| nfabg 2913 | Bound-variable hypothesis ... |
| nfaba1 2914 | Bound-variable hypothesis ... |
| nfaba1g 2915 | Bound-variable hypothesis ... |
| nfeqd 2916 | Hypothesis builder for equ... |
| nfeld 2917 | Hypothesis builder for ele... |
| nfnfc 2918 | Hypothesis builder for ` F... |
| nfeq 2919 | Hypothesis builder for equ... |
| nfel 2920 | Hypothesis builder for ele... |
| nfeq1 2921 | Hypothesis builder for equ... |
| nfel1 2922 | Hypothesis builder for ele... |
| nfeq2 2923 | Hypothesis builder for equ... |
| nfel2 2924 | Hypothesis builder for ele... |
| drnfc1 2925 | Formula-building lemma for... |
| drnfc1OLD 2926 | Obsolete version of ~ drnf... |
| drnfc2 2927 | Formula-building lemma for... |
| drnfc2OLD 2928 | Obsolete version of ~ drnf... |
| nfabdw 2929 | Bound-variable hypothesis ... |
| nfabdwOLD 2930 | Obsolete version of ~ nfab... |
| nfabd 2931 | Bound-variable hypothesis ... |
| nfabd2 2932 | Bound-variable hypothesis ... |
| dvelimdc 2933 | Deduction form of ~ dvelim... |
| dvelimc 2934 | Version of ~ dvelim for cl... |
| nfcvf 2935 | If ` x ` and ` y ` are dis... |
| nfcvf2 2936 | If ` x ` and ` y ` are dis... |
| cleqf 2937 | Establish equality between... |
| abid2f 2938 | A simplification of class ... |
| abeq2f 2939 | Equality of a class variab... |
| sbabel 2940 | Theorem to move a substitu... |
| sbabelOLD 2941 | Obsolete version of ~ sbab... |
| neii 2944 | Inference associated with ... |
| neir 2945 | Inference associated with ... |
| nne 2946 | Negation of inequality. (... |
| neneqd 2947 | Deduction eliminating ineq... |
| neneq 2948 | From inequality to non-equ... |
| neqned 2949 | If it is not the case that... |
| neqne 2950 | From non-equality to inequ... |
| neirr 2951 | No class is unequal to its... |
| exmidne 2952 | Excluded middle with equal... |
| eqneqall 2953 | A contradiction concerning... |
| nonconne 2954 | Law of noncontradiction wi... |
| necon3ad 2955 | Contrapositive law deducti... |
| necon3bd 2956 | Contrapositive law deducti... |
| necon2ad 2957 | Contrapositive inference f... |
| necon2bd 2958 | Contrapositive inference f... |
| necon1ad 2959 | Contrapositive deduction f... |
| necon1bd 2960 | Contrapositive deduction f... |
| necon4ad 2961 | Contrapositive inference f... |
| necon4bd 2962 | Contrapositive inference f... |
| necon3d 2963 | Contrapositive law deducti... |
| necon1d 2964 | Contrapositive law deducti... |
| necon2d 2965 | Contrapositive inference f... |
| necon4d 2966 | Contrapositive inference f... |
| necon3ai 2967 | Contrapositive inference f... |
| necon3aiOLD 2968 | Obsolete version of ~ neco... |
| necon3bi 2969 | Contrapositive inference f... |
| necon1ai 2970 | Contrapositive inference f... |
| necon1bi 2971 | Contrapositive inference f... |
| necon2ai 2972 | Contrapositive inference f... |
| necon2bi 2973 | Contrapositive inference f... |
| necon4ai 2974 | Contrapositive inference f... |
| necon3i 2975 | Contrapositive inference f... |
| necon1i 2976 | Contrapositive inference f... |
| necon2i 2977 | Contrapositive inference f... |
| necon4i 2978 | Contrapositive inference f... |
| necon3abid 2979 | Deduction from equality to... |
| necon3bbid 2980 | Deduction from equality to... |
| necon1abid 2981 | Contrapositive deduction f... |
| necon1bbid 2982 | Contrapositive inference f... |
| necon4abid 2983 | Contrapositive law deducti... |
| necon4bbid 2984 | Contrapositive law deducti... |
| necon2abid 2985 | Contrapositive deduction f... |
| necon2bbid 2986 | Contrapositive deduction f... |
| necon3bid 2987 | Deduction from equality to... |
| necon4bid 2988 | Contrapositive law deducti... |
| necon3abii 2989 | Deduction from equality to... |
| necon3bbii 2990 | Deduction from equality to... |
| necon1abii 2991 | Contrapositive inference f... |
| necon1bbii 2992 | Contrapositive inference f... |
| necon2abii 2993 | Contrapositive inference f... |
| necon2bbii 2994 | Contrapositive inference f... |
| necon3bii 2995 | Inference from equality to... |
| necom 2996 | Commutation of inequality.... |
| necomi 2997 | Inference from commutative... |
| necomd 2998 | Deduction from commutative... |
| nesym 2999 | Characterization of inequa... |
| nesymi 3000 | Inference associated with ... |
| nesymir 3001 | Inference associated with ... |
| neeq1d 3002 | Deduction for inequality. ... |
| neeq2d 3003 | Deduction for inequality. ... |
| neeq12d 3004 | Deduction for inequality. ... |
| neeq1 3005 | Equality theorem for inequ... |
| neeq2 3006 | Equality theorem for inequ... |
| neeq1i 3007 | Inference for inequality. ... |
| neeq2i 3008 | Inference for inequality. ... |
| neeq12i 3009 | Inference for inequality. ... |
| eqnetrd 3010 | Substitution of equal clas... |
| eqnetrrd 3011 | Substitution of equal clas... |
| neeqtrd 3012 | Substitution of equal clas... |
| eqnetri 3013 | Substitution of equal clas... |
| eqnetrri 3014 | Substitution of equal clas... |
| neeqtri 3015 | Substitution of equal clas... |
| neeqtrri 3016 | Substitution of equal clas... |
| neeqtrrd 3017 | Substitution of equal clas... |
| eqnetrrid 3018 | A chained equality inferen... |
| 3netr3d 3019 | Substitution of equality i... |
| 3netr4d 3020 | Substitution of equality i... |
| 3netr3g 3021 | Substitution of equality i... |
| 3netr4g 3022 | Substitution of equality i... |
| nebi 3023 | Contraposition law for ine... |
| pm13.18 3024 | Theorem *13.18 in [Whitehe... |
| pm13.181 3025 | Theorem *13.181 in [Whiteh... |
| pm13.181OLD 3026 | Obsolete version of ~ pm13... |
| pm2.61ine 3027 | Inference eliminating an i... |
| pm2.21ddne 3028 | A contradiction implies an... |
| pm2.61ne 3029 | Deduction eliminating an i... |
| pm2.61dne 3030 | Deduction eliminating an i... |
| pm2.61dane 3031 | Deduction eliminating an i... |
| pm2.61da2ne 3032 | Deduction eliminating two ... |
| pm2.61da3ne 3033 | Deduction eliminating thre... |
| pm2.61iine 3034 | Equality version of ~ pm2.... |
| neor 3035 | Logical OR with an equalit... |
| neanior 3036 | A De Morgan's law for ineq... |
| ne3anior 3037 | A De Morgan's law for ineq... |
| neorian 3038 | A De Morgan's law for ineq... |
| nemtbir 3039 | An inference from an inequ... |
| nelne1 3040 | Two classes are different ... |
| nelne2 3041 | Two classes are different ... |
| nelelne 3042 | Two classes are different ... |
| neneor 3043 | If two classes are differe... |
| nfne 3044 | Bound-variable hypothesis ... |
| nfned 3045 | Bound-variable hypothesis ... |
| nabbi 3046 | Not equivalent wff's corre... |
| mteqand 3047 | A modus tollens deduction ... |
| neli 3050 | Inference associated with ... |
| nelir 3051 | Inference associated with ... |
| neleq12d 3052 | Equality theorem for negat... |
| neleq1 3053 | Equality theorem for negat... |
| neleq2 3054 | Equality theorem for negat... |
| nfnel 3055 | Bound-variable hypothesis ... |
| nfneld 3056 | Bound-variable hypothesis ... |
| nnel 3057 | Negation of negated member... |
| elnelne1 3058 | Two classes are different ... |
| elnelne2 3059 | Two classes are different ... |
| nelcon3d 3060 | Contrapositive law deducti... |
| elnelall 3061 | A contradiction concerning... |
| pm2.61danel 3062 | Deduction eliminating an e... |
| rgen 3073 | Generalization rule for re... |
| ralel 3074 | All elements of a class ar... |
| rgenw 3075 | Generalization rule for re... |
| rgen2w 3076 | Generalization rule for re... |
| mprg 3077 | Modus ponens combined with... |
| mprgbir 3078 | Modus ponens on biconditio... |
| alral 3079 | Universal quantification i... |
| raln 3080 | Restricted universally qua... |
| ral2imi 3081 | Inference quantifying ante... |
| ralim 3082 | Distribution of restricted... |
| ralimi2 3083 | Inference quantifying both... |
| ralimia 3084 | Inference quantifying both... |
| ralimiaa 3085 | Inference quantifying both... |
| ralimi 3086 | Inference quantifying both... |
| 2ralimi 3087 | Inference quantifying both... |
| ralbii2 3088 | Inference adding different... |
| ralbiia 3089 | Inference adding restricte... |
| ralbii 3090 | Inference adding restricte... |
| 2ralbii 3091 | Inference adding two restr... |
| ralbi 3092 | Distribute a restricted un... |
| ralanid 3093 | Cancellation law for restr... |
| r19.26 3094 | Restricted quantifier vers... |
| r19.26-2 3095 | Restricted quantifier vers... |
| r19.26-3 3096 | Version of ~ r19.26 with t... |
| r19.26m 3097 | Version of ~ 19.26 and ~ r... |
| ralbiim 3098 | Split a biconditional and ... |
| 2ralbiim 3099 | Split a biconditional and ... |
| r19.21v 3100 | Restricted quantifier vers... |
| ralimdv2 3101 | Inference quantifying both... |
| ralimdva 3102 | Deduction quantifying both... |
| ralimdv 3103 | Deduction quantifying both... |
| ralimdvva 3104 | Deduction doubly quantifyi... |
| hbralrimi 3105 | Inference from Theorem 19.... |
| ralrimiv 3106 | Inference from Theorem 19.... |
| ralrimiva 3107 | Inference from Theorem 19.... |
| ralrimivw 3108 | Inference from Theorem 19.... |
| r19.27v 3109 | Restricted quantitifer ver... |
| r19.28v 3110 | Restricted quantifier vers... |
| ralrimdv 3111 | Inference from Theorem 19.... |
| ralrimdva 3112 | Inference from Theorem 19.... |
| ralrimivv 3113 | Inference from Theorem 19.... |
| ralrimivva 3114 | Inference from Theorem 19.... |
| ralrimivvva 3115 | Inference from Theorem 19.... |
| ralrimdvv 3116 | Inference from Theorem 19.... |
| ralrimdvva 3117 | Inference from Theorem 19.... |
| ralbidv2 3118 | Formula-building rule for ... |
| ralbidva 3119 | Formula-building rule for ... |
| ralbidv 3120 | Formula-building rule for ... |
| 2ralbidva 3121 | Formula-building rule for ... |
| 2ralbidv 3122 | Formula-building rule for ... |
| r2allem 3123 | Lemma factoring out common... |
| r2al 3124 | Double restricted universa... |
| r3al 3125 | Triple restricted universa... |
| rgen2 3126 | Generalization rule for re... |
| rgen3 3127 | Generalization rule for re... |
| rspw 3128 | Restricted specialization.... |
| rsp 3129 | Restricted specialization.... |
| rspa 3130 | Restricted specialization.... |
| rspec 3131 | Specialization rule for re... |
| r19.21bi 3132 | Inference from Theorem 19.... |
| r19.21be 3133 | Inference from Theorem 19.... |
| rspec2 3134 | Specialization rule for re... |
| rspec3 3135 | Specialization rule for re... |
| rsp2 3136 | Restricted specialization,... |
| r19.21t 3137 | Restricted quantifier vers... |
| r19.21 3138 | Restricted quantifier vers... |
| ralrimi 3139 | Inference from Theorem 19.... |
| ralimdaa 3140 | Deduction quantifying both... |
| ralrimd 3141 | Inference from Theorem 19.... |
| nfra1 3142 | The setvar ` x ` is not fr... |
| hbra1 3143 | The setvar ` x ` is not fr... |
| hbral 3144 | Bound-variable hypothesis ... |
| r2alf 3145 | Double restricted universa... |
| nfraldw 3146 | Deduction version of ~ nfr... |
| nfraldwOLD 3147 | Obsolete version of ~ nfra... |
| nfrald 3148 | Deduction version of ~ nfr... |
| nfralw 3149 | Bound-variable hypothesis ... |
| nfral 3150 | Bound-variable hypothesis ... |
| nfra2w 3151 | Similar to Lemma 24 of [Mo... |
| nfra2wOLD 3152 | Obsolete version of ~ nfra... |
| nfra2wOLDOLD 3153 | Obsolete version of ~ nfra... |
| nfra2 3154 | Similar to Lemma 24 of [Mo... |
| rgen2a 3155 | Generalization rule for re... |
| ralbida 3156 | Formula-building rule for ... |
| ralbidaOLD 3157 | Obsolete version of ~ ralb... |
| ralbid 3158 | Formula-building rule for ... |
| 2ralbida 3159 | Formula-building rule for ... |
| raleqbii 3160 | Equality deduction for res... |
| ralcom4 3161 | Commutation of restricted ... |
| ralcom4OLD 3162 | Obsolete version of ~ ralc... |
| ralnex 3163 | Relationship between restr... |
| dfral2 3164 | Relationship between restr... |
| rexnal 3165 | Relationship between restr... |
| dfrex2 3166 | Relationship between restr... |
| rexex 3167 | Restricted existence impli... |
| rexim 3168 | Theorem 19.22 of [Margaris... |
| rexbi 3169 | Distribute restricted quan... |
| rexbiOLD 3170 | Obsolete version of ~ rexb... |
| reximi2 3171 | Inference quantifying both... |
| reximia 3172 | Inference quantifying both... |
| reximiaOLD 3173 | Obsolete version of ~ rexi... |
| reximi 3174 | Inference quantifying both... |
| rexbii2 3175 | Inference adding different... |
| rexbiia 3176 | Inference adding restricte... |
| rexbii 3177 | Inference adding restricte... |
| 2rexbii 3178 | Inference adding two restr... |
| rexcom4 3179 | Commutation of restricted ... |
| 2ex2rexrot 3180 | Rotate two existential qua... |
| rexcom4a 3181 | Specialized existential co... |
| rexanid 3182 | Cancellation law for restr... |
| r19.29 3183 | Restricted quantifier vers... |
| r19.29r 3184 | Restricted quantifier vers... |
| r19.29imd 3185 | Theorem 19.29 of [Margaris... |
| rexnal2 3186 | Relationship between two r... |
| rexnal3 3187 | Relationship between three... |
| ralnex2 3188 | Relationship between two r... |
| ralnex3 3189 | Relationship between three... |
| ralinexa 3190 | A transformation of restri... |
| rexanali 3191 | A transformation of restri... |
| nrexralim 3192 | Negation of a complex pred... |
| risset 3193 | Two ways to say " ` A ` be... |
| nelb 3194 | A definition of ` -. A e. ... |
| nelbOLD 3195 | Obsolete version of ~ nelb... |
| nrex 3196 | Inference adding restricte... |
| nrexdv 3197 | Deduction adding restricte... |
| reximdv2 3198 | Deduction quantifying both... |
| reximdvai 3199 | Deduction quantifying both... |
| reximdvaiOLD 3200 | Obsolete version of ~ rexi... |
| reximdv 3201 | Deduction from Theorem 19.... |
| reximdva 3202 | Deduction quantifying both... |
| reximddv 3203 | Deduction from Theorem 19.... |
| reximssdv 3204 | Derivation of a restricted... |
| reximdvva 3205 | Deduction doubly quantifyi... |
| reximddv2 3206 | Double deduction from Theo... |
| r19.23v 3207 | Restricted quantifier vers... |
| rexlimiv 3208 | Inference from Theorem 19.... |
| rexlimiva 3209 | Inference from Theorem 19.... |
| rexlimivw 3210 | Weaker version of ~ rexlim... |
| rexlimdv 3211 | Inference from Theorem 19.... |
| rexlimdva 3212 | Inference from Theorem 19.... |
| rexlimdvaa 3213 | Inference from Theorem 19.... |
| rexlimdv3a 3214 | Inference from Theorem 19.... |
| rexlimdva2 3215 | Inference from Theorem 19.... |
| r19.29an 3216 | A commonly used pattern in... |
| r19.29a 3217 | A commonly used pattern in... |
| rexlimdvw 3218 | Inference from Theorem 19.... |
| rexlimddv 3219 | Restricted existential eli... |
| rexlimivv 3220 | Inference from Theorem 19.... |
| rexlimdvv 3221 | Inference from Theorem 19.... |
| rexlimdvva 3222 | Inference from Theorem 19.... |
| rexbidv2 3223 | Formula-building rule for ... |
| rexbidva 3224 | Formula-building rule for ... |
| rexbidv 3225 | Formula-building rule for ... |
| 2rexbiia 3226 | Inference adding two restr... |
| 2rexbidva 3227 | Formula-building rule for ... |
| 2rexbidv 3228 | Formula-building rule for ... |
| rexralbidv 3229 | Formula-building rule for ... |
| r2exlem 3230 | Lemma factoring out common... |
| r2ex 3231 | Double restricted existent... |
| rspe 3232 | Restricted specialization.... |
| rsp2e 3233 | Restricted specialization.... |
| nfre1 3234 | The setvar ` x ` is not fr... |
| nfrexd 3235 | Deduction version of ~ nfr... |
| nfrexdg 3236 | Deduction version of ~ nfr... |
| nfrex 3237 | Bound-variable hypothesis ... |
| nfrexg 3238 | Bound-variable hypothesis ... |
| reximdai 3239 | Deduction from Theorem 19.... |
| reximd2a 3240 | Deduction quantifying both... |
| r19.23t 3241 | Closed theorem form of ~ r... |
| r19.23 3242 | Restricted quantifier vers... |
| rexlimi 3243 | Restricted quantifier vers... |
| rexlimd2 3244 | Version of ~ rexlimd with ... |
| rexlimd 3245 | Deduction form of ~ rexlim... |
| rexbida 3246 | Formula-building rule for ... |
| rexbidvaALT 3247 | Alternate proof of ~ rexbi... |
| rexbid 3248 | Formula-building rule for ... |
| rexbidvALT 3249 | Alternate proof of ~ rexbi... |
| ralrexbid 3250 | Formula-building rule for ... |
| ralrexbidOLD 3251 | Obsolete version of ~ ralr... |
| r19.12 3252 | Restricted quantifier vers... |
| r19.12OLD 3253 | Obsolete version of ~ 19.1... |
| r2exf 3254 | Double restricted existent... |
| rexeqbii 3255 | Equality deduction for res... |
| reuanid 3256 | Cancellation law for restr... |
| rmoanid 3257 | Cancellation law for restr... |
| r19.29af2 3258 | A commonly used pattern ba... |
| r19.29af 3259 | A commonly used pattern ba... |
| 2r19.29 3260 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3261 | Theorem 19.29 of [Margaris... |
| r19.29d2rOLD 3262 | Obsolete version of ~ r19.... |
| r19.29vva 3263 | A commonly used pattern ba... |
| r19.29vvaOLD 3264 | Obsolete version of ~ r19.... |
| r19.30 3265 | Restricted quantifier vers... |
| r19.30OLD 3266 | Obsolete version of ~ 19.3... |
| r19.32v 3267 | Restricted quantifier vers... |
| r19.35 3268 | Restricted quantifier vers... |
| r19.36v 3269 | Restricted quantifier vers... |
| r19.37 3270 | Restricted quantifier vers... |
| r19.37v 3271 | Restricted quantifier vers... |
| r19.40 3272 | Restricted quantifier vers... |
| r19.41v 3273 | Restricted quantifier vers... |
| r19.41 3274 | Restricted quantifier vers... |
| r19.41vv 3275 | Version of ~ r19.41v with ... |
| r19.42v 3276 | Restricted quantifier vers... |
| r19.43 3277 | Restricted quantifier vers... |
| r19.44v 3278 | One direction of a restric... |
| r19.45v 3279 | Restricted quantifier vers... |
| ralcom 3280 | Commutation of restricted ... |
| rexcom 3281 | Commutation of restricted ... |
| ralcomf 3282 | Commutation of restricted ... |
| rexcomf 3283 | Commutation of restricted ... |
| ralcom13 3284 | Swap first and third restr... |
| rexcom13 3285 | Swap first and third restr... |
| ralrot3 3286 | Rotate three restricted un... |
| rexrot4 3287 | Rotate four restricted exi... |
| ralcom2 3288 | Commutation of restricted ... |
| ralcom3 3289 | A commutation law for rest... |
| reeanlem 3290 | Lemma factoring out common... |
| reean 3291 | Rearrange restricted exist... |
| reeanv 3292 | Rearrange restricted exist... |
| 3reeanv 3293 | Rearrange three restricted... |
| 2ralor 3294 | Distribute restricted univ... |
| 2ralorOLD 3295 | Obsolete version of ~ 2ral... |
| nfreu1 3296 | The setvar ` x ` is not fr... |
| nfrmo1 3297 | The setvar ` x ` is not fr... |
| nfreud 3298 | Deduction version of ~ nfr... |
| nfrmod 3299 | Deduction version of ~ nfr... |
| nfreuw 3300 | Bound-variable hypothesis ... |
| nfrmow 3301 | Bound-variable hypothesis ... |
| nfreu 3302 | Bound-variable hypothesis ... |
| nfrmo 3303 | Bound-variable hypothesis ... |
| rabid 3304 | An "identity" law of concr... |
| rabrab 3305 | Abstract builder restricte... |
| rabidim1 3306 | Membership in a restricted... |
| rabid2 3307 | An "identity" law for rest... |
| rabid2f 3308 | An "identity" law for rest... |
| rabbi 3309 | Equivalent wff's correspon... |
| nfrab1 3310 | The abstraction variable i... |
| nfrabw 3311 | A variable not free in a w... |
| nfrab 3312 | A variable not free in a w... |
| reubida 3313 | Formula-building rule for ... |
| reubidva 3314 | Formula-building rule for ... |
| reubidv 3315 | Formula-building rule for ... |
| reubiia 3316 | Formula-building rule for ... |
| reubii 3317 | Formula-building rule for ... |
| rmobida 3318 | Formula-building rule for ... |
| rmobidva 3319 | Formula-building rule for ... |
| rmobidv 3320 | Formula-building rule for ... |
| rmobiia 3321 | Formula-building rule for ... |
| rmobii 3322 | Formula-building rule for ... |
| raleqf 3323 | Equality theorem for restr... |
| rexeqf 3324 | Equality theorem for restr... |
| reueq1f 3325 | Equality theorem for restr... |
| rmoeq1f 3326 | Equality theorem for restr... |
| raleqbidv 3327 | Equality deduction for res... |
| rexeqbidv 3328 | Equality deduction for res... |
| raleqbidvv 3329 | Version of ~ raleqbidv wit... |
| rexeqbidvv 3330 | Version of ~ rexeqbidv wit... |
| raleqbi1dv 3331 | Equality deduction for res... |
| rexeqbi1dv 3332 | Equality deduction for res... |
| raleq 3333 | Equality theorem for restr... |
| rexeq 3334 | Equality theorem for restr... |
| reueq1 3335 | Equality theorem for restr... |
| rmoeq1 3336 | Equality theorem for restr... |
| raleqi 3337 | Equality inference for res... |
| rexeqi 3338 | Equality inference for res... |
| raleqdv 3339 | Equality deduction for res... |
| rexeqdv 3340 | Equality deduction for res... |
| reueqd 3341 | Equality deduction for res... |
| rmoeqd 3342 | Equality deduction for res... |
| raleqbid 3343 | Equality deduction for res... |
| rexeqbid 3344 | Equality deduction for res... |
| raleqbidva 3345 | Equality deduction for res... |
| rexeqbidva 3346 | Equality deduction for res... |
| raleleq 3347 | All elements of a class ar... |
| raleleqALT 3348 | Alternate proof of ~ ralel... |
| moel 3349 | "At most one" element in a... |
| mormo 3350 | Unrestricted "at most one"... |
| reu5 3351 | Restricted uniqueness in t... |
| reurex 3352 | Restricted unique existenc... |
| 2reu2rex 3353 | Double restricted existent... |
| reurmo 3354 | Restricted existential uni... |
| rmo5 3355 | Restricted "at most one" i... |
| nrexrmo 3356 | Nonexistence implies restr... |
| reueubd 3357 | Restricted existential uni... |
| cbvralfw 3358 | Rule used to change bound ... |
| cbvralfwOLD 3359 | Obsolete version of ~ cbvr... |
| cbvrexfw 3360 | Rule used to change bound ... |
| cbvralf 3361 | Rule used to change bound ... |
| cbvrexf 3362 | Rule used to change bound ... |
| cbvralw 3363 | Rule used to change bound ... |
| cbvrexw 3364 | Rule used to change bound ... |
| cbvreuw 3365 | Change the bound variable ... |
| cbvrmow 3366 | Change the bound variable ... |
| cbvrmowOLD 3367 | Obsolete version of ~ cbvr... |
| cbvral 3368 | Rule used to change bound ... |
| cbvrex 3369 | Rule used to change bound ... |
| cbvreu 3370 | Change the bound variable ... |
| cbvrmo 3371 | Change the bound variable ... |
| cbvralvw 3372 | Change the bound variable ... |
| cbvrexvw 3373 | Change the bound variable ... |
| cbvrmovw 3374 | Change the bound variable ... |
| cbvreuvw 3375 | Change the bound variable ... |
| cbvreuvwOLD 3376 | Obsolete version of ~ cbvr... |
| cbvralv 3377 | Change the bound variable ... |
| cbvrexv 3378 | Change the bound variable ... |
| cbvreuv 3379 | Change the bound variable ... |
| cbvrmov 3380 | Change the bound variable ... |
| cbvraldva2 3381 | Rule used to change the bo... |
| cbvrexdva2 3382 | Rule used to change the bo... |
| cbvraldva 3383 | Rule used to change the bo... |
| cbvrexdva 3384 | Rule used to change the bo... |
| cbvral2vw 3385 | Change bound variables of ... |
| cbvrex2vw 3386 | Change bound variables of ... |
| cbvral3vw 3387 | Change bound variables of ... |
| cbvral2v 3388 | Change bound variables of ... |
| cbvrex2v 3389 | Change bound variables of ... |
| cbvral3v 3390 | Change bound variables of ... |
| cbvralsvw 3391 | Change bound variable by u... |
| cbvrexsvw 3392 | Change bound variable by u... |
| cbvralsv 3393 | Change bound variable by u... |
| cbvrexsv 3394 | Change bound variable by u... |
| sbralie 3395 | Implicit to explicit subst... |
| rabbiia 3396 | Equivalent formulas yield ... |
| rabbii 3397 | Equivalent wff's correspon... |
| rabbida 3398 | Equivalent wff's yield equ... |
| rabbid 3399 | Version of ~ rabbidv with ... |
| rabbidva2 3400 | Equivalent wff's yield equ... |
| rabbia2 3401 | Equivalent wff's yield equ... |
| rabbidva 3402 | Equivalent wff's yield equ... |
| rabbidvaOLD 3403 | Obsolete proof of ~ rabbid... |
| rabbidv 3404 | Equivalent wff's yield equ... |
| rabeqf 3405 | Equality theorem for restr... |
| rabeqi 3406 | Equality theorem for restr... |
| rabeqiOLD 3407 | Obsolete version of ~ rabe... |
| rabeq 3408 | Equality theorem for restr... |
| rabeqdv 3409 | Equality of restricted cla... |
| rabeqbidv 3410 | Equality of restricted cla... |
| rabeqbidva 3411 | Equality of restricted cla... |
| rabeq2i 3412 | Inference from equality of... |
| rabswap 3413 | Swap with a membership rel... |
| cbvrabw 3414 | Rule to change the bound v... |
| cbvrab 3415 | Rule to change the bound v... |
| cbvrabv 3416 | Rule to change the bound v... |
| rabrabi 3417 | Abstract builder restricte... |
| rabrabiOLD 3418 | Obsolete version of ~ rabr... |
| rabeqcda 3419 | When ` ps ` is always true... |
| ralrimia 3420 | Inference from Theorem 19.... |
| ralimda 3421 | Deduction quantifying both... |
| vjust 3423 | Justification theorem for ... |
| dfv2 3425 | Alternate definition of th... |
| vex 3426 | All setvar variables are s... |
| vexOLD 3427 | Obsolete version of ~ vex ... |
| elv 3428 | If a proposition is implie... |
| elvd 3429 | If a proposition is implie... |
| el2v 3430 | If a proposition is implie... |
| eqv 3431 | The universe contains ever... |
| eqvf 3432 | The universe contains ever... |
| abv 3433 | The class of sets verifyin... |
| abvALT 3434 | Alternate proof of ~ abv ,... |
| isset 3435 | Two ways to express that "... |
| issetf 3436 | A version of ~ isset that ... |
| isseti 3437 | A way to say " ` A ` is a ... |
| issetri 3438 | A way to say " ` A ` is a ... |
| eqvisset 3439 | A class equal to a variabl... |
| elex 3440 | If a class is a member of ... |
| elexi 3441 | If a class is a member of ... |
| elexd 3442 | If a class is a member of ... |
| elex2 3443 | If a class contains anothe... |
| elex22 3444 | If two classes each contai... |
| prcnel 3445 | A proper class doesn't bel... |
| ralv 3446 | A universal quantifier res... |
| rexv 3447 | An existential quantifier ... |
| reuv 3448 | A unique existential quant... |
| rmov 3449 | An at-most-one quantifier ... |
| rabab 3450 | A class abstraction restri... |
| rexcom4b 3451 | Specialized existential co... |
| ceqsalt 3452 | Closed theorem version of ... |
| ceqsralt 3453 | Restricted quantifier vers... |
| ceqsalg 3454 | A representation of explic... |
| ceqsalgALT 3455 | Alternate proof of ~ ceqsa... |
| ceqsal 3456 | A representation of explic... |
| ceqsalv 3457 | A representation of explic... |
| ceqsalvOLD 3458 | Obsolete version of ~ ceqs... |
| ceqsralv 3459 | Restricted quantifier vers... |
| ceqsralvOLD 3460 | Obsolete version of ~ ceqs... |
| gencl 3461 | Implicit substitution for ... |
| 2gencl 3462 | Implicit substitution for ... |
| 3gencl 3463 | Implicit substitution for ... |
| cgsexg 3464 | Implicit substitution infe... |
| cgsex2g 3465 | Implicit substitution infe... |
| cgsex4g 3466 | An implicit substitution i... |
| cgsex4gOLD 3467 | Obsolete version of ~ cgse... |
| ceqsex 3468 | Elimination of an existent... |
| ceqsexv 3469 | Elimination of an existent... |
| ceqsexvOLD 3470 | Obsolete version of ~ ceqs... |
| ceqsexv2d 3471 | Elimination of an existent... |
| ceqsex2 3472 | Elimination of two existen... |
| ceqsex2v 3473 | Elimination of two existen... |
| ceqsex3v 3474 | Elimination of three exist... |
| ceqsex4v 3475 | Elimination of four existe... |
| ceqsex6v 3476 | Elimination of six existen... |
| ceqsex8v 3477 | Elimination of eight exist... |
| gencbvex 3478 | Change of bound variable u... |
| gencbvex2 3479 | Restatement of ~ gencbvex ... |
| gencbval 3480 | Change of bound variable u... |
| sbhypf 3481 | Introduce an explicit subs... |
| vtoclgft 3482 | Closed theorem form of ~ v... |
| vtocldf 3483 | Implicit substitution of a... |
| vtocld 3484 | Implicit substitution of a... |
| vtocldOLD 3485 | Obsolete version of ~ vtoc... |
| vtocl2d 3486 | Implicit substitution of t... |
| vtoclf 3487 | Implicit substitution of a... |
| vtocl 3488 | Implicit substitution of a... |
| vtoclALT 3489 | Alternate proof of ~ vtocl... |
| vtocl2 3490 | Implicit substitution of c... |
| vtocl3 3491 | Implicit substitution of c... |
| vtoclb 3492 | Implicit substitution of a... |
| vtoclgf 3493 | Implicit substitution of a... |
| vtoclg1f 3494 | Version of ~ vtoclgf with ... |
| vtoclg 3495 | Implicit substitution of a... |
| vtoclgOLD 3496 | Obsolete version of ~ vtoc... |
| vtoclbg 3497 | Implicit substitution of a... |
| vtocl2gf 3498 | Implicit substitution of a... |
| vtocl3gf 3499 | Implicit substitution of a... |
| vtocl2g 3500 | Implicit substitution of 2... |
| vtocl3g 3501 | Implicit substitution of a... |
| vtoclgaf 3502 | Implicit substitution of a... |
| vtoclga 3503 | Implicit substitution of a... |
| vtocl2ga 3504 | Implicit substitution of 2... |
| vtocl2gaf 3505 | Implicit substitution of 2... |
| vtocl3gaf 3506 | Implicit substitution of 3... |
| vtocl3ga 3507 | Implicit substitution of 3... |
| vtocl3gaOLD 3508 | Obsolete version of ~ vtoc... |
| vtocl4g 3509 | Implicit substitution of 4... |
| vtocl4ga 3510 | Implicit substitution of 4... |
| vtocleg 3511 | Implicit substitution of a... |
| vtoclegft 3512 | Implicit substitution of a... |
| vtoclef 3513 | Implicit substitution of a... |
| vtocle 3514 | Implicit substitution of a... |
| vtoclri 3515 | Implicit substitution of a... |
| spcimgft 3516 | A closed version of ~ spci... |
| spcgft 3517 | A closed version of ~ spcg... |
| spcimgf 3518 | Rule of specialization, us... |
| spcimegf 3519 | Existential specialization... |
| spcgf 3520 | Rule of specialization, us... |
| spcegf 3521 | Existential specialization... |
| spcimdv 3522 | Restricted specialization,... |
| spcdv 3523 | Rule of specialization, us... |
| spcimedv 3524 | Restricted existential spe... |
| spcgv 3525 | Rule of specialization, us... |
| spcegv 3526 | Existential specialization... |
| spcedv 3527 | Existential specialization... |
| spc2egv 3528 | Existential specialization... |
| spc2gv 3529 | Specialization with two qu... |
| spc2ed 3530 | Existential specialization... |
| spc2d 3531 | Specialization with 2 quan... |
| spc3egv 3532 | Existential specialization... |
| spc3gv 3533 | Specialization with three ... |
| spcv 3534 | Rule of specialization, us... |
| spcev 3535 | Existential specialization... |
| spc2ev 3536 | Existential specialization... |
| rspct 3537 | A closed version of ~ rspc... |
| rspcdf 3538 | Restricted specialization,... |
| rspc 3539 | Restricted specialization,... |
| rspce 3540 | Restricted existential spe... |
| rspcimdv 3541 | Restricted specialization,... |
| rspcimedv 3542 | Restricted existential spe... |
| rspcdv 3543 | Restricted specialization,... |
| rspcedv 3544 | Restricted existential spe... |
| rspcebdv 3545 | Restricted existential spe... |
| rspcdv2 3546 | Restricted specialization,... |
| rspcv 3547 | Restricted specialization,... |
| rspcvOLD 3548 | Obsolete version of ~ rspc... |
| rspccv 3549 | Restricted specialization,... |
| rspcva 3550 | Restricted specialization,... |
| rspccva 3551 | Restricted specialization,... |
| rspcev 3552 | Restricted existential spe... |
| rspcevOLD 3553 | Obsolete version of ~ rspc... |
| rspcdva 3554 | Restricted specialization,... |
| rspcedvd 3555 | Restricted existential spe... |
| rspcime 3556 | Prove a restricted existen... |
| rspceaimv 3557 | Restricted existential spe... |
| rspcedeq1vd 3558 | Restricted existential spe... |
| rspcedeq2vd 3559 | Restricted existential spe... |
| rspc2 3560 | Restricted specialization ... |
| rspc2gv 3561 | Restricted specialization ... |
| rspc2v 3562 | 2-variable restricted spec... |
| rspc2va 3563 | 2-variable restricted spec... |
| rspc2ev 3564 | 2-variable restricted exis... |
| rspc3v 3565 | 3-variable restricted spec... |
| rspc3ev 3566 | 3-variable restricted exis... |
| rspceeqv 3567 | Restricted existential spe... |
| ralxpxfr2d 3568 | Transfer a universal quant... |
| rexraleqim 3569 | Statement following from e... |
| eqvincg 3570 | A variable introduction la... |
| eqvinc 3571 | A variable introduction la... |
| eqvincf 3572 | A variable introduction la... |
| alexeqg 3573 | Two ways to express substi... |
| ceqex 3574 | Equality implies equivalen... |
| ceqsexg 3575 | A representation of explic... |
| ceqsexgv 3576 | Elimination of an existent... |
| ceqsexgvOLD 3577 | Obsolete version of ~ ceqs... |
| ceqsrexv 3578 | Elimination of a restricte... |
| ceqsrexbv 3579 | Elimination of a restricte... |
| ceqsrex2v 3580 | Elimination of a restricte... |
| clel2g 3581 | Alternate definition of me... |
| clel2gOLD 3582 | Obsolete version of ~ clel... |
| clel2 3583 | Alternate definition of me... |
| clel3g 3584 | Alternate definition of me... |
| clel3 3585 | Alternate definition of me... |
| clel4g 3586 | Alternate definition of me... |
| clel4 3587 | Alternate definition of me... |
| clel4OLD 3588 | Obsolete version of ~ clel... |
| clel5 3589 | Alternate definition of cl... |
| pm13.183 3590 | Compare theorem *13.183 in... |
| rr19.3v 3591 | Restricted quantifier vers... |
| rr19.28v 3592 | Restricted quantifier vers... |
| elab6g 3593 | Membership in a class abst... |
| elabd2 3594 | Membership in a class abst... |
| elabd3 3595 | Membership in a class abst... |
| elabgt 3596 | Membership in a class abst... |
| elabgtOLD 3597 | Obsolete version of ~ elab... |
| elabgf 3598 | Membership in a class abst... |
| elabf 3599 | Membership in a class abst... |
| elabg 3600 | Membership in a class abst... |
| elabgOLD 3601 | Obsolete version of ~ elab... |
| elab 3602 | Membership in a class abst... |
| elabOLD 3603 | Obsolete version of ~ elab... |
| elab2g 3604 | Membership in a class abst... |
| elabd 3605 | Explicit demonstration the... |
| elab2 3606 | Membership in a class abst... |
| elab4g 3607 | Membership in a class abst... |
| elab3gf 3608 | Membership in a class abst... |
| elab3g 3609 | Membership in a class abst... |
| elab3 3610 | Membership in a class abst... |
| elrabi 3611 | Implication for the member... |
| elrabiOLD 3612 | Obsolete version of ~ elra... |
| elrabf 3613 | Membership in a restricted... |
| rabtru 3614 | Abstract builder using the... |
| rabeqc 3615 | A restricted class abstrac... |
| elrab3t 3616 | Membership in a restricted... |
| elrab 3617 | Membership in a restricted... |
| elrab3 3618 | Membership in a restricted... |
| elrabd 3619 | Membership in a restricted... |
| elrab2 3620 | Membership in a restricted... |
| ralab 3621 | Universal quantification o... |
| ralabOLD 3622 | Obsolete version of ~ rala... |
| ralrab 3623 | Universal quantification o... |
| rexab 3624 | Existential quantification... |
| rexabOLD 3625 | Obsolete version of ~ rexa... |
| rexrab 3626 | Existential quantification... |
| ralab2 3627 | Universal quantification o... |
| ralab2OLD 3628 | Obsolete version of ~ rala... |
| ralrab2 3629 | Universal quantification o... |
| rexab2 3630 | Existential quantification... |
| rexab2OLD 3631 | Obsolete version of ~ rexa... |
| rexrab2 3632 | Existential quantification... |
| abidnf 3633 | Identity used to create cl... |
| dedhb 3634 | A deduction theorem for co... |
| nelrdva 3635 | Deduce negative membership... |
| eqeu 3636 | A condition which implies ... |
| moeq 3637 | There exists at most one s... |
| eueq 3638 | A class is a set if and on... |
| eueqi 3639 | There exists a unique set ... |
| eueq2 3640 | Equality has existential u... |
| eueq3 3641 | Equality has existential u... |
| moeq3 3642 | "At most one" property of ... |
| mosub 3643 | "At most one" remains true... |
| mo2icl 3644 | Theorem for inferring "at ... |
| mob2 3645 | Consequence of "at most on... |
| moi2 3646 | Consequence of "at most on... |
| mob 3647 | Equality implied by "at mo... |
| moi 3648 | Equality implied by "at mo... |
| morex 3649 | Derive membership from uni... |
| euxfr2w 3650 | Transfer existential uniqu... |
| euxfrw 3651 | Transfer existential uniqu... |
| euxfr2 3652 | Transfer existential uniqu... |
| euxfr 3653 | Transfer existential uniqu... |
| euind 3654 | Existential uniqueness via... |
| reu2 3655 | A way to express restricte... |
| reu6 3656 | A way to express restricte... |
| reu3 3657 | A way to express restricte... |
| reu6i 3658 | A condition which implies ... |
| eqreu 3659 | A condition which implies ... |
| rmo4 3660 | Restricted "at most one" u... |
| reu4 3661 | Restricted uniqueness usin... |
| reu7 3662 | Restricted uniqueness usin... |
| reu8 3663 | Restricted uniqueness usin... |
| rmo3f 3664 | Restricted "at most one" u... |
| rmo4f 3665 | Restricted "at most one" u... |
| reu2eqd 3666 | Deduce equality from restr... |
| reueq 3667 | Equality has existential u... |
| rmoeq 3668 | Equality's restricted exis... |
| rmoan 3669 | Restricted "at most one" s... |
| rmoim 3670 | Restricted "at most one" i... |
| rmoimia 3671 | Restricted "at most one" i... |
| rmoimi 3672 | Restricted "at most one" i... |
| rmoimi2 3673 | Restricted "at most one" i... |
| 2reu5a 3674 | Double restricted existent... |
| reuimrmo 3675 | Restricted uniqueness impl... |
| 2reuswap 3676 | A condition allowing swap ... |
| 2reuswap2 3677 | A condition allowing swap ... |
| reuxfrd 3678 | Transfer existential uniqu... |
| reuxfr 3679 | Transfer existential uniqu... |
| reuxfr1d 3680 | Transfer existential uniqu... |
| reuxfr1ds 3681 | Transfer existential uniqu... |
| reuxfr1 3682 | Transfer existential uniqu... |
| reuind 3683 | Existential uniqueness via... |
| 2rmorex 3684 | Double restricted quantifi... |
| 2reu5lem1 3685 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3686 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3687 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3688 | Double restricted existent... |
| 2reurmo 3689 | Double restricted quantifi... |
| 2reurex 3690 | Double restricted quantifi... |
| 2rmoswap 3691 | A condition allowing to sw... |
| 2rexreu 3692 | Double restricted existent... |
| cdeqi 3695 | Deduce conditional equalit... |
| cdeqri 3696 | Property of conditional eq... |
| cdeqth 3697 | Deduce conditional equalit... |
| cdeqnot 3698 | Distribute conditional equ... |
| cdeqal 3699 | Distribute conditional equ... |
| cdeqab 3700 | Distribute conditional equ... |
| cdeqal1 3701 | Distribute conditional equ... |
| cdeqab1 3702 | Distribute conditional equ... |
| cdeqim 3703 | Distribute conditional equ... |
| cdeqcv 3704 | Conditional equality for s... |
| cdeqeq 3705 | Distribute conditional equ... |
| cdeqel 3706 | Distribute conditional equ... |
| nfcdeq 3707 | If we have a conditional e... |
| nfccdeq 3708 | Variation of ~ nfcdeq for ... |
| rru 3709 | Relative version of Russel... |
| ru 3710 | Russell's Paradox. Propos... |
| dfsbcq 3713 | Proper substitution of a c... |
| dfsbcq2 3714 | This theorem, which is sim... |
| sbsbc 3715 | Show that ~ df-sb and ~ df... |
| sbceq1d 3716 | Equality theorem for class... |
| sbceq1dd 3717 | Equality theorem for class... |
| sbceqbid 3718 | Equality theorem for class... |
| sbc8g 3719 | This is the closest we can... |
| sbc2or 3720 | The disjunction of two equ... |
| sbcex 3721 | By our definition of prope... |
| sbceq1a 3722 | Equality theorem for class... |
| sbceq2a 3723 | Equality theorem for class... |
| spsbc 3724 | Specialization: if a formu... |
| spsbcd 3725 | Specialization: if a formu... |
| sbcth 3726 | A substitution into a theo... |
| sbcthdv 3727 | Deduction version of ~ sbc... |
| sbcid 3728 | An identity theorem for su... |
| nfsbc1d 3729 | Deduction version of ~ nfs... |
| nfsbc1 3730 | Bound-variable hypothesis ... |
| nfsbc1v 3731 | Bound-variable hypothesis ... |
| nfsbcdw 3732 | Deduction version of ~ nfs... |
| nfsbcw 3733 | Bound-variable hypothesis ... |
| sbccow 3734 | A composition law for clas... |
| nfsbcd 3735 | Deduction version of ~ nfs... |
| nfsbc 3736 | Bound-variable hypothesis ... |
| sbcco 3737 | A composition law for clas... |
| sbcco2 3738 | A composition law for clas... |
| sbc5 3739 | An equivalence for class s... |
| sbc5ALT 3740 | Alternate proof of ~ sbc5 ... |
| sbc6g 3741 | An equivalence for class s... |
| sbc6gOLD 3742 | Obsolete version of ~ sbc6... |
| sbc6 3743 | An equivalence for class s... |
| sbc7 3744 | An equivalence for class s... |
| cbvsbcw 3745 | Change bound variables in ... |
| cbvsbcvw 3746 | Change the bound variable ... |
| cbvsbc 3747 | Change bound variables in ... |
| cbvsbcv 3748 | Change the bound variable ... |
| sbciegft 3749 | Conversion of implicit sub... |
| sbciegf 3750 | Conversion of implicit sub... |
| sbcieg 3751 | Conversion of implicit sub... |
| sbciegOLD 3752 | Obsolete version of ~ sbci... |
| sbcie2g 3753 | Conversion of implicit sub... |
| sbcie 3754 | Conversion of implicit sub... |
| sbciedf 3755 | Conversion of implicit sub... |
| sbcied 3756 | Conversion of implicit sub... |
| sbciedOLD 3757 | Obsolete version of ~ sbci... |
| sbcied2 3758 | Conversion of implicit sub... |
| elrabsf 3759 | Membership in a restricted... |
| eqsbc1 3760 | Substitution for the left-... |
| sbcng 3761 | Move negation in and out o... |
| sbcimg 3762 | Distribution of class subs... |
| sbcan 3763 | Distribution of class subs... |
| sbcor 3764 | Distribution of class subs... |
| sbcbig 3765 | Distribution of class subs... |
| sbcn1 3766 | Move negation in and out o... |
| sbcim1 3767 | Distribution of class subs... |
| sbcim1OLD 3768 | Obsolete version of ~ sbci... |
| sbcbid 3769 | Formula-building deduction... |
| sbcbidv 3770 | Formula-building deduction... |
| sbcbidvOLD 3771 | Obsolete version of ~ sbcb... |
| sbcbii 3772 | Formula-building inference... |
| sbcbi1 3773 | Distribution of class subs... |
| sbcbi2 3774 | Substituting into equivale... |
| sbcbi2OLD 3775 | Obsolete proof of ~ sbcbi2... |
| sbcal 3776 | Move universal quantifier ... |
| sbcex2 3777 | Move existential quantifie... |
| sbceqal 3778 | Class version of one impli... |
| sbceqalOLD 3779 | Obsolete version of ~ sbce... |
| sbeqalb 3780 | Theorem *14.121 in [Whiteh... |
| eqsbc2 3781 | Substitution for the right... |
| sbc3an 3782 | Distribution of class subs... |
| sbcel1v 3783 | Class substitution into a ... |
| sbcel2gv 3784 | Class substitution into a ... |
| sbcel21v 3785 | Class substitution into a ... |
| sbcimdv 3786 | Substitution analogue of T... |
| sbcimdvOLD 3787 | Obsolete version of ~ sbci... |
| sbctt 3788 | Substitution for a variabl... |
| sbcgf 3789 | Substitution for a variabl... |
| sbc19.21g 3790 | Substitution for a variabl... |
| sbcg 3791 | Substitution for a variabl... |
| sbcgOLD 3792 | Obsolete version of ~ sbcg... |
| sbcgfi 3793 | Substitution for a variabl... |
| sbc2iegf 3794 | Conversion of implicit sub... |
| sbc2ie 3795 | Conversion of implicit sub... |
| sbc2ieOLD 3796 | Obsolete version of ~ sbc2... |
| sbc2iedv 3797 | Conversion of implicit sub... |
| sbc3ie 3798 | Conversion of implicit sub... |
| sbccomlem 3799 | Lemma for ~ sbccom . (Con... |
| sbccom 3800 | Commutative law for double... |
| sbcralt 3801 | Interchange class substitu... |
| sbcrext 3802 | Interchange class substitu... |
| sbcralg 3803 | Interchange class substitu... |
| sbcrex 3804 | Interchange class substitu... |
| sbcreu 3805 | Interchange class substitu... |
| reu8nf 3806 | Restricted uniqueness usin... |
| sbcabel 3807 | Interchange class substitu... |
| rspsbc 3808 | Restricted quantifier vers... |
| rspsbca 3809 | Restricted quantifier vers... |
| rspesbca 3810 | Existence form of ~ rspsbc... |
| spesbc 3811 | Existence form of ~ spsbc ... |
| spesbcd 3812 | form of ~ spsbc . (Contri... |
| sbcth2 3813 | A substitution into a theo... |
| ra4v 3814 | Version of ~ ra4 with a di... |
| ra4 3815 | Restricted quantifier vers... |
| rmo2 3816 | Alternate definition of re... |
| rmo2i 3817 | Condition implying restric... |
| rmo3 3818 | Restricted "at most one" u... |
| rmob 3819 | Consequence of "at most on... |
| rmoi 3820 | Consequence of "at most on... |
| rmob2 3821 | Consequence of "restricted... |
| rmoi2 3822 | Consequence of "restricted... |
| rmoanim 3823 | Introduction of a conjunct... |
| rmoanimALT 3824 | Alternate proof of ~ rmoan... |
| reuan 3825 | Introduction of a conjunct... |
| 2reu1 3826 | Double restricted existent... |
| 2reu2 3827 | Double restricted existent... |
| csb2 3830 | Alternate expression for t... |
| csbeq1 3831 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3832 | Equality deduction for pro... |
| csbeq2 3833 | Substituting into equivale... |
| csbeq2d 3834 | Formula-building deduction... |
| csbeq2dv 3835 | Formula-building deduction... |
| csbeq2i 3836 | Formula-building inference... |
| csbeq12dv 3837 | Formula-building inference... |
| cbvcsbw 3838 | Change bound variables in ... |
| cbvcsb 3839 | Change bound variables in ... |
| cbvcsbv 3840 | Change the bound variable ... |
| csbid 3841 | Analogue of ~ sbid for pro... |
| csbeq1a 3842 | Equality theorem for prope... |
| csbcow 3843 | Composition law for chaine... |
| csbco 3844 | Composition law for chaine... |
| csbtt 3845 | Substitution doesn't affec... |
| csbconstgf 3846 | Substitution doesn't affec... |
| csbconstg 3847 | Substitution doesn't affec... |
| csbconstgOLD 3848 | Obsolete version of ~ csbc... |
| csbgfi 3849 | Substitution for a variabl... |
| csbconstgi 3850 | The proper substitution of... |
| nfcsb1d 3851 | Bound-variable hypothesis ... |
| nfcsb1 3852 | Bound-variable hypothesis ... |
| nfcsb1v 3853 | Bound-variable hypothesis ... |
| nfcsbd 3854 | Deduction version of ~ nfc... |
| nfcsbw 3855 | Bound-variable hypothesis ... |
| nfcsb 3856 | Bound-variable hypothesis ... |
| csbhypf 3857 | Introduce an explicit subs... |
| csbiebt 3858 | Conversion of implicit sub... |
| csbiedf 3859 | Conversion of implicit sub... |
| csbieb 3860 | Bidirectional conversion b... |
| csbiebg 3861 | Bidirectional conversion b... |
| csbiegf 3862 | Conversion of implicit sub... |
| csbief 3863 | Conversion of implicit sub... |
| csbie 3864 | Conversion of implicit sub... |
| csbieOLD 3865 | Obsolete version of ~ csbi... |
| csbied 3866 | Conversion of implicit sub... |
| csbiedOLD 3867 | Obsolete version of ~ csbi... |
| csbied2 3868 | Conversion of implicit sub... |
| csbie2t 3869 | Conversion of implicit sub... |
| csbie2 3870 | Conversion of implicit sub... |
| csbie2g 3871 | Conversion of implicit sub... |
| cbvrabcsfw 3872 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3873 | A more general version of ... |
| cbvrexcsf 3874 | A more general version of ... |
| cbvreucsf 3875 | A more general version of ... |
| cbvrabcsf 3876 | A more general version of ... |
| cbvralv2 3877 | Rule used to change the bo... |
| cbvrexv2 3878 | Rule used to change the bo... |
| rspc2vd 3879 | Deduction version of 2-var... |
| difjust 3885 | Soundness justification th... |
| unjust 3887 | Soundness justification th... |
| injust 3889 | Soundness justification th... |
| dfin5 3891 | Alternate definition for t... |
| dfdif2 3892 | Alternate definition of cl... |
| eldif 3893 | Expansion of membership in... |
| eldifd 3894 | If a class is in one class... |
| eldifad 3895 | If a class is in the diffe... |
| eldifbd 3896 | If a class is in the diffe... |
| elneeldif 3897 | The elements of a set diff... |
| velcomp 3898 | Characterization of setvar... |
| elin 3899 | Expansion of membership in... |
| dfss 3901 | Variant of subclass defini... |
| dfss2 3903 | Alternate definition of th... |
| dfss2OLD 3904 | Obsolete version of ~ dfss... |
| dfss3 3905 | Alternate definition of su... |
| dfss6 3906 | Alternate definition of su... |
| dfss2f 3907 | Equivalence for subclass r... |
| dfss3f 3908 | Equivalence for subclass r... |
| nfss 3909 | If ` x ` is not free in ` ... |
| ssel 3910 | Membership relationships f... |
| sselOLD 3911 | Obsolete version of ~ ssel... |
| ssel2 3912 | Membership relationships f... |
| sseli 3913 | Membership implication fro... |
| sselii 3914 | Membership inference from ... |
| sselid 3915 | Membership inference from ... |
| sseld 3916 | Membership deduction from ... |
| sselda 3917 | Membership deduction from ... |
| sseldd 3918 | Membership inference from ... |
| ssneld 3919 | If a class is not in anoth... |
| ssneldd 3920 | If an element is not in a ... |
| ssriv 3921 | Inference based on subclas... |
| ssrd 3922 | Deduction based on subclas... |
| ssrdv 3923 | Deduction based on subclas... |
| sstr2 3924 | Transitivity of subclass r... |
| sstr 3925 | Transitivity of subclass r... |
| sstri 3926 | Subclass transitivity infe... |
| sstrd 3927 | Subclass transitivity dedu... |
| sstrid 3928 | Subclass transitivity dedu... |
| sstrdi 3929 | Subclass transitivity dedu... |
| sylan9ss 3930 | A subclass transitivity de... |
| sylan9ssr 3931 | A subclass transitivity de... |
| eqss 3932 | The subclass relationship ... |
| eqssi 3933 | Infer equality from two su... |
| eqssd 3934 | Equality deduction from tw... |
| sssseq 3935 | If a class is a subclass o... |
| eqrd 3936 | Deduce equality of classes... |
| eqri 3937 | Infer equality of classes ... |
| eqelssd 3938 | Equality deduction from su... |
| ssid 3939 | Any class is a subclass of... |
| ssidd 3940 | Weakening of ~ ssid . (Co... |
| ssv 3941 | Any class is a subclass of... |
| sseq1 3942 | Equality theorem for subcl... |
| sseq2 3943 | Equality theorem for the s... |
| sseq12 3944 | Equality theorem for the s... |
| sseq1i 3945 | An equality inference for ... |
| sseq2i 3946 | An equality inference for ... |
| sseq12i 3947 | An equality inference for ... |
| sseq1d 3948 | An equality deduction for ... |
| sseq2d 3949 | An equality deduction for ... |
| sseq12d 3950 | An equality deduction for ... |
| eqsstri 3951 | Substitution of equality i... |
| eqsstrri 3952 | Substitution of equality i... |
| sseqtri 3953 | Substitution of equality i... |
| sseqtrri 3954 | Substitution of equality i... |
| eqsstrd 3955 | Substitution of equality i... |
| eqsstrrd 3956 | Substitution of equality i... |
| sseqtrd 3957 | Substitution of equality i... |
| sseqtrrd 3958 | Substitution of equality i... |
| 3sstr3i 3959 | Substitution of equality i... |
| 3sstr4i 3960 | Substitution of equality i... |
| 3sstr3g 3961 | Substitution of equality i... |
| 3sstr4g 3962 | Substitution of equality i... |
| 3sstr3d 3963 | Substitution of equality i... |
| 3sstr4d 3964 | Substitution of equality i... |
| eqsstrid 3965 | A chained subclass and equ... |
| eqsstrrid 3966 | A chained subclass and equ... |
| sseqtrdi 3967 | A chained subclass and equ... |
| sseqtrrdi 3968 | A chained subclass and equ... |
| sseqtrid 3969 | Subclass transitivity dedu... |
| sseqtrrid 3970 | Subclass transitivity dedu... |
| eqsstrdi 3971 | A chained subclass and equ... |
| eqsstrrdi 3972 | A chained subclass and equ... |
| eqimss 3973 | Equality implies inclusion... |
| eqimss2 3974 | Equality implies inclusion... |
| eqimssi 3975 | Infer subclass relationshi... |
| eqimss2i 3976 | Infer subclass relationshi... |
| nssne1 3977 | Two classes are different ... |
| nssne2 3978 | Two classes are different ... |
| nss 3979 | Negation of subclass relat... |
| nelss 3980 | Demonstrate by witnesses t... |
| ssrexf 3981 | Restricted existential qua... |
| ssrmof 3982 | "At most one" existential ... |
| ssralv 3983 | Quantification restricted ... |
| ssrexv 3984 | Existential quantification... |
| ss2ralv 3985 | Two quantifications restri... |
| ss2rexv 3986 | Two existential quantifica... |
| ralss 3987 | Restricted universal quant... |
| rexss 3988 | Restricted existential qua... |
| ss2ab 3989 | Class abstractions in a su... |
| abss 3990 | Class abstraction in a sub... |
| ssab 3991 | Subclass of a class abstra... |
| ssabral 3992 | The relation for a subclas... |
| ss2abdv 3993 | Deduction of abstraction s... |
| ss2abdvALT 3994 | Alternate proof of ~ ss2ab... |
| ss2abdvOLD 3995 | Obsolete version of ~ ss2a... |
| ss2abi 3996 | Inference of abstraction s... |
| ss2abiOLD 3997 | Obsolete version of ~ ss2a... |
| abssdv 3998 | Deduction of abstraction s... |
| abssi 3999 | Inference of abstraction s... |
| ss2rab 4000 | Restricted abstraction cla... |
| rabss 4001 | Restricted class abstracti... |
| ssrab 4002 | Subclass of a restricted c... |
| ssrabdv 4003 | Subclass of a restricted c... |
| rabssdv 4004 | Subclass of a restricted c... |
| ss2rabdv 4005 | Deduction of restricted ab... |
| ss2rabi 4006 | Inference of restricted ab... |
| rabss2 4007 | Subclass law for restricte... |
| ssab2 4008 | Subclass relation for the ... |
| ssrab2 4009 | Subclass relation for a re... |
| ssrab2OLD 4010 | Obsolete version of ~ ssra... |
| ssrab3 4011 | Subclass relation for a re... |
| rabssrabd 4012 | Subclass of a restricted c... |
| ssrabeq 4013 | If the restricting class o... |
| rabssab 4014 | A restricted class is a su... |
| uniiunlem 4015 | A subset relationship usef... |
| dfpss2 4016 | Alternate definition of pr... |
| dfpss3 4017 | Alternate definition of pr... |
| psseq1 4018 | Equality theorem for prope... |
| psseq2 4019 | Equality theorem for prope... |
| psseq1i 4020 | An equality inference for ... |
| psseq2i 4021 | An equality inference for ... |
| psseq12i 4022 | An equality inference for ... |
| psseq1d 4023 | An equality deduction for ... |
| psseq2d 4024 | An equality deduction for ... |
| psseq12d 4025 | An equality deduction for ... |
| pssss 4026 | A proper subclass is a sub... |
| pssne 4027 | Two classes in a proper su... |
| pssssd 4028 | Deduce subclass from prope... |
| pssned 4029 | Proper subclasses are uneq... |
| sspss 4030 | Subclass in terms of prope... |
| pssirr 4031 | Proper subclass is irrefle... |
| pssn2lp 4032 | Proper subclass has no 2-c... |
| sspsstri 4033 | Two ways of stating tricho... |
| ssnpss 4034 | Partial trichotomy law for... |
| psstr 4035 | Transitive law for proper ... |
| sspsstr 4036 | Transitive law for subclas... |
| psssstr 4037 | Transitive law for subclas... |
| psstrd 4038 | Proper subclass inclusion ... |
| sspsstrd 4039 | Transitivity involving sub... |
| psssstrd 4040 | Transitivity involving sub... |
| npss 4041 | A class is not a proper su... |
| ssnelpss 4042 | A subclass missing a membe... |
| ssnelpssd 4043 | Subclass inclusion with on... |
| ssexnelpss 4044 | If there is an element of ... |
| dfdif3 4045 | Alternate definition of cl... |
| difeq1 4046 | Equality theorem for class... |
| difeq2 4047 | Equality theorem for class... |
| difeq12 4048 | Equality theorem for class... |
| difeq1i 4049 | Inference adding differenc... |
| difeq2i 4050 | Inference adding differenc... |
| difeq12i 4051 | Equality inference for cla... |
| difeq1d 4052 | Deduction adding differenc... |
| difeq2d 4053 | Deduction adding differenc... |
| difeq12d 4054 | Equality deduction for cla... |
| difeqri 4055 | Inference from membership ... |
| nfdif 4056 | Bound-variable hypothesis ... |
| eldifi 4057 | Implication of membership ... |
| eldifn 4058 | Implication of membership ... |
| elndif 4059 | A set does not belong to a... |
| neldif 4060 | Implication of membership ... |
| difdif 4061 | Double class difference. ... |
| difss 4062 | Subclass relationship for ... |
| difssd 4063 | A difference of two classe... |
| difss2 4064 | If a class is contained in... |
| difss2d 4065 | If a class is contained in... |
| ssdifss 4066 | Preservation of a subclass... |
| ddif 4067 | Double complement under un... |
| ssconb 4068 | Contraposition law for sub... |
| sscon 4069 | Contraposition law for sub... |
| ssdif 4070 | Difference law for subsets... |
| ssdifd 4071 | If ` A ` is contained in `... |
| sscond 4072 | If ` A ` is contained in `... |
| ssdifssd 4073 | If ` A ` is contained in `... |
| ssdif2d 4074 | If ` A ` is contained in `... |
| raldifb 4075 | Restricted universal quant... |
| rexdifi 4076 | Restricted existential qua... |
| complss 4077 | Complementation reverses i... |
| compleq 4078 | Two classes are equal if a... |
| elun 4079 | Expansion of membership in... |
| elunnel1 4080 | A member of a union that i... |
| uneqri 4081 | Inference from membership ... |
| unidm 4082 | Idempotent law for union o... |
| uncom 4083 | Commutative law for union ... |
| equncom 4084 | If a class equals the unio... |
| equncomi 4085 | Inference form of ~ equnco... |
| uneq1 4086 | Equality theorem for the u... |
| uneq2 4087 | Equality theorem for the u... |
| uneq12 4088 | Equality theorem for the u... |
| uneq1i 4089 | Inference adding union to ... |
| uneq2i 4090 | Inference adding union to ... |
| uneq12i 4091 | Equality inference for the... |
| uneq1d 4092 | Deduction adding union to ... |
| uneq2d 4093 | Deduction adding union to ... |
| uneq12d 4094 | Equality deduction for the... |
| nfun 4095 | Bound-variable hypothesis ... |
| unass 4096 | Associative law for union ... |
| un12 4097 | A rearrangement of union. ... |
| un23 4098 | A rearrangement of union. ... |
| un4 4099 | A rearrangement of the uni... |
| unundi 4100 | Union distributes over its... |
| unundir 4101 | Union distributes over its... |
| ssun1 4102 | Subclass relationship for ... |
| ssun2 4103 | Subclass relationship for ... |
| ssun3 4104 | Subclass law for union of ... |
| ssun4 4105 | Subclass law for union of ... |
| elun1 4106 | Membership law for union o... |
| elun2 4107 | Membership law for union o... |
| elunant 4108 | A statement is true for ev... |
| unss1 4109 | Subclass law for union of ... |
| ssequn1 4110 | A relationship between sub... |
| unss2 4111 | Subclass law for union of ... |
| unss12 4112 | Subclass law for union of ... |
| ssequn2 4113 | A relationship between sub... |
| unss 4114 | The union of two subclasse... |
| unssi 4115 | An inference showing the u... |
| unssd 4116 | A deduction showing the un... |
| unssad 4117 | If ` ( A u. B ) ` is conta... |
| unssbd 4118 | If ` ( A u. B ) ` is conta... |
| ssun 4119 | A condition that implies i... |
| rexun 4120 | Restricted existential qua... |
| ralunb 4121 | Restricted quantification ... |
| ralun 4122 | Restricted quantification ... |
| elini 4123 | Membership in an intersect... |
| elind 4124 | Deduce membership in an in... |
| elinel1 4125 | Membership in an intersect... |
| elinel2 4126 | Membership in an intersect... |
| elin2 4127 | Membership in a class defi... |
| elin1d 4128 | Elementhood in the first s... |
| elin2d 4129 | Elementhood in the first s... |
| elin3 4130 | Membership in a class defi... |
| incom 4131 | Commutative law for inters... |
| incomOLD 4132 | Obsolete version of ~ inco... |
| ineqcom 4133 | Two ways of expressing tha... |
| ineqcomi 4134 | Two ways of expressing tha... |
| ineqri 4135 | Inference from membership ... |
| ineq1 4136 | Equality theorem for inter... |
| ineq2 4137 | Equality theorem for inter... |
| ineq12 4138 | Equality theorem for inter... |
| ineq1i 4139 | Equality inference for int... |
| ineq2i 4140 | Equality inference for int... |
| ineq12i 4141 | Equality inference for int... |
| ineq1d 4142 | Equality deduction for int... |
| ineq2d 4143 | Equality deduction for int... |
| ineq12d 4144 | Equality deduction for int... |
| ineqan12d 4145 | Equality deduction for int... |
| sseqin2 4146 | A relationship between sub... |
| nfin 4147 | Bound-variable hypothesis ... |
| rabbi2dva 4148 | Deduction from a wff to a ... |
| inidm 4149 | Idempotent law for interse... |
| inass 4150 | Associative law for inters... |
| in12 4151 | A rearrangement of interse... |
| in32 4152 | A rearrangement of interse... |
| in13 4153 | A rearrangement of interse... |
| in31 4154 | A rearrangement of interse... |
| inrot 4155 | Rotate the intersection of... |
| in4 4156 | Rearrangement of intersect... |
| inindi 4157 | Intersection distributes o... |
| inindir 4158 | Intersection distributes o... |
| inss1 4159 | The intersection of two cl... |
| inss2 4160 | The intersection of two cl... |
| ssin 4161 | Subclass of intersection. ... |
| ssini 4162 | An inference showing that ... |
| ssind 4163 | A deduction showing that a... |
| ssrin 4164 | Add right intersection to ... |
| sslin 4165 | Add left intersection to s... |
| ssrind 4166 | Add right intersection to ... |
| ss2in 4167 | Intersection of subclasses... |
| ssinss1 4168 | Intersection preserves sub... |
| inss 4169 | Inclusion of an intersecti... |
| rexin 4170 | Restricted existential qua... |
| dfss7 4171 | Alternate definition of su... |
| symdifcom 4174 | Symmetric difference commu... |
| symdifeq1 4175 | Equality theorem for symme... |
| symdifeq2 4176 | Equality theorem for symme... |
| nfsymdif 4177 | Hypothesis builder for sym... |
| elsymdif 4178 | Membership in a symmetric ... |
| dfsymdif4 4179 | Alternate definition of th... |
| elsymdifxor 4180 | Membership in a symmetric ... |
| dfsymdif2 4181 | Alternate definition of th... |
| symdifass 4182 | Symmetric difference is as... |
| difsssymdif 4183 | The symmetric difference c... |
| difsymssdifssd 4184 | If the symmetric differenc... |
| unabs 4185 | Absorption law for union. ... |
| inabs 4186 | Absorption law for interse... |
| nssinpss 4187 | Negation of subclass expre... |
| nsspssun 4188 | Negation of subclass expre... |
| dfss4 4189 | Subclass defined in terms ... |
| dfun2 4190 | An alternate definition of... |
| dfin2 4191 | An alternate definition of... |
| difin 4192 | Difference with intersecti... |
| ssdifim 4193 | Implication of a class dif... |
| ssdifsym 4194 | Symmetric class difference... |
| dfss5 4195 | Alternate definition of su... |
| dfun3 4196 | Union defined in terms of ... |
| dfin3 4197 | Intersection defined in te... |
| dfin4 4198 | Alternate definition of th... |
| invdif 4199 | Intersection with universa... |
| indif 4200 | Intersection with class di... |
| indif2 4201 | Bring an intersection in a... |
| indif1 4202 | Bring an intersection in a... |
| indifcom 4203 | Commutation law for inters... |
| indi 4204 | Distributive law for inter... |
| undi 4205 | Distributive law for union... |
| indir 4206 | Distributive law for inter... |
| undir 4207 | Distributive law for union... |
| unineq 4208 | Infer equality from equali... |
| uneqin 4209 | Equality of union and inte... |
| difundi 4210 | Distributive law for class... |
| difundir 4211 | Distributive law for class... |
| difindi 4212 | Distributive law for class... |
| difindir 4213 | Distributive law for class... |
| indifdi 4214 | Distribute intersection ov... |
| indifdir 4215 | Distribute intersection ov... |
| indifdirOLD 4216 | Obsolete version of ~ indi... |
| difdif2 4217 | Class difference by a clas... |
| undm 4218 | De Morgan's law for union.... |
| indm 4219 | De Morgan's law for inters... |
| difun1 4220 | A relationship involving d... |
| undif3 4221 | An equality involving clas... |
| difin2 4222 | Represent a class differen... |
| dif32 4223 | Swap second and third argu... |
| difabs 4224 | Absorption-like law for cl... |
| sscon34b 4225 | Relative complementation r... |
| rcompleq 4226 | Two subclasses are equal i... |
| dfsymdif3 4227 | Alternate definition of th... |
| unabw 4228 | Union of two class abstrac... |
| unab 4229 | Union of two class abstrac... |
| inab 4230 | Intersection of two class ... |
| difab 4231 | Difference of two class ab... |
| abanssl 4232 | A class abstraction with a... |
| abanssr 4233 | A class abstraction with a... |
| notabw 4234 | A class abstraction define... |
| notab 4235 | A class abstraction define... |
| unrab 4236 | Union of two restricted cl... |
| inrab 4237 | Intersection of two restri... |
| inrab2 4238 | Intersection with a restri... |
| difrab 4239 | Difference of two restrict... |
| dfrab3 4240 | Alternate definition of re... |
| dfrab2 4241 | Alternate definition of re... |
| notrab 4242 | Complementation of restric... |
| dfrab3ss 4243 | Restricted class abstracti... |
| rabun2 4244 | Abstraction restricted to ... |
| reuun2 4245 | Transfer uniqueness to a s... |
| reuss2 4246 | Transfer uniqueness to a s... |
| reuss 4247 | Transfer uniqueness to a s... |
| reuun1 4248 | Transfer uniqueness to a s... |
| reupick 4249 | Restricted uniqueness "pic... |
| reupick3 4250 | Restricted uniqueness "pic... |
| reupick2 4251 | Restricted uniqueness "pic... |
| euelss 4252 | Transfer uniqueness of an ... |
| dfnul4 4255 | Alternate definition of th... |
| dfnul2 4256 | Alternate definition of th... |
| dfnul3 4257 | Alternate definition of th... |
| dfnul2OLD 4258 | Obsolete version of ~ dfnu... |
| dfnul3OLD 4259 | Obsolete version of ~ dfnu... |
| dfnul4OLD 4260 | Obsolete version of ~ dfnu... |
| noel 4261 | The empty set has no eleme... |
| noelOLD 4262 | Obsolete version of ~ noel... |
| nel02 4263 | The empty set has no eleme... |
| n0i 4264 | If a class has elements, t... |
| ne0i 4265 | If a class has elements, t... |
| ne0d 4266 | Deduction form of ~ ne0i .... |
| n0ii 4267 | If a class has elements, t... |
| ne0ii 4268 | If a class has elements, t... |
| vn0 4269 | The universal class is not... |
| vn0ALT 4270 | Alternate proof of ~ vn0 .... |
| eq0f 4271 | A class is equal to the em... |
| neq0f 4272 | A class is not empty if an... |
| n0f 4273 | A class is nonempty if and... |
| eq0 4274 | A class is equal to the em... |
| eq0ALT 4275 | Alternate proof of ~ eq0 .... |
| neq0 4276 | A class is not empty if an... |
| n0 4277 | A class is nonempty if and... |
| eq0OLDOLD 4278 | Obsolete version of ~ eq0 ... |
| neq0OLD 4279 | Obsolete version of ~ neq0... |
| n0OLD 4280 | Obsolete version of ~ n0 a... |
| nel0 4281 | From the general negation ... |
| reximdva0 4282 | Restricted existence deduc... |
| rspn0 4283 | Specialization for restric... |
| rspn0OLD 4284 | Obsolete version of ~ rspn... |
| n0rex 4285 | There is an element in a n... |
| ssn0rex 4286 | There is an element in a c... |
| n0moeu 4287 | A case of equivalence of "... |
| rex0 4288 | Vacuous restricted existen... |
| reu0 4289 | Vacuous restricted uniquen... |
| rmo0 4290 | Vacuous restricted at-most... |
| 0el 4291 | Membership of the empty se... |
| n0el 4292 | Negated membership of the ... |
| eqeuel 4293 | A condition which implies ... |
| ssdif0 4294 | Subclass expressed in term... |
| difn0 4295 | If the difference of two s... |
| pssdifn0 4296 | A proper subclass has a no... |
| pssdif 4297 | A proper subclass has a no... |
| ndisj 4298 | Express that an intersecti... |
| difin0ss 4299 | Difference, intersection, ... |
| inssdif0 4300 | Intersection, subclass, an... |
| difid 4301 | The difference between a c... |
| difidALT 4302 | Alternate proof of ~ difid... |
| dif0 4303 | The difference between a c... |
| ab0w 4304 | The class of sets verifyin... |
| ab0 4305 | The class of sets verifyin... |
| ab0OLD 4306 | Obsolete version of ~ ab0 ... |
| ab0ALT 4307 | Alternate proof of ~ ab0 ,... |
| dfnf5 4308 | Characterization of nonfre... |
| ab0orv 4309 | The class abstraction defi... |
| ab0orvALT 4310 | Alternate proof of ~ ab0or... |
| abn0 4311 | Nonempty class abstraction... |
| abn0OLD 4312 | Obsolete version of ~ abn0... |
| rab0 4313 | Any restricted class abstr... |
| rabeq0w 4314 | Condition for a restricted... |
| rabeq0 4315 | Condition for a restricted... |
| rabn0 4316 | Nonempty restricted class ... |
| rabxm 4317 | Law of excluded middle, in... |
| rabnc 4318 | Law of noncontradiction, i... |
| elneldisj 4319 | The set of elements ` s ` ... |
| elnelun 4320 | The union of the set of el... |
| un0 4321 | The union of a class with ... |
| in0 4322 | The intersection of a clas... |
| 0un 4323 | The union of the empty set... |
| 0in 4324 | The intersection of the em... |
| inv1 4325 | The intersection of a clas... |
| unv 4326 | The union of a class with ... |
| 0ss 4327 | The null set is a subset o... |
| ss0b 4328 | Any subset of the empty se... |
| ss0 4329 | Any subset of the empty se... |
| sseq0 4330 | A subclass of an empty cla... |
| ssn0 4331 | A class with a nonempty su... |
| 0dif 4332 | The difference between the... |
| abf 4333 | A class abstraction determ... |
| abfOLD 4334 | Obsolete version of ~ abf ... |
| eq0rdv 4335 | Deduction for equality to ... |
| eq0rdvALT 4336 | Alternate proof of ~ eq0rd... |
| csbprc 4337 | The proper substitution of... |
| csb0 4338 | The proper substitution of... |
| sbcel12 4339 | Distribute proper substitu... |
| sbceqg 4340 | Distribute proper substitu... |
| sbceqi 4341 | Distribution of class subs... |
| sbcnel12g 4342 | Distribute proper substitu... |
| sbcne12 4343 | Distribute proper substitu... |
| sbcel1g 4344 | Move proper substitution i... |
| sbceq1g 4345 | Move proper substitution t... |
| sbcel2 4346 | Move proper substitution i... |
| sbceq2g 4347 | Move proper substitution t... |
| csbcom 4348 | Commutative law for double... |
| sbcnestgfw 4349 | Nest the composition of tw... |
| csbnestgfw 4350 | Nest the composition of tw... |
| sbcnestgw 4351 | Nest the composition of tw... |
| csbnestgw 4352 | Nest the composition of tw... |
| sbcco3gw 4353 | Composition of two substit... |
| sbcnestgf 4354 | Nest the composition of tw... |
| csbnestgf 4355 | Nest the composition of tw... |
| sbcnestg 4356 | Nest the composition of tw... |
| csbnestg 4357 | Nest the composition of tw... |
| sbcco3g 4358 | Composition of two substit... |
| csbco3g 4359 | Composition of two class s... |
| csbnest1g 4360 | Nest the composition of tw... |
| csbidm 4361 | Idempotent law for class s... |
| csbvarg 4362 | The proper substitution of... |
| csbvargi 4363 | The proper substitution of... |
| sbccsb 4364 | Substitution into a wff ex... |
| sbccsb2 4365 | Substitution into a wff ex... |
| rspcsbela 4366 | Special case related to ~ ... |
| sbnfc2 4367 | Two ways of expressing " `... |
| csbab 4368 | Move substitution into a c... |
| csbun 4369 | Distribution of class subs... |
| csbin 4370 | Distribute proper substitu... |
| csbie2df 4371 | Conversion of implicit sub... |
| 2nreu 4372 | If there are two different... |
| un00 4373 | Two classes are empty iff ... |
| vss 4374 | Only the universal class h... |
| 0pss 4375 | The null set is a proper s... |
| npss0 4376 | No set is a proper subset ... |
| pssv 4377 | Any non-universal class is... |
| disj 4378 | Two ways of saying that tw... |
| disjOLD 4379 | Obsolete version of ~ disj... |
| disjr 4380 | Two ways of saying that tw... |
| disj1 4381 | Two ways of saying that tw... |
| reldisj 4382 | Two ways of saying that tw... |
| reldisjOLD 4383 | Obsolete version of ~ reld... |
| disj3 4384 | Two ways of saying that tw... |
| disjne 4385 | Members of disjoint sets a... |
| disjeq0 4386 | Two disjoint sets are equa... |
| disjel 4387 | A set can't belong to both... |
| disj2 4388 | Two ways of saying that tw... |
| disj4 4389 | Two ways of saying that tw... |
| ssdisj 4390 | Intersection with a subcla... |
| disjpss 4391 | A class is a proper subset... |
| undisj1 4392 | The union of disjoint clas... |
| undisj2 4393 | The union of disjoint clas... |
| ssindif0 4394 | Subclass expressed in term... |
| inelcm 4395 | The intersection of classe... |
| minel 4396 | A minimum element of a cla... |
| undif4 4397 | Distribute union over diff... |
| disjssun 4398 | Subset relation for disjoi... |
| vdif0 4399 | Universal class equality i... |
| difrab0eq 4400 | If the difference between ... |
| pssnel 4401 | A proper subclass has a me... |
| disjdif 4402 | A class and its relative c... |
| disjdifr 4403 | A class and its relative c... |
| difin0 4404 | The difference of a class ... |
| unvdif 4405 | The union of a class and i... |
| undif1 4406 | Absorption of difference b... |
| undif2 4407 | Absorption of difference b... |
| undifabs 4408 | Absorption of difference b... |
| inundif 4409 | The intersection and class... |
| disjdif2 4410 | The difference of a class ... |
| difun2 4411 | Absorption of union by dif... |
| undif 4412 | Union of complementary par... |
| ssdifin0 4413 | A subset of a difference d... |
| ssdifeq0 4414 | A class is a subclass of i... |
| ssundif 4415 | A condition equivalent to ... |
| difcom 4416 | Swap the arguments of a cl... |
| pssdifcom1 4417 | Two ways to express overla... |
| pssdifcom2 4418 | Two ways to express non-co... |
| difdifdir 4419 | Distributive law for class... |
| uneqdifeq 4420 | Two ways to say that ` A `... |
| raldifeq 4421 | Equality theorem for restr... |
| r19.2z 4422 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4423 | A response to the notion t... |
| r19.3rz 4424 | Restricted quantification ... |
| r19.28z 4425 | Restricted quantifier vers... |
| r19.3rzv 4426 | Restricted quantification ... |
| r19.9rzv 4427 | Restricted quantification ... |
| r19.28zv 4428 | Restricted quantifier vers... |
| r19.37zv 4429 | Restricted quantifier vers... |
| r19.45zv 4430 | Restricted version of Theo... |
| r19.44zv 4431 | Restricted version of Theo... |
| r19.27z 4432 | Restricted quantifier vers... |
| r19.27zv 4433 | Restricted quantifier vers... |
| r19.36zv 4434 | Restricted quantifier vers... |
| ralidmw 4435 | Idempotent law for restric... |
| rzal 4436 | Vacuous quantification is ... |
| rzalALT 4437 | Alternate proof of ~ rzal ... |
| rexn0 4438 | Restricted existential qua... |
| ralidm 4439 | Idempotent law for restric... |
| ral0 4440 | Vacuous universal quantifi... |
| ralf0 4441 | The quantification of a fa... |
| rexn0OLD 4442 | Obsolete version of ~ rexn... |
| ralidmOLD 4443 | Obsolete version of ~ rali... |
| ral0OLD 4444 | Obsolete version of ~ ral0... |
| ralf0OLD 4445 | Obsolete version of ~ ralf... |
| ralnralall 4446 | A contradiction concerning... |
| falseral0 4447 | A false statement can only... |
| raaan 4448 | Rearrange restricted quant... |
| raaanv 4449 | Rearrange restricted quant... |
| sbss 4450 | Set substitution into the ... |
| sbcssg 4451 | Distribute proper substitu... |
| raaan2 4452 | Rearrange restricted quant... |
| 2reu4lem 4453 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4454 | Definition of double restr... |
| csbdif 4455 | Distribution of class subs... |
| dfif2 4458 | An alternate definition of... |
| dfif6 4459 | An alternate definition of... |
| ifeq1 4460 | Equality theorem for condi... |
| ifeq2 4461 | Equality theorem for condi... |
| iftrue 4462 | Value of the conditional o... |
| iftruei 4463 | Inference associated with ... |
| iftrued 4464 | Value of the conditional o... |
| iffalse 4465 | Value of the conditional o... |
| iffalsei 4466 | Inference associated with ... |
| iffalsed 4467 | Value of the conditional o... |
| ifnefalse 4468 | When values are unequal, b... |
| ifsb 4469 | Distribute a function over... |
| dfif3 4470 | Alternate definition of th... |
| dfif4 4471 | Alternate definition of th... |
| dfif5 4472 | Alternate definition of th... |
| ifssun 4473 | A conditional class is inc... |
| ifeq12 4474 | Equality theorem for condi... |
| ifeq1d 4475 | Equality deduction for con... |
| ifeq2d 4476 | Equality deduction for con... |
| ifeq12d 4477 | Equality deduction for con... |
| ifbi 4478 | Equivalence theorem for co... |
| ifbid 4479 | Equivalence deduction for ... |
| ifbieq1d 4480 | Equivalence/equality deduc... |
| ifbieq2i 4481 | Equivalence/equality infer... |
| ifbieq2d 4482 | Equivalence/equality deduc... |
| ifbieq12i 4483 | Equivalence deduction for ... |
| ifbieq12d 4484 | Equivalence deduction for ... |
| nfifd 4485 | Deduction form of ~ nfif .... |
| nfif 4486 | Bound-variable hypothesis ... |
| ifeq1da 4487 | Conditional equality. (Co... |
| ifeq2da 4488 | Conditional equality. (Co... |
| ifeq12da 4489 | Equivalence deduction for ... |
| ifbieq12d2 4490 | Equivalence deduction for ... |
| ifclda 4491 | Conditional closure. (Con... |
| ifeqda 4492 | Separation of the values o... |
| elimif 4493 | Elimination of a condition... |
| ifbothda 4494 | A wff ` th ` containing a ... |
| ifboth 4495 | A wff ` th ` containing a ... |
| ifid 4496 | Identical true and false a... |
| eqif 4497 | Expansion of an equality w... |
| ifval 4498 | Another expression of the ... |
| elif 4499 | Membership in a conditiona... |
| ifel 4500 | Membership of a conditiona... |
| ifcl 4501 | Membership (closure) of a ... |
| ifcld 4502 | Membership (closure) of a ... |
| ifcli 4503 | Inference associated with ... |
| ifexd 4504 | Existence of the condition... |
| ifexg 4505 | Existence of the condition... |
| ifex 4506 | Existence of the condition... |
| ifeqor 4507 | The possible values of a c... |
| ifnot 4508 | Negating the first argumen... |
| ifan 4509 | Rewrite a conjunction in a... |
| ifor 4510 | Rewrite a disjunction in a... |
| 2if2 4511 | Resolve two nested conditi... |
| ifcomnan 4512 | Commute the conditions in ... |
| csbif 4513 | Distribute proper substitu... |
| dedth 4514 | Weak deduction theorem tha... |
| dedth2h 4515 | Weak deduction theorem eli... |
| dedth3h 4516 | Weak deduction theorem eli... |
| dedth4h 4517 | Weak deduction theorem eli... |
| dedth2v 4518 | Weak deduction theorem for... |
| dedth3v 4519 | Weak deduction theorem for... |
| dedth4v 4520 | Weak deduction theorem for... |
| elimhyp 4521 | Eliminate a hypothesis con... |
| elimhyp2v 4522 | Eliminate a hypothesis con... |
| elimhyp3v 4523 | Eliminate a hypothesis con... |
| elimhyp4v 4524 | Eliminate a hypothesis con... |
| elimel 4525 | Eliminate a membership hyp... |
| elimdhyp 4526 | Version of ~ elimhyp where... |
| keephyp 4527 | Transform a hypothesis ` p... |
| keephyp2v 4528 | Keep a hypothesis containi... |
| keephyp3v 4529 | Keep a hypothesis containi... |
| pwjust 4531 | Soundness justification th... |
| elpwg 4533 | Membership in a power clas... |
| elpw 4534 | Membership in a power clas... |
| velpw 4535 | Setvar variable membership... |
| elpwOLD 4536 | Obsolete proof of ~ elpw a... |
| elpwgOLD 4537 | Obsolete proof of ~ elpwg ... |
| elpwd 4538 | Membership in a power clas... |
| elpwi 4539 | Subset relation implied by... |
| elpwb 4540 | Characterization of the el... |
| elpwid 4541 | An element of a power clas... |
| elelpwi 4542 | If ` A ` belongs to a part... |
| sspw 4543 | The powerclass preserves i... |
| sspwi 4544 | The powerclass preserves i... |
| sspwd 4545 | The powerclass preserves i... |
| pweq 4546 | Equality theorem for power... |
| pweqALT 4547 | Alternate proof of ~ pweq ... |
| pweqi 4548 | Equality inference for pow... |
| pweqd 4549 | Equality deduction for pow... |
| pwunss 4550 | The power class of the uni... |
| 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 ... |
| pwundif 4556 | Break up the power class o... |
| snjust 4557 | Soundness justification th... |
| sneq 4568 | Equality theorem for singl... |
| sneqi 4569 | Equality inference for sin... |
| sneqd 4570 | Equality deduction for sin... |
| dfsn2 4571 | Alternate definition of si... |
| elsng 4572 | There is exactly one eleme... |
| elsn 4573 | There is exactly one eleme... |
| velsn 4574 | There is only one element ... |
| elsni 4575 | There is at most one eleme... |
| absn 4576 | Condition for a class abst... |
| dfpr2 4577 | Alternate definition of a ... |
| dfsn2ALT 4578 | Alternate definition of si... |
| elprg 4579 | A member of a pair of clas... |
| elpri 4580 | If a class is an element o... |
| elpr 4581 | A member of a pair of clas... |
| elpr2g 4582 | A member of a pair of sets... |
| elpr2 4583 | A member of a pair of sets... |
| elpr2OLD 4584 | Obsolete version of ~ elpr... |
| nelpr2 4585 | If a class is not an eleme... |
| nelpr1 4586 | If a class is not an eleme... |
| nelpri 4587 | If an element doesn't matc... |
| prneli 4588 | If an element doesn't matc... |
| nelprd 4589 | If an element doesn't matc... |
| eldifpr 4590 | Membership in a set with t... |
| rexdifpr 4591 | Restricted existential qua... |
| snidg 4592 | A set is a member of its s... |
| snidb 4593 | A class is a set iff it is... |
| snid 4594 | A set is a member of its s... |
| vsnid 4595 | A setvar variable is a mem... |
| elsn2g 4596 | There is exactly one eleme... |
| elsn2 4597 | There is exactly one eleme... |
| nelsn 4598 | If a class is not equal to... |
| rabeqsn 4599 | Conditions for a restricte... |
| rabsssn 4600 | Conditions for a restricte... |
| ralsnsg 4601 | Substitution expressed in ... |
| rexsns 4602 | Restricted existential qua... |
| rexsngf 4603 | Restricted existential qua... |
| ralsngf 4604 | Restricted universal quant... |
| reusngf 4605 | Restricted existential uni... |
| ralsng 4606 | Substitution expressed in ... |
| rexsng 4607 | Restricted existential qua... |
| reusng 4608 | Restricted existential uni... |
| 2ralsng 4609 | Substitution expressed in ... |
| ralsngOLD 4610 | Obsolete version of ~ rals... |
| rexsngOLD 4611 | Obsolete version of ~ rexs... |
| rexreusng 4612 | Restricted existential uni... |
| exsnrex 4613 | There is a set being the e... |
| ralsn 4614 | Convert a universal quanti... |
| rexsn 4615 | Convert an existential qua... |
| elpwunsn 4616 | Membership in an extension... |
| eqoreldif 4617 | An element of a set is eit... |
| eltpg 4618 | Members of an unordered tr... |
| eldiftp 4619 | Membership in a set with t... |
| eltpi 4620 | A member of an unordered t... |
| eltp 4621 | A member of an unordered t... |
| dftp2 4622 | Alternate definition of un... |
| nfpr 4623 | Bound-variable hypothesis ... |
| ifpr 4624 | Membership of a conditiona... |
| ralprgf 4625 | Convert a restricted unive... |
| rexprgf 4626 | Convert a restricted exist... |
| ralprg 4627 | Convert a restricted unive... |
| ralprgOLD 4628 | Obsolete version of ~ ralp... |
| rexprg 4629 | Convert a restricted exist... |
| rexprgOLD 4630 | Obsolete version of ~ rexp... |
| raltpg 4631 | Convert a restricted unive... |
| rextpg 4632 | Convert a restricted exist... |
| ralpr 4633 | Convert a restricted unive... |
| rexpr 4634 | Convert a restricted exist... |
| reuprg0 4635 | Convert a restricted exist... |
| reuprg 4636 | Convert a restricted exist... |
| reurexprg 4637 | Convert a restricted exist... |
| raltp 4638 | Convert a universal quanti... |
| rextp 4639 | Convert an existential qua... |
| nfsn 4640 | Bound-variable hypothesis ... |
| csbsng 4641 | Distribute proper substitu... |
| csbprg 4642 | Distribute proper substitu... |
| elinsn 4643 | If the intersection of two... |
| disjsn 4644 | Intersection with the sing... |
| disjsn2 4645 | Two distinct singletons ar... |
| disjpr2 4646 | Two completely distinct un... |
| disjprsn 4647 | The disjoint intersection ... |
| disjtpsn 4648 | The disjoint intersection ... |
| disjtp2 4649 | Two completely distinct un... |
| snprc 4650 | The singleton of a proper ... |
| snnzb 4651 | A singleton is nonempty if... |
| rmosn 4652 | A restricted at-most-one q... |
| r19.12sn 4653 | Special case of ~ r19.12 w... |
| rabsn 4654 | Condition where a restrict... |
| rabsnifsb 4655 | A restricted class abstrac... |
| rabsnif 4656 | A restricted class abstrac... |
| rabrsn 4657 | A restricted class abstrac... |
| euabsn2 4658 | Another way to express exi... |
| euabsn 4659 | Another way to express exi... |
| reusn 4660 | A way to express restricte... |
| absneu 4661 | Restricted existential uni... |
| rabsneu 4662 | Restricted existential uni... |
| eusn 4663 | Two ways to express " ` A ... |
| rabsnt 4664 | Truth implied by equality ... |
| prcom 4665 | Commutative law for unorde... |
| preq1 4666 | Equality theorem for unord... |
| preq2 4667 | Equality theorem for unord... |
| preq12 4668 | Equality theorem for unord... |
| preq1i 4669 | Equality inference for uno... |
| preq2i 4670 | Equality inference for uno... |
| preq12i 4671 | Equality inference for uno... |
| preq1d 4672 | Equality deduction for uno... |
| preq2d 4673 | Equality deduction for uno... |
| preq12d 4674 | Equality deduction for uno... |
| tpeq1 4675 | Equality theorem for unord... |
| tpeq2 4676 | Equality theorem for unord... |
| tpeq3 4677 | Equality theorem for unord... |
| tpeq1d 4678 | Equality theorem for unord... |
| tpeq2d 4679 | Equality theorem for unord... |
| tpeq3d 4680 | Equality theorem for unord... |
| tpeq123d 4681 | Equality theorem for unord... |
| tprot 4682 | Rotation of the elements o... |
| tpcoma 4683 | Swap 1st and 2nd members o... |
| tpcomb 4684 | Swap 2nd and 3rd members o... |
| tpass 4685 | Split off the first elemen... |
| qdass 4686 | Two ways to write an unord... |
| qdassr 4687 | Two ways to write an unord... |
| tpidm12 4688 | Unordered triple ` { A , A... |
| tpidm13 4689 | Unordered triple ` { A , B... |
| tpidm23 4690 | Unordered triple ` { A , B... |
| tpidm 4691 | Unordered triple ` { A , A... |
| tppreq3 4692 | An unordered triple is an ... |
| prid1g 4693 | An unordered pair contains... |
| prid2g 4694 | An unordered pair contains... |
| prid1 4695 | An unordered pair contains... |
| prid2 4696 | An unordered pair contains... |
| ifpprsnss 4697 | An unordered pair is a sin... |
| prprc1 4698 | A proper class vanishes in... |
| prprc2 4699 | A proper class vanishes in... |
| prprc 4700 | An unordered pair containi... |
| tpid1 4701 | One of the three elements ... |
| tpid1g 4702 | Closed theorem form of ~ t... |
| tpid2 4703 | One of the three elements ... |
| tpid2g 4704 | Closed theorem form of ~ t... |
| tpid3g 4705 | Closed theorem form of ~ t... |
| tpid3 4706 | One of the three elements ... |
| snnzg 4707 | The singleton of a set is ... |
| snn0d 4708 | The singleton of a set is ... |
| snnz 4709 | The singleton of a set is ... |
| prnz 4710 | A pair containing a set is... |
| prnzg 4711 | A pair containing a set is... |
| tpnz 4712 | An unordered triple contai... |
| tpnzd 4713 | An unordered triple contai... |
| raltpd 4714 | Convert a universal quanti... |
| snssg 4715 | The singleton of an elemen... |
| snss 4716 | The singleton of an elemen... |
| eldifsn 4717 | Membership in a set with a... |
| ssdifsn 4718 | Subset of a set with an el... |
| elpwdifsn 4719 | A subset of a set is an el... |
| eldifsni 4720 | Membership in a set with a... |
| eldifsnneq 4721 | An element of a difference... |
| neldifsn 4722 | The class ` A ` is not in ... |
| neldifsnd 4723 | The class ` A ` is not in ... |
| rexdifsn 4724 | Restricted existential qua... |
| raldifsni 4725 | Rearrangement of a propert... |
| raldifsnb 4726 | Restricted universal quant... |
| eldifvsn 4727 | A set is an element of the... |
| difsn 4728 | An element not in a set ca... |
| difprsnss 4729 | Removal of a singleton fro... |
| difprsn1 4730 | Removal of a singleton fro... |
| difprsn2 4731 | Removal of a singleton fro... |
| diftpsn3 4732 | Removal of a singleton fro... |
| difpr 4733 | Removing two elements as p... |
| tpprceq3 4734 | An unordered triple is an ... |
| tppreqb 4735 | An unordered triple is an ... |
| difsnb 4736 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4737 | ` ( B \ { A } ) ` is a pro... |
| snssi 4738 | The singleton of an elemen... |
| snssd 4739 | The singleton of an elemen... |
| difsnid 4740 | If we remove a single elem... |
| eldifeldifsn 4741 | An element of a difference... |
| pw0 4742 | Compute the power set of t... |
| pwpw0 4743 | Compute the power set of t... |
| snsspr1 4744 | A singleton is a subset of... |
| snsspr2 4745 | A singleton is a subset of... |
| snsstp1 4746 | A singleton is a subset of... |
| snsstp2 4747 | A singleton is a subset of... |
| snsstp3 4748 | A singleton is a subset of... |
| prssg 4749 | A pair of elements of a cl... |
| prss 4750 | A pair of elements of a cl... |
| prssi 4751 | A pair of elements of a cl... |
| prssd 4752 | Deduction version of ~ prs... |
| prsspwg 4753 | An unordered pair belongs ... |
| ssprss 4754 | A pair as subset of a pair... |
| ssprsseq 4755 | A proper pair is a subset ... |
| sssn 4756 | The subsets of a singleton... |
| ssunsn2 4757 | The property of being sand... |
| ssunsn 4758 | Possible values for a set ... |
| eqsn 4759 | Two ways to express that a... |
| issn 4760 | A sufficient condition for... |
| n0snor2el 4761 | A nonempty set is either a... |
| ssunpr 4762 | Possible values for a set ... |
| sspr 4763 | The subsets of a pair. (C... |
| sstp 4764 | The subsets of an unordere... |
| tpss 4765 | An unordered triple of ele... |
| tpssi 4766 | An unordered triple of ele... |
| sneqrg 4767 | Closed form of ~ sneqr . ... |
| sneqr 4768 | If the singletons of two s... |
| snsssn 4769 | If a singleton is a subset... |
| mosneq 4770 | There exists at most one s... |
| sneqbg 4771 | Two singletons of sets are... |
| snsspw 4772 | The singleton of a class i... |
| prsspw 4773 | An unordered pair belongs ... |
| preq1b 4774 | Biconditional equality lem... |
| preq2b 4775 | Biconditional equality lem... |
| preqr1 4776 | Reverse equality lemma for... |
| preqr2 4777 | Reverse equality lemma for... |
| preq12b 4778 | Equality relationship for ... |
| opthpr 4779 | An unordered pair has the ... |
| preqr1g 4780 | Reverse equality lemma for... |
| preq12bg 4781 | Closed form of ~ preq12b .... |
| prneimg 4782 | Two pairs are not equal if... |
| prnebg 4783 | A (proper) pair is not equ... |
| pr1eqbg 4784 | A (proper) pair is equal t... |
| pr1nebg 4785 | A (proper) pair is not equ... |
| preqsnd 4786 | Equivalence for a pair equ... |
| prnesn 4787 | A proper unordered pair is... |
| prneprprc 4788 | A proper unordered pair is... |
| preqsn 4789 | Equivalence for a pair equ... |
| preq12nebg 4790 | Equality relationship for ... |
| prel12g 4791 | Equality of two unordered ... |
| opthprneg 4792 | An unordered pair has the ... |
| elpreqprlem 4793 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4794 | Equality and membership ru... |
| elpreqprb 4795 | A set is an element of an ... |
| elpr2elpr 4796 | For an element ` A ` of an... |
| dfopif 4797 | Rewrite ~ df-op using ` if... |
| dfopifOLD 4798 | Obsolete version of ~ dfop... |
| dfopg 4799 | Value of the ordered pair ... |
| dfop 4800 | Value of an ordered pair w... |
| opeq1 4801 | Equality theorem for order... |
| opeq2 4802 | Equality theorem for order... |
| opeq12 4803 | Equality theorem for order... |
| opeq1i 4804 | Equality inference for ord... |
| opeq2i 4805 | Equality inference for ord... |
| opeq12i 4806 | Equality inference for ord... |
| opeq1d 4807 | Equality deduction for ord... |
| opeq2d 4808 | Equality deduction for ord... |
| opeq12d 4809 | Equality deduction for ord... |
| oteq1 4810 | Equality theorem for order... |
| oteq2 4811 | Equality theorem for order... |
| oteq3 4812 | Equality theorem for order... |
| oteq1d 4813 | Equality deduction for ord... |
| oteq2d 4814 | Equality deduction for ord... |
| oteq3d 4815 | Equality deduction for ord... |
| oteq123d 4816 | Equality deduction for ord... |
| nfop 4817 | Bound-variable hypothesis ... |
| nfopd 4818 | Deduction version of bound... |
| csbopg 4819 | Distribution of class subs... |
| opidg 4820 | The ordered pair ` <. A , ... |
| opid 4821 | The ordered pair ` <. A , ... |
| ralunsn 4822 | Restricted quantification ... |
| 2ralunsn 4823 | Double restricted quantifi... |
| opprc 4824 | Expansion of an ordered pa... |
| opprc1 4825 | Expansion of an ordered pa... |
| opprc2 4826 | Expansion of an ordered pa... |
| oprcl 4827 | If an ordered pair has an ... |
| pwsn 4828 | The power set of a singlet... |
| pwsnOLD 4829 | Obsolete version of ~ pwsn... |
| pwpr 4830 | The power set of an unorde... |
| pwtp 4831 | The power set of an unorde... |
| pwpwpw0 4832 | Compute the power set of t... |
| pwv 4833 | The power class of the uni... |
| prproe 4834 | For an element of a proper... |
| 3elpr2eq 4835 | If there are three element... |
| dfuni2 4838 | Alternate definition of cl... |
| eluni 4839 | Membership in class union.... |
| eluni2 4840 | Membership in class union.... |
| elunii 4841 | Membership in class union.... |
| nfunid 4842 | Deduction version of ~ nfu... |
| nfuni 4843 | Bound-variable hypothesis ... |
| uniss 4844 | Subclass relationship for ... |
| unissi 4845 | Subclass relationship for ... |
| unissd 4846 | Subclass relationship for ... |
| unieq 4847 | Equality theorem for class... |
| unieqOLD 4848 | Obsolete version of ~ unie... |
| unieqi 4849 | Inference of equality of t... |
| unieqd 4850 | Deduction of equality of t... |
| eluniab 4851 | Membership in union of a c... |
| elunirab 4852 | Membership in union of a c... |
| uniprg 4853 | The union of a pair is the... |
| unipr 4854 | The union of a pair is the... |
| uniprOLD 4855 | Obsolete version of ~ unip... |
| uniprgOLD 4856 | Obsolete version of ~ unip... |
| unisng 4857 | A set equals the union of ... |
| unisn 4858 | A set equals the union of ... |
| unisn3 4859 | Union of a singleton in th... |
| dfnfc2 4860 | An alternative statement o... |
| uniun 4861 | The class union of the uni... |
| uniin 4862 | The class union of the int... |
| ssuni 4863 | Subclass relationship for ... |
| uni0b 4864 | The union of a set is empt... |
| uni0c 4865 | The union of a set is empt... |
| uni0 4866 | The union of the empty set... |
| csbuni 4867 | Distribute proper substitu... |
| elssuni 4868 | An element of a class is a... |
| unissel 4869 | Condition turning a subcla... |
| unissb 4870 | Relationship involving mem... |
| uniss2 4871 | A subclass condition on th... |
| unidif 4872 | If the difference ` A \ B ... |
| ssunieq 4873 | Relationship implying unio... |
| unimax 4874 | Any member of a class is t... |
| pwuni 4875 | A class is a subclass of t... |
| dfint2 4878 | Alternate definition of cl... |
| inteq 4879 | Equality law for intersect... |
| inteqi 4880 | Equality inference for cla... |
| inteqd 4881 | Equality deduction for cla... |
| elint 4882 | Membership in class inters... |
| elint2 4883 | Membership in class inters... |
| elintg 4884 | Membership in class inters... |
| elinti 4885 | Membership in class inters... |
| nfint 4886 | Bound-variable hypothesis ... |
| elintab 4887 | Membership in the intersec... |
| elintrab 4888 | Membership in the intersec... |
| elintrabg 4889 | Membership in the intersec... |
| int0 4890 | The intersection of the em... |
| intss1 4891 | An element of a class incl... |
| ssint 4892 | Subclass of a class inters... |
| ssintab 4893 | Subclass of the intersecti... |
| ssintub 4894 | Subclass of the least uppe... |
| ssmin 4895 | Subclass of the minimum va... |
| intmin 4896 | Any member of a class is t... |
| intss 4897 | Intersection of subclasses... |
| intssuni 4898 | The intersection of a none... |
| ssintrab 4899 | Subclass of the intersecti... |
| unissint 4900 | If the union of a class is... |
| intssuni2 4901 | Subclass relationship for ... |
| intminss 4902 | Under subset ordering, the... |
| intmin2 4903 | Any set is the smallest of... |
| intmin3 4904 | Under subset ordering, the... |
| intmin4 4905 | Elimination of a conjunct ... |
| intab 4906 | The intersection of a spec... |
| int0el 4907 | The intersection of a clas... |
| intun 4908 | The class intersection of ... |
| intprg 4909 | The intersection of a pair... |
| intpr 4910 | The intersection of a pair... |
| intprOLD 4911 | Obsolete version of ~ intp... |
| intprgOLD 4912 | Obsolete version of ~ intp... |
| intsng 4913 | Intersection of a singleto... |
| intsn 4914 | The intersection of a sing... |
| uniintsn 4915 | Two ways to express " ` A ... |
| uniintab 4916 | The union and the intersec... |
| intunsn 4917 | Theorem joining a singleto... |
| rint0 4918 | Relative intersection of a... |
| elrint 4919 | Membership in a restricted... |
| elrint2 4920 | Membership in a restricted... |
| eliun 4925 | Membership in indexed unio... |
| eliin 4926 | Membership in indexed inte... |
| eliuni 4927 | Membership in an indexed u... |
| iuncom 4928 | Commutation of indexed uni... |
| iuncom4 4929 | Commutation of union with ... |
| iunconst 4930 | Indexed union of a constan... |
| iinconst 4931 | Indexed intersection of a ... |
| iuneqconst 4932 | Indexed union of identical... |
| iuniin 4933 | Law combining indexed unio... |
| iinssiun 4934 | An indexed intersection is... |
| iunss1 4935 | Subclass theorem for index... |
| iinss1 4936 | Subclass theorem for index... |
| iuneq1 4937 | Equality theorem for index... |
| iineq1 4938 | Equality theorem for index... |
| ss2iun 4939 | Subclass theorem for index... |
| iuneq2 4940 | Equality theorem for index... |
| iineq2 4941 | Equality theorem for index... |
| iuneq2i 4942 | Equality inference for ind... |
| iineq2i 4943 | Equality inference for ind... |
| iineq2d 4944 | Equality deduction for ind... |
| iuneq2dv 4945 | Equality deduction for ind... |
| iineq2dv 4946 | Equality deduction for ind... |
| iuneq12df 4947 | Equality deduction for ind... |
| iuneq1d 4948 | Equality theorem for index... |
| iuneq12d 4949 | Equality deduction for ind... |
| iuneq2d 4950 | Equality deduction for ind... |
| nfiun 4951 | Bound-variable hypothesis ... |
| nfiin 4952 | Bound-variable hypothesis ... |
| nfiung 4953 | Bound-variable hypothesis ... |
| nfiing 4954 | Bound-variable hypothesis ... |
| nfiu1 4955 | Bound-variable hypothesis ... |
| nfii1 4956 | Bound-variable hypothesis ... |
| dfiun2g 4957 | Alternate definition of in... |
| dfiin2g 4958 | Alternate definition of in... |
| dfiun2 4959 | Alternate definition of in... |
| dfiin2 4960 | Alternate definition of in... |
| dfiunv2 4961 | Define double indexed unio... |
| cbviun 4962 | Rule used to change the bo... |
| cbviin 4963 | Change bound variables in ... |
| cbviung 4964 | Rule used to change the bo... |
| cbviing 4965 | Change bound variables in ... |
| cbviunv 4966 | Rule used to change the bo... |
| cbviinv 4967 | Change bound variables in ... |
| cbviunvg 4968 | Rule used to change the bo... |
| cbviinvg 4969 | Change bound variables in ... |
| iunssf 4970 | Subset theorem for an inde... |
| iunss 4971 | Subset theorem for an inde... |
| ssiun 4972 | Subset implication for an ... |
| ssiun2 4973 | Identity law for subset of... |
| ssiun2s 4974 | Subset relationship for an... |
| iunss2 4975 | A subclass condition on th... |
| iunssd 4976 | Subset theorem for an inde... |
| iunab 4977 | The indexed union of a cla... |
| iunrab 4978 | The indexed union of a res... |
| iunxdif2 4979 | Indexed union with a class... |
| ssiinf 4980 | Subset theorem for an inde... |
| ssiin 4981 | Subset theorem for an inde... |
| iinss 4982 | Subset implication for an ... |
| iinss2 4983 | An indexed intersection is... |
| uniiun 4984 | Class union in terms of in... |
| intiin 4985 | Class intersection in term... |
| iunid 4986 | An indexed union of single... |
| iun0 4987 | An indexed union of the em... |
| 0iun 4988 | An empty indexed union is ... |
| 0iin 4989 | An empty indexed intersect... |
| viin 4990 | Indexed intersection with ... |
| iunsn 4991 | Indexed union of a singlet... |
| iunn0 4992 | There is a nonempty class ... |
| iinab 4993 | Indexed intersection of a ... |
| iinrab 4994 | Indexed intersection of a ... |
| iinrab2 4995 | Indexed intersection of a ... |
| iunin2 4996 | Indexed union of intersect... |
| iunin1 4997 | Indexed union of intersect... |
| iinun2 4998 | Indexed intersection of un... |
| iundif2 4999 | Indexed union of class dif... |
| iindif1 5000 | Indexed intersection of cl... |
| 2iunin 5001 | Rearrange indexed unions o... |
| iindif2 5002 | Indexed intersection of cl... |
| iinin2 5003 | Indexed intersection of in... |
| iinin1 5004 | Indexed intersection of in... |
| iinvdif 5005 | The indexed intersection o... |
| elriin 5006 | Elementhood in a relative ... |
| riin0 5007 | Relative intersection of a... |
| riinn0 5008 | Relative intersection of a... |
| riinrab 5009 | Relative intersection of a... |
| symdif0 5010 | Symmetric difference with ... |
| symdifv 5011 | The symmetric difference w... |
| symdifid 5012 | The symmetric difference o... |
| iinxsng 5013 | A singleton index picks ou... |
| iinxprg 5014 | Indexed intersection with ... |
| iunxsng 5015 | A singleton index picks ou... |
| iunxsn 5016 | A singleton index picks ou... |
| iunxsngf 5017 | A singleton index picks ou... |
| iunun 5018 | Separate a union in an ind... |
| iunxun 5019 | Separate a union in the in... |
| iunxdif3 5020 | An indexed union where som... |
| iunxprg 5021 | A pair index picks out two... |
| iunxiun 5022 | Separate an indexed union ... |
| iinuni 5023 | A relationship involving u... |
| iununi 5024 | A relationship involving u... |
| sspwuni 5025 | Subclass relationship for ... |
| pwssb 5026 | Two ways to express a coll... |
| elpwpw 5027 | Characterization of the el... |
| pwpwab 5028 | The double power class wri... |
| pwpwssunieq 5029 | The class of sets whose un... |
| elpwuni 5030 | Relationship for power cla... |
| iinpw 5031 | The power class of an inte... |
| iunpwss 5032 | Inclusion of an indexed un... |
| intss2 5033 | A nonempty intersection of... |
| rintn0 5034 | Relative intersection of a... |
| dfdisj2 5037 | Alternate definition for d... |
| disjss2 5038 | If each element of a colle... |
| disjeq2 5039 | Equality theorem for disjo... |
| disjeq2dv 5040 | Equality deduction for dis... |
| disjss1 5041 | A subset of a disjoint col... |
| disjeq1 5042 | Equality theorem for disjo... |
| disjeq1d 5043 | Equality theorem for disjo... |
| disjeq12d 5044 | Equality theorem for disjo... |
| cbvdisj 5045 | Change bound variables in ... |
| cbvdisjv 5046 | Change bound variables in ... |
| nfdisjw 5047 | Bound-variable hypothesis ... |
| nfdisj 5048 | Bound-variable hypothesis ... |
| nfdisj1 5049 | Bound-variable hypothesis ... |
| disjor 5050 | Two ways to say that a col... |
| disjors 5051 | Two ways to say that a col... |
| disji2 5052 | Property of a disjoint col... |
| disji 5053 | Property of a disjoint col... |
| invdisj 5054 | If there is a function ` C... |
| invdisjrabw 5055 | Version of ~ invdisjrab wi... |
| invdisjrab 5056 | The restricted class abstr... |
| disjiun 5057 | A disjoint collection yiel... |
| disjord 5058 | Conditions for a collectio... |
| disjiunb 5059 | Two ways to say that a col... |
| disjiund 5060 | Conditions for a collectio... |
| sndisj 5061 | Any collection of singleto... |
| 0disj 5062 | Any collection of empty se... |
| disjxsn 5063 | A singleton collection is ... |
| disjx0 5064 | An empty collection is dis... |
| disjprgw 5065 | Version of ~ disjprg with ... |
| disjprg 5066 | A pair collection is disjo... |
| disjxiun 5067 | An indexed union of a disj... |
| disjxun 5068 | The union of two disjoint ... |
| disjss3 5069 | Expand a disjoint collecti... |
| breq 5072 | Equality theorem for binar... |
| breq1 5073 | Equality theorem for a bin... |
| breq2 5074 | Equality theorem for a bin... |
| breq12 5075 | Equality theorem for a bin... |
| breqi 5076 | Equality inference for bin... |
| breq1i 5077 | Equality inference for a b... |
| breq2i 5078 | Equality inference for a b... |
| breq12i 5079 | Equality inference for a b... |
| breq1d 5080 | Equality deduction for a b... |
| breqd 5081 | Equality deduction for a b... |
| breq2d 5082 | Equality deduction for a b... |
| breq12d 5083 | Equality deduction for a b... |
| breq123d 5084 | Equality deduction for a b... |
| breqdi 5085 | Equality deduction for a b... |
| breqan12d 5086 | Equality deduction for a b... |
| breqan12rd 5087 | Equality deduction for a b... |
| eqnbrtrd 5088 | Substitution of equal clas... |
| nbrne1 5089 | Two classes are different ... |
| nbrne2 5090 | Two classes are different ... |
| eqbrtri 5091 | Substitution of equal clas... |
| eqbrtrd 5092 | Substitution of equal clas... |
| eqbrtrri 5093 | Substitution of equal clas... |
| eqbrtrrd 5094 | Substitution of equal clas... |
| breqtri 5095 | Substitution of equal clas... |
| breqtrd 5096 | Substitution of equal clas... |
| breqtrri 5097 | Substitution of equal clas... |
| breqtrrd 5098 | Substitution of equal clas... |
| 3brtr3i 5099 | Substitution of equality i... |
| 3brtr4i 5100 | Substitution of equality i... |
| 3brtr3d 5101 | Substitution of equality i... |
| 3brtr4d 5102 | Substitution of equality i... |
| 3brtr3g 5103 | Substitution of equality i... |
| 3brtr4g 5104 | Substitution of equality i... |
| eqbrtrid 5105 | A chained equality inferen... |
| eqbrtrrid 5106 | A chained equality inferen... |
| breqtrid 5107 | A chained equality inferen... |
| breqtrrid 5108 | A chained equality inferen... |
| eqbrtrdi 5109 | A chained equality inferen... |
| eqbrtrrdi 5110 | A chained equality inferen... |
| breqtrdi 5111 | A chained equality inferen... |
| breqtrrdi 5112 | A chained equality inferen... |
| ssbrd 5113 | Deduction from a subclass ... |
| ssbr 5114 | Implication from a subclas... |
| ssbri 5115 | Inference from a subclass ... |
| nfbrd 5116 | Deduction version of bound... |
| nfbr 5117 | Bound-variable hypothesis ... |
| brab1 5118 | Relationship between a bin... |
| br0 5119 | The empty binary relation ... |
| brne0 5120 | If two sets are in a binar... |
| brun 5121 | The union of two binary re... |
| brin 5122 | The intersection of two re... |
| brdif 5123 | The difference of two bina... |
| sbcbr123 5124 | Move substitution in and o... |
| sbcbr 5125 | Move substitution in and o... |
| sbcbr12g 5126 | Move substitution in and o... |
| sbcbr1g 5127 | Move substitution in and o... |
| sbcbr2g 5128 | Move substitution in and o... |
| brsymdif 5129 | Characterization of the sy... |
| brralrspcev 5130 | Restricted existential spe... |
| brimralrspcev 5131 | Restricted existential spe... |
| opabss 5134 | The collection of ordered ... |
| opabbid 5135 | Equivalent wff's yield equ... |
| opabbidv 5136 | Equivalent wff's yield equ... |
| opabbii 5137 | Equivalent wff's yield equ... |
| nfopabd 5138 | Bound-variable hypothesis ... |
| nfopab 5139 | Bound-variable hypothesis ... |
| nfopab1 5140 | The first abstraction vari... |
| nfopab2 5141 | The second abstraction var... |
| cbvopab 5142 | Rule used to change bound ... |
| cbvopabv 5143 | Rule used to change bound ... |
| cbvopabvOLD 5144 | Obsolete version of ~ cbvo... |
| cbvopab1 5145 | Change first bound variabl... |
| cbvopab1g 5146 | Change first bound variabl... |
| cbvopab2 5147 | Change second bound variab... |
| cbvopab1s 5148 | Change first bound variabl... |
| cbvopab1v 5149 | Rule used to change the fi... |
| cbvopab1vOLD 5150 | Obsolete version of ~ cbvo... |
| cbvopab2v 5151 | Rule used to change the se... |
| unopab 5152 | Union of two ordered pair ... |
| mpteq12da 5155 | An equality inference for ... |
| mpteq12df 5156 | An equality inference for ... |
| mpteq12dfOLD 5157 | Obsolete version of ~ mpte... |
| mpteq12f 5158 | An equality theorem for th... |
| mpteq12dva 5159 | An equality inference for ... |
| mpteq12dvaOLD 5160 | Obsolete version of ~ mpte... |
| mpteq12dv 5161 | An equality inference for ... |
| mpteq12 5162 | An equality theorem for th... |
| mpteq1 5163 | An equality theorem for th... |
| mpteq1OLD 5164 | Obsolete version of ~ mpte... |
| mpteq1d 5165 | An equality theorem for th... |
| mpteq1i 5166 | An equality theorem for th... |
| mpteq1iOLD 5167 | An equality theorem for th... |
| mpteq2da 5168 | Slightly more general equa... |
| mpteq2daOLD 5169 | Obsolete version of ~ mpte... |
| mpteq2dva 5170 | Slightly more general equa... |
| mpteq2dvaOLD 5171 | Obsolete version of ~ mpte... |
| mpteq2dv 5172 | An equality inference for ... |
| mpteq2ia 5173 | An equality inference for ... |
| mpteq2iaOLD 5174 | Obsolete version of ~ mpte... |
| mpteq2i 5175 | An equality inference for ... |
| mpteq12i 5176 | An equality inference for ... |
| nfmpt 5177 | Bound-variable hypothesis ... |
| nfmpt1 5178 | Bound-variable hypothesis ... |
| cbvmptf 5179 | Rule to change the bound v... |
| cbvmptfg 5180 | Rule to change the bound v... |
| cbvmpt 5181 | Rule to change the bound v... |
| cbvmptg 5182 | Rule to change the bound v... |
| cbvmptv 5183 | Rule to change the bound v... |
| cbvmptvOLD 5184 | Obsolete version of ~ cbvm... |
| cbvmptvg 5185 | Rule to change the bound v... |
| mptv 5186 | Function with universal do... |
| dftr2 5189 | An alternate way of defini... |
| dftr5 5190 | An alternate way of defini... |
| dftr3 5191 | An alternate way of defini... |
| dftr4 5192 | An alternate way of defini... |
| treq 5193 | Equality theorem for the t... |
| trel 5194 | In a transitive class, the... |
| trel3 5195 | In a transitive class, the... |
| trss 5196 | An element of a transitive... |
| trin 5197 | The intersection of transi... |
| tr0 5198 | The empty set is transitiv... |
| trv 5199 | The universe is transitive... |
| triun 5200 | An indexed union of a clas... |
| truni 5201 | The union of a class of tr... |
| triin 5202 | An indexed intersection of... |
| trint 5203 | The intersection of a clas... |
| trintss 5204 | Any nonempty transitive cl... |
| axrep1 5206 | The version of the Axiom o... |
| axreplem 5207 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5208 | Axiom of Replacement expre... |
| axrep3 5209 | Axiom of Replacement sligh... |
| axrep4 5210 | A more traditional version... |
| axrep5 5211 | Axiom of Replacement (simi... |
| axrep6 5212 | A condensed form of ~ ax-r... |
| zfrepclf 5213 | An inference based on the ... |
| zfrep3cl 5214 | An inference based on the ... |
| zfrep4 5215 | A version of Replacement u... |
| axsepgfromrep 5216 | A more general version ~ a... |
| axsep 5217 | Axiom scheme of separation... |
| axsepg 5219 | A more general version of ... |
| zfauscl 5220 | Separation Scheme (Aussond... |
| bm1.3ii 5221 | Convert implication to equ... |
| ax6vsep 5222 | Derive ~ ax6v (a weakened ... |
| axnulALT 5223 | Alternate proof of ~ axnul... |
| axnul 5224 | The Null Set Axiom of ZF s... |
| 0ex 5226 | The Null Set Axiom of ZF s... |
| al0ssb 5227 | The empty set is the uniqu... |
| sseliALT 5228 | Alternate proof of ~ sseli... |
| csbexg 5229 | The existence of proper su... |
| csbex 5230 | The existence of proper su... |
| unisn2 5231 | A version of ~ unisn witho... |
| nalset 5232 | No set contains all sets. ... |
| vnex 5233 | The universal class does n... |
| vprc 5234 | The universal class is not... |
| nvel 5235 | The universal class does n... |
| inex1 5236 | Separation Scheme (Aussond... |
| inex2 5237 | Separation Scheme (Aussond... |
| inex1g 5238 | Closed-form, generalized S... |
| inex2g 5239 | Sufficient condition for a... |
| ssex 5240 | The subset of a set is als... |
| ssexi 5241 | The subset of a set is als... |
| ssexg 5242 | The subset of a set is als... |
| ssexd 5243 | A subclass of a set is a s... |
| prcssprc 5244 | The superclass of a proper... |
| sselpwd 5245 | Elementhood to a power set... |
| difexg 5246 | Existence of a difference.... |
| difexi 5247 | Existence of a difference,... |
| difexd 5248 | Existence of a difference.... |
| zfausab 5249 | Separation Scheme (Aussond... |
| rabexg 5250 | Separation Scheme in terms... |
| rabex 5251 | Separation Scheme in terms... |
| rabexd 5252 | Separation Scheme in terms... |
| rabex2 5253 | Separation Scheme in terms... |
| rab2ex 5254 | A class abstraction based ... |
| elssabg 5255 | Membership in a class abst... |
| intex 5256 | The intersection of a none... |
| intnex 5257 | If a class intersection is... |
| intexab 5258 | The intersection of a none... |
| intexrab 5259 | The intersection of a none... |
| iinexg 5260 | The existence of a class i... |
| intabs 5261 | Absorption of a redundant ... |
| inuni 5262 | The intersection of a unio... |
| elpw2g 5263 | Membership in a power clas... |
| elpw2 5264 | Membership in a power clas... |
| elpwi2 5265 | Membership in a power clas... |
| elpwi2OLD 5266 | Obsolete version of ~ elpw... |
| pwnss 5267 | The power set of a set is ... |
| pwne 5268 | No set equals its power se... |
| difelpw 5269 | A difference is an element... |
| rabelpw 5270 | A restricted class abstrac... |
| class2set 5271 | Construct, from any class ... |
| class2seteq 5272 | Equality theorem based on ... |
| 0elpw 5273 | Every power class contains... |
| pwne0 5274 | A power class is never emp... |
| 0nep0 5275 | The empty set and its powe... |
| 0inp0 5276 | Something cannot be equal ... |
| unidif0 5277 | The removal of the empty s... |
| eqsnuniex 5278 | If a class is equal to the... |
| iin0 5279 | An indexed intersection of... |
| notzfaus 5280 | In the Separation Scheme ~... |
| intv 5281 | The intersection of the un... |
| axpweq 5282 | Two equivalent ways to exp... |
| zfpow 5284 | Axiom of Power Sets expres... |
| axpow2 5285 | A variant of the Axiom of ... |
| axpow3 5286 | A variant of the Axiom of ... |
| el 5287 | Every set is an element of... |
| dtru 5288 | At least two sets exist (o... |
| dtrucor 5289 | Corollary of ~ dtru . Thi... |
| dtrucor2 5290 | The theorem form of the de... |
| dvdemo1 5291 | Demonstration of a theorem... |
| dvdemo2 5292 | Demonstration of a theorem... |
| nfnid 5293 | A setvar variable is not f... |
| nfcvb 5294 | The "distinctor" expressio... |
| vpwex 5295 | Power set axiom: the power... |
| pwexg 5296 | Power set axiom expressed ... |
| pwexd 5297 | Deduction version of the p... |
| pwex 5298 | Power set axiom expressed ... |
| pwel 5299 | Quantitative version of ~ ... |
| abssexg 5300 | Existence of a class of su... |
| snexALT 5301 | Alternate proof of ~ snex ... |
| p0ex 5302 | The power set of the empty... |
| p0exALT 5303 | Alternate proof of ~ p0ex ... |
| pp0ex 5304 | The power set of the power... |
| ord3ex 5305 | The ordinal number 3 is a ... |
| dtruALT 5306 | Alternate proof of ~ dtru ... |
| axc16b 5307 | This theorem shows that Ax... |
| eunex 5308 | Existential uniqueness imp... |
| eusv1 5309 | Two ways to express single... |
| eusvnf 5310 | Even if ` x ` is free in `... |
| eusvnfb 5311 | Two ways to say that ` A (... |
| eusv2i 5312 | Two ways to express single... |
| eusv2nf 5313 | Two ways to express single... |
| eusv2 5314 | Two ways to express single... |
| reusv1 5315 | Two ways to express single... |
| reusv2lem1 5316 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5317 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5318 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5319 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5320 | Lemma for ~ reusv2 . (Con... |
| reusv2 5321 | Two ways to express single... |
| reusv3i 5322 | Two ways of expressing exi... |
| reusv3 5323 | Two ways to express single... |
| eusv4 5324 | Two ways to express single... |
| alxfr 5325 | Transfer universal quantif... |
| ralxfrd 5326 | Transfer universal quantif... |
| rexxfrd 5327 | Transfer universal quantif... |
| ralxfr2d 5328 | Transfer universal quantif... |
| rexxfr2d 5329 | Transfer universal quantif... |
| ralxfrd2 5330 | Transfer universal quantif... |
| rexxfrd2 5331 | Transfer existence from a ... |
| ralxfr 5332 | Transfer universal quantif... |
| ralxfrALT 5333 | Alternate proof of ~ ralxf... |
| rexxfr 5334 | Transfer existence from a ... |
| rabxfrd 5335 | Membership in a restricted... |
| rabxfr 5336 | Membership in a restricted... |
| reuhypd 5337 | A theorem useful for elimi... |
| reuhyp 5338 | A theorem useful for elimi... |
| zfpair 5339 | The Axiom of Pairing of Ze... |
| axprALT 5340 | Alternate proof of ~ axpr ... |
| axprlem1 5341 | Lemma for ~ axpr . There ... |
| axprlem2 5342 | Lemma for ~ axpr . There ... |
| axprlem3 5343 | Lemma for ~ axpr . Elimin... |
| axprlem4 5344 | Lemma for ~ axpr . The fi... |
| axprlem5 5345 | Lemma for ~ axpr . The se... |
| axpr 5346 | Unabbreviated version of t... |
| zfpair2 5348 | Derive the abbreviated ver... |
| snex 5349 | A singleton is a set. The... |
| prex 5350 | The Axiom of Pairing using... |
| sels 5351 | If a class is a set, then ... |
| elALT 5352 | Alternate proof of ~ el , ... |
| dtruALT2 5353 | Alternate proof of ~ dtru ... |
| snelpwi 5354 | A singleton of a set belon... |
| snelpw 5355 | A singleton of a set belon... |
| prelpw 5356 | A pair of two sets belongs... |
| prelpwi 5357 | A pair of two sets belongs... |
| rext 5358 | A theorem similar to exten... |
| sspwb 5359 | The powerclass constructio... |
| unipw 5360 | A class equals the union o... |
| univ 5361 | The union of the universe ... |
| pwtr 5362 | A class is transitive iff ... |
| ssextss 5363 | An extensionality-like pri... |
| ssext 5364 | An extensionality-like pri... |
| nssss 5365 | Negation of subclass relat... |
| pweqb 5366 | Classes are equal if and o... |
| intid 5367 | The intersection of all se... |
| moabex 5368 | "At most one" existence im... |
| rmorabex 5369 | Restricted "at most one" e... |
| euabex 5370 | The abstraction of a wff w... |
| nnullss 5371 | A nonempty class (even if ... |
| exss 5372 | Restricted existence in a ... |
| opex 5373 | An ordered pair of classes... |
| otex 5374 | An ordered triple of class... |
| elopg 5375 | Characterization of the el... |
| elop 5376 | Characterization of the el... |
| opi1 5377 | One of the two elements in... |
| opi2 5378 | One of the two elements of... |
| opeluu 5379 | Each member of an ordered ... |
| op1stb 5380 | Extract the first member o... |
| brv 5381 | Two classes are always in ... |
| opnz 5382 | An ordered pair is nonempt... |
| opnzi 5383 | An ordered pair is nonempt... |
| opth1 5384 | Equality of the first memb... |
| opth 5385 | The ordered pair theorem. ... |
| opthg 5386 | Ordered pair theorem. ` C ... |
| opth1g 5387 | Equality of the first memb... |
| opthg2 5388 | Ordered pair theorem. (Co... |
| opth2 5389 | Ordered pair theorem. (Co... |
| opthneg 5390 | Two ordered pairs are not ... |
| opthne 5391 | Two ordered pairs are not ... |
| otth2 5392 | Ordered triple theorem, wi... |
| otth 5393 | Ordered triple theorem. (... |
| otthg 5394 | Ordered triple theorem, cl... |
| eqvinop 5395 | A variable introduction la... |
| sbcop1 5396 | The proper substitution of... |
| sbcop 5397 | The proper substitution of... |
| copsexgw 5398 | Version of ~ copsexg with ... |
| copsexg 5399 | Substitution of class ` A ... |
| copsex2t 5400 | Closed theorem form of ~ c... |
| copsex2g 5401 | Implicit substitution infe... |
| copsex2gOLD 5402 | Obsolete version of ~ cops... |
| copsex4g 5403 | An implicit substitution i... |
| 0nelop 5404 | A property of ordered pair... |
| opwo0id 5405 | An ordered pair is equal t... |
| opeqex 5406 | Equivalence of existence i... |
| oteqex2 5407 | Equivalence of existence i... |
| oteqex 5408 | Equivalence of existence i... |
| opcom 5409 | An ordered pair commutes i... |
| moop2 5410 | "At most one" property of ... |
| opeqsng 5411 | Equivalence for an ordered... |
| opeqsn 5412 | Equivalence for an ordered... |
| opeqpr 5413 | Equivalence for an ordered... |
| snopeqop 5414 | Equivalence for an ordered... |
| propeqop 5415 | Equivalence for an ordered... |
| propssopi 5416 | If a pair of ordered pairs... |
| snopeqopsnid 5417 | Equivalence for an ordered... |
| mosubopt 5418 | "At most one" remains true... |
| mosubop 5419 | "At most one" remains true... |
| euop2 5420 | Transfer existential uniqu... |
| euotd 5421 | Prove existential uniquene... |
| opthwiener 5422 | Justification theorem for ... |
| uniop 5423 | The union of an ordered pa... |
| uniopel 5424 | Ordered pair membership is... |
| opthhausdorff 5425 | Justification theorem for ... |
| opthhausdorff0 5426 | Justification theorem for ... |
| otsndisj 5427 | The singletons consisting ... |
| otiunsndisj 5428 | The union of singletons co... |
| iunopeqop 5429 | Implication of an ordered ... |
| brsnop 5430 | Binary relation for an ord... |
| opabidw 5431 | The law of concretion. Sp... |
| opabid 5432 | The law of concretion. Sp... |
| elopab 5433 | Membership in a class abst... |
| rexopabb 5434 | Restricted existential qua... |
| vopelopabsb 5435 | The law of concretion in t... |
| opelopabsb 5436 | The law of concretion in t... |
| brabsb 5437 | The law of concretion in t... |
| opelopabt 5438 | Closed theorem form of ~ o... |
| opelopabga 5439 | The law of concretion. Th... |
| brabga 5440 | The law of concretion for ... |
| opelopab2a 5441 | Ordered pair membership in... |
| opelopaba 5442 | The law of concretion. Th... |
| braba 5443 | The law of concretion for ... |
| opelopabg 5444 | The law of concretion. Th... |
| brabg 5445 | The law of concretion for ... |
| opelopabgf 5446 | The law of concretion. Th... |
| opelopab2 5447 | Ordered pair membership in... |
| opelopab 5448 | The law of concretion. Th... |
| brab 5449 | The law of concretion for ... |
| opelopabaf 5450 | The law of concretion. Th... |
| opelopabf 5451 | The law of concretion. Th... |
| ssopab2 5452 | Equivalence of ordered pai... |
| ssopab2bw 5453 | Equivalence of ordered pai... |
| eqopab2bw 5454 | Equivalence of ordered pai... |
| ssopab2b 5455 | Equivalence of ordered pai... |
| ssopab2i 5456 | Inference of ordered pair ... |
| ssopab2dv 5457 | Inference of ordered pair ... |
| eqopab2b 5458 | Equivalence of ordered pai... |
| opabn0 5459 | Nonempty ordered pair clas... |
| opab0 5460 | Empty ordered pair class a... |
| csbopab 5461 | Move substitution into a c... |
| csbopabgALT 5462 | Move substitution into a c... |
| csbmpt12 5463 | Move substitution into a m... |
| csbmpt2 5464 | Move substitution into the... |
| iunopab 5465 | Move indexed union inside ... |
| elopabr 5466 | Membership in an ordered-p... |
| elopabran 5467 | Membership in an ordered-p... |
| rbropapd 5468 | Properties of a pair in an... |
| rbropap 5469 | Properties of a pair in a ... |
| 2rbropap 5470 | Properties of a pair in a ... |
| 0nelopab 5471 | The empty set is never an ... |
| 0nelopabOLD 5472 | Obsolete version of ~ 0nel... |
| brabv 5473 | If two classes are in a re... |
| pwin 5474 | The power class of the int... |
| pwunssOLD 5475 | Obsolete version of ~ pwun... |
| pwssun 5476 | The power class of the uni... |
| pwundifOLD 5477 | Obsolete proof of ~ pwundi... |
| pwun 5478 | The power class of the uni... |
| dfid4 5481 | The identity function expr... |
| dfid2 5482 | Alternate definition of th... |
| dfid3 5483 | A stronger version of ~ df... |
| dfid2OLD 5484 | Obsolete version of ~ dfid... |
| epelg 5487 | The membership relation an... |
| epeli 5488 | The membership relation an... |
| epel 5489 | The membership relation an... |
| 0sn0ep 5490 | An example for the members... |
| epn0 5491 | The membership relation is... |
| poss 5496 | Subset theorem for the par... |
| poeq1 5497 | Equality theorem for parti... |
| poeq2 5498 | Equality theorem for parti... |
| nfpo 5499 | Bound-variable hypothesis ... |
| nfso 5500 | Bound-variable hypothesis ... |
| pocl 5501 | Characteristic properties ... |
| poclOLD 5502 | Obsolete version of ~ pocl... |
| ispod 5503 | Sufficient conditions for ... |
| swopolem 5504 | Perform the substitutions ... |
| swopo 5505 | A strict weak order is a p... |
| poirr 5506 | A partial order is irrefle... |
| potr 5507 | A partial order is a trans... |
| po2nr 5508 | A partial order has no 2-c... |
| po3nr 5509 | A partial order has no 3-c... |
| po2ne 5510 | Two sets related by a part... |
| po0 5511 | Any relation is a partial ... |
| pofun 5512 | The inverse image of a par... |
| sopo 5513 | A strict linear order is a... |
| soss 5514 | Subset theorem for the str... |
| soeq1 5515 | Equality theorem for the s... |
| soeq2 5516 | Equality theorem for the s... |
| sonr 5517 | A strict order relation is... |
| sotr 5518 | A strict order relation is... |
| solin 5519 | A strict order relation is... |
| so2nr 5520 | A strict order relation ha... |
| so3nr 5521 | A strict order relation ha... |
| sotric 5522 | A strict order relation sa... |
| sotrieq 5523 | Trichotomy law for strict ... |
| sotrieq2 5524 | Trichotomy law for strict ... |
| soasym 5525 | Asymmetry law for strict o... |
| sotr2 5526 | A transitivity relation. ... |
| issod 5527 | An irreflexive, transitive... |
| issoi 5528 | An irreflexive, transitive... |
| isso2i 5529 | Deduce strict ordering fro... |
| so0 5530 | Any relation is a strict o... |
| somo 5531 | A totally ordered set has ... |
| dffr6 5538 | Alternate definition of ~ ... |
| frd 5539 | A nonempty subset of an ` ... |
| fri 5540 | A nonempty subset of an ` ... |
| friOLD 5541 | Obsolete version of ~ fri ... |
| seex 5542 | The ` R ` -preimage of an ... |
| exse 5543 | Any relation on a set is s... |
| dffr2 5544 | Alternate definition of we... |
| dffr2ALT 5545 | Alternate proof of ~ dffr2... |
| frc 5546 | Property of well-founded r... |
| frss 5547 | Subset theorem for the wel... |
| sess1 5548 | Subset theorem for the set... |
| sess2 5549 | Subset theorem for the set... |
| freq1 5550 | Equality theorem for the w... |
| freq2 5551 | Equality theorem for the w... |
| seeq1 5552 | Equality theorem for the s... |
| seeq2 5553 | Equality theorem for the s... |
| nffr 5554 | Bound-variable hypothesis ... |
| nfse 5555 | Bound-variable hypothesis ... |
| nfwe 5556 | Bound-variable hypothesis ... |
| frirr 5557 | A well-founded relation is... |
| fr2nr 5558 | A well-founded relation ha... |
| fr0 5559 | Any relation is well-found... |
| frminex 5560 | If an element of a well-fo... |
| efrirr 5561 | A well-founded class does ... |
| efrn2lp 5562 | A well-founded class conta... |
| epse 5563 | The membership relation is... |
| tz7.2 5564 | Similar to Theorem 7.2 of ... |
| dfepfr 5565 | An alternate way of saying... |
| epfrc 5566 | A subset of a well-founded... |
| wess 5567 | Subset theorem for the wel... |
| weeq1 5568 | Equality theorem for the w... |
| weeq2 5569 | Equality theorem for the w... |
| wefr 5570 | A well-ordering is well-fo... |
| weso 5571 | A well-ordering is a stric... |
| wecmpep 5572 | The elements of a class we... |
| wetrep 5573 | On a class well-ordered by... |
| wefrc 5574 | A nonempty subclass of a c... |
| we0 5575 | Any relation is a well-ord... |
| wereu 5576 | A nonempty subset of an ` ... |
| wereu2 5577 | A nonempty subclass of an ... |
| xpeq1 5594 | Equality theorem for Carte... |
| xpss12 5595 | Subset theorem for Cartesi... |
| xpss 5596 | A Cartesian product is inc... |
| inxpssres 5597 | Intersection with a Cartes... |
| relxp 5598 | A Cartesian product is a r... |
| xpss1 5599 | Subset relation for Cartes... |
| xpss2 5600 | Subset relation for Cartes... |
| xpeq2 5601 | Equality theorem for Carte... |
| elxpi 5602 | Membership in a Cartesian ... |
| elxp 5603 | Membership in a Cartesian ... |
| elxp2 5604 | Membership in a Cartesian ... |
| xpeq12 5605 | Equality theorem for Carte... |
| xpeq1i 5606 | Equality inference for Car... |
| xpeq2i 5607 | Equality inference for Car... |
| xpeq12i 5608 | Equality inference for Car... |
| xpeq1d 5609 | Equality deduction for Car... |
| xpeq2d 5610 | Equality deduction for Car... |
| xpeq12d 5611 | Equality deduction for Car... |
| sqxpeqd 5612 | Equality deduction for a C... |
| nfxp 5613 | Bound-variable hypothesis ... |
| 0nelxp 5614 | The empty set is not a mem... |
| 0nelelxp 5615 | A member of a Cartesian pr... |
| opelxp 5616 | Ordered pair membership in... |
| opelxpi 5617 | Ordered pair membership in... |
| opelxpd 5618 | Ordered pair membership in... |
| opelvv 5619 | Ordered pair membership in... |
| opelvvg 5620 | Ordered pair membership in... |
| opelxp1 5621 | The first member of an ord... |
| opelxp2 5622 | The second member of an or... |
| otelxp1 5623 | The first member of an ord... |
| otel3xp 5624 | An ordered triple is an el... |
| opabssxpd 5625 | An ordered-pair class abst... |
| rabxp 5626 | Class abstraction restrict... |
| brxp 5627 | Binary relation on a Carte... |
| pwvrel 5628 | A set is a binary relation... |
| pwvabrel 5629 | The powerclass of the cart... |
| brrelex12 5630 | Two classes related by a b... |
| brrelex1 5631 | If two classes are related... |
| brrelex2 5632 | If two classes are related... |
| brrelex12i 5633 | Two classes that are relat... |
| brrelex1i 5634 | The first argument of a bi... |
| brrelex2i 5635 | The second argument of a b... |
| nprrel12 5636 | Proper classes are not rel... |
| nprrel 5637 | No proper class is related... |
| 0nelrel0 5638 | A binary relation does not... |
| 0nelrel 5639 | A binary relation does not... |
| fconstmpt 5640 | Representation of a consta... |
| vtoclr 5641 | Variable to class conversi... |
| opthprc 5642 | Justification theorem for ... |
| brel 5643 | Two things in a binary rel... |
| elxp3 5644 | Membership in a Cartesian ... |
| opeliunxp 5645 | Membership in a union of C... |
| xpundi 5646 | Distributive law for Carte... |
| xpundir 5647 | Distributive law for Carte... |
| xpiundi 5648 | Distributive law for Carte... |
| xpiundir 5649 | Distributive law for Carte... |
| iunxpconst 5650 | Membership in a union of C... |
| xpun 5651 | The Cartesian product of t... |
| elvv 5652 | Membership in universal cl... |
| elvvv 5653 | Membership in universal cl... |
| elvvuni 5654 | An ordered pair contains i... |
| brinxp2 5655 | Intersection of binary rel... |
| brinxp 5656 | Intersection of binary rel... |
| opelinxp 5657 | Ordered pair element in an... |
| poinxp 5658 | Intersection of partial or... |
| soinxp 5659 | Intersection of total orde... |
| frinxp 5660 | Intersection of well-found... |
| seinxp 5661 | Intersection of set-like r... |
| weinxp 5662 | Intersection of well-order... |
| posn 5663 | Partial ordering of a sing... |
| sosn 5664 | Strict ordering on a singl... |
| frsn 5665 | Founded relation on a sing... |
| wesn 5666 | Well-ordering of a singlet... |
| elopaelxp 5667 | Membership in an ordered-p... |
| bropaex12 5668 | Two classes related by an ... |
| opabssxp 5669 | An abstraction relation is... |
| brab2a 5670 | The law of concretion for ... |
| optocl 5671 | Implicit substitution of c... |
| 2optocl 5672 | Implicit substitution of c... |
| 3optocl 5673 | Implicit substitution of c... |
| opbrop 5674 | Ordered pair membership in... |
| 0xp 5675 | The Cartesian product with... |
| csbxp 5676 | Distribute proper substitu... |
| releq 5677 | Equality theorem for the r... |
| releqi 5678 | Equality inference for the... |
| releqd 5679 | Equality deduction for the... |
| nfrel 5680 | Bound-variable hypothesis ... |
| sbcrel 5681 | Distribute proper substitu... |
| relss 5682 | Subclass theorem for relat... |
| ssrel 5683 | A subclass relationship de... |
| eqrel 5684 | Extensionality principle f... |
| ssrel2 5685 | A subclass relationship de... |
| relssi 5686 | Inference from subclass pr... |
| relssdv 5687 | Deduction from subclass pr... |
| eqrelriv 5688 | Inference from extensional... |
| eqrelriiv 5689 | Inference from extensional... |
| eqbrriv 5690 | Inference from extensional... |
| eqrelrdv 5691 | Deduce equality of relatio... |
| eqbrrdv 5692 | Deduction from extensional... |
| eqbrrdiv 5693 | Deduction from extensional... |
| eqrelrdv2 5694 | A version of ~ eqrelrdv . ... |
| ssrelrel 5695 | A subclass relationship de... |
| eqrelrel 5696 | Extensionality principle f... |
| elrel 5697 | A member of a relation is ... |
| rel0 5698 | The empty set is a relatio... |
| nrelv 5699 | The universal class is not... |
| relsng 5700 | A singleton is a relation ... |
| relsnb 5701 | An at-most-singleton is a ... |
| relsnopg 5702 | A singleton of an ordered ... |
| relsn 5703 | A singleton is a relation ... |
| relsnop 5704 | A singleton of an ordered ... |
| copsex2gb 5705 | Implicit substitution infe... |
| copsex2ga 5706 | Implicit substitution infe... |
| elopaba 5707 | Membership in an ordered-p... |
| xpsspw 5708 | A Cartesian product is inc... |
| unixpss 5709 | The double class union of ... |
| relun 5710 | The union of two relations... |
| relin1 5711 | The intersection with a re... |
| relin2 5712 | The intersection with a re... |
| relinxp 5713 | Intersection with a Cartes... |
| reldif 5714 | A difference cutting down ... |
| reliun 5715 | An indexed union is a rela... |
| reliin 5716 | An indexed intersection is... |
| reluni 5717 | The union of a class is a ... |
| relint 5718 | The intersection of a clas... |
| relopabiv 5719 | A class of ordered pairs i... |
| relopabv 5720 | A class of ordered pairs i... |
| relopabi 5721 | A class of ordered pairs i... |
| relopabiALT 5722 | Alternate proof of ~ relop... |
| relopab 5723 | A class of ordered pairs i... |
| mptrel 5724 | The maps-to notation alway... |
| reli 5725 | The identity relation is a... |
| rele 5726 | The membership relation is... |
| opabid2 5727 | A relation expressed as an... |
| inopab 5728 | Intersection of two ordere... |
| difopab 5729 | Difference of two ordered-... |
| inxp 5730 | Intersection of two Cartes... |
| xpindi 5731 | Distributive law for Carte... |
| xpindir 5732 | Distributive law for Carte... |
| xpiindi 5733 | Distributive law for Carte... |
| xpriindi 5734 | Distributive law for Carte... |
| eliunxp 5735 | Membership in a union of C... |
| opeliunxp2 5736 | Membership in a union of C... |
| raliunxp 5737 | Write a double restricted ... |
| rexiunxp 5738 | Write a double restricted ... |
| ralxp 5739 | Universal quantification r... |
| rexxp 5740 | Existential quantification... |
| exopxfr 5741 | Transfer ordered-pair exis... |
| exopxfr2 5742 | Transfer ordered-pair exis... |
| djussxp 5743 | Disjoint union is a subset... |
| ralxpf 5744 | Version of ~ ralxp with bo... |
| rexxpf 5745 | Version of ~ rexxp with bo... |
| iunxpf 5746 | Indexed union on a Cartesi... |
| opabbi2dv 5747 | Deduce equality of a relat... |
| relop 5748 | A necessary and sufficient... |
| ideqg 5749 | For sets, the identity rel... |
| ideq 5750 | For sets, the identity rel... |
| ididg 5751 | A set is identical to itse... |
| issetid 5752 | Two ways of expressing set... |
| coss1 5753 | Subclass theorem for compo... |
| coss2 5754 | Subclass theorem for compo... |
| coeq1 5755 | Equality theorem for compo... |
| coeq2 5756 | Equality theorem for compo... |
| coeq1i 5757 | Equality inference for com... |
| coeq2i 5758 | Equality inference for com... |
| coeq1d 5759 | Equality deduction for com... |
| coeq2d 5760 | Equality deduction for com... |
| coeq12i 5761 | Equality inference for com... |
| coeq12d 5762 | Equality deduction for com... |
| nfco 5763 | Bound-variable hypothesis ... |
| brcog 5764 | Ordered pair membership in... |
| opelco2g 5765 | Ordered pair membership in... |
| brcogw 5766 | Ordered pair membership in... |
| eqbrrdva 5767 | Deduction from extensional... |
| brco 5768 | Binary relation on a compo... |
| opelco 5769 | Ordered pair membership in... |
| cnvss 5770 | Subset theorem for convers... |
| cnveq 5771 | Equality theorem for conve... |
| cnveqi 5772 | Equality inference for con... |
| cnveqd 5773 | Equality deduction for con... |
| elcnv 5774 | Membership in a converse r... |
| elcnv2 5775 | Membership in a converse r... |
| nfcnv 5776 | Bound-variable hypothesis ... |
| brcnvg 5777 | The converse of a binary r... |
| opelcnvg 5778 | Ordered-pair membership in... |
| opelcnv 5779 | Ordered-pair membership in... |
| brcnv 5780 | The converse of a binary r... |
| csbcnv 5781 | Move class substitution in... |
| csbcnvgALT 5782 | Move class substitution in... |
| cnvco 5783 | Distributive law of conver... |
| cnvuni 5784 | The converse of a class un... |
| dfdm3 5785 | Alternate definition of do... |
| dfrn2 5786 | Alternate definition of ra... |
| dfrn3 5787 | Alternate definition of ra... |
| elrn2g 5788 | Membership in a range. (C... |
| elrng 5789 | Membership in a range. (C... |
| elrn2 5790 | Membership in a range. (C... |
| elrn 5791 | Membership in a range. (C... |
| ssrelrn 5792 | If a relation is a subset ... |
| dfdm4 5793 | Alternate definition of do... |
| dfdmf 5794 | Definition of domain, usin... |
| csbdm 5795 | Distribute proper substitu... |
| eldmg 5796 | Domain membership. Theore... |
| eldm2g 5797 | Domain membership. Theore... |
| eldm 5798 | Membership in a domain. T... |
| eldm2 5799 | Membership in a domain. T... |
| dmss 5800 | Subset theorem for domain.... |
| dmeq 5801 | Equality theorem for domai... |
| dmeqi 5802 | Equality inference for dom... |
| dmeqd 5803 | Equality deduction for dom... |
| opeldmd 5804 | Membership of first of an ... |
| opeldm 5805 | Membership of first of an ... |
| breldm 5806 | Membership of first of a b... |
| breldmg 5807 | Membership of first of a b... |
| dmun 5808 | The domain of a union is t... |
| dmin 5809 | The domain of an intersect... |
| breldmd 5810 | Membership of first of a b... |
| dmiun 5811 | The domain of an indexed u... |
| dmuni 5812 | The domain of a union. Pa... |
| dmopab 5813 | The domain of a class of o... |
| dmopabelb 5814 | A set is an element of the... |
| dmopab2rex 5815 | The domain of an ordered p... |
| dmopabss 5816 | Upper bound for the domain... |
| dmopab3 5817 | The domain of a restricted... |
| dm0 5818 | The domain of the empty se... |
| dmi 5819 | The domain of the identity... |
| dmv 5820 | The domain of the universe... |
| dmep 5821 | The domain of the membersh... |
| domepOLD 5822 | Obsolete proof of ~ dmep a... |
| dm0rn0 5823 | An empty domain is equival... |
| rn0 5824 | The range of the empty set... |
| rnep 5825 | The range of the membershi... |
| reldm0 5826 | A relation is empty iff it... |
| dmxp 5827 | The domain of a Cartesian ... |
| dmxpid 5828 | The domain of a Cartesian ... |
| dmxpin 5829 | The domain of the intersec... |
| xpid11 5830 | The Cartesian square is a ... |
| dmcnvcnv 5831 | The domain of the double c... |
| rncnvcnv 5832 | The range of the double co... |
| elreldm 5833 | The first member of an ord... |
| rneq 5834 | Equality theorem for range... |
| rneqi 5835 | Equality inference for ran... |
| rneqd 5836 | Equality deduction for ran... |
| rnss 5837 | Subset theorem for range. ... |
| rnssi 5838 | Subclass inference for ran... |
| brelrng 5839 | The second argument of a b... |
| brelrn 5840 | The second argument of a b... |
| opelrn 5841 | Membership of second membe... |
| releldm 5842 | The first argument of a bi... |
| relelrn 5843 | The second argument of a b... |
| releldmb 5844 | Membership in a domain. (... |
| relelrnb 5845 | Membership in a range. (C... |
| releldmi 5846 | The first argument of a bi... |
| relelrni 5847 | The second argument of a b... |
| dfrnf 5848 | Definition of range, using... |
| nfdm 5849 | Bound-variable hypothesis ... |
| nfrn 5850 | Bound-variable hypothesis ... |
| dmiin 5851 | Domain of an intersection.... |
| rnopab 5852 | The range of a class of or... |
| rnmpt 5853 | The range of a function in... |
| elrnmpt 5854 | The range of a function in... |
| elrnmpt1s 5855 | Elementhood in an image se... |
| elrnmpt1 5856 | Elementhood in an image se... |
| elrnmptg 5857 | Membership in the range of... |
| elrnmpti 5858 | Membership in the range of... |
| elrnmptd 5859 | The range of a function in... |
| elrnmptdv 5860 | Elementhood in the range o... |
| elrnmpt2d 5861 | Elementhood in the range o... |
| dfiun3g 5862 | Alternate definition of in... |
| dfiin3g 5863 | Alternate definition of in... |
| dfiun3 5864 | Alternate definition of in... |
| dfiin3 5865 | Alternate definition of in... |
| riinint 5866 | Express a relative indexed... |
| relrn0 5867 | A relation is empty iff it... |
| dmrnssfld 5868 | The domain and range of a ... |
| dmcoss 5869 | Domain of a composition. ... |
| rncoss 5870 | Range of a composition. (... |
| dmcosseq 5871 | Domain of a composition. ... |
| dmcoeq 5872 | Domain of a composition. ... |
| rncoeq 5873 | Range of a composition. (... |
| reseq1 5874 | Equality theorem for restr... |
| reseq2 5875 | Equality theorem for restr... |
| reseq1i 5876 | Equality inference for res... |
| reseq2i 5877 | Equality inference for res... |
| reseq12i 5878 | Equality inference for res... |
| reseq1d 5879 | Equality deduction for res... |
| reseq2d 5880 | Equality deduction for res... |
| reseq12d 5881 | Equality deduction for res... |
| nfres 5882 | Bound-variable hypothesis ... |
| csbres 5883 | Distribute proper substitu... |
| res0 5884 | A restriction to the empty... |
| dfres3 5885 | Alternate definition of re... |
| opelres 5886 | Ordered pair elementhood i... |
| brres 5887 | Binary relation on a restr... |
| opelresi 5888 | Ordered pair membership in... |
| brresi 5889 | Binary relation on a restr... |
| opres 5890 | Ordered pair membership in... |
| resieq 5891 | A restricted identity rela... |
| opelidres 5892 | ` <. A , A >. ` belongs to... |
| resres 5893 | The restriction of a restr... |
| resundi 5894 | Distributive law for restr... |
| resundir 5895 | Distributive law for restr... |
| resindi 5896 | Class restriction distribu... |
| resindir 5897 | Class restriction distribu... |
| inres 5898 | Move intersection into cla... |
| resdifcom 5899 | Commutative law for restri... |
| resiun1 5900 | Distribution of restrictio... |
| resiun2 5901 | Distribution of restrictio... |
| dmres 5902 | The domain of a restrictio... |
| ssdmres 5903 | A domain restricted to a s... |
| dmresexg 5904 | The domain of a restrictio... |
| resss 5905 | A class includes its restr... |
| rescom 5906 | Commutative law for restri... |
| ssres 5907 | Subclass theorem for restr... |
| ssres2 5908 | Subclass theorem for restr... |
| relres 5909 | A restriction is a relatio... |
| resabs1 5910 | Absorption law for restric... |
| resabs1d 5911 | Absorption law for restric... |
| resabs2 5912 | Absorption law for restric... |
| residm 5913 | Idempotent law for restric... |
| resima 5914 | A restriction to an image.... |
| resima2 5915 | Image under a restricted c... |
| rnresss 5916 | The range of a restriction... |
| xpssres 5917 | Restriction of a constant ... |
| elinxp 5918 | Membership in an intersect... |
| elres 5919 | Membership in a restrictio... |
| elsnres 5920 | Membership in restriction ... |
| relssres 5921 | Simplification law for res... |
| dmressnsn 5922 | The domain of a restrictio... |
| eldmressnsn 5923 | The element of the domain ... |
| eldmeldmressn 5924 | An element of the domain (... |
| resdm 5925 | A relation restricted to i... |
| resexg 5926 | The restriction of a set i... |
| resexd 5927 | The restriction of a set i... |
| resex 5928 | The restriction of a set i... |
| resindm 5929 | When restricting a relatio... |
| resdmdfsn 5930 | Restricting a relation to ... |
| resopab 5931 | Restriction of a class abs... |
| iss 5932 | A subclass of the identity... |
| resopab2 5933 | Restriction of a class abs... |
| resmpt 5934 | Restriction of the mapping... |
| resmpt3 5935 | Unconditional restriction ... |
| resmptf 5936 | Restriction of the mapping... |
| resmptd 5937 | Restriction of the mapping... |
| dfres2 5938 | Alternate definition of th... |
| mptss 5939 | Sufficient condition for i... |
| elidinxp 5940 | Characterization of the el... |
| elidinxpid 5941 | Characterization of the el... |
| elrid 5942 | Characterization of the el... |
| idinxpres 5943 | The intersection of the id... |
| idinxpresid 5944 | The intersection of the id... |
| idssxp 5945 | A diagonal set as a subset... |
| opabresid 5946 | The restricted identity re... |
| mptresid 5947 | The restricted identity re... |
| opabresidOLD 5948 | Obsolete version of ~ opab... |
| mptresidOLD 5949 | Obsolete version of ~ mptr... |
| dmresi 5950 | The domain of a restricted... |
| restidsing 5951 | Restriction of the identit... |
| iresn0n0 5952 | The identity function rest... |
| imaeq1 5953 | Equality theorem for image... |
| imaeq2 5954 | Equality theorem for image... |
| imaeq1i 5955 | Equality theorem for image... |
| imaeq2i 5956 | Equality theorem for image... |
| imaeq1d 5957 | Equality theorem for image... |
| imaeq2d 5958 | Equality theorem for image... |
| imaeq12d 5959 | Equality theorem for image... |
| dfima2 5960 | Alternate definition of im... |
| dfima3 5961 | Alternate definition of im... |
| elimag 5962 | Membership in an image. T... |
| elima 5963 | Membership in an image. T... |
| elima2 5964 | Membership in an image. T... |
| elima3 5965 | Membership in an image. T... |
| nfima 5966 | Bound-variable hypothesis ... |
| nfimad 5967 | Deduction version of bound... |
| imadmrn 5968 | The image of the domain of... |
| imassrn 5969 | The image of a class is a ... |
| mptima 5970 | Image of a function in map... |
| imai 5971 | Image under the identity r... |
| rnresi 5972 | The range of the restricte... |
| resiima 5973 | The image of a restriction... |
| ima0 5974 | Image of the empty set. T... |
| 0ima 5975 | Image under the empty rela... |
| csbima12 5976 | Move class substitution in... |
| imadisj 5977 | A class whose image under ... |
| cnvimass 5978 | A preimage under any class... |
| cnvimarndm 5979 | The preimage of the range ... |
| imasng 5980 | The image of a singleton. ... |
| relimasn 5981 | The image of a singleton. ... |
| elrelimasn 5982 | Elementhood in the image o... |
| elimasng1 5983 | Membership in an image of ... |
| elimasn1 5984 | Membership in an image of ... |
| elimasng 5985 | Membership in an image of ... |
| elimasn 5986 | Membership in an image of ... |
| elimasngOLD 5987 | Obsolete version of ~ elim... |
| elimasni 5988 | Membership in an image of ... |
| args 5989 | Two ways to express the cl... |
| elinisegg 5990 | Membership in the inverse ... |
| eliniseg 5991 | Membership in the inverse ... |
| epin 5992 | Any set is equal to its pr... |
| epini 5993 | Any set is equal to its pr... |
| iniseg 5994 | An idiom that signifies an... |
| inisegn0 5995 | Nonemptiness of an initial... |
| dffr3 5996 | Alternate definition of we... |
| dfse2 5997 | Alternate definition of se... |
| imass1 5998 | Subset theorem for image. ... |
| imass2 5999 | Subset theorem for image. ... |
| ndmima 6000 | The image of a singleton o... |
| relcnv 6001 | A converse is a relation. ... |
| relbrcnvg 6002 | When ` R ` is a relation, ... |
| eliniseg2 6003 | Eliminate the class existe... |
| relbrcnv 6004 | When ` R ` is a relation, ... |
| cotrg 6005 | Two ways of saying that th... |
| cotr 6006 | Two ways of saying a relat... |
| idrefALT 6007 | Alternate proof of ~ idref... |
| cnvsym 6008 | Two ways of saying a relat... |
| intasym 6009 | Two ways of saying a relat... |
| asymref 6010 | Two ways of saying a relat... |
| asymref2 6011 | Two ways of saying a relat... |
| intirr 6012 | Two ways of saying a relat... |
| brcodir 6013 | Two ways of saying that tw... |
| codir 6014 | Two ways of saying a relat... |
| qfto 6015 | A quantifier-free way of e... |
| xpidtr 6016 | A Cartesian square is a tr... |
| trin2 6017 | The intersection of two tr... |
| poirr2 6018 | A partial order is irrefle... |
| trinxp 6019 | The relation induced by a ... |
| soirri 6020 | A strict order relation is... |
| sotri 6021 | A strict order relation is... |
| son2lpi 6022 | A strict order relation ha... |
| sotri2 6023 | A transitivity relation. ... |
| sotri3 6024 | A transitivity relation. ... |
| poleloe 6025 | Express "less than or equa... |
| poltletr 6026 | Transitive law for general... |
| somin1 6027 | Property of a minimum in a... |
| somincom 6028 | Commutativity of minimum i... |
| somin2 6029 | Property of a minimum in a... |
| soltmin 6030 | Being less than a minimum,... |
| cnvopab 6031 | The converse of a class ab... |
| mptcnv 6032 | The converse of a mapping ... |
| cnv0 6033 | The converse of the empty ... |
| cnvi 6034 | The converse of the identi... |
| cnvun 6035 | The converse of a union is... |
| cnvdif 6036 | Distributive law for conve... |
| cnvin 6037 | Distributive law for conve... |
| rnun 6038 | Distributive law for range... |
| rnin 6039 | The range of an intersecti... |
| rniun 6040 | The range of an indexed un... |
| rnuni 6041 | The range of a union. Par... |
| imaundi 6042 | Distributive law for image... |
| imaundir 6043 | The image of a union. (Co... |
| cnvimassrndm 6044 | The preimage of a superset... |
| dminss 6045 | An upper bound for interse... |
| imainss 6046 | An upper bound for interse... |
| inimass 6047 | The image of an intersecti... |
| inimasn 6048 | The intersection of the im... |
| cnvxp 6049 | The converse of a Cartesia... |
| xp0 6050 | The Cartesian product with... |
| xpnz 6051 | The Cartesian product of n... |
| xpeq0 6052 | At least one member of an ... |
| xpdisj1 6053 | Cartesian products with di... |
| xpdisj2 6054 | Cartesian products with di... |
| xpsndisj 6055 | Cartesian products with tw... |
| difxp 6056 | Difference of Cartesian pr... |
| difxp1 6057 | Difference law for Cartesi... |
| difxp2 6058 | Difference law for Cartesi... |
| djudisj 6059 | Disjoint unions with disjo... |
| xpdifid 6060 | The set of distinct couple... |
| resdisj 6061 | A double restriction to di... |
| rnxp 6062 | The range of a Cartesian p... |
| dmxpss 6063 | The domain of a Cartesian ... |
| rnxpss 6064 | The range of a Cartesian p... |
| rnxpid 6065 | The range of a Cartesian s... |
| ssxpb 6066 | A Cartesian product subcla... |
| xp11 6067 | The Cartesian product of n... |
| xpcan 6068 | Cancellation law for Carte... |
| xpcan2 6069 | Cancellation law for Carte... |
| ssrnres 6070 | Two ways to express surjec... |
| rninxp 6071 | Two ways to express surjec... |
| dminxp 6072 | Two ways to express totali... |
| imainrect 6073 | Image by a restricted and ... |
| xpima 6074 | Direct image by a Cartesia... |
| xpima1 6075 | Direct image by a Cartesia... |
| xpima2 6076 | Direct image by a Cartesia... |
| xpimasn 6077 | Direct image of a singleto... |
| sossfld 6078 | The base set of a strict o... |
| sofld 6079 | The base set of a nonempty... |
| cnvcnv3 6080 | The set of all ordered pai... |
| dfrel2 6081 | Alternate definition of re... |
| dfrel4v 6082 | A relation can be expresse... |
| dfrel4 6083 | A relation can be expresse... |
| cnvcnv 6084 | The double converse of a c... |
| cnvcnv2 6085 | The double converse of a c... |
| cnvcnvss 6086 | The double converse of a c... |
| cnvrescnv 6087 | Two ways to express the co... |
| cnveqb 6088 | Equality theorem for conve... |
| cnveq0 6089 | A relation empty iff its c... |
| dfrel3 6090 | Alternate definition of re... |
| elid 6091 | Characterization of the el... |
| dmresv 6092 | The domain of a universal ... |
| rnresv 6093 | The range of a universal r... |
| dfrn4 6094 | Range defined in terms of ... |
| csbrn 6095 | Distribute proper substitu... |
| rescnvcnv 6096 | The restriction of the dou... |
| cnvcnvres 6097 | The double converse of the... |
| imacnvcnv 6098 | The image of the double co... |
| dmsnn0 6099 | The domain of a singleton ... |
| rnsnn0 6100 | The range of a singleton i... |
| dmsn0 6101 | The domain of the singleto... |
| cnvsn0 6102 | The converse of the single... |
| dmsn0el 6103 | The domain of a singleton ... |
| relsn2 6104 | A singleton is a relation ... |
| dmsnopg 6105 | The domain of a singleton ... |
| dmsnopss 6106 | The domain of a singleton ... |
| dmpropg 6107 | The domain of an unordered... |
| dmsnop 6108 | The domain of a singleton ... |
| dmprop 6109 | The domain of an unordered... |
| dmtpop 6110 | The domain of an unordered... |
| cnvcnvsn 6111 | Double converse of a singl... |
| dmsnsnsn 6112 | The domain of the singleto... |
| rnsnopg 6113 | The range of a singleton o... |
| rnpropg 6114 | The range of a pair of ord... |
| cnvsng 6115 | Converse of a singleton of... |
| rnsnop 6116 | The range of a singleton o... |
| op1sta 6117 | Extract the first member o... |
| cnvsn 6118 | Converse of a singleton of... |
| op2ndb 6119 | Extract the second member ... |
| op2nda 6120 | Extract the second member ... |
| opswap 6121 | Swap the members of an ord... |
| cnvresima 6122 | An image under the convers... |
| resdm2 6123 | A class restricted to its ... |
| resdmres 6124 | Restriction to the domain ... |
| resresdm 6125 | A restriction by an arbitr... |
| imadmres 6126 | The image of the domain of... |
| resdmss 6127 | Subset relationship for th... |
| resdifdi 6128 | Distributive law for restr... |
| resdifdir 6129 | Distributive law for restr... |
| mptpreima 6130 | The preimage of a function... |
| mptiniseg 6131 | Converse singleton image o... |
| dmmpt 6132 | The domain of the mapping ... |
| dmmptss 6133 | The domain of a mapping is... |
| dmmptg 6134 | The domain of the mapping ... |
| rnmpt0f 6135 | The range of a function in... |
| rnmptn0 6136 | The range of a function in... |
| relco 6137 | A composition is a relatio... |
| dfco2 6138 | Alternate definition of a ... |
| dfco2a 6139 | Generalization of ~ dfco2 ... |
| coundi 6140 | Class composition distribu... |
| coundir 6141 | Class composition distribu... |
| cores 6142 | Restricted first member of... |
| resco 6143 | Associative law for the re... |
| imaco 6144 | Image of the composition o... |
| rnco 6145 | The range of the compositi... |
| rnco2 6146 | The range of the compositi... |
| dmco 6147 | The domain of a compositio... |
| coeq0 6148 | A composition of two relat... |
| coiun 6149 | Composition with an indexe... |
| cocnvcnv1 6150 | A composition is not affec... |
| cocnvcnv2 6151 | A composition is not affec... |
| cores2 6152 | Absorption of a reverse (p... |
| co02 6153 | Composition with the empty... |
| co01 6154 | Composition with the empty... |
| coi1 6155 | Composition with the ident... |
| coi2 6156 | Composition with the ident... |
| coires1 6157 | Composition with a restric... |
| coass 6158 | Associative law for class ... |
| relcnvtrg 6159 | General form of ~ relcnvtr... |
| relcnvtr 6160 | A relation is transitive i... |
| relssdmrn 6161 | A relation is included in ... |
| resssxp 6162 | If the ` R ` -image of a c... |
| cnvssrndm 6163 | The converse is a subset o... |
| cossxp 6164 | Composition as a subset of... |
| relrelss 6165 | Two ways to describe the s... |
| unielrel 6166 | The membership relation fo... |
| relfld 6167 | The double union of a rela... |
| relresfld 6168 | Restriction of a relation ... |
| relcoi2 6169 | Composition with the ident... |
| relcoi1 6170 | Composition with the ident... |
| unidmrn 6171 | The double union of the co... |
| relcnvfld 6172 | if ` R ` is a relation, it... |
| dfdm2 6173 | Alternate definition of do... |
| unixp 6174 | The double class union of ... |
| unixp0 6175 | A Cartesian product is emp... |
| unixpid 6176 | Field of a Cartesian squar... |
| ressn 6177 | Restriction of a class to ... |
| cnviin 6178 | The converse of an interse... |
| cnvpo 6179 | The converse of a partial ... |
| cnvso 6180 | The converse of a strict o... |
| xpco 6181 | Composition of two Cartesi... |
| xpcoid 6182 | Composition of two Cartesi... |
| elsnxp 6183 | Membership in a Cartesian ... |
| reu3op 6184 | There is a unique ordered ... |
| reuop 6185 | There is a unique ordered ... |
| opreu2reurex 6186 | There is a unique ordered ... |
| opreu2reu 6187 | If there is a unique order... |
| dfpo2 6188 | Quantifier-free definition... |
| csbcog 6189 | Distribute proper substitu... |
| predeq123 6192 | Equality theorem for the p... |
| predeq1 6193 | Equality theorem for the p... |
| predeq2 6194 | Equality theorem for the p... |
| predeq3 6195 | Equality theorem for the p... |
| nfpred 6196 | Bound-variable hypothesis ... |
| csbpredg 6197 | Move class substitution in... |
| predpredss 6198 | If ` A ` is a subset of ` ... |
| predss 6199 | The predecessor class of `... |
| sspred 6200 | Another subset/predecessor... |
| dfpred2 6201 | An alternate definition of... |
| dfpred3 6202 | An alternate definition of... |
| dfpred3g 6203 | An alternate definition of... |
| elpredgg 6204 | Membership in a predecesso... |
| elpredg 6205 | Membership in a predecesso... |
| elpredimg 6206 | Membership in a predecesso... |
| elpredim 6207 | Membership in a predecesso... |
| elpred 6208 | Membership in a predecesso... |
| predexg 6209 | The predecessor class exis... |
| predasetexOLD 6210 | Obsolete form of ~ predexg... |
| dffr4 6211 | Alternate definition of we... |
| predel 6212 | Membership in the predeces... |
| predbrg 6213 | Closed form of ~ elpredim ... |
| predtrss 6214 | If ` R ` is transitive ove... |
| predpo 6215 | Property of the predecesso... |
| predso 6216 | Property of the predecesso... |
| setlikespec 6217 | If ` R ` is set-like in ` ... |
| predidm 6218 | Idempotent law for the pre... |
| predin 6219 | Intersection law for prede... |
| predun 6220 | Union law for predecessor ... |
| preddif 6221 | Difference law for predece... |
| predep 6222 | The predecessor under the ... |
| trpred 6223 | The class of predecessors ... |
| preddowncl 6224 | A property of classes that... |
| predpoirr 6225 | Given a partial ordering, ... |
| predfrirr 6226 | Given a well-founded relat... |
| pred0 6227 | The predecessor class over... |
| frpomin 6228 | Every nonempty (possibly p... |
| frpomin2 6229 | Every nonempty (possibly p... |
| frpoind 6230 | The principle of well-foun... |
| frpoinsg 6231 | Well-Founded Induction Sch... |
| frpoins2fg 6232 | Well-Founded Induction sch... |
| frpoins2g 6233 | Well-Founded Induction sch... |
| frpoins3g 6234 | Well-Founded Induction sch... |
| tz6.26 6235 | All nonempty subclasses of... |
| tz6.26OLD 6236 | Obsolete proof of ~ tz6.26... |
| tz6.26i 6237 | All nonempty subclasses of... |
| wfi 6238 | The Principle of Well-Orde... |
| wfiOLD 6239 | Obsolete proof of ~ wfi as... |
| wfii 6240 | The Principle of Well-Orde... |
| wfisg 6241 | Well-Ordered Induction Sch... |
| wfisgOLD 6242 | Obsolete proof of ~ wfisg ... |
| wfis 6243 | Well-Ordered Induction Sch... |
| wfis2fg 6244 | Well-Ordered Induction Sch... |
| wfis2fgOLD 6245 | Obsolete proof of ~ wfis2f... |
| wfis2f 6246 | Well-Ordered Induction sch... |
| wfis2g 6247 | Well-Ordered Induction Sch... |
| wfis2 6248 | Well-Ordered Induction sch... |
| wfis3 6249 | Well-Ordered Induction sch... |
| ordeq 6258 | Equality theorem for the o... |
| elong 6259 | An ordinal number is an or... |
| elon 6260 | An ordinal number is an or... |
| eloni 6261 | An ordinal number has the ... |
| elon2 6262 | An ordinal number is an or... |
| limeq 6263 | Equality theorem for the l... |
| ordwe 6264 | Membership well-orders eve... |
| ordtr 6265 | An ordinal class is transi... |
| ordfr 6266 | Membership is well-founded... |
| ordelss 6267 | An element of an ordinal c... |
| trssord 6268 | A transitive subclass of a... |
| ordirr 6269 | No ordinal class is a memb... |
| nordeq 6270 | A member of an ordinal cla... |
| ordn2lp 6271 | An ordinal class cannot be... |
| tz7.5 6272 | A nonempty subclass of an ... |
| ordelord 6273 | An element of an ordinal c... |
| tron 6274 | The class of all ordinal n... |
| ordelon 6275 | An element of an ordinal c... |
| onelon 6276 | An element of an ordinal n... |
| tz7.7 6277 | A transitive class belongs... |
| ordelssne 6278 | For ordinal classes, membe... |
| ordelpss 6279 | For ordinal classes, membe... |
| ordsseleq 6280 | For ordinal classes, inclu... |
| ordin 6281 | The intersection of two or... |
| onin 6282 | The intersection of two or... |
| ordtri3or 6283 | A trichotomy law for ordin... |
| ordtri1 6284 | A trichotomy law for ordin... |
| ontri1 6285 | A trichotomy law for ordin... |
| ordtri2 6286 | A trichotomy law for ordin... |
| ordtri3 6287 | A trichotomy law for ordin... |
| ordtri4 6288 | A trichotomy law for ordin... |
| orddisj 6289 | An ordinal class and its s... |
| onfr 6290 | The ordinal class is well-... |
| onelpss 6291 | Relationship between membe... |
| onsseleq 6292 | Relationship between subse... |
| onelss 6293 | An element of an ordinal n... |
| ordtr1 6294 | Transitive law for ordinal... |
| ordtr2 6295 | Transitive law for ordinal... |
| ordtr3 6296 | Transitive law for ordinal... |
| ontr1 6297 | Transitive law for ordinal... |
| ontr2 6298 | Transitive law for ordinal... |
| ordunidif 6299 | The union of an ordinal st... |
| ordintdif 6300 | If ` B ` is smaller than `... |
| onintss 6301 | If a property is true for ... |
| oneqmini 6302 | A way to show that an ordi... |
| ord0 6303 | The empty set is an ordina... |
| 0elon 6304 | The empty set is an ordina... |
| ord0eln0 6305 | A nonempty ordinal contain... |
| on0eln0 6306 | An ordinal number contains... |
| dflim2 6307 | An alternate definition of... |
| inton 6308 | The intersection of the cl... |
| nlim0 6309 | The empty set is not a lim... |
| limord 6310 | A limit ordinal is ordinal... |
| limuni 6311 | A limit ordinal is its own... |
| limuni2 6312 | The union of a limit ordin... |
| 0ellim 6313 | A limit ordinal contains t... |
| limelon 6314 | A limit ordinal class that... |
| onn0 6315 | The class of all ordinal n... |
| suceq 6316 | Equality of successors. (... |
| elsuci 6317 | Membership in a successor.... |
| elsucg 6318 | Membership in a successor.... |
| elsuc2g 6319 | Variant of membership in a... |
| elsuc 6320 | Membership in a successor.... |
| elsuc2 6321 | Membership in a successor.... |
| nfsuc 6322 | Bound-variable hypothesis ... |
| elelsuc 6323 | Membership in a successor.... |
| sucel 6324 | Membership of a successor ... |
| suc0 6325 | The successor of the empty... |
| sucprc 6326 | A proper class is its own ... |
| unisuc 6327 | A transitive class is equa... |
| sssucid 6328 | A class is included in its... |
| sucidg 6329 | Part of Proposition 7.23 o... |
| sucid 6330 | A set belongs to its succe... |
| nsuceq0 6331 | No successor is empty. (C... |
| eqelsuc 6332 | A set belongs to the succe... |
| iunsuc 6333 | Inductive definition for t... |
| suctr 6334 | The successor of a transit... |
| trsuc 6335 | A set whose successor belo... |
| trsucss 6336 | A member of the successor ... |
| ordsssuc 6337 | An ordinal is a subset of ... |
| onsssuc 6338 | A subset of an ordinal num... |
| ordsssuc2 6339 | An ordinal subset of an or... |
| onmindif 6340 | When its successor is subt... |
| ordnbtwn 6341 | There is no set between an... |
| onnbtwn 6342 | There is no set between an... |
| sucssel 6343 | A set whose successor is a... |
| orddif 6344 | Ordinal derived from its s... |
| orduniss 6345 | An ordinal class includes ... |
| ordtri2or 6346 | A trichotomy law for ordin... |
| ordtri2or2 6347 | A trichotomy law for ordin... |
| ordtri2or3 6348 | A consequence of total ord... |
| ordelinel 6349 | The intersection of two or... |
| ordssun 6350 | Property of a subclass of ... |
| ordequn 6351 | The maximum (i.e. union) o... |
| ordun 6352 | The maximum (i.e. union) o... |
| ordunisssuc 6353 | A subclass relationship fo... |
| suc11 6354 | The successor operation be... |
| onun2 6355 | The union of two ordinals ... |
| onordi 6356 | An ordinal number is an or... |
| ontrci 6357 | An ordinal number is a tra... |
| onirri 6358 | An ordinal number is not a... |
| oneli 6359 | A member of an ordinal num... |
| onelssi 6360 | A member of an ordinal num... |
| onssneli 6361 | An ordering law for ordina... |
| onssnel2i 6362 | An ordering law for ordina... |
| onelini 6363 | An element of an ordinal n... |
| oneluni 6364 | An ordinal number equals i... |
| onunisuci 6365 | An ordinal number is equal... |
| onsseli 6366 | Subset is equivalent to me... |
| onun2i 6367 | The union of two ordinal n... |
| unizlim 6368 | An ordinal equal to its ow... |
| on0eqel 6369 | An ordinal number either e... |
| snsn0non 6370 | The singleton of the singl... |
| onxpdisj 6371 | Ordinal numbers and ordere... |
| onnev 6372 | The class of ordinal numbe... |
| onnevOLD 6373 | Obsolete version of ~ onne... |
| iotajust 6375 | Soundness justification th... |
| dfiota2 6377 | Alternate definition for d... |
| nfiota1 6378 | Bound-variable hypothesis ... |
| nfiotadw 6379 | Deduction version of ~ nfi... |
| nfiotaw 6380 | Bound-variable hypothesis ... |
| nfiotad 6381 | Deduction version of ~ nfi... |
| nfiota 6382 | Bound-variable hypothesis ... |
| cbviotaw 6383 | Change bound variables in ... |
| cbviotavw 6384 | Change bound variables in ... |
| cbviotavwOLD 6385 | Obsolete version of ~ cbvi... |
| cbviota 6386 | Change bound variables in ... |
| cbviotav 6387 | Change bound variables in ... |
| sb8iota 6388 | Variable substitution in d... |
| iotaeq 6389 | Equality theorem for descr... |
| iotabi 6390 | Equivalence theorem for de... |
| uniabio 6391 | Part of Theorem 8.17 in [Q... |
| iotaval 6392 | Theorem 8.19 in [Quine] p.... |
| iotauni 6393 | Equivalence between two di... |
| iotaint 6394 | Equivalence between two di... |
| iota1 6395 | Property of iota. (Contri... |
| iotanul 6396 | Theorem 8.22 in [Quine] p.... |
| iotassuni 6397 | The ` iota ` class is a su... |
| iotaex 6398 | Theorem 8.23 in [Quine] p.... |
| iota4 6399 | Theorem *14.22 in [Whitehe... |
| iota4an 6400 | Theorem *14.23 in [Whitehe... |
| iota5 6401 | A method for computing iot... |
| iotabidv 6402 | Formula-building deduction... |
| iotabii 6403 | Formula-building deduction... |
| iotacl 6404 | Membership law for descrip... |
| iota2df 6405 | A condition that allows us... |
| iota2d 6406 | A condition that allows us... |
| iota2 6407 | The unique element such th... |
| iotan0 6408 | Representation of "the uni... |
| sniota 6409 | A class abstraction with a... |
| dfiota4 6410 | The ` iota ` operation usi... |
| csbiota 6411 | Class substitution within ... |
| dffun2 6428 | Alternate definition of a ... |
| dffun3 6429 | Alternate definition of fu... |
| dffun4 6430 | Alternate definition of a ... |
| dffun5 6431 | Alternate definition of fu... |
| dffun6f 6432 | Definition of function, us... |
| dffun6 6433 | Alternate definition of a ... |
| funmo 6434 | A function has at most one... |
| funrel 6435 | A function is a relation. ... |
| 0nelfun 6436 | A function does not contai... |
| funss 6437 | Subclass theorem for funct... |
| funeq 6438 | Equality theorem for funct... |
| funeqi 6439 | Equality inference for the... |
| funeqd 6440 | Equality deduction for the... |
| nffun 6441 | Bound-variable hypothesis ... |
| sbcfung 6442 | Distribute proper substitu... |
| funeu 6443 | There is exactly one value... |
| funeu2 6444 | There is exactly one value... |
| dffun7 6445 | Alternate definition of a ... |
| dffun8 6446 | Alternate definition of a ... |
| dffun9 6447 | Alternate definition of a ... |
| funfn 6448 | A class is a function if a... |
| funfnd 6449 | A function is a function o... |
| funi 6450 | The identity relation is a... |
| nfunv 6451 | The universal class is not... |
| funopg 6452 | A Kuratowski ordered pair ... |
| funopab 6453 | A class of ordered pairs i... |
| funopabeq 6454 | A class of ordered pairs o... |
| funopab4 6455 | A class of ordered pairs o... |
| funmpt 6456 | A function in maps-to nota... |
| funmpt2 6457 | Functionality of a class g... |
| funco 6458 | The composition of two fun... |
| funresfunco 6459 | Composition of two functio... |
| funres 6460 | A restriction of a functio... |
| funresd 6461 | A restriction of a functio... |
| funssres 6462 | The restriction of a funct... |
| fun2ssres 6463 | Equality of restrictions o... |
| funun 6464 | The union of functions wit... |
| fununmo 6465 | If the union of classes is... |
| fununfun 6466 | If the union of classes is... |
| fundif 6467 | A function with removed el... |
| funcnvsn 6468 | The converse singleton of ... |
| funsng 6469 | A singleton of an ordered ... |
| fnsng 6470 | Functionality and domain o... |
| funsn 6471 | A singleton of an ordered ... |
| funprg 6472 | A set of two pairs is a fu... |
| funtpg 6473 | A set of three pairs is a ... |
| funpr 6474 | A function with a domain o... |
| funtp 6475 | A function with a domain o... |
| fnsn 6476 | Functionality and domain o... |
| fnprg 6477 | Function with a domain of ... |
| fntpg 6478 | Function with a domain of ... |
| fntp 6479 | A function with a domain o... |
| funcnvpr 6480 | The converse pair of order... |
| funcnvtp 6481 | The converse triple of ord... |
| funcnvqp 6482 | The converse quadruple of ... |
| fun0 6483 | The empty set is a functio... |
| funcnv0 6484 | The converse of the empty ... |
| funcnvcnv 6485 | The double converse of a f... |
| funcnv2 6486 | A simpler equivalence for ... |
| funcnv 6487 | The converse of a class is... |
| funcnv3 6488 | A condition showing a clas... |
| fun2cnv 6489 | The double converse of a c... |
| svrelfun 6490 | A single-valued relation i... |
| fncnv 6491 | Single-rootedness (see ~ f... |
| fun11 6492 | Two ways of stating that `... |
| fununi 6493 | The union of a chain (with... |
| funin 6494 | The intersection with a fu... |
| funres11 6495 | The restriction of a one-t... |
| funcnvres 6496 | The converse of a restrict... |
| cnvresid 6497 | Converse of a restricted i... |
| funcnvres2 6498 | The converse of a restrict... |
| funimacnv 6499 | The image of the preimage ... |
| funimass1 6500 | A kind of contraposition l... |
| funimass2 6501 | A kind of contraposition l... |
| imadif 6502 | The image of a difference ... |
| imain 6503 | The image of an intersecti... |
| funimaexg 6504 | Axiom of Replacement using... |
| funimaex 6505 | The image of a set under a... |
| isarep1 6506 | Part of a study of the Axi... |
| isarep2 6507 | Part of a study of the Axi... |
| fneq1 6508 | Equality theorem for funct... |
| fneq2 6509 | Equality theorem for funct... |
| fneq1d 6510 | Equality deduction for fun... |
| fneq2d 6511 | Equality deduction for fun... |
| fneq12d 6512 | Equality deduction for fun... |
| fneq12 6513 | Equality theorem for funct... |
| fneq1i 6514 | Equality inference for fun... |
| fneq2i 6515 | Equality inference for fun... |
| nffn 6516 | Bound-variable hypothesis ... |
| fnfun 6517 | A function with domain is ... |
| fnfund 6518 | A function with domain is ... |
| fnrel 6519 | A function with domain is ... |
| fndm 6520 | The domain of a function. ... |
| fndmi 6521 | The domain of a function. ... |
| fndmd 6522 | The domain of a function. ... |
| funfni 6523 | Inference to convert a fun... |
| fndmu 6524 | A function has a unique do... |
| fnbr 6525 | The first argument of bina... |
| fnop 6526 | The first argument of an o... |
| fneu 6527 | There is exactly one value... |
| fneu2 6528 | There is exactly one value... |
| fnun 6529 | The union of two functions... |
| fnund 6530 | The union of two functions... |
| fnunop 6531 | Extension of a function wi... |
| fncofn 6532 | Composition of a function ... |
| fnco 6533 | Composition of two functio... |
| fncoOLD 6534 | Obsolete version of ~ fnco... |
| fnresdm 6535 | A function does not change... |
| fnresdisj 6536 | A function restricted to a... |
| 2elresin 6537 | Membership in two function... |
| fnssresb 6538 | Restriction of a function ... |
| fnssres 6539 | Restriction of a function ... |
| fnssresd 6540 | Restriction of a function ... |
| fnresin1 6541 | Restriction of a function'... |
| fnresin2 6542 | Restriction of a function'... |
| fnres 6543 | An equivalence for functio... |
| idfn 6544 | The identity relation is a... |
| fnresi 6545 | The restricted identity re... |
| fnresiOLD 6546 | Obsolete proof of ~ fnresi... |
| fnima 6547 | The image of a function's ... |
| fn0 6548 | A function with empty doma... |
| fnimadisj 6549 | A class that is disjoint w... |
| fnimaeq0 6550 | Images under a function ne... |
| dfmpt3 6551 | Alternate definition for t... |
| mptfnf 6552 | The maps-to notation defin... |
| fnmptf 6553 | The maps-to notation defin... |
| fnopabg 6554 | Functionality and domain o... |
| fnopab 6555 | Functionality and domain o... |
| mptfng 6556 | The maps-to notation defin... |
| fnmpt 6557 | The maps-to notation defin... |
| fnmptd 6558 | The maps-to notation defin... |
| mpt0 6559 | A mapping operation with e... |
| fnmpti 6560 | Functionality and domain o... |
| dmmpti 6561 | Domain of the mapping oper... |
| dmmptd 6562 | The domain of the mapping ... |
| mptun 6563 | Union of mappings which ar... |
| partfun 6564 | Rewrite a function defined... |
| feq1 6565 | Equality theorem for funct... |
| feq2 6566 | Equality theorem for funct... |
| feq3 6567 | Equality theorem for funct... |
| feq23 6568 | Equality theorem for funct... |
| feq1d 6569 | Equality deduction for fun... |
| feq2d 6570 | Equality deduction for fun... |
| feq3d 6571 | Equality deduction for fun... |
| feq12d 6572 | Equality deduction for fun... |
| feq123d 6573 | Equality deduction for fun... |
| feq123 6574 | Equality theorem for funct... |
| feq1i 6575 | Equality inference for fun... |
| feq2i 6576 | Equality inference for fun... |
| feq12i 6577 | Equality inference for fun... |
| feq23i 6578 | Equality inference for fun... |
| feq23d 6579 | Equality deduction for fun... |
| nff 6580 | Bound-variable hypothesis ... |
| sbcfng 6581 | Distribute proper substitu... |
| sbcfg 6582 | Distribute proper substitu... |
| elimf 6583 | Eliminate a mapping hypoth... |
| ffn 6584 | A mapping is a function wi... |
| ffnd 6585 | A mapping is a function wi... |
| dffn2 6586 | Any function is a mapping ... |
| ffun 6587 | A mapping is a function. ... |
| ffund 6588 | A mapping is a function, d... |
| frel 6589 | A mapping is a relation. ... |
| freld 6590 | A mapping is a relation. ... |
| frn 6591 | The range of a mapping. (... |
| frnd 6592 | Deduction form of ~ frn . ... |
| fdm 6593 | The domain of a mapping. ... |
| fdmOLD 6594 | Obsolete version of ~ fdm ... |
| fdmd 6595 | Deduction form of ~ fdm . ... |
| fdmi 6596 | Inference associated with ... |
| dffn3 6597 | A function maps to its ran... |
| ffrn 6598 | A function maps to its ran... |
| ffrnb 6599 | Characterization of a func... |
| ffrnbd 6600 | A function maps to its ran... |
| fss 6601 | Expanding the codomain of ... |
| fssd 6602 | Expanding the codomain of ... |
| fssdmd 6603 | Expressing that a class is... |
| fssdm 6604 | Expressing that a class is... |
| fimass 6605 | The image of a class under... |
| fimacnv 6606 | The preimage of the codoma... |
| fcof 6607 | Composition of a function ... |
| fco 6608 | Composition of two functio... |
| fcoOLD 6609 | Obsolete version of ~ fco ... |
| fcod 6610 | Composition of two mapping... |
| fco2 6611 | Functionality of a composi... |
| fssxp 6612 | A mapping is a class of or... |
| funssxp 6613 | Two ways of specifying a p... |
| ffdm 6614 | A mapping is a partial fun... |
| ffdmd 6615 | The domain of a function. ... |
| fdmrn 6616 | A different way to write `... |
| funcofd 6617 | Composition of two functio... |
| fco3OLD 6618 | Obsolete version of ~ func... |
| opelf 6619 | The members of an ordered ... |
| fun 6620 | The union of two functions... |
| fun2 6621 | The union of two functions... |
| fun2d 6622 | The union of functions wit... |
| fnfco 6623 | Composition of two functio... |
| fssres 6624 | Restriction of a function ... |
| fssresd 6625 | Restriction of a function ... |
| fssres2 6626 | Restriction of a restricte... |
| fresin 6627 | An identity for the mappin... |
| resasplit 6628 | If two functions agree on ... |
| fresaun 6629 | The union of two functions... |
| fresaunres2 6630 | From the union of two func... |
| fresaunres1 6631 | From the union of two func... |
| fcoi1 6632 | Composition of a mapping a... |
| fcoi2 6633 | Composition of restricted ... |
| feu 6634 | There is exactly one value... |
| fcnvres 6635 | The converse of a restrict... |
| fimacnvdisj 6636 | The preimage of a class di... |
| fint 6637 | Function into an intersect... |
| fin 6638 | Mapping into an intersecti... |
| f0 6639 | The empty function. (Cont... |
| f00 6640 | A class is a function with... |
| f0bi 6641 | A function with empty doma... |
| f0dom0 6642 | A function is empty iff it... |
| f0rn0 6643 | If there is no element in ... |
| fconst 6644 | A Cartesian product with a... |
| fconstg 6645 | A Cartesian product with a... |
| fnconstg 6646 | A Cartesian product with a... |
| fconst6g 6647 | Constant function with loo... |
| fconst6 6648 | A constant function as a m... |
| f1eq1 6649 | Equality theorem for one-t... |
| f1eq2 6650 | Equality theorem for one-t... |
| f1eq3 6651 | Equality theorem for one-t... |
| nff1 6652 | Bound-variable hypothesis ... |
| dff12 6653 | Alternate definition of a ... |
| f1f 6654 | A one-to-one mapping is a ... |
| f1fn 6655 | A one-to-one mapping is a ... |
| f1fun 6656 | A one-to-one mapping is a ... |
| f1rel 6657 | A one-to-one onto mapping ... |
| f1dm 6658 | The domain of a one-to-one... |
| f1dmOLD 6659 | Obsolete version of ~ f1dm... |
| f1ss 6660 | A function that is one-to-... |
| f1ssr 6661 | A function that is one-to-... |
| f1ssres 6662 | A function that is one-to-... |
| f1resf1 6663 | The restriction of an inje... |
| f1cnvcnv 6664 | Two ways to express that a... |
| f1cof1 6665 | Composition of two one-to-... |
| f1co 6666 | Composition of one-to-one ... |
| f1coOLD 6667 | Obsolete version of ~ f1co... |
| foeq1 6668 | Equality theorem for onto ... |
| foeq2 6669 | Equality theorem for onto ... |
| foeq3 6670 | Equality theorem for onto ... |
| nffo 6671 | Bound-variable hypothesis ... |
| fof 6672 | An onto mapping is a mappi... |
| fofun 6673 | An onto mapping is a funct... |
| fofn 6674 | An onto mapping is a funct... |
| forn 6675 | The codomain of an onto fu... |
| dffo2 6676 | Alternate definition of an... |
| foima 6677 | The image of the domain of... |
| dffn4 6678 | A function maps onto its r... |
| funforn 6679 | A function maps its domain... |
| fodmrnu 6680 | An onto function has uniqu... |
| fimadmfo 6681 | A function is a function o... |
| fores 6682 | Restriction of an onto fun... |
| fimadmfoALT 6683 | Alternate proof of ~ fimad... |
| focnvimacdmdm 6684 | The preimage of the codoma... |
| focofo 6685 | Composition of onto functi... |
| foco 6686 | Composition of onto functi... |
| foconst 6687 | A nonzero constant functio... |
| f1oeq1 6688 | Equality theorem for one-t... |
| f1oeq2 6689 | Equality theorem for one-t... |
| f1oeq3 6690 | Equality theorem for one-t... |
| f1oeq23 6691 | Equality theorem for one-t... |
| f1eq123d 6692 | Equality deduction for one... |
| foeq123d 6693 | Equality deduction for ont... |
| f1oeq123d 6694 | Equality deduction for one... |
| f1oeq1d 6695 | Equality deduction for one... |
| f1oeq2d 6696 | Equality deduction for one... |
| f1oeq3d 6697 | Equality deduction for one... |
| nff1o 6698 | Bound-variable hypothesis ... |
| f1of1 6699 | A one-to-one onto mapping ... |
| f1of 6700 | A one-to-one onto mapping ... |
| f1ofn 6701 | A one-to-one onto mapping ... |
| f1ofun 6702 | A one-to-one onto mapping ... |
| f1orel 6703 | A one-to-one onto mapping ... |
| f1odm 6704 | The domain of a one-to-one... |
| dff1o2 6705 | Alternate definition of on... |
| dff1o3 6706 | Alternate definition of on... |
| f1ofo 6707 | A one-to-one onto function... |
| dff1o4 6708 | Alternate definition of on... |
| dff1o5 6709 | Alternate definition of on... |
| f1orn 6710 | A one-to-one function maps... |
| f1f1orn 6711 | A one-to-one function maps... |
| f1ocnv 6712 | The converse of a one-to-o... |
| f1ocnvb 6713 | A relation is a one-to-one... |
| f1ores 6714 | The restriction of a one-t... |
| f1orescnv 6715 | The converse of a one-to-o... |
| f1imacnv 6716 | Preimage of an image. (Co... |
| foimacnv 6717 | A reverse version of ~ f1i... |
| foun 6718 | The union of two onto func... |
| f1oun 6719 | The union of two one-to-on... |
| resdif 6720 | The restriction of a one-t... |
| resin 6721 | The restriction of a one-t... |
| f1oco 6722 | Composition of one-to-one ... |
| f1cnv 6723 | The converse of an injecti... |
| funcocnv2 6724 | Composition with the conve... |
| fococnv2 6725 | The composition of an onto... |
| f1ococnv2 6726 | The composition of a one-t... |
| f1cocnv2 6727 | Composition of an injectiv... |
| f1ococnv1 6728 | The composition of a one-t... |
| f1cocnv1 6729 | Composition of an injectiv... |
| funcoeqres 6730 | Express a constraint on a ... |
| f1ssf1 6731 | A subset of an injective f... |
| f10 6732 | The empty set maps one-to-... |
| f10d 6733 | The empty set maps one-to-... |
| f1o00 6734 | One-to-one onto mapping of... |
| fo00 6735 | Onto mapping of the empty ... |
| f1o0 6736 | One-to-one onto mapping of... |
| f1oi 6737 | A restriction of the ident... |
| f1ovi 6738 | The identity relation is a... |
| f1osn 6739 | A singleton of an ordered ... |
| f1osng 6740 | A singleton of an ordered ... |
| f1sng 6741 | A singleton of an ordered ... |
| fsnd 6742 | A singleton of an ordered ... |
| f1oprswap 6743 | A two-element swap is a bi... |
| f1oprg 6744 | An unordered pair of order... |
| tz6.12-2 6745 | Function value when ` F ` ... |
| fveu 6746 | The value of a function at... |
| brprcneu 6747 | If ` A ` is a proper class... |
| fvprc 6748 | A function's value at a pr... |
| fvprcALT 6749 | Alternate proof of ~ fvprc... |
| rnfvprc 6750 | The range of a function va... |
| fv2 6751 | Alternate definition of fu... |
| dffv3 6752 | A definition of function v... |
| dffv4 6753 | The previous definition of... |
| elfv 6754 | Membership in a function v... |
| fveq1 6755 | Equality theorem for funct... |
| fveq2 6756 | Equality theorem for funct... |
| fveq1i 6757 | Equality inference for fun... |
| fveq1d 6758 | Equality deduction for fun... |
| fveq2i 6759 | Equality inference for fun... |
| fveq2d 6760 | Equality deduction for fun... |
| 2fveq3 6761 | Equality theorem for neste... |
| fveq12i 6762 | Equality deduction for fun... |
| fveq12d 6763 | Equality deduction for fun... |
| fveqeq2d 6764 | Equality deduction for fun... |
| fveqeq2 6765 | Equality deduction for fun... |
| nffv 6766 | Bound-variable hypothesis ... |
| nffvmpt1 6767 | Bound-variable hypothesis ... |
| nffvd 6768 | Deduction version of bound... |
| fvex 6769 | The value of a class exist... |
| fvexi 6770 | The value of a class exist... |
| fvexd 6771 | The value of a class exist... |
| fvif 6772 | Move a conditional outside... |
| iffv 6773 | Move a conditional outside... |
| fv3 6774 | Alternate definition of th... |
| fvres 6775 | The value of a restricted ... |
| fvresd 6776 | The value of a restricted ... |
| funssfv 6777 | The value of a member of t... |
| tz6.12-1 6778 | Function value. Theorem 6... |
| tz6.12 6779 | Function value. Theorem 6... |
| tz6.12f 6780 | Function value, using boun... |
| tz6.12c 6781 | Corollary of Theorem 6.12(... |
| tz6.12i 6782 | Corollary of Theorem 6.12(... |
| fvbr0 6783 | Two possibilities for the ... |
| fvrn0 6784 | A function value is a memb... |
| fvssunirn 6785 | The result of a function v... |
| ndmfv 6786 | The value of a class outsi... |
| ndmfvrcl 6787 | Reverse closure law for fu... |
| elfvdm 6788 | If a function value has a ... |
| elfvex 6789 | If a function value has a ... |
| elfvexd 6790 | If a function value has a ... |
| eliman0 6791 | A nonempty function value ... |
| nfvres 6792 | The value of a non-member ... |
| nfunsn 6793 | If the restriction of a cl... |
| fvfundmfvn0 6794 | If the "value of a class" ... |
| 0fv 6795 | Function value of the empt... |
| fv2prc 6796 | A function value of a func... |
| elfv2ex 6797 | If a function value of a f... |
| fveqres 6798 | Equal values imply equal v... |
| csbfv12 6799 | Move class substitution in... |
| csbfv2g 6800 | Move class substitution in... |
| csbfv 6801 | Substitution for a functio... |
| funbrfv 6802 | The second argument of a b... |
| funopfv 6803 | The second element in an o... |
| fnbrfvb 6804 | Equivalence of function va... |
| fnopfvb 6805 | Equivalence of function va... |
| funbrfvb 6806 | Equivalence of function va... |
| funopfvb 6807 | Equivalence of function va... |
| fnbrfvb2 6808 | Version of ~ fnbrfvb for f... |
| funbrfv2b 6809 | Function value in terms of... |
| dffn5 6810 | Representation of a functi... |
| fnrnfv 6811 | The range of a function ex... |
| fvelrnb 6812 | A member of a function's r... |
| foelrni 6813 | A member of a surjective f... |
| dfimafn 6814 | Alternate definition of th... |
| dfimafn2 6815 | Alternate definition of th... |
| funimass4 6816 | Membership relation for th... |
| fvelima 6817 | Function value in an image... |
| fvelimad 6818 | Function value in an image... |
| feqmptd 6819 | Deduction form of ~ dffn5 ... |
| feqresmpt 6820 | Express a restricted funct... |
| feqmptdf 6821 | Deduction form of ~ dffn5f... |
| dffn5f 6822 | Representation of a functi... |
| fvelimab 6823 | Function value in an image... |
| fvelimabd 6824 | Deduction form of ~ fvelim... |
| unima 6825 | Image of a union. (Contri... |
| fvi 6826 | The value of the identity ... |
| fviss 6827 | The value of the identity ... |
| fniinfv 6828 | The indexed intersection o... |
| fnsnfv 6829 | Singleton of function valu... |
| fnsnfvOLD 6830 | Obsolete version of ~ fnsn... |
| opabiotafun 6831 | Define a function whose va... |
| opabiotadm 6832 | Define a function whose va... |
| opabiota 6833 | Define a function whose va... |
| fnimapr 6834 | The image of a pair under ... |
| ssimaex 6835 | The existence of a subimag... |
| ssimaexg 6836 | The existence of a subimag... |
| funfv 6837 | A simplified expression fo... |
| funfv2 6838 | The value of a function. ... |
| funfv2f 6839 | The value of a function. ... |
| fvun 6840 | Value of the union of two ... |
| fvun1 6841 | The value of a union when ... |
| fvun2 6842 | The value of a union when ... |
| fvun1d 6843 | The value of a union when ... |
| fvun2d 6844 | The value of a union when ... |
| dffv2 6845 | Alternate definition of fu... |
| dmfco 6846 | Domains of a function comp... |
| fvco2 6847 | Value of a function compos... |
| fvco 6848 | Value of a function compos... |
| fvco3 6849 | Value of a function compos... |
| fvco3d 6850 | Value of a function compos... |
| fvco4i 6851 | Conditions for a compositi... |
| fvopab3g 6852 | Value of a function given ... |
| fvopab3ig 6853 | Value of a function given ... |
| brfvopabrbr 6854 | The binary relation of a f... |
| fvmptg 6855 | Value of a function given ... |
| fvmpti 6856 | Value of a function given ... |
| fvmpt 6857 | Value of a function given ... |
| fvmpt2f 6858 | Value of a function given ... |
| fvtresfn 6859 | Functionality of a tuple-r... |
| fvmpts 6860 | Value of a function given ... |
| fvmpt3 6861 | Value of a function given ... |
| fvmpt3i 6862 | Value of a function given ... |
| fvmptdf 6863 | Deduction version of ~ fvm... |
| fvmptd 6864 | Deduction version of ~ fvm... |
| fvmptd2 6865 | Deduction version of ~ fvm... |
| mptrcl 6866 | Reverse closure for a mapp... |
| fvmpt2i 6867 | Value of a function given ... |
| fvmpt2 6868 | Value of a function given ... |
| fvmptss 6869 | If all the values of the m... |
| fvmpt2d 6870 | Deduction version of ~ fvm... |
| fvmptex 6871 | Express a function ` F ` w... |
| fvmptd3f 6872 | Alternate deduction versio... |
| fvmptd2f 6873 | Alternate deduction versio... |
| fvmptdv 6874 | Alternate deduction versio... |
| fvmptdv2 6875 | Alternate deduction versio... |
| mpteqb 6876 | Bidirectional equality the... |
| fvmptt 6877 | Closed theorem form of ~ f... |
| fvmptf 6878 | Value of a function given ... |
| fvmptnf 6879 | The value of a function gi... |
| fvmptd3 6880 | Deduction version of ~ fvm... |
| fvmptn 6881 | This somewhat non-intuitiv... |
| fvmptss2 6882 | A mapping always evaluates... |
| elfvmptrab1w 6883 | Implications for the value... |
| elfvmptrab1 6884 | Implications for the value... |
| elfvmptrab 6885 | Implications for the value... |
| fvopab4ndm 6886 | Value of a function given ... |
| fvmptndm 6887 | Value of a function given ... |
| fvmptrabfv 6888 | Value of a function mappin... |
| fvopab5 6889 | The value of a function th... |
| fvopab6 6890 | Value of a function given ... |
| eqfnfv 6891 | Equality of functions is d... |
| eqfnfv2 6892 | Equality of functions is d... |
| eqfnfv3 6893 | Derive equality of functio... |
| eqfnfvd 6894 | Deduction for equality of ... |
| eqfnfv2f 6895 | Equality of functions is d... |
| eqfunfv 6896 | Equality of functions is d... |
| fvreseq0 6897 | Equality of restricted fun... |
| fvreseq1 6898 | Equality of a function res... |
| fvreseq 6899 | Equality of restricted fun... |
| fnmptfvd 6900 | A function with a given do... |
| fndmdif 6901 | Two ways to express the lo... |
| fndmdifcom 6902 | The difference set between... |
| fndmdifeq0 6903 | The difference set of two ... |
| fndmin 6904 | Two ways to express the lo... |
| fneqeql 6905 | Two functions are equal if... |
| fneqeql2 6906 | Two functions are equal if... |
| fnreseql 6907 | Two functions are equal on... |
| chfnrn 6908 | The range of a choice func... |
| funfvop 6909 | Ordered pair with function... |
| funfvbrb 6910 | Two ways to say that ` A `... |
| fvimacnvi 6911 | A member of a preimage is ... |
| fvimacnv 6912 | The argument of a function... |
| funimass3 6913 | A kind of contraposition l... |
| funimass5 6914 | A subclass of a preimage i... |
| funconstss 6915 | Two ways of specifying tha... |
| fvimacnvALT 6916 | Alternate proof of ~ fvima... |
| elpreima 6917 | Membership in the preimage... |
| elpreimad 6918 | Membership in the preimage... |
| fniniseg 6919 | Membership in the preimage... |
| fncnvima2 6920 | Inverse images under funct... |
| fniniseg2 6921 | Inverse point images under... |
| unpreima 6922 | Preimage of a union. (Con... |
| inpreima 6923 | Preimage of an intersectio... |
| difpreima 6924 | Preimage of a difference. ... |
| respreima 6925 | The preimage of a restrict... |
| cnvimainrn 6926 | The preimage of the inters... |
| sspreima 6927 | The preimage of a subset i... |
| iinpreima 6928 | Preimage of an intersectio... |
| intpreima 6929 | Preimage of an intersectio... |
| fimacnvOLD 6930 | Obsolete version of ~ fima... |
| fimacnvinrn 6931 | Taking the converse image ... |
| fimacnvinrn2 6932 | Taking the converse image ... |
| rescnvimafod 6933 | The restriction of a funct... |
| fvn0ssdmfun 6934 | If a class' function value... |
| fnopfv 6935 | Ordered pair with function... |
| fvelrn 6936 | A function's value belongs... |
| nelrnfvne 6937 | A function value cannot be... |
| fveqdmss 6938 | If the empty set is not co... |
| fveqressseq 6939 | If the empty set is not co... |
| fnfvelrn 6940 | A function's value belongs... |
| ffvelrn 6941 | A function's value belongs... |
| ffvelrni 6942 | A function's value belongs... |
| ffvelrnda 6943 | A function's value belongs... |
| ffvelrnd 6944 | A function's value belongs... |
| rexrn 6945 | Restricted existential qua... |
| ralrn 6946 | Restricted universal quant... |
| elrnrexdm 6947 | For any element in the ran... |
| elrnrexdmb 6948 | For any element in the ran... |
| eldmrexrn 6949 | For any element in the dom... |
| eldmrexrnb 6950 | For any element in the dom... |
| fvcofneq 6951 | The values of two function... |
| ralrnmptw 6952 | A restricted quantifier ov... |
| rexrnmptw 6953 | A restricted quantifier ov... |
| ralrnmpt 6954 | A restricted quantifier ov... |
| rexrnmpt 6955 | A restricted quantifier ov... |
| f0cli 6956 | Unconditional closure of a... |
| dff2 6957 | Alternate definition of a ... |
| dff3 6958 | Alternate definition of a ... |
| dff4 6959 | Alternate definition of a ... |
| dffo3 6960 | An onto mapping expressed ... |
| dffo4 6961 | Alternate definition of an... |
| dffo5 6962 | Alternate definition of an... |
| exfo 6963 | A relation equivalent to t... |
| foelrn 6964 | Property of a surjective f... |
| foco2 6965 | If a composition of two fu... |
| fmpt 6966 | Functionality of the mappi... |
| f1ompt 6967 | Express bijection for a ma... |
| fmpti 6968 | Functionality of the mappi... |
| fvmptelrn 6969 | The value of a function at... |
| fmptd 6970 | Domain and codomain of the... |
| fmpttd 6971 | Version of ~ fmptd with in... |
| fmpt3d 6972 | Domain and codomain of the... |
| fmptdf 6973 | A version of ~ fmptd using... |
| ffnfv 6974 | A function maps to a class... |
| ffnfvf 6975 | A function maps to a class... |
| fnfvrnss 6976 | An upper bound for range d... |
| frnssb 6977 | A function is a function i... |
| rnmptss 6978 | The range of an operation ... |
| fmpt2d 6979 | Domain and codomain of the... |
| ffvresb 6980 | A necessary and sufficient... |
| f1oresrab 6981 | Build a bijection between ... |
| f1ossf1o 6982 | Restricting a bijection, w... |
| fmptco 6983 | Composition of two functio... |
| fmptcof 6984 | Version of ~ fmptco where ... |
| fmptcos 6985 | Composition of two functio... |
| cofmpt 6986 | Express composition of a m... |
| fcompt 6987 | Express composition of two... |
| fcoconst 6988 | Composition with a constan... |
| fsn 6989 | A function maps a singleto... |
| fsn2 6990 | A function that maps a sin... |
| fsng 6991 | A function maps a singleto... |
| fsn2g 6992 | A function that maps a sin... |
| xpsng 6993 | The Cartesian product of t... |
| xpprsng 6994 | The Cartesian product of a... |
| xpsn 6995 | The Cartesian product of t... |
| f1o2sn 6996 | A singleton consisting in ... |
| residpr 6997 | Restriction of the identit... |
| dfmpt 6998 | Alternate definition for t... |
| fnasrn 6999 | A function expressed as th... |
| idref 7000 | Two ways to state that a r... |
| funiun 7001 | A function is a union of s... |
| funopsn 7002 | If a function is an ordere... |
| funop 7003 | An ordered pair is a funct... |
| funopdmsn 7004 | The domain of a function w... |
| funsndifnop 7005 | A singleton of an ordered ... |
| funsneqopb 7006 | A singleton of an ordered ... |
| ressnop0 7007 | If ` A ` is not in ` C ` ,... |
| fpr 7008 | A function with a domain o... |
| fprg 7009 | A function with a domain o... |
| ftpg 7010 | A function with a domain o... |
| ftp 7011 | A function with a domain o... |
| fnressn 7012 | A function restricted to a... |
| funressn 7013 | A function restricted to a... |
| fressnfv 7014 | The value of a function re... |
| fvrnressn 7015 | If the value of a function... |
| fvressn 7016 | The value of a function re... |
| fvn0fvelrn 7017 | If the value of a function... |
| fvconst 7018 | The value of a constant fu... |
| fnsnr 7019 | If a class belongs to a fu... |
| fnsnb 7020 | A function whose domain is... |
| fmptsn 7021 | Express a singleton functi... |
| fmptsng 7022 | Express a singleton functi... |
| fmptsnd 7023 | Express a singleton functi... |
| fmptap 7024 | Append an additional value... |
| fmptapd 7025 | Append an additional value... |
| fmptpr 7026 | Express a pair function in... |
| fvresi 7027 | The value of a restricted ... |
| fninfp 7028 | Express the class of fixed... |
| fnelfp 7029 | Property of a fixed point ... |
| fndifnfp 7030 | Express the class of non-f... |
| fnelnfp 7031 | Property of a non-fixed po... |
| fnnfpeq0 7032 | A function is the identity... |
| fvunsn 7033 | Remove an ordered pair not... |
| fvsng 7034 | The value of a singleton o... |
| fvsn 7035 | The value of a singleton o... |
| fvsnun1 7036 | The value of a function wi... |
| fvsnun2 7037 | The value of a function wi... |
| fnsnsplit 7038 | Split a function into a si... |
| fsnunf 7039 | Adjoining a point to a fun... |
| fsnunf2 7040 | Adjoining a point to a pun... |
| fsnunfv 7041 | Recover the added point fr... |
| fsnunres 7042 | Recover the original funct... |
| funresdfunsn 7043 | Restricting a function to ... |
| fvpr1g 7044 | The value of a function wi... |
| fvpr2g 7045 | The value of a function wi... |
| fvpr2gOLD 7046 | Obsolete version of ~ fvpr... |
| fvpr1 7047 | The value of a function wi... |
| fvpr1OLD 7048 | Obsolete version of ~ fvpr... |
| fvpr2 7049 | The value of a function wi... |
| fvpr2OLD 7050 | Obsolete version of ~ fvpr... |
| fprb 7051 | A condition for functionho... |
| fvtp1 7052 | The first value of a funct... |
| fvtp2 7053 | The second value of a func... |
| fvtp3 7054 | The third value of a funct... |
| fvtp1g 7055 | The value of a function wi... |
| fvtp2g 7056 | The value of a function wi... |
| fvtp3g 7057 | The value of a function wi... |
| tpres 7058 | An unordered triple of ord... |
| fvconst2g 7059 | The value of a constant fu... |
| fconst2g 7060 | A constant function expres... |
| fvconst2 7061 | The value of a constant fu... |
| fconst2 7062 | A constant function expres... |
| fconst5 7063 | Two ways to express that a... |
| rnmptc 7064 | Range of a constant functi... |
| rnmptcOLD 7065 | Obsolete version of ~ rnmp... |
| fnprb 7066 | A function whose domain ha... |
| fntpb 7067 | A function whose domain ha... |
| fnpr2g 7068 | A function whose domain ha... |
| fpr2g 7069 | A function that maps a pai... |
| fconstfv 7070 | A constant function expres... |
| fconst3 7071 | Two ways to express a cons... |
| fconst4 7072 | Two ways to express a cons... |
| resfunexg 7073 | The restriction of a funct... |
| resiexd 7074 | The restriction of the ide... |
| fnex 7075 | If the domain of a functio... |
| fnexd 7076 | If the domain of a functio... |
| funex 7077 | If the domain of a functio... |
| opabex 7078 | Existence of a function ex... |
| mptexg 7079 | If the domain of a functio... |
| mptexgf 7080 | If the domain of a functio... |
| mptex 7081 | If the domain of a functio... |
| mptexd 7082 | If the domain of a functio... |
| mptrabex 7083 | If the domain of a functio... |
| fex 7084 | If the domain of a mapping... |
| fexd 7085 | If the domain of a mapping... |
| mptfvmpt 7086 | A function in maps-to nota... |
| eufnfv 7087 | A function is uniquely det... |
| funfvima 7088 | A function's value in a pr... |
| funfvima2 7089 | A function's value in an i... |
| funfvima2d 7090 | A function's value in a pr... |
| fnfvima 7091 | The function value of an o... |
| fnfvimad 7092 | A function's value belongs... |
| resfvresima 7093 | The value of the function ... |
| funfvima3 7094 | A class including a functi... |
| rexima 7095 | Existential quantification... |
| ralima 7096 | Universal quantification u... |
| fvclss 7097 | Upper bound for the class ... |
| elabrex 7098 | Elementhood in an image se... |
| abrexco 7099 | Composition of two image m... |
| imaiun 7100 | The image of an indexed un... |
| imauni 7101 | The image of a union is th... |
| fniunfv 7102 | The indexed union of a fun... |
| funiunfv 7103 | The indexed union of a fun... |
| funiunfvf 7104 | The indexed union of a fun... |
| eluniima 7105 | Membership in the union of... |
| elunirn 7106 | Membership in the union of... |
| elunirnALT 7107 | Alternate proof of ~ eluni... |
| fnunirn 7108 | Membership in a union of s... |
| dff13 7109 | A one-to-one function in t... |
| dff13f 7110 | A one-to-one function in t... |
| f1veqaeq 7111 | If the values of a one-to-... |
| f1cofveqaeq 7112 | If the values of a composi... |
| f1cofveqaeqALT 7113 | Alternate proof of ~ f1cof... |
| 2f1fvneq 7114 | If two one-to-one function... |
| f1mpt 7115 | Express injection for a ma... |
| f1fveq 7116 | Equality of function value... |
| f1elima 7117 | Membership in the image of... |
| f1imass 7118 | Taking images under a one-... |
| f1imaeq 7119 | Taking images under a one-... |
| f1imapss 7120 | Taking images under a one-... |
| fpropnf1 7121 | A function, given by an un... |
| f1dom3fv3dif 7122 | The function values for a ... |
| f1dom3el3dif 7123 | The range of a 1-1 functio... |
| dff14a 7124 | A one-to-one function in t... |
| dff14b 7125 | A one-to-one function in t... |
| f12dfv 7126 | A one-to-one function with... |
| f13dfv 7127 | A one-to-one function with... |
| dff1o6 7128 | A one-to-one onto function... |
| f1ocnvfv1 7129 | The converse value of the ... |
| f1ocnvfv2 7130 | The value of the converse ... |
| f1ocnvfv 7131 | Relationship between the v... |
| f1ocnvfvb 7132 | Relationship between the v... |
| nvof1o 7133 | An involution is a bijecti... |
| nvocnv 7134 | The converse of an involut... |
| fsnex 7135 | Relate a function with a s... |
| f1prex 7136 | Relate a one-to-one functi... |
| f1ocnvdm 7137 | The value of the converse ... |
| f1ocnvfvrneq 7138 | If the values of a one-to-... |
| fcof1 7139 | An application is injectiv... |
| fcofo 7140 | An application is surjecti... |
| cbvfo 7141 | Change bound variable betw... |
| cbvexfo 7142 | Change bound variable betw... |
| cocan1 7143 | An injection is left-cance... |
| cocan2 7144 | A surjection is right-canc... |
| fcof1oinvd 7145 | Show that a function is th... |
| fcof1od 7146 | A function is bijective if... |
| 2fcoidinvd 7147 | Show that a function is th... |
| fcof1o 7148 | Show that two functions ar... |
| 2fvcoidd 7149 | Show that the composition ... |
| 2fvidf1od 7150 | A function is bijective if... |
| 2fvidinvd 7151 | Show that two functions ar... |
| foeqcnvco 7152 | Condition for function equ... |
| f1eqcocnv 7153 | Condition for function equ... |
| f1eqcocnvOLD 7154 | Obsolete version of ~ f1eq... |
| fveqf1o 7155 | Given a bijection ` F ` , ... |
| nf1const 7156 | A constant function from a... |
| nf1oconst 7157 | A constant function from a... |
| f1ofvswap 7158 | Swapping two values in a b... |
| fliftrel 7159 | ` F ` , a function lift, i... |
| fliftel 7160 | Elementhood in the relatio... |
| fliftel1 7161 | Elementhood in the relatio... |
| fliftcnv 7162 | Converse of the relation `... |
| fliftfun 7163 | The function ` F ` is the ... |
| fliftfund 7164 | The function ` F ` is the ... |
| fliftfuns 7165 | The function ` F ` is the ... |
| fliftf 7166 | The domain and range of th... |
| fliftval 7167 | The value of the function ... |
| isoeq1 7168 | Equality theorem for isomo... |
| isoeq2 7169 | Equality theorem for isomo... |
| isoeq3 7170 | Equality theorem for isomo... |
| isoeq4 7171 | Equality theorem for isomo... |
| isoeq5 7172 | Equality theorem for isomo... |
| nfiso 7173 | Bound-variable hypothesis ... |
| isof1o 7174 | An isomorphism is a one-to... |
| isof1oidb 7175 | A function is a bijection ... |
| isof1oopb 7176 | A function is a bijection ... |
| isorel 7177 | An isomorphism connects bi... |
| soisores 7178 | Express the condition of i... |
| soisoi 7179 | Infer isomorphism from one... |
| isoid 7180 | Identity law for isomorphi... |
| isocnv 7181 | Converse law for isomorphi... |
| isocnv2 7182 | Converse law for isomorphi... |
| isocnv3 7183 | Complementation law for is... |
| isores2 7184 | An isomorphism from one we... |
| isores1 7185 | An isomorphism from one we... |
| isores3 7186 | Induced isomorphism on a s... |
| isotr 7187 | Composition (transitive) l... |
| isomin 7188 | Isomorphisms preserve mini... |
| isoini 7189 | Isomorphisms preserve init... |
| isoini2 7190 | Isomorphisms are isomorphi... |
| isofrlem 7191 | Lemma for ~ isofr . (Cont... |
| isoselem 7192 | Lemma for ~ isose . (Cont... |
| isofr 7193 | An isomorphism preserves w... |
| isose 7194 | An isomorphism preserves s... |
| isofr2 7195 | A weak form of ~ isofr tha... |
| isopolem 7196 | Lemma for ~ isopo . (Cont... |
| isopo 7197 | An isomorphism preserves t... |
| isosolem 7198 | Lemma for ~ isoso . (Cont... |
| isoso 7199 | An isomorphism preserves t... |
| isowe 7200 | An isomorphism preserves t... |
| isowe2 7201 | A weak form of ~ isowe tha... |
| f1oiso 7202 | Any one-to-one onto functi... |
| f1oiso2 7203 | Any one-to-one onto functi... |
| f1owe 7204 | Well-ordering of isomorphi... |
| weniso 7205 | A set-like well-ordering h... |
| weisoeq 7206 | Thus, there is at most one... |
| weisoeq2 7207 | Thus, there is at most one... |
| knatar 7208 | The Knaster-Tarski theorem... |
| canth 7209 | No set ` A ` is equinumero... |
| ncanth 7210 | Cantor's theorem fails for... |
| riotaeqdv 7213 | Formula-building deduction... |
| riotabidv 7214 | Formula-building deduction... |
| riotaeqbidv 7215 | Equality deduction for res... |
| riotaex 7216 | Restricted iota is a set. ... |
| riotav 7217 | An iota restricted to the ... |
| riotauni 7218 | Restricted iota in terms o... |
| nfriota1 7219 | The abstraction variable i... |
| nfriotadw 7220 | Deduction version of ~ nfr... |
| cbvriotaw 7221 | Change bound variable in a... |
| cbvriotavw 7222 | Change bound variable in a... |
| cbvriotavwOLD 7223 | Obsolete version of ~ cbvr... |
| nfriotad 7224 | Deduction version of ~ nfr... |
| nfriota 7225 | A variable not free in a w... |
| cbvriota 7226 | Change bound variable in a... |
| cbvriotav 7227 | Change bound variable in a... |
| csbriota 7228 | Interchange class substitu... |
| riotacl2 7229 | Membership law for "the un... |
| riotacl 7230 | Closure of restricted iota... |
| riotasbc 7231 | Substitution law for descr... |
| riotabidva 7232 | Equivalent wff's yield equ... |
| riotabiia 7233 | Equivalent wff's yield equ... |
| riota1 7234 | Property of restricted iot... |
| riota1a 7235 | Property of iota. (Contri... |
| riota2df 7236 | A deduction version of ~ r... |
| riota2f 7237 | This theorem shows a condi... |
| riota2 7238 | This theorem shows a condi... |
| riotaeqimp 7239 | If two restricted iota des... |
| riotaprop 7240 | Properties of a restricted... |
| riota5f 7241 | A method for computing res... |
| riota5 7242 | A method for computing res... |
| riotass2 7243 | Restriction of a unique el... |
| riotass 7244 | Restriction of a unique el... |
| moriotass 7245 | Restriction of a unique el... |
| snriota 7246 | A restricted class abstrac... |
| riotaxfrd 7247 | Change the variable ` x ` ... |
| eusvobj2 7248 | Specify the same property ... |
| eusvobj1 7249 | Specify the same object in... |
| f1ofveu 7250 | There is one domain elemen... |
| f1ocnvfv3 7251 | Value of the converse of a... |
| riotaund 7252 | Restricted iota equals the... |
| riotassuni 7253 | The restricted iota class ... |
| riotaclb 7254 | Bidirectional closure of r... |
| oveq 7261 | Equality theorem for opera... |
| oveq1 7262 | Equality theorem for opera... |
| oveq2 7263 | Equality theorem for opera... |
| oveq12 7264 | Equality theorem for opera... |
| oveq1i 7265 | Equality inference for ope... |
| oveq2i 7266 | Equality inference for ope... |
| oveq12i 7267 | Equality inference for ope... |
| oveqi 7268 | Equality inference for ope... |
| oveq123i 7269 | Equality inference for ope... |
| oveq1d 7270 | Equality deduction for ope... |
| oveq2d 7271 | Equality deduction for ope... |
| oveqd 7272 | Equality deduction for ope... |
| oveq12d 7273 | Equality deduction for ope... |
| oveqan12d 7274 | Equality deduction for ope... |
| oveqan12rd 7275 | Equality deduction for ope... |
| oveq123d 7276 | Equality deduction for ope... |
| fvoveq1d 7277 | Equality deduction for nes... |
| fvoveq1 7278 | Equality theorem for neste... |
| ovanraleqv 7279 | Equality theorem for a con... |
| imbrov2fvoveq 7280 | Equality theorem for neste... |
| ovrspc2v 7281 | If an operation value is e... |
| oveqrspc2v 7282 | Restricted specialization ... |
| oveqdr 7283 | Equality of two operations... |
| nfovd 7284 | Deduction version of bound... |
| nfov 7285 | Bound-variable hypothesis ... |
| oprabidw 7286 | The law of concretion. Sp... |
| oprabid 7287 | The law of concretion. Sp... |
| ovex 7288 | The result of an operation... |
| ovexi 7289 | The result of an operation... |
| ovexd 7290 | The result of an operation... |
| ovssunirn 7291 | The result of an operation... |
| 0ov 7292 | Operation value of the emp... |
| ovprc 7293 | The value of an operation ... |
| ovprc1 7294 | The value of an operation ... |
| ovprc2 7295 | The value of an operation ... |
| ovrcl 7296 | Reverse closure for an ope... |
| csbov123 7297 | Move class substitution in... |
| csbov 7298 | Move class substitution in... |
| csbov12g 7299 | Move class substitution in... |
| csbov1g 7300 | Move class substitution in... |
| csbov2g 7301 | Move class substitution in... |
| rspceov 7302 | A frequently used special ... |
| elovimad 7303 | Elementhood of the image s... |
| fnbrovb 7304 | Value of a binary operatio... |
| fnotovb 7305 | Equivalence of operation v... |
| opabbrex 7306 | A collection of ordered pa... |
| opabresex2d 7307 | Restrictions of a collecti... |
| fvmptopab 7308 | The function value of a ma... |
| f1opr 7309 | Condition for an operation... |
| brfvopab 7310 | The classes involved in a ... |
| dfoprab2 7311 | Class abstraction for oper... |
| reloprab 7312 | An operation class abstrac... |
| oprabv 7313 | If a pair and a class are ... |
| nfoprab1 7314 | The abstraction variables ... |
| nfoprab2 7315 | The abstraction variables ... |
| nfoprab3 7316 | The abstraction variables ... |
| nfoprab 7317 | Bound-variable hypothesis ... |
| oprabbid 7318 | Equivalent wff's yield equ... |
| oprabbidv 7319 | Equivalent wff's yield equ... |
| oprabbii 7320 | Equivalent wff's yield equ... |
| ssoprab2 7321 | Equivalence of ordered pai... |
| ssoprab2b 7322 | Equivalence of ordered pai... |
| eqoprab2bw 7323 | Equivalence of ordered pai... |
| eqoprab2b 7324 | Equivalence of ordered pai... |
| mpoeq123 7325 | An equality theorem for th... |
| mpoeq12 7326 | An equality theorem for th... |
| mpoeq123dva 7327 | An equality deduction for ... |
| mpoeq123dv 7328 | An equality deduction for ... |
| mpoeq123i 7329 | An equality inference for ... |
| mpoeq3dva 7330 | Slightly more general equa... |
| mpoeq3ia 7331 | An equality inference for ... |
| mpoeq3dv 7332 | An equality deduction for ... |
| nfmpo1 7333 | Bound-variable hypothesis ... |
| nfmpo2 7334 | Bound-variable hypothesis ... |
| nfmpo 7335 | Bound-variable hypothesis ... |
| 0mpo0 7336 | A mapping operation with e... |
| mpo0v 7337 | A mapping operation with e... |
| mpo0 7338 | A mapping operation with e... |
| oprab4 7339 | Two ways to state the doma... |
| cbvoprab1 7340 | Rule used to change first ... |
| cbvoprab2 7341 | Change the second bound va... |
| cbvoprab12 7342 | Rule used to change first ... |
| cbvoprab12v 7343 | Rule used to change first ... |
| cbvoprab3 7344 | Rule used to change the th... |
| cbvoprab3v 7345 | Rule used to change the th... |
| cbvmpox 7346 | Rule to change the bound v... |
| cbvmpo 7347 | Rule to change the bound v... |
| cbvmpov 7348 | Rule to change the bound v... |
| elimdelov 7349 | Eliminate a hypothesis whi... |
| ovif 7350 | Move a conditional outside... |
| ovif2 7351 | Move a conditional outside... |
| ovif12 7352 | Move a conditional outside... |
| ifov 7353 | Move a conditional outside... |
| dmoprab 7354 | The domain of an operation... |
| dmoprabss 7355 | The domain of an operation... |
| rnoprab 7356 | The range of an operation ... |
| rnoprab2 7357 | The range of a restricted ... |
| reldmoprab 7358 | The domain of an operation... |
| oprabss 7359 | Structure of an operation ... |
| eloprabga 7360 | The law of concretion for ... |
| eloprabgaOLD 7361 | Obsolete version of ~ elop... |
| eloprabg 7362 | The law of concretion for ... |
| ssoprab2i 7363 | Inference of operation cla... |
| mpov 7364 | Operation with universal d... |
| mpomptx 7365 | Express a two-argument fun... |
| mpompt 7366 | Express a two-argument fun... |
| mpodifsnif 7367 | A mapping with two argumen... |
| mposnif 7368 | A mapping with two argumen... |
| fconstmpo 7369 | Representation of a consta... |
| resoprab 7370 | Restriction of an operatio... |
| resoprab2 7371 | Restriction of an operator... |
| resmpo 7372 | Restriction of the mapping... |
| funoprabg 7373 | "At most one" is a suffici... |
| funoprab 7374 | "At most one" is a suffici... |
| fnoprabg 7375 | Functionality and domain o... |
| mpofun 7376 | The maps-to notation for a... |
| mpofunOLD 7377 | Obsolete version of ~ mpof... |
| fnoprab 7378 | Functionality and domain o... |
| ffnov 7379 | An operation maps to a cla... |
| fovcl 7380 | Closure law for an operati... |
| eqfnov 7381 | Equality of two operations... |
| eqfnov2 7382 | Two operators with the sam... |
| fnov 7383 | Representation of a functi... |
| mpo2eqb 7384 | Bidirectional equality the... |
| rnmpo 7385 | The range of an operation ... |
| reldmmpo 7386 | The domain of an operation... |
| elrnmpog 7387 | Membership in the range of... |
| elrnmpo 7388 | Membership in the range of... |
| elrnmpores 7389 | Membership in the range of... |
| ralrnmpo 7390 | A restricted quantifier ov... |
| rexrnmpo 7391 | A restricted quantifier ov... |
| ovid 7392 | The value of an operation ... |
| ovidig 7393 | The value of an operation ... |
| ovidi 7394 | The value of an operation ... |
| ov 7395 | The value of an operation ... |
| ovigg 7396 | The value of an operation ... |
| ovig 7397 | The value of an operation ... |
| ovmpt4g 7398 | Value of a function given ... |
| ovmpos 7399 | Value of a function given ... |
| ov2gf 7400 | The value of an operation ... |
| ovmpodxf 7401 | Value of an operation give... |
| ovmpodx 7402 | Value of an operation give... |
| ovmpod 7403 | Value of an operation give... |
| ovmpox 7404 | The value of an operation ... |
| ovmpoga 7405 | Value of an operation give... |
| ovmpoa 7406 | Value of an operation give... |
| ovmpodf 7407 | Alternate deduction versio... |
| ovmpodv 7408 | Alternate deduction versio... |
| ovmpodv2 7409 | Alternate deduction versio... |
| ovmpog 7410 | Value of an operation give... |
| ovmpo 7411 | Value of an operation give... |
| fvmpopr2d 7412 | Value of an operation give... |
| ov3 7413 | The value of an operation ... |
| ov6g 7414 | The value of an operation ... |
| ovg 7415 | The value of an operation ... |
| ovres 7416 | The value of a restricted ... |
| ovresd 7417 | Lemma for converting metri... |
| oprres 7418 | The restriction of an oper... |
| oprssov 7419 | The value of a member of t... |
| fovrn 7420 | An operation's value belon... |
| fovrnda 7421 | An operation's value belon... |
| fovrnd 7422 | An operation's value belon... |
| fnrnov 7423 | The range of an operation ... |
| foov 7424 | An onto mapping of an oper... |
| fnovrn 7425 | An operation's value belon... |
| ovelrn 7426 | A member of an operation's... |
| funimassov 7427 | Membership relation for th... |
| ovelimab 7428 | Operation value in an imag... |
| ovima0 7429 | An operation value is a me... |
| ovconst2 7430 | The value of a constant op... |
| oprssdm 7431 | Domain of closure of an op... |
| nssdmovg 7432 | The value of an operation ... |
| ndmovg 7433 | The value of an operation ... |
| ndmov 7434 | The value of an operation ... |
| ndmovcl 7435 | The closure of an operatio... |
| ndmovrcl 7436 | Reverse closure law, when ... |
| ndmovcom 7437 | Any operation is commutati... |
| ndmovass 7438 | Any operation is associati... |
| ndmovdistr 7439 | Any operation is distribut... |
| ndmovord 7440 | Elimination of redundant a... |
| ndmovordi 7441 | Elimination of redundant a... |
| caovclg 7442 | Convert an operation closu... |
| caovcld 7443 | Convert an operation closu... |
| caovcl 7444 | Convert an operation closu... |
| caovcomg 7445 | Convert an operation commu... |
| caovcomd 7446 | Convert an operation commu... |
| caovcom 7447 | Convert an operation commu... |
| caovassg 7448 | Convert an operation assoc... |
| caovassd 7449 | Convert an operation assoc... |
| caovass 7450 | Convert an operation assoc... |
| caovcang 7451 | Convert an operation cance... |
| caovcand 7452 | Convert an operation cance... |
| caovcanrd 7453 | Commute the arguments of a... |
| caovcan 7454 | Convert an operation cance... |
| caovordig 7455 | Convert an operation order... |
| caovordid 7456 | Convert an operation order... |
| caovordg 7457 | Convert an operation order... |
| caovordd 7458 | Convert an operation order... |
| caovord2d 7459 | Operation ordering law wit... |
| caovord3d 7460 | Ordering law. (Contribute... |
| caovord 7461 | Convert an operation order... |
| caovord2 7462 | Operation ordering law wit... |
| caovord3 7463 | Ordering law. (Contribute... |
| caovdig 7464 | Convert an operation distr... |
| caovdid 7465 | Convert an operation distr... |
| caovdir2d 7466 | Convert an operation distr... |
| caovdirg 7467 | Convert an operation rever... |
| caovdird 7468 | Convert an operation distr... |
| caovdi 7469 | Convert an operation distr... |
| caov32d 7470 | Rearrange arguments in a c... |
| caov12d 7471 | Rearrange arguments in a c... |
| caov31d 7472 | Rearrange arguments in a c... |
| caov13d 7473 | Rearrange arguments in a c... |
| caov4d 7474 | Rearrange arguments in a c... |
| caov411d 7475 | Rearrange arguments in a c... |
| caov42d 7476 | Rearrange arguments in a c... |
| caov32 7477 | Rearrange arguments in a c... |
| caov12 7478 | Rearrange arguments in a c... |
| caov31 7479 | Rearrange arguments in a c... |
| caov13 7480 | Rearrange arguments in a c... |
| caov4 7481 | Rearrange arguments in a c... |
| caov411 7482 | Rearrange arguments in a c... |
| caov42 7483 | Rearrange arguments in a c... |
| caovdir 7484 | Reverse distributive law. ... |
| caovdilem 7485 | Lemma used by real number ... |
| caovlem2 7486 | Lemma used in real number ... |
| caovmo 7487 | Uniqueness of inverse elem... |
| mpondm0 7488 | The value of an operation ... |
| elmpocl 7489 | If a two-parameter class i... |
| elmpocl1 7490 | If a two-parameter class i... |
| elmpocl2 7491 | If a two-parameter class i... |
| elovmpo 7492 | Utility lemma for two-para... |
| elovmporab 7493 | Implications for the value... |
| elovmporab1w 7494 | Implications for the value... |
| elovmporab1 7495 | Implications for the value... |
| 2mpo0 7496 | If the operation value of ... |
| relmptopab 7497 | Any function to sets of or... |
| f1ocnvd 7498 | Describe an implicit one-t... |
| f1od 7499 | Describe an implicit one-t... |
| f1ocnv2d 7500 | Describe an implicit one-t... |
| f1o2d 7501 | Describe an implicit one-t... |
| f1opw2 7502 | A one-to-one mapping induc... |
| f1opw 7503 | A one-to-one mapping induc... |
| elovmpt3imp 7504 | If the value of a function... |
| ovmpt3rab1 7505 | The value of an operation ... |
| ovmpt3rabdm 7506 | If the value of a function... |
| elovmpt3rab1 7507 | Implications for the value... |
| elovmpt3rab 7508 | Implications for the value... |
| ofeqd 7513 | Equality theorem for funct... |
| ofeq 7514 | Equality theorem for funct... |
| ofreq 7515 | Equality theorem for funct... |
| ofexg 7516 | A function operation restr... |
| nfof 7517 | Hypothesis builder for fun... |
| nfofr 7518 | Hypothesis builder for fun... |
| ofrfvalg 7519 | Value of a relation applie... |
| offval 7520 | Value of an operation appl... |
| ofrfval 7521 | Value of a relation applie... |
| ofval 7522 | Evaluate a function operat... |
| ofrval 7523 | Exhibit a function relatio... |
| offn 7524 | The function operation pro... |
| offun 7525 | The function operation pro... |
| offval2f 7526 | The function operation exp... |
| ofmresval 7527 | Value of a restriction of ... |
| fnfvof 7528 | Function value of a pointw... |
| off 7529 | The function operation pro... |
| ofres 7530 | Restrict the operands of a... |
| offval2 7531 | The function operation exp... |
| ofrfval2 7532 | The function relation acti... |
| ofmpteq 7533 | Value of a pointwise opera... |
| ofco 7534 | The composition of a funct... |
| offveq 7535 | Convert an identity of the... |
| offveqb 7536 | Equivalent expressions for... |
| ofc1 7537 | Left operation by a consta... |
| ofc2 7538 | Right operation by a const... |
| ofc12 7539 | Function operation on two ... |
| caofref 7540 | Transfer a reflexive law t... |
| caofinvl 7541 | Transfer a left inverse la... |
| caofid0l 7542 | Transfer a left identity l... |
| caofid0r 7543 | Transfer a right identity ... |
| caofid1 7544 | Transfer a right absorptio... |
| caofid2 7545 | Transfer a right absorptio... |
| caofcom 7546 | Transfer a commutative law... |
| caofrss 7547 | Transfer a relation subset... |
| caofass 7548 | Transfer an associative la... |
| caoftrn 7549 | Transfer a transitivity la... |
| caofdi 7550 | Transfer a distributive la... |
| caofdir 7551 | Transfer a reverse distrib... |
| caonncan 7552 | Transfer ~ nncan -shaped l... |
| relrpss 7555 | The proper subset relation... |
| brrpssg 7556 | The proper subset relation... |
| brrpss 7557 | The proper subset relation... |
| porpss 7558 | Every class is partially o... |
| sorpss 7559 | Express strict ordering un... |
| sorpssi 7560 | Property of a chain of set... |
| sorpssun 7561 | A chain of sets is closed ... |
| sorpssin 7562 | A chain of sets is closed ... |
| sorpssuni 7563 | In a chain of sets, a maxi... |
| sorpssint 7564 | In a chain of sets, a mini... |
| sorpsscmpl 7565 | The componentwise compleme... |
| zfun 7567 | Axiom of Union expressed w... |
| axun2 7568 | A variant of the Axiom of ... |
| uniex2 7569 | The Axiom of Union using t... |
| vuniex 7570 | The union of a setvar is a... |
| uniexg 7571 | The ZF Axiom of Union in c... |
| uniex 7572 | The Axiom of Union in clas... |
| uniexd 7573 | Deduction version of the Z... |
| unex 7574 | The union of two sets is a... |
| tpex 7575 | An unordered triple of cla... |
| unexb 7576 | Existence of union is equi... |
| unexg 7577 | A union of two sets is a s... |
| xpexg 7578 | The Cartesian product of t... |
| xpexd 7579 | The Cartesian product of t... |
| 3xpexg 7580 | The Cartesian product of t... |
| xpex 7581 | The Cartesian product of t... |
| unexd 7582 | The union of two sets is a... |
| sqxpexg 7583 | The Cartesian square of a ... |
| abnexg 7584 | Sufficient condition for a... |
| abnex 7585 | Sufficient condition for a... |
| snnex 7586 | The class of all singleton... |
| pwnex 7587 | The class of all power set... |
| difex2 7588 | If the subtrahend of a cla... |
| difsnexi 7589 | If the difference of a cla... |
| uniuni 7590 | Expression for double unio... |
| uniexr 7591 | Converse of the Axiom of U... |
| uniexb 7592 | The Axiom of Union and its... |
| pwexr 7593 | Converse of the Axiom of P... |
| pwexb 7594 | The Axiom of Power Sets an... |
| elpwpwel 7595 | A class belongs to a doubl... |
| eldifpw 7596 | Membership in a power clas... |
| elpwun 7597 | Membership in the power cl... |
| pwuncl 7598 | Power classes are closed u... |
| iunpw 7599 | An indexed union of a powe... |
| fr3nr 7600 | A well-founded relation ha... |
| epne3 7601 | A well-founded class conta... |
| dfwe2 7602 | Alternate definition of we... |
| epweon 7603 | The membership relation we... |
| ordon 7604 | The class of all ordinal n... |
| onprc 7605 | No set contains all ordina... |
| ssorduni 7606 | The union of a class of or... |
| ssonuni 7607 | The union of a set of ordi... |
| ssonunii 7608 | The union of a set of ordi... |
| ordeleqon 7609 | A way to express the ordin... |
| ordsson 7610 | Any ordinal class is a sub... |
| onss 7611 | An ordinal number is a sub... |
| predon 7612 | The predecessor of an ordi... |
| predonOLD 7613 | Obsolete version of ~ pred... |
| ssonprc 7614 | Two ways of saying a class... |
| onuni 7615 | The union of an ordinal nu... |
| orduni 7616 | The union of an ordinal cl... |
| onint 7617 | The intersection (infimum)... |
| onint0 7618 | The intersection of a clas... |
| onssmin 7619 | A nonempty class of ordina... |
| onminesb 7620 | If a property is true for ... |
| onminsb 7621 | If a property is true for ... |
| oninton 7622 | The intersection of a none... |
| onintrab 7623 | The intersection of a clas... |
| onintrab2 7624 | An existence condition equ... |
| onnmin 7625 | No member of a set of ordi... |
| onnminsb 7626 | An ordinal number smaller ... |
| oneqmin 7627 | A way to show that an ordi... |
| uniordint 7628 | The union of a set of ordi... |
| onminex 7629 | If a wff is true for an or... |
| sucon 7630 | The class of all ordinal n... |
| sucexb 7631 | A successor exists iff its... |
| sucexg 7632 | The successor of a set is ... |
| sucex 7633 | The successor of a set is ... |
| onmindif2 7634 | The minimum of a class of ... |
| suceloni 7635 | The successor of an ordina... |
| ordsuc 7636 | The successor of an ordina... |
| ordpwsuc 7637 | The collection of ordinals... |
| onpwsuc 7638 | The collection of ordinal ... |
| sucelon 7639 | The successor of an ordina... |
| ordsucss 7640 | The successor of an elemen... |
| onpsssuc 7641 | An ordinal number is a pro... |
| ordelsuc 7642 | A set belongs to an ordina... |
| onsucmin 7643 | The successor of an ordina... |
| ordsucelsuc 7644 | Membership is inherited by... |
| ordsucsssuc 7645 | The subclass relationship ... |
| ordsucuniel 7646 | Given an element ` A ` of ... |
| ordsucun 7647 | The successor of the maxim... |
| ordunpr 7648 | The maximum of two ordinal... |
| ordunel 7649 | The maximum of two ordinal... |
| onsucuni 7650 | A class of ordinal numbers... |
| ordsucuni 7651 | An ordinal class is a subc... |
| orduniorsuc 7652 | An ordinal class is either... |
| unon 7653 | The class of all ordinal n... |
| ordunisuc 7654 | An ordinal class is equal ... |
| orduniss2 7655 | The union of the ordinal s... |
| onsucuni2 7656 | A successor ordinal is the... |
| 0elsuc 7657 | The successor of an ordina... |
| limon 7658 | The class of ordinal numbe... |
| onssi 7659 | An ordinal number is a sub... |
| onsuci 7660 | The successor of an ordina... |
| onuniorsuci 7661 | An ordinal number is eithe... |
| onuninsuci 7662 | A limit ordinal is not a s... |
| onsucssi 7663 | A set belongs to an ordina... |
| nlimsucg 7664 | A successor is not a limit... |
| orduninsuc 7665 | An ordinal equal to its un... |
| ordunisuc2 7666 | An ordinal equal to its un... |
| ordzsl 7667 | An ordinal is zero, a succ... |
| onzsl 7668 | An ordinal number is zero,... |
| dflim3 7669 | An alternate definition of... |
| dflim4 7670 | An alternate definition of... |
| limsuc 7671 | The successor of a member ... |
| limsssuc 7672 | A class includes a limit o... |
| nlimon 7673 | Two ways to express the cl... |
| limuni3 7674 | The union of a nonempty cl... |
| tfi 7675 | The Principle of Transfini... |
| tfis 7676 | Transfinite Induction Sche... |
| tfis2f 7677 | Transfinite Induction Sche... |
| tfis2 7678 | Transfinite Induction Sche... |
| tfis3 7679 | Transfinite Induction Sche... |
| tfisi 7680 | A transfinite induction sc... |
| tfinds 7681 | Principle of Transfinite I... |
| tfindsg 7682 | Transfinite Induction (inf... |
| tfindsg2 7683 | Transfinite Induction (inf... |
| tfindes 7684 | Transfinite Induction with... |
| tfinds2 7685 | Transfinite Induction (inf... |
| tfinds3 7686 | Principle of Transfinite I... |
| dfom2 7689 | An alternate definition of... |
| elom 7690 | Membership in omega. The ... |
| omsson 7691 | Omega is a subset of ` On ... |
| limomss 7692 | The class of natural numbe... |
| nnon 7693 | A natural number is an ord... |
| nnoni 7694 | A natural number is an ord... |
| nnord 7695 | A natural number is ordina... |
| trom 7696 | The class of finite ordina... |
| ordom 7697 | The class of finite ordina... |
| elnn 7698 | A member of a natural numb... |
| omon 7699 | The class of natural numbe... |
| omelon2 7700 | Omega is an ordinal number... |
| nnlim 7701 | A natural number is not a ... |
| omssnlim 7702 | The class of natural numbe... |
| limom 7703 | Omega is a limit ordinal. ... |
| peano2b 7704 | A class belongs to omega i... |
| nnsuc 7705 | A nonzero natural number i... |
| omsucne 7706 | A natural number is not th... |
| ssnlim 7707 | An ordinal subclass of non... |
| omsinds 7708 | Strong (or "total") induct... |
| omsindsOLD 7709 | Obsolete version of ~ omsi... |
| peano1 7710 | Zero is a natural number. ... |
| peano2 7711 | The successor of any natur... |
| peano3 7712 | The successor of any natur... |
| peano4 7713 | Two natural numbers are eq... |
| peano5 7714 | The induction postulate: a... |
| peano5OLD 7715 | Obsolete version of ~ pean... |
| nn0suc 7716 | A natural number is either... |
| find 7717 | The Principle of Finite In... |
| findOLD 7718 | Obsolete version of ~ find... |
| finds 7719 | Principle of Finite Induct... |
| findsg 7720 | Principle of Finite Induct... |
| finds2 7721 | Principle of Finite Induct... |
| finds1 7722 | Principle of Finite Induct... |
| findes 7723 | Finite induction with expl... |
| dmexg 7724 | The domain of a set is a s... |
| rnexg 7725 | The range of a set is a se... |
| dmexd 7726 | The domain of a set is a s... |
| fndmexd 7727 | If a function is a set, it... |
| dmfex 7728 | If a mapping is a set, its... |
| fndmexb 7729 | The domain of a function i... |
| fdmexb 7730 | The domain of a function i... |
| dmfexALT 7731 | Alternate proof of ~ dmfex... |
| dmex 7732 | The domain of a set is a s... |
| rnex 7733 | The range of a set is a se... |
| iprc 7734 | The identity function is a... |
| resiexg 7735 | The existence of a restric... |
| imaexg 7736 | The image of a set is a se... |
| imaex 7737 | The image of a set is a se... |
| exse2 7738 | Any set relation is set-li... |
| xpexr 7739 | If a Cartesian product is ... |
| xpexr2 7740 | If a nonempty Cartesian pr... |
| xpexcnv 7741 | A condition where the conv... |
| soex 7742 | If the relation in a stric... |
| elxp4 7743 | Membership in a Cartesian ... |
| elxp5 7744 | Membership in a Cartesian ... |
| cnvexg 7745 | The converse of a set is a... |
| cnvex 7746 | The converse of a set is a... |
| relcnvexb 7747 | A relation is a set iff it... |
| f1oexrnex 7748 | If the range of a 1-1 onto... |
| f1oexbi 7749 | There is a one-to-one onto... |
| coexg 7750 | The composition of two set... |
| coex 7751 | The composition of two set... |
| funcnvuni 7752 | The union of a chain (with... |
| fun11uni 7753 | The union of a chain (with... |
| fex2 7754 | A function with bounded do... |
| fabexg 7755 | Existence of a set of func... |
| fabex 7756 | Existence of a set of func... |
| f1oabexg 7757 | The class of all 1-1-onto ... |
| fiunlem 7758 | Lemma for ~ fiun and ~ f1i... |
| fiun 7759 | The union of a chain (with... |
| f1iun 7760 | The union of a chain (with... |
| fviunfun 7761 | The function value of an i... |
| ffoss 7762 | Relationship between a map... |
| f11o 7763 | Relationship between one-t... |
| resfunexgALT 7764 | Alternate proof of ~ resfu... |
| cofunexg 7765 | Existence of a composition... |
| cofunex2g 7766 | Existence of a composition... |
| fnexALT 7767 | Alternate proof of ~ fnex ... |
| funexw 7768 | Weak version of ~ funex th... |
| mptexw 7769 | Weak version of ~ mptex th... |
| funrnex 7770 | If the domain of a functio... |
| zfrep6 7771 | A version of the Axiom of ... |
| fornex 7772 | If the domain of an onto f... |
| f1dmex 7773 | If the codomain of a one-t... |
| f1ovv 7774 | The range of a 1-1 onto fu... |
| fvclex 7775 | Existence of the class of ... |
| fvresex 7776 | Existence of the class of ... |
| abrexexg 7777 | Existence of a class abstr... |
| abrexex 7778 | Existence of a class abstr... |
| iunexg 7779 | The existence of an indexe... |
| abrexex2g 7780 | Existence of an existentia... |
| opabex3d 7781 | Existence of an ordered pa... |
| opabex3rd 7782 | Existence of an ordered pa... |
| opabex3 7783 | Existence of an ordered pa... |
| iunex 7784 | The existence of an indexe... |
| abrexex2 7785 | Existence of an existentia... |
| abexssex 7786 | Existence of a class abstr... |
| abexex 7787 | A condition where a class ... |
| f1oweALT 7788 | Alternate proof of ~ f1owe... |
| wemoiso 7789 | Thus, there is at most one... |
| wemoiso2 7790 | Thus, there is at most one... |
| oprabexd 7791 | Existence of an operator a... |
| oprabex 7792 | Existence of an operation ... |
| oprabex3 7793 | Existence of an operation ... |
| oprabrexex2 7794 | Existence of an existentia... |
| ab2rexex 7795 | Existence of a class abstr... |
| ab2rexex2 7796 | Existence of an existentia... |
| xpexgALT 7797 | Alternate proof of ~ xpexg... |
| offval3 7798 | General value of ` ( F oF ... |
| offres 7799 | Pointwise combination comm... |
| ofmres 7800 | Equivalent expressions for... |
| ofmresex 7801 | Existence of a restriction... |
| 1stval 7806 | The value of the function ... |
| 2ndval 7807 | The value of the function ... |
| 1stnpr 7808 | Value of the first-member ... |
| 2ndnpr 7809 | Value of the second-member... |
| 1st0 7810 | The value of the first-mem... |
| 2nd0 7811 | The value of the second-me... |
| op1st 7812 | Extract the first member o... |
| op2nd 7813 | Extract the second member ... |
| op1std 7814 | Extract the first member o... |
| op2ndd 7815 | Extract the second member ... |
| op1stg 7816 | Extract the first member o... |
| op2ndg 7817 | Extract the second member ... |
| ot1stg 7818 | Extract the first member o... |
| ot2ndg 7819 | Extract the second member ... |
| ot3rdg 7820 | Extract the third member o... |
| 1stval2 7821 | Alternate value of the fun... |
| 2ndval2 7822 | Alternate value of the fun... |
| oteqimp 7823 | The components of an order... |
| fo1st 7824 | The ` 1st ` function maps ... |
| fo2nd 7825 | The ` 2nd ` function maps ... |
| br1steqg 7826 | Uniqueness condition for t... |
| br2ndeqg 7827 | Uniqueness condition for t... |
| f1stres 7828 | Mapping of a restriction o... |
| f2ndres 7829 | Mapping of a restriction o... |
| fo1stres 7830 | Onto mapping of a restrict... |
| fo2ndres 7831 | Onto mapping of a restrict... |
| 1st2val 7832 | Value of an alternate defi... |
| 2nd2val 7833 | Value of an alternate defi... |
| 1stcof 7834 | Composition of the first m... |
| 2ndcof 7835 | Composition of the second ... |
| xp1st 7836 | Location of the first elem... |
| xp2nd 7837 | Location of the second ele... |
| elxp6 7838 | Membership in a Cartesian ... |
| elxp7 7839 | Membership in a Cartesian ... |
| eqopi 7840 | Equality with an ordered p... |
| xp2 7841 | Representation of Cartesia... |
| unielxp 7842 | The membership relation fo... |
| 1st2nd2 7843 | Reconstruction of a member... |
| 1st2ndb 7844 | Reconstruction of an order... |
| xpopth 7845 | An ordered pair theorem fo... |
| eqop 7846 | Two ways to express equali... |
| eqop2 7847 | Two ways to express equali... |
| op1steq 7848 | Two ways of expressing tha... |
| opreuopreu 7849 | There is a unique ordered ... |
| el2xptp 7850 | A member of a nested Carte... |
| el2xptp0 7851 | A member of a nested Carte... |
| 2nd1st 7852 | Swap the members of an ord... |
| 1st2nd 7853 | Reconstruction of a member... |
| 1stdm 7854 | The first ordered pair com... |
| 2ndrn 7855 | The second ordered pair co... |
| 1st2ndbr 7856 | Express an element of a re... |
| releldm2 7857 | Two ways of expressing mem... |
| reldm 7858 | An expression for the doma... |
| releldmdifi 7859 | One way of expressing memb... |
| funfv1st2nd 7860 | The function value for the... |
| funelss 7861 | If the first component of ... |
| funeldmdif 7862 | Two ways of expressing mem... |
| sbcopeq1a 7863 | Equality theorem for subst... |
| csbopeq1a 7864 | Equality theorem for subst... |
| dfopab2 7865 | A way to define an ordered... |
| dfoprab3s 7866 | A way to define an operati... |
| dfoprab3 7867 | Operation class abstractio... |
| dfoprab4 7868 | Operation class abstractio... |
| dfoprab4f 7869 | Operation class abstractio... |
| opabex2 7870 | Condition for an operation... |
| opabn1stprc 7871 | An ordered-pair class abst... |
| opiota 7872 | The property of a uniquely... |
| cnvoprab 7873 | The converse of a class ab... |
| dfxp3 7874 | Define the Cartesian produ... |
| elopabi 7875 | A consequence of membershi... |
| eloprabi 7876 | A consequence of membershi... |
| mpomptsx 7877 | Express a two-argument fun... |
| mpompts 7878 | Express a two-argument fun... |
| dmmpossx 7879 | The domain of a mapping is... |
| fmpox 7880 | Functionality, domain and ... |
| fmpo 7881 | Functionality, domain and ... |
| fnmpo 7882 | Functionality and domain o... |
| fnmpoi 7883 | Functionality and domain o... |
| dmmpo 7884 | Domain of a class given by... |
| ovmpoelrn 7885 | An operation's value belon... |
| dmmpoga 7886 | Domain of an operation giv... |
| dmmpogaOLD 7887 | Obsolete version of ~ dmmp... |
| dmmpog 7888 | Domain of an operation giv... |
| mpoexxg 7889 | Existence of an operation ... |
| mpoexg 7890 | Existence of an operation ... |
| mpoexga 7891 | If the domain of an operat... |
| mpoexw 7892 | Weak version of ~ mpoex th... |
| mpoex 7893 | If the domain of an operat... |
| mptmpoopabbrd 7894 | The operation value of a f... |
| mptmpoopabovd 7895 | The operation value of a f... |
| el2mpocsbcl 7896 | If the operation value of ... |
| el2mpocl 7897 | If the operation value of ... |
| fnmpoovd 7898 | A function with a Cartesia... |
| offval22 7899 | The function operation exp... |
| brovpreldm 7900 | If a binary relation holds... |
| bropopvvv 7901 | If a binary relation holds... |
| bropfvvvvlem 7902 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 7903 | If a binary relation holds... |
| ovmptss 7904 | If all the values of the m... |
| relmpoopab 7905 | Any function to sets of or... |
| fmpoco 7906 | Composition of two functio... |
| oprabco 7907 | Composition of a function ... |
| oprab2co 7908 | Composition of operator ab... |
| df1st2 7909 | An alternate possible defi... |
| df2nd2 7910 | An alternate possible defi... |
| 1stconst 7911 | The mapping of a restricti... |
| 2ndconst 7912 | The mapping of a restricti... |
| dfmpo 7913 | Alternate definition for t... |
| mposn 7914 | An operation (in maps-to n... |
| curry1 7915 | Composition with ` ``' ( 2... |
| curry1val 7916 | The value of a curried fun... |
| curry1f 7917 | Functionality of a curried... |
| curry2 7918 | Composition with ` ``' ( 1... |
| curry2f 7919 | Functionality of a curried... |
| curry2val 7920 | The value of a curried fun... |
| cnvf1olem 7921 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 7922 | Describe a function that m... |
| fparlem1 7923 | Lemma for ~ fpar . (Contr... |
| fparlem2 7924 | Lemma for ~ fpar . (Contr... |
| fparlem3 7925 | Lemma for ~ fpar . (Contr... |
| fparlem4 7926 | Lemma for ~ fpar . (Contr... |
| fpar 7927 | Merge two functions in par... |
| fsplit 7928 | A function that can be use... |
| fsplitOLD 7929 | Obsolete proof of ~ fsplit... |
| fsplitfpar 7930 | Merge two functions with a... |
| offsplitfpar 7931 | Express the function opera... |
| f2ndf 7932 | The ` 2nd ` (second compon... |
| fo2ndf 7933 | The ` 2nd ` (second compon... |
| f1o2ndf1 7934 | The ` 2nd ` (second compon... |
| opco1 7935 | Value of an operation prec... |
| opco2 7936 | Value of an operation prec... |
| opco1i 7937 | Inference form of ~ opco1 ... |
| frxp 7938 | A lexicographical ordering... |
| xporderlem 7939 | Lemma for lexicographical ... |
| poxp 7940 | A lexicographical ordering... |
| soxp 7941 | A lexicographical ordering... |
| wexp 7942 | A lexicographical ordering... |
| fnwelem 7943 | Lemma for ~ fnwe . (Contr... |
| fnwe 7944 | A variant on lexicographic... |
| fnse 7945 | Condition for the well-ord... |
| fvproj 7946 | Value of a function on ord... |
| fimaproj 7947 | Image of a cartesian produ... |
| suppval 7950 | The value of the operation... |
| supp0prc 7951 | The support of a class is ... |
| suppvalbr 7952 | The value of the operation... |
| supp0 7953 | The support of the empty s... |
| suppval1 7954 | The value of the operation... |
| suppvalfng 7955 | The value of the operation... |
| suppvalfn 7956 | The value of the operation... |
| elsuppfng 7957 | An element of the support ... |
| elsuppfn 7958 | An element of the support ... |
| cnvimadfsn 7959 | The support of functions "... |
| suppimacnvss 7960 | The support of functions "... |
| suppimacnv 7961 | Support sets of functions ... |
| frnsuppeq 7962 | Two ways of writing the su... |
| frnsuppeqg 7963 | Version of ~ frnsuppeq avo... |
| suppssdm 7964 | The support of a function ... |
| suppsnop 7965 | The support of a singleton... |
| snopsuppss 7966 | The support of a singleton... |
| fvn0elsupp 7967 | If the function value for ... |
| fvn0elsuppb 7968 | The function value for a g... |
| rexsupp 7969 | Existential quantification... |
| ressuppss 7970 | The support of the restric... |
| suppun 7971 | The support of a class/fun... |
| ressuppssdif 7972 | The support of the restric... |
| mptsuppdifd 7973 | The support of a function ... |
| mptsuppd 7974 | The support of a function ... |
| extmptsuppeq 7975 | The support of an extended... |
| suppfnss 7976 | The support of a function ... |
| funsssuppss 7977 | The support of a function ... |
| fnsuppres 7978 | Two ways to express restri... |
| fnsuppeq0 7979 | The support of a function ... |
| fczsupp0 7980 | The support of a constant ... |
| suppss 7981 | Show that the support of a... |
| suppssOLD 7982 | Obsolete version of ~ supp... |
| suppssr 7983 | A function is zero outside... |
| suppssrg 7984 | A function is zero outside... |
| suppssov1 7985 | Formula building theorem f... |
| suppssof1 7986 | Formula building theorem f... |
| suppss2 7987 | Show that the support of a... |
| suppsssn 7988 | Show that the support of a... |
| suppssfv 7989 | Formula building theorem f... |
| suppofssd 7990 | Condition for the support ... |
| suppofss1d 7991 | Condition for the support ... |
| suppofss2d 7992 | Condition for the support ... |
| suppco 7993 | The support of the composi... |
| suppcoss 7994 | The support of the composi... |
| supp0cosupp0 7995 | The support of the composi... |
| imacosupp 7996 | The image of the support o... |
| opeliunxp2f 7997 | Membership in a union of C... |
| mpoxeldm 7998 | If there is an element of ... |
| mpoxneldm 7999 | If the first argument of a... |
| mpoxopn0yelv 8000 | If there is an element of ... |
| mpoxopynvov0g 8001 | If the second argument of ... |
| mpoxopxnop0 8002 | If the first argument of a... |
| mpoxopx0ov0 8003 | If the first argument of a... |
| mpoxopxprcov0 8004 | If the components of the f... |
| mpoxopynvov0 8005 | If the second argument of ... |
| mpoxopoveq 8006 | Value of an operation give... |
| mpoxopovel 8007 | Element of the value of an... |
| mpoxopoveqd 8008 | Value of an operation give... |
| brovex 8009 | A binary relation of the v... |
| brovmpoex 8010 | A binary relation of the v... |
| sprmpod 8011 | The extension of a binary ... |
| tposss 8014 | Subset theorem for transpo... |
| tposeq 8015 | Equality theorem for trans... |
| tposeqd 8016 | Equality theorem for trans... |
| tposssxp 8017 | The transposition is a sub... |
| reltpos 8018 | The transposition is a rel... |
| brtpos2 8019 | Value of the transposition... |
| brtpos0 8020 | The behavior of ` tpos ` w... |
| reldmtpos 8021 | Necessary and sufficient c... |
| brtpos 8022 | The transposition swaps ar... |
| ottpos 8023 | The transposition swaps th... |
| relbrtpos 8024 | The transposition swaps ar... |
| dmtpos 8025 | The domain of ` tpos F ` w... |
| rntpos 8026 | The range of ` tpos F ` wh... |
| tposexg 8027 | The transposition of a set... |
| ovtpos 8028 | The transposition swaps th... |
| tposfun 8029 | The transposition of a fun... |
| dftpos2 8030 | Alternate definition of ` ... |
| dftpos3 8031 | Alternate definition of ` ... |
| dftpos4 8032 | Alternate definition of ` ... |
| tpostpos 8033 | Value of the double transp... |
| tpostpos2 8034 | Value of the double transp... |
| tposfn2 8035 | The domain of a transposit... |
| tposfo2 8036 | Condition for a surjective... |
| tposf2 8037 | The domain and range of a ... |
| tposf12 8038 | Condition for an injective... |
| tposf1o2 8039 | Condition of a bijective t... |
| tposfo 8040 | The domain and range of a ... |
| tposf 8041 | The domain and range of a ... |
| tposfn 8042 | Functionality of a transpo... |
| tpos0 8043 | Transposition of the empty... |
| tposco 8044 | Transposition of a composi... |
| tpossym 8045 | Two ways to say a function... |
| tposeqi 8046 | Equality theorem for trans... |
| tposex 8047 | A transposition is a set. ... |
| nftpos 8048 | Hypothesis builder for tra... |
| tposoprab 8049 | Transposition of a class o... |
| tposmpo 8050 | Transposition of a two-arg... |
| tposconst 8051 | The transposition of a con... |
| mpocurryd 8056 | The currying of an operati... |
| mpocurryvald 8057 | The value of a curried ope... |
| fvmpocurryd 8058 | The value of the value of ... |
| pwuninel2 8061 | Direct proof of ~ pwuninel... |
| pwuninel 8062 | The power set of the union... |
| undefval 8063 | Value of the undefined val... |
| undefnel2 8064 | The undefined value genera... |
| undefnel 8065 | The undefined value genera... |
| undefne0 8066 | The undefined value genera... |
| frecseq123 8069 | Equality theorem for the w... |
| nffrecs 8070 | Bound-variable hypothesis ... |
| csbfrecsg 8071 | Move class substitution in... |
| fpr3g 8072 | Functions defined by well-... |
| frrlem1 8073 | Lemma for well-founded rec... |
| frrlem2 8074 | Lemma for well-founded rec... |
| frrlem3 8075 | Lemma for well-founded rec... |
| frrlem4 8076 | Lemma for well-founded rec... |
| frrlem5 8077 | Lemma for well-founded rec... |
| frrlem6 8078 | Lemma for well-founded rec... |
| frrlem7 8079 | Lemma for well-founded rec... |
| frrlem8 8080 | Lemma for well-founded rec... |
| frrlem9 8081 | Lemma for well-founded rec... |
| frrlem10 8082 | Lemma for well-founded rec... |
| frrlem11 8083 | Lemma for well-founded rec... |
| frrlem12 8084 | Lemma for well-founded rec... |
| frrlem13 8085 | Lemma for well-founded rec... |
| frrlem14 8086 | Lemma for well-founded rec... |
| fprlem1 8087 | Lemma for well-founded rec... |
| fprlem2 8088 | Lemma for well-founded rec... |
| fpr2a 8089 | Weak version of ~ fpr2 whi... |
| fpr1 8090 | Law of well-founded recurs... |
| fpr2 8091 | Law of well-founded recurs... |
| fpr3 8092 | Law of well-founded recurs... |
| frrrel 8093 | Show without using the axi... |
| frrdmss 8094 | Show without using the axi... |
| frrdmcl 8095 | Show without using the axi... |
| fprfung 8096 | A "function" defined by we... |
| fprresex 8097 | The restriction of a funct... |
| dfwrecsOLD 8100 | Obsolete definition of the... |
| wrecseq123 8101 | General equality theorem f... |
| wrecseq123OLD 8102 | Obsolete proof of ~ wrecse... |
| nfwrecs 8103 | Bound-variable hypothesis ... |
| nfwrecsOLD 8104 | Obsolete proof of ~ nfwrec... |
| wrecseq1 8105 | Equality theorem for the w... |
| wrecseq2 8106 | Equality theorem for the w... |
| wrecseq3 8107 | Equality theorem for the w... |
| csbwrecsg 8108 | Move class substitution in... |
| wfr3g 8109 | Functions defined by well-... |
| wfrlem1OLD 8110 | Lemma for well-ordered rec... |
| wfrlem2OLD 8111 | Lemma for well-ordered rec... |
| wfrlem3OLD 8112 | Lemma for well-ordered rec... |
| wfrlem3OLDa 8113 | Lemma for well-ordered rec... |
| wfrlem4OLD 8114 | Lemma for well-ordered rec... |
| wfrlem5OLD 8115 | Lemma for well-ordered rec... |
| wfrrelOLD 8116 | Obsolete proof of ~ wfrrel... |
| wfrdmssOLD 8117 | Obsolete proof of ~ wfrdms... |
| wfrlem8OLD 8118 | Lemma for well-ordered rec... |
| wfrdmclOLD 8119 | Obsolete proof of ~ wfrdmc... |
| wfrlem10OLD 8120 | Lemma for well-ordered rec... |
| wfrfunOLD 8121 | Obsolete proof of ~ wfrfun... |
| wfrlem12OLD 8122 | Lemma for well-ordered rec... |
| wfrlem13OLD 8123 | Lemma for well-ordered rec... |
| wfrlem14OLD 8124 | Lemma for well-ordered rec... |
| wfrlem15OLD 8125 | Lemma for well-ordered rec... |
| wfrlem16OLD 8126 | Lemma for well-ordered rec... |
| wfrlem17OLD 8127 | Without using ~ ax-rep , s... |
| wfr2aOLD 8128 | Obsolete proof of ~ wfr2a ... |
| wfr1OLD 8129 | Obsolete proof of ~ wfr1 a... |
| wfr2OLD 8130 | Obsolete proof of ~ wfr2 a... |
| wfrrel 8131 | The well-ordered recursion... |
| wfrdmss 8132 | The domain of the well-ord... |
| wfrdmcl 8133 | The predecessor class of a... |
| wfrfun 8134 | The "function" generated b... |
| wfrresex 8135 | Show without using the axi... |
| wfr2a 8136 | A weak version of ~ wfr2 w... |
| wfr1 8137 | The Principle of Well-Orde... |
| wfr2 8138 | The Principle of Well-Orde... |
| wfr3 8139 | The principle of Well-Orde... |
| wfr3OLD 8140 | Obsolete form of ~ wfr3 as... |
| iunon 8141 | The indexed union of a set... |
| iinon 8142 | The nonempty indexed inter... |
| onfununi 8143 | A property of functions on... |
| onovuni 8144 | A variant of ~ onfununi fo... |
| onoviun 8145 | A variant of ~ onovuni wit... |
| onnseq 8146 | There are no length ` _om ... |
| dfsmo2 8149 | Alternate definition of a ... |
| issmo 8150 | Conditions for which ` A `... |
| issmo2 8151 | Alternate definition of a ... |
| smoeq 8152 | Equality theorem for stric... |
| smodm 8153 | The domain of a strictly m... |
| smores 8154 | A strictly monotone functi... |
| smores3 8155 | A strictly monotone functi... |
| smores2 8156 | A strictly monotone ordina... |
| smodm2 8157 | The domain of a strictly m... |
| smofvon2 8158 | The function values of a s... |
| iordsmo 8159 | The identity relation rest... |
| smo0 8160 | The null set is a strictly... |
| smofvon 8161 | If ` B ` is a strictly mon... |
| smoel 8162 | If ` x ` is less than ` y ... |
| smoiun 8163 | The value of a strictly mo... |
| smoiso 8164 | If ` F ` is an isomorphism... |
| smoel2 8165 | A strictly monotone ordina... |
| smo11 8166 | A strictly monotone ordina... |
| smoord 8167 | A strictly monotone ordina... |
| smoword 8168 | A strictly monotone ordina... |
| smogt 8169 | A strictly monotone ordina... |
| smorndom 8170 | The range of a strictly mo... |
| smoiso2 8171 | The strictly monotone ordi... |
| dfrecs3 8174 | The old definition of tran... |
| dfrecs3OLD 8175 | Obsolete proof of ~ dfrecs... |
| recseq 8176 | Equality theorem for ` rec... |
| nfrecs 8177 | Bound-variable hypothesis ... |
| tfrlem1 8178 | A technical lemma for tran... |
| tfrlem3a 8179 | Lemma for transfinite recu... |
| tfrlem3 8180 | Lemma for transfinite recu... |
| tfrlem4 8181 | Lemma for transfinite recu... |
| tfrlem5 8182 | Lemma for transfinite recu... |
| recsfval 8183 | Lemma for transfinite recu... |
| tfrlem6 8184 | Lemma for transfinite recu... |
| tfrlem7 8185 | Lemma for transfinite recu... |
| tfrlem8 8186 | Lemma for transfinite recu... |
| tfrlem9 8187 | Lemma for transfinite recu... |
| tfrlem9a 8188 | Lemma for transfinite recu... |
| tfrlem10 8189 | Lemma for transfinite recu... |
| tfrlem11 8190 | Lemma for transfinite recu... |
| tfrlem12 8191 | Lemma for transfinite recu... |
| tfrlem13 8192 | Lemma for transfinite recu... |
| tfrlem14 8193 | Lemma for transfinite recu... |
| tfrlem15 8194 | Lemma for transfinite recu... |
| tfrlem16 8195 | Lemma for finite recursion... |
| tfr1a 8196 | A weak version of ~ tfr1 w... |
| tfr2a 8197 | A weak version of ~ tfr2 w... |
| tfr2b 8198 | Without assuming ~ ax-rep ... |
| tfr1 8199 | Principle of Transfinite R... |
| tfr2 8200 | Principle of Transfinite R... |
| tfr3 8201 | Principle of Transfinite R... |
| tfr1ALT 8202 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8203 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8204 | Alternate proof of ~ tfr3 ... |
| recsfnon 8205 | Strong transfinite recursi... |
| recsval 8206 | Strong transfinite recursi... |
| tz7.44lem1 8207 | The ordered pair abstracti... |
| tz7.44-1 8208 | The value of ` F ` at ` (/... |
| tz7.44-2 8209 | The value of ` F ` at a su... |
| tz7.44-3 8210 | The value of ` F ` at a li... |
| rdgeq1 8213 | Equality theorem for the r... |
| rdgeq2 8214 | Equality theorem for the r... |
| rdgeq12 8215 | Equality theorem for the r... |
| nfrdg 8216 | Bound-variable hypothesis ... |
| rdglem1 8217 | Lemma used with the recurs... |
| rdgfun 8218 | The recursive definition g... |
| rdgdmlim 8219 | The domain of the recursiv... |
| rdgfnon 8220 | The recursive definition g... |
| rdgvalg 8221 | Value of the recursive def... |
| rdgval 8222 | Value of the recursive def... |
| rdg0 8223 | The initial value of the r... |
| rdgseg 8224 | The initial segments of th... |
| rdgsucg 8225 | The value of the recursive... |
| rdgsuc 8226 | The value of the recursive... |
| rdglimg 8227 | The value of the recursive... |
| rdglim 8228 | The value of the recursive... |
| rdg0g 8229 | The initial value of the r... |
| rdgsucmptf 8230 | The value of the recursive... |
| rdgsucmptnf 8231 | The value of the recursive... |
| rdgsucmpt2 8232 | This version of ~ rdgsucmp... |
| rdgsucmpt 8233 | The value of the recursive... |
| rdglim2 8234 | The value of the recursive... |
| rdglim2a 8235 | The value of the recursive... |
| frfnom 8236 | The function generated by ... |
| fr0g 8237 | The initial value resultin... |
| frsuc 8238 | The successor value result... |
| frsucmpt 8239 | The successor value result... |
| frsucmptn 8240 | The value of the finite re... |
| frsucmpt2 8241 | The successor value result... |
| tz7.48lem 8242 | A way of showing an ordina... |
| tz7.48-2 8243 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8244 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8245 | Proposition 7.48(3) of [Ta... |
| tz7.49 8246 | Proposition 7.49 of [Takeu... |
| tz7.49c 8247 | Corollary of Proposition 7... |
| seqomlem0 8250 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8251 | Lemma for ` seqom ` . The... |
| seqomlem2 8252 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8253 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8254 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8255 | Equality theorem for ` seq... |
| fnseqom 8256 | An index-aware recursive d... |
| seqom0g 8257 | Value of an index-aware re... |
| seqomsuc 8258 | Value of an index-aware re... |
| omsucelsucb 8259 | Membership is inherited by... |
| 1on 8274 | Ordinal 1 is an ordinal nu... |
| 2on 8275 | Ordinal 2 is an ordinal nu... |
| 2on0 8276 | Ordinal two is not zero. ... |
| 3on 8277 | Ordinal 3 is an ordinal nu... |
| 4on 8278 | Ordinal 3 is an ordinal nu... |
| df1o2 8279 | Expanded value of the ordi... |
| 1oex 8280 | Ordinal 1 is a set. (Cont... |
| 1oexOLD 8281 | Obsolete version of ~ 1oex... |
| df2o3 8282 | Expanded value of the ordi... |
| df2o2 8283 | Expanded value of the ordi... |
| 2oex 8284 | ` 2o ` is a set. (Contrib... |
| 2oexOLD 8285 | Obsolete version of ~ 2oex... |
| 1n0 8286 | Ordinal one is not equal t... |
| xp01disj 8287 | Cartesian products with th... |
| xp01disjl 8288 | Cartesian products with th... |
| ordgt0ge1 8289 | Two ways to express that a... |
| ordge1n0 8290 | An ordinal greater than or... |
| el1o 8291 | Membership in ordinal one.... |
| dif1o 8292 | Two ways to say that ` A `... |
| ondif1 8293 | Two ways to say that ` A `... |
| ondif2 8294 | Two ways to say that ` A `... |
| 2oconcl 8295 | Closure of the pair swappi... |
| 0lt1o 8296 | Ordinal zero is less than ... |
| dif20el 8297 | An ordinal greater than on... |
| 0we1 8298 | The empty set is a well-or... |
| brwitnlem 8299 | Lemma for relations which ... |
| fnoa 8300 | Functionality and domain o... |
| fnom 8301 | Functionality and domain o... |
| fnoe 8302 | Functionality and domain o... |
| oav 8303 | Value of ordinal addition.... |
| omv 8304 | Value of ordinal multiplic... |
| oe0lem 8305 | A helper lemma for ~ oe0 a... |
| oev 8306 | Value of ordinal exponenti... |
| oevn0 8307 | Value of ordinal exponenti... |
| oa0 8308 | Addition with zero. Propo... |
| om0 8309 | Ordinal multiplication wit... |
| oe0m 8310 | Value of zero raised to an... |
| om0x 8311 | Ordinal multiplication wit... |
| oe0m0 8312 | Ordinal exponentiation wit... |
| oe0m1 8313 | Ordinal exponentiation wit... |
| oe0 8314 | Ordinal exponentiation wit... |
| oev2 8315 | Alternate value of ordinal... |
| oasuc 8316 | Addition with successor. ... |
| oesuclem 8317 | Lemma for ~ oesuc . (Cont... |
| omsuc 8318 | Multiplication with succes... |
| oesuc 8319 | Ordinal exponentiation wit... |
| onasuc 8320 | Addition with successor. ... |
| onmsuc 8321 | Multiplication with succes... |
| onesuc 8322 | Exponentiation with a succ... |
| oa1suc 8323 | Addition with 1 is same as... |
| oalim 8324 | Ordinal addition with a li... |
| omlim 8325 | Ordinal multiplication wit... |
| oelim 8326 | Ordinal exponentiation wit... |
| oacl 8327 | Closure law for ordinal ad... |
| omcl 8328 | Closure law for ordinal mu... |
| oecl 8329 | Closure law for ordinal ex... |
| oa0r 8330 | Ordinal addition with zero... |
| om0r 8331 | Ordinal multiplication wit... |
| o1p1e2 8332 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8333 | 2 + 2 = 4 for ordinal numb... |
| o2p2e4OLD 8334 | Obsolete version of ~ o2p2... |
| om1 8335 | Ordinal multiplication wit... |
| om1r 8336 | Ordinal multiplication wit... |
| oe1 8337 | Ordinal exponentiation wit... |
| oe1m 8338 | Ordinal exponentiation wit... |
| oaordi 8339 | Ordering property of ordin... |
| oaord 8340 | Ordering property of ordin... |
| oacan 8341 | Left cancellation law for ... |
| oaword 8342 | Weak ordering property of ... |
| oawordri 8343 | Weak ordering property of ... |
| oaord1 8344 | An ordinal is less than it... |
| oaword1 8345 | An ordinal is less than or... |
| oaword2 8346 | An ordinal is less than or... |
| oawordeulem 8347 | Lemma for ~ oawordex . (C... |
| oawordeu 8348 | Existence theorem for weak... |
| oawordexr 8349 | Existence theorem for weak... |
| oawordex 8350 | Existence theorem for weak... |
| oaordex 8351 | Existence theorem for orde... |
| oa00 8352 | An ordinal sum is zero iff... |
| oalimcl 8353 | The ordinal sum with a lim... |
| oaass 8354 | Ordinal addition is associ... |
| oarec 8355 | Recursive definition of or... |
| oaf1o 8356 | Left addition by a constan... |
| oacomf1olem 8357 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8358 | Define a bijection from ` ... |
| omordi 8359 | Ordering property of ordin... |
| omord2 8360 | Ordering property of ordin... |
| omord 8361 | Ordering property of ordin... |
| omcan 8362 | Left cancellation law for ... |
| omword 8363 | Weak ordering property of ... |
| omwordi 8364 | Weak ordering property of ... |
| omwordri 8365 | Weak ordering property of ... |
| omword1 8366 | An ordinal is less than or... |
| omword2 8367 | An ordinal is less than or... |
| om00 8368 | The product of two ordinal... |
| om00el 8369 | The product of two nonzero... |
| omordlim 8370 | Ordering involving the pro... |
| omlimcl 8371 | The product of any nonzero... |
| odi 8372 | Distributive law for ordin... |
| omass 8373 | Multiplication of ordinal ... |
| oneo 8374 | If an ordinal number is ev... |
| omeulem1 8375 | Lemma for ~ omeu : existen... |
| omeulem2 8376 | Lemma for ~ omeu : uniquen... |
| omopth2 8377 | An ordered pair-like theor... |
| omeu 8378 | The division algorithm for... |
| oen0 8379 | Ordinal exponentiation wit... |
| oeordi 8380 | Ordering law for ordinal e... |
| oeord 8381 | Ordering property of ordin... |
| oecan 8382 | Left cancellation law for ... |
| oeword 8383 | Weak ordering property of ... |
| oewordi 8384 | Weak ordering property of ... |
| oewordri 8385 | Weak ordering property of ... |
| oeworde 8386 | Ordinal exponentiation com... |
| oeordsuc 8387 | Ordering property of ordin... |
| oelim2 8388 | Ordinal exponentiation wit... |
| oeoalem 8389 | Lemma for ~ oeoa . (Contr... |
| oeoa 8390 | Sum of exponents law for o... |
| oeoelem 8391 | Lemma for ~ oeoe . (Contr... |
| oeoe 8392 | Product of exponents law f... |
| oelimcl 8393 | The ordinal exponential wi... |
| oeeulem 8394 | Lemma for ~ oeeu . (Contr... |
| oeeui 8395 | The division algorithm for... |
| oeeu 8396 | The division algorithm for... |
| nna0 8397 | Addition with zero. Theor... |
| nnm0 8398 | Multiplication with zero. ... |
| nnasuc 8399 | Addition with successor. ... |
| nnmsuc 8400 | Multiplication with succes... |
| nnesuc 8401 | Exponentiation with a succ... |
| nna0r 8402 | Addition to zero. Remark ... |
| nnm0r 8403 | Multiplication with zero. ... |
| nnacl 8404 | Closure of addition of nat... |
| nnmcl 8405 | Closure of multiplication ... |
| nnecl 8406 | Closure of exponentiation ... |
| nnacli 8407 | ` _om ` is closed under ad... |
| nnmcli 8408 | ` _om ` is closed under mu... |
| nnarcl 8409 | Reverse closure law for ad... |
| nnacom 8410 | Addition of natural number... |
| nnaordi 8411 | Ordering property of addit... |
| nnaord 8412 | Ordering property of addit... |
| nnaordr 8413 | Ordering property of addit... |
| nnawordi 8414 | Adding to both sides of an... |
| nnaass 8415 | Addition of natural number... |
| nndi 8416 | Distributive law for natur... |
| nnmass 8417 | Multiplication of natural ... |
| nnmsucr 8418 | Multiplication with succes... |
| nnmcom 8419 | Multiplication of natural ... |
| nnaword 8420 | Weak ordering property of ... |
| nnacan 8421 | Cancellation law for addit... |
| nnaword1 8422 | Weak ordering property of ... |
| nnaword2 8423 | Weak ordering property of ... |
| nnmordi 8424 | Ordering property of multi... |
| nnmord 8425 | Ordering property of multi... |
| nnmword 8426 | Weak ordering property of ... |
| nnmcan 8427 | Cancellation law for multi... |
| nnmwordi 8428 | Weak ordering property of ... |
| nnmwordri 8429 | Weak ordering property of ... |
| nnawordex 8430 | Equivalence for weak order... |
| nnaordex 8431 | Equivalence for ordering. ... |
| 1onn 8432 | One is a natural number. ... |
| 2onn 8433 | The ordinal 2 is a natural... |
| 3onn 8434 | The ordinal 3 is a natural... |
| 4onn 8435 | The ordinal 4 is a natural... |
| 1one2o 8436 | Ordinal one is not ordinal... |
| oaabslem 8437 | Lemma for ~ oaabs . (Cont... |
| oaabs 8438 | Ordinal addition absorbs a... |
| oaabs2 8439 | The absorption law ~ oaabs... |
| omabslem 8440 | Lemma for ~ omabs . (Cont... |
| omabs 8441 | Ordinal multiplication is ... |
| nnm1 8442 | Multiply an element of ` _... |
| nnm2 8443 | Multiply an element of ` _... |
| nn2m 8444 | Multiply an element of ` _... |
| nnneo 8445 | If a natural number is eve... |
| nneob 8446 | A natural number is even i... |
| omsmolem 8447 | Lemma for ~ omsmo . (Cont... |
| omsmo 8448 | A strictly monotonic ordin... |
| omopthlem1 8449 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8450 | Lemma for ~ omopthi . (Co... |
| omopthi 8451 | An ordered pair theorem fo... |
| omopth 8452 | An ordered pair theorem fo... |
| dfer2 8457 | Alternate definition of eq... |
| dfec2 8459 | Alternate definition of ` ... |
| ecexg 8460 | An equivalence class modul... |
| ecexr 8461 | A nonempty equivalence cla... |
| ereq1 8463 | Equality theorem for equiv... |
| ereq2 8464 | Equality theorem for equiv... |
| errel 8465 | An equivalence relation is... |
| erdm 8466 | The domain of an equivalen... |
| ercl 8467 | Elementhood in the field o... |
| ersym 8468 | An equivalence relation is... |
| ercl2 8469 | Elementhood in the field o... |
| ersymb 8470 | An equivalence relation is... |
| ertr 8471 | An equivalence relation is... |
| ertrd 8472 | A transitivity relation fo... |
| ertr2d 8473 | A transitivity relation fo... |
| ertr3d 8474 | A transitivity relation fo... |
| ertr4d 8475 | A transitivity relation fo... |
| erref 8476 | An equivalence relation is... |
| ercnv 8477 | The converse of an equival... |
| errn 8478 | The range and domain of an... |
| erssxp 8479 | An equivalence relation is... |
| erex 8480 | An equivalence relation is... |
| erexb 8481 | An equivalence relation is... |
| iserd 8482 | A reflexive, symmetric, tr... |
| iseri 8483 | A reflexive, symmetric, tr... |
| iseriALT 8484 | Alternate proof of ~ iseri... |
| brdifun 8485 | Evaluate the incomparabili... |
| swoer 8486 | Incomparability under a st... |
| swoord1 8487 | The incomparability equiva... |
| swoord2 8488 | The incomparability equiva... |
| swoso 8489 | If the incomparability rel... |
| eqerlem 8490 | Lemma for ~ eqer . (Contr... |
| eqer 8491 | Equivalence relation invol... |
| ider 8492 | The identity relation is a... |
| 0er 8493 | The empty set is an equiva... |
| eceq1 8494 | Equality theorem for equiv... |
| eceq1d 8495 | Equality theorem for equiv... |
| eceq2 8496 | Equality theorem for equiv... |
| eceq2i 8497 | Equality theorem for the `... |
| eceq2d 8498 | Equality theorem for the `... |
| elecg 8499 | Membership in an equivalen... |
| elec 8500 | Membership in an equivalen... |
| relelec 8501 | Membership in an equivalen... |
| ecss 8502 | An equivalence class is a ... |
| ecdmn0 8503 | A representative of a none... |
| ereldm 8504 | Equality of equivalence cl... |
| erth 8505 | Basic property of equivale... |
| erth2 8506 | Basic property of equivale... |
| erthi 8507 | Basic property of equivale... |
| erdisj 8508 | Equivalence classes do not... |
| ecidsn 8509 | An equivalence class modul... |
| qseq1 8510 | Equality theorem for quoti... |
| qseq2 8511 | Equality theorem for quoti... |
| qseq2i 8512 | Equality theorem for quoti... |
| qseq2d 8513 | Equality theorem for quoti... |
| qseq12 8514 | Equality theorem for quoti... |
| elqsg 8515 | Closed form of ~ elqs . (... |
| elqs 8516 | Membership in a quotient s... |
| elqsi 8517 | Membership in a quotient s... |
| elqsecl 8518 | Membership in a quotient s... |
| ecelqsg 8519 | Membership of an equivalen... |
| ecelqsi 8520 | Membership of an equivalen... |
| ecopqsi 8521 | "Closure" law for equivale... |
| qsexg 8522 | A quotient set exists. (C... |
| qsex 8523 | A quotient set exists. (C... |
| uniqs 8524 | The union of a quotient se... |
| qsss 8525 | A quotient set is a set of... |
| uniqs2 8526 | The union of a quotient se... |
| snec 8527 | The singleton of an equiva... |
| ecqs 8528 | Equivalence class in terms... |
| ecid 8529 | A set is equal to its cose... |
| qsid 8530 | A set is equal to its quot... |
| ectocld 8531 | Implicit substitution of c... |
| ectocl 8532 | Implicit substitution of c... |
| elqsn0 8533 | A quotient set does not co... |
| ecelqsdm 8534 | Membership of an equivalen... |
| xpider 8535 | A Cartesian square is an e... |
| iiner 8536 | The intersection of a none... |
| riiner 8537 | The relative intersection ... |
| erinxp 8538 | A restricted equivalence r... |
| ecinxp 8539 | Restrict the relation in a... |
| qsinxp 8540 | Restrict the equivalence r... |
| qsdisj 8541 | Members of a quotient set ... |
| qsdisj2 8542 | A quotient set is a disjoi... |
| qsel 8543 | If an element of a quotien... |
| uniinqs 8544 | Class union distributes ov... |
| qliftlem 8545 | Lemma for theorems about a... |
| qliftrel 8546 | ` F ` , a function lift, i... |
| qliftel 8547 | Elementhood in the relatio... |
| qliftel1 8548 | Elementhood in the relatio... |
| qliftfun 8549 | The function ` F ` is the ... |
| qliftfund 8550 | The function ` F ` is the ... |
| qliftfuns 8551 | The function ` F ` is the ... |
| qliftf 8552 | The domain and range of th... |
| qliftval 8553 | The value of the function ... |
| ecoptocl 8554 | Implicit substitution of c... |
| 2ecoptocl 8555 | Implicit substitution of c... |
| 3ecoptocl 8556 | Implicit substitution of c... |
| brecop 8557 | Binary relation on a quoti... |
| brecop2 8558 | Binary relation on a quoti... |
| eroveu 8559 | Lemma for ~ erov and ~ ero... |
| erovlem 8560 | Lemma for ~ erov and ~ ero... |
| erov 8561 | The value of an operation ... |
| eroprf 8562 | Functionality of an operat... |
| erov2 8563 | The value of an operation ... |
| eroprf2 8564 | Functionality of an operat... |
| ecopoveq 8565 | This is the first of sever... |
| ecopovsym 8566 | Assuming the operation ` F... |
| ecopovtrn 8567 | Assuming that operation ` ... |
| ecopover 8568 | Assuming that operation ` ... |
| eceqoveq 8569 | Equality of equivalence re... |
| ecovcom 8570 | Lemma used to transfer a c... |
| ecovass 8571 | Lemma used to transfer an ... |
| ecovdi 8572 | Lemma used to transfer a d... |
| mapprc 8577 | When ` A ` is a proper cla... |
| pmex 8578 | The class of all partial f... |
| mapex 8579 | The class of all functions... |
| fnmap 8580 | Set exponentiation has a u... |
| fnpm 8581 | Partial function exponenti... |
| reldmmap 8582 | Set exponentiation is a we... |
| mapvalg 8583 | The value of set exponenti... |
| pmvalg 8584 | The value of the partial m... |
| mapval 8585 | The value of set exponenti... |
| elmapg 8586 | Membership relation for se... |
| elmapd 8587 | Deduction form of ~ elmapg... |
| mapdm0 8588 | The empty set is the only ... |
| elpmg 8589 | The predicate "is a partia... |
| elpm2g 8590 | The predicate "is a partia... |
| elpm2r 8591 | Sufficient condition for b... |
| elpmi 8592 | A partial function is a fu... |
| pmfun 8593 | A partial function is a fu... |
| elmapex 8594 | Eliminate antecedent for m... |
| elmapi 8595 | A mapping is a function, f... |
| mapfset 8596 | If ` B ` is a set, the val... |
| mapssfset 8597 | The value of the set expon... |
| mapfoss 8598 | The value of the set expon... |
| fsetsspwxp 8599 | The class of all functions... |
| fset0 8600 | The set of functions from ... |
| fsetdmprc0 8601 | The set of functions with ... |
| fsetex 8602 | The set of functions betwe... |
| f1setex 8603 | The set of injections betw... |
| fosetex 8604 | The set of surjections bet... |
| f1osetex 8605 | The set of bijections betw... |
| fsetfcdm 8606 | The class of functions wit... |
| fsetfocdm 8607 | The class of functions wit... |
| fsetprcnex 8608 | The class of all functions... |
| fsetcdmex 8609 | The class of all functions... |
| fsetexb 8610 | The class of all functions... |
| elmapfn 8611 | A mapping is a function wi... |
| elmapfun 8612 | A mapping is always a func... |
| elmapssres 8613 | A restricted mapping is a ... |
| fpmg 8614 | A total function is a part... |
| pmss12g 8615 | Subset relation for the se... |
| pmresg 8616 | Elementhood of a restricte... |
| elmap 8617 | Membership relation for se... |
| mapval2 8618 | Alternate expression for t... |
| elpm 8619 | The predicate "is a partia... |
| elpm2 8620 | The predicate "is a partia... |
| fpm 8621 | A total function is a part... |
| mapsspm 8622 | Set exponentiation is a su... |
| pmsspw 8623 | Partial maps are a subset ... |
| mapsspw 8624 | Set exponentiation is a su... |
| mapfvd 8625 | The value of a function th... |
| elmapresaun 8626 | ~ fresaun transposed to ma... |
| fvmptmap 8627 | Special case of ~ fvmpt fo... |
| map0e 8628 | Set exponentiation with an... |
| map0b 8629 | Set exponentiation with an... |
| map0g 8630 | Set exponentiation is empt... |
| 0map0sn0 8631 | The set of mappings of the... |
| mapsnd 8632 | The value of set exponenti... |
| map0 8633 | Set exponentiation is empt... |
| mapsn 8634 | The value of set exponenti... |
| mapss 8635 | Subset inheritance for set... |
| fdiagfn 8636 | Functionality of the diago... |
| fvdiagfn 8637 | Functionality of the diago... |
| mapsnconst 8638 | Every singleton map is a c... |
| mapsncnv 8639 | Expression for the inverse... |
| mapsnf1o2 8640 | Explicit bijection between... |
| mapsnf1o3 8641 | Explicit bijection in the ... |
| ralxpmap 8642 | Quantification over functi... |
| dfixp 8645 | Eliminate the expression `... |
| ixpsnval 8646 | The value of an infinite C... |
| elixp2 8647 | Membership in an infinite ... |
| fvixp 8648 | Projection of a factor of ... |
| ixpfn 8649 | A nuple is a function. (C... |
| elixp 8650 | Membership in an infinite ... |
| elixpconst 8651 | Membership in an infinite ... |
| ixpconstg 8652 | Infinite Cartesian product... |
| ixpconst 8653 | Infinite Cartesian product... |
| ixpeq1 8654 | Equality theorem for infin... |
| ixpeq1d 8655 | Equality theorem for infin... |
| ss2ixp 8656 | Subclass theorem for infin... |
| ixpeq2 8657 | Equality theorem for infin... |
| ixpeq2dva 8658 | Equality theorem for infin... |
| ixpeq2dv 8659 | Equality theorem for infin... |
| cbvixp 8660 | Change bound variable in a... |
| cbvixpv 8661 | Change bound variable in a... |
| nfixpw 8662 | Bound-variable hypothesis ... |
| nfixp 8663 | Bound-variable hypothesis ... |
| nfixp1 8664 | The index variable in an i... |
| ixpprc 8665 | A cartesian product of pro... |
| ixpf 8666 | A member of an infinite Ca... |
| uniixp 8667 | The union of an infinite C... |
| ixpexg 8668 | The existence of an infini... |
| ixpin 8669 | The intersection of two in... |
| ixpiin 8670 | The indexed intersection o... |
| ixpint 8671 | The intersection of a coll... |
| ixp0x 8672 | An infinite Cartesian prod... |
| ixpssmap2g 8673 | An infinite Cartesian prod... |
| ixpssmapg 8674 | An infinite Cartesian prod... |
| 0elixp 8675 | Membership of the empty se... |
| ixpn0 8676 | The infinite Cartesian pro... |
| ixp0 8677 | The infinite Cartesian pro... |
| ixpssmap 8678 | An infinite Cartesian prod... |
| resixp 8679 | Restriction of an element ... |
| undifixp 8680 | Union of two projections o... |
| mptelixpg 8681 | Condition for an explicit ... |
| resixpfo 8682 | Restriction of elements of... |
| elixpsn 8683 | Membership in a class of s... |
| ixpsnf1o 8684 | A bijection between a clas... |
| mapsnf1o 8685 | A bijection between a set ... |
| boxriin 8686 | A rectangular subset of a ... |
| boxcutc 8687 | The relative complement of... |
| relen 8696 | Equinumerosity is a relati... |
| reldom 8697 | Dominance is a relation. ... |
| relsdom 8698 | Strict dominance is a rela... |
| encv 8699 | If two classes are equinum... |
| breng 8700 | Equinumerosity relation. ... |
| bren 8701 | Equinumerosity relation. ... |
| brenOLD 8702 | Obsolete version of ~ bren... |
| brdomg 8703 | Dominance relation. (Cont... |
| brdomi 8704 | Dominance relation. (Cont... |
| brdom 8705 | Dominance relation. (Cont... |
| domen 8706 | Dominance in terms of equi... |
| domeng 8707 | Dominance in terms of equi... |
| ctex 8708 | A countable set is a set. ... |
| f1oen3g 8709 | The domain and range of a ... |
| f1dom3g 8710 | The domain of a one-to-one... |
| f1oen2g 8711 | The domain and range of a ... |
| f1dom2g 8712 | The domain of a one-to-one... |
| f1dom2gOLD 8713 | Obsolete version of ~ f1do... |
| f1oeng 8714 | The domain and range of a ... |
| f1domg 8715 | The domain of a one-to-one... |
| f1oen 8716 | The domain and range of a ... |
| f1dom 8717 | The domain of a one-to-one... |
| brsdom 8718 | Strict dominance relation,... |
| isfi 8719 | Express " ` A ` is finite"... |
| enssdom 8720 | Equinumerosity implies dom... |
| dfdom2 8721 | Alternate definition of do... |
| endom 8722 | Equinumerosity implies dom... |
| sdomdom 8723 | Strict dominance implies d... |
| sdomnen 8724 | Strict dominance implies n... |
| brdom2 8725 | Dominance in terms of stri... |
| bren2 8726 | Equinumerosity expressed i... |
| enrefg 8727 | Equinumerosity is reflexiv... |
| enref 8728 | Equinumerosity is reflexiv... |
| eqeng 8729 | Equality implies equinumer... |
| domrefg 8730 | Dominance is reflexive. (... |
| en2d 8731 | Equinumerosity inference f... |
| en3d 8732 | Equinumerosity inference f... |
| en2i 8733 | Equinumerosity inference f... |
| en3i 8734 | Equinumerosity inference f... |
| dom2lem 8735 | A mapping (first hypothesi... |
| dom2d 8736 | A mapping (first hypothesi... |
| dom3d 8737 | A mapping (first hypothesi... |
| dom2 8738 | A mapping (first hypothesi... |
| dom3 8739 | A mapping (first hypothesi... |
| idssen 8740 | Equality implies equinumer... |
| ssdomg 8741 | A set dominates its subset... |
| ener 8742 | Equinumerosity is an equiv... |
| ensymb 8743 | Symmetry of equinumerosity... |
| ensym 8744 | Symmetry of equinumerosity... |
| ensymi 8745 | Symmetry of equinumerosity... |
| ensymd 8746 | Symmetry of equinumerosity... |
| entr 8747 | Transitivity of equinumero... |
| domtr 8748 | Transitivity of dominance ... |
| entri 8749 | A chained equinumerosity i... |
| entr2i 8750 | A chained equinumerosity i... |
| entr3i 8751 | A chained equinumerosity i... |
| entr4i 8752 | A chained equinumerosity i... |
| endomtr 8753 | Transitivity of equinumero... |
| domentr 8754 | Transitivity of dominance ... |
| f1imaeng 8755 | If a function is one-to-on... |
| f1imaen2g 8756 | If a function is one-to-on... |
| f1imaen 8757 | If a function is one-to-on... |
| en0 8758 | The empty set is equinumer... |
| en0OLD 8759 | Obsolete version of ~ en0 ... |
| en0ALT 8760 | Shorter proof of ~ en0 , d... |
| ensn1 8761 | A singleton is equinumerou... |
| ensn1OLD 8762 | Obsolete version of ~ ensn... |
| ensn1g 8763 | A singleton is equinumerou... |
| enpr1g 8764 | ` { A , A } ` has only one... |
| en1 8765 | A set is equinumerous to o... |
| en1OLD 8766 | Obsolete version of ~ en1 ... |
| en1b 8767 | A set is equinumerous to o... |
| en1bOLD 8768 | Obsolete version of ~ en1b... |
| reuen1 8769 | Two ways to express "exact... |
| euen1 8770 | Two ways to express "exact... |
| euen1b 8771 | Two ways to express " ` A ... |
| en1uniel 8772 | A singleton contains its s... |
| en1unielOLD 8773 | Obsolete version of ~ en1u... |
| 2dom 8774 | A set that dominates ordin... |
| fundmen 8775 | A function is equinumerous... |
| fundmeng 8776 | A function is equinumerous... |
| cnven 8777 | A relational set is equinu... |
| cnvct 8778 | If a set is countable, so ... |
| fndmeng 8779 | A function is equinumerate... |
| mapsnend 8780 | Set exponentiation to a si... |
| mapsnen 8781 | Set exponentiation to a si... |
| snmapen 8782 | Set exponentiation: a sing... |
| snmapen1 8783 | Set exponentiation: a sing... |
| map1 8784 | Set exponentiation: ordina... |
| en2sn 8785 | Two singletons are equinum... |
| en2snOLD 8786 | Obsolete version of ~ en2s... |
| en2snOLDOLD 8787 | Obsolete version of ~ en2s... |
| snfi 8788 | A singleton is finite. (C... |
| fiprc 8789 | The class of finite sets i... |
| unen 8790 | Equinumerosity of union of... |
| enrefnn 8791 | Equinumerosity is reflexiv... |
| enpr2d 8792 | A pair with distinct eleme... |
| ssct 8793 | Any subset of a countable ... |
| difsnen 8794 | All decrements of a set ar... |
| domdifsn 8795 | Dominance over a set with ... |
| xpsnen 8796 | A set is equinumerous to i... |
| xpsneng 8797 | A set is equinumerous to i... |
| xp1en 8798 | One times a cardinal numbe... |
| endisj 8799 | Any two sets are equinumer... |
| undom 8800 | Dominance law for union. ... |
| xpcomf1o 8801 | The canonical bijection fr... |
| xpcomco 8802 | Composition with the bijec... |
| xpcomen 8803 | Commutative law for equinu... |
| xpcomeng 8804 | Commutative law for equinu... |
| xpsnen2g 8805 | A set is equinumerous to i... |
| xpassen 8806 | Associative law for equinu... |
| xpdom2 8807 | Dominance law for Cartesia... |
| xpdom2g 8808 | Dominance law for Cartesia... |
| xpdom1g 8809 | Dominance law for Cartesia... |
| xpdom3 8810 | A set is dominated by its ... |
| xpdom1 8811 | Dominance law for Cartesia... |
| domunsncan 8812 | A singleton cancellation l... |
| omxpenlem 8813 | Lemma for ~ omxpen . (Con... |
| omxpen 8814 | The cardinal and ordinal p... |
| omf1o 8815 | Construct an explicit bije... |
| pw2f1olem 8816 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 8817 | The power set of a set is ... |
| pw2eng 8818 | The power set of a set is ... |
| pw2en 8819 | The power set of a set is ... |
| fopwdom 8820 | Covering implies injection... |
| enfixsn 8821 | Given two equipollent sets... |
| sucdom2 8822 | Strict dominance of a set ... |
| sbthlem1 8823 | Lemma for ~ sbth . (Contr... |
| sbthlem2 8824 | Lemma for ~ sbth . (Contr... |
| sbthlem3 8825 | Lemma for ~ sbth . (Contr... |
| sbthlem4 8826 | Lemma for ~ sbth . (Contr... |
| sbthlem5 8827 | Lemma for ~ sbth . (Contr... |
| sbthlem6 8828 | Lemma for ~ sbth . (Contr... |
| sbthlem7 8829 | Lemma for ~ sbth . (Contr... |
| sbthlem8 8830 | Lemma for ~ sbth . (Contr... |
| sbthlem9 8831 | Lemma for ~ sbth . (Contr... |
| sbthlem10 8832 | Lemma for ~ sbth . (Contr... |
| sbth 8833 | Schroeder-Bernstein Theore... |
| sbthb 8834 | Schroeder-Bernstein Theore... |
| sbthcl 8835 | Schroeder-Bernstein Theore... |
| dfsdom2 8836 | Alternate definition of st... |
| brsdom2 8837 | Alternate definition of st... |
| sdomnsym 8838 | Strict dominance is asymme... |
| domnsym 8839 | Theorem 22(i) of [Suppes] ... |
| 0domg 8840 | Any set dominates the empt... |
| dom0 8841 | A set dominated by the emp... |
| 0sdomg 8842 | A set strictly dominates t... |
| 0dom 8843 | Any set dominates the empt... |
| 0sdom 8844 | A set strictly dominates t... |
| sdom0 8845 | The empty set does not str... |
| sdomdomtr 8846 | Transitivity of strict dom... |
| sdomentr 8847 | Transitivity of strict dom... |
| domsdomtr 8848 | Transitivity of dominance ... |
| ensdomtr 8849 | Transitivity of equinumero... |
| sdomirr 8850 | Strict dominance is irrefl... |
| sdomtr 8851 | Strict dominance is transi... |
| sdomn2lp 8852 | Strict dominance has no 2-... |
| enen1 8853 | Equality-like theorem for ... |
| enen2 8854 | Equality-like theorem for ... |
| domen1 8855 | Equality-like theorem for ... |
| domen2 8856 | Equality-like theorem for ... |
| sdomen1 8857 | Equality-like theorem for ... |
| sdomen2 8858 | Equality-like theorem for ... |
| domtriord 8859 | Dominance is trichotomous ... |
| sdomel 8860 | For ordinals, strict domin... |
| sdomdif 8861 | The difference of a set fr... |
| onsdominel 8862 | An ordinal with more eleme... |
| domunsn 8863 | Dominance over a set with ... |
| fodomr 8864 | There exists a mapping fro... |
| pwdom 8865 | Injection of sets implies ... |
| canth2 8866 | Cantor's Theorem. No set ... |
| canth2g 8867 | Cantor's theorem with the ... |
| 2pwuninel 8868 | The power set of the power... |
| 2pwne 8869 | No set equals the power se... |
| disjen 8870 | A stronger form of ~ pwuni... |
| disjenex 8871 | Existence version of ~ dis... |
| domss2 8872 | A corollary of ~ disjenex ... |
| domssex2 8873 | A corollary of ~ disjenex ... |
| domssex 8874 | Weakening of ~ domssex2 to... |
| xpf1o 8875 | Construct a bijection on a... |
| xpen 8876 | Equinumerosity law for Car... |
| mapen 8877 | Two set exponentiations ar... |
| mapdom1 8878 | Order-preserving property ... |
| mapxpen 8879 | Equinumerosity law for dou... |
| xpmapenlem 8880 | Lemma for ~ xpmapen . (Co... |
| xpmapen 8881 | Equinumerosity law for set... |
| mapunen 8882 | Equinumerosity law for set... |
| map2xp 8883 | A cardinal power with expo... |
| mapdom2 8884 | Order-preserving property ... |
| mapdom3 8885 | Set exponentiation dominat... |
| pwen 8886 | If two sets are equinumero... |
| ssenen 8887 | Equinumerosity of equinume... |
| limenpsi 8888 | A limit ordinal is equinum... |
| limensuci 8889 | A limit ordinal is equinum... |
| limensuc 8890 | A limit ordinal is equinum... |
| infensuc 8891 | Any infinite ordinal is eq... |
| phplem1 8892 | Lemma for Pigeonhole Princ... |
| phplem2 8893 | Lemma for Pigeonhole Princ... |
| phplem3 8894 | Lemma for Pigeonhole Princ... |
| phplem4 8895 | Lemma for Pigeonhole Princ... |
| nneneq 8896 | Two equinumerous natural n... |
| php 8897 | Pigeonhole Principle. A n... |
| php2 8898 | Corollary of Pigeonhole Pr... |
| php3 8899 | Corollary of Pigeonhole Pr... |
| php4 8900 | Corollary of the Pigeonhol... |
| php5 8901 | Corollary of the Pigeonhol... |
| phpeqd 8902 | Corollary of the Pigeonhol... |
| snnen2o 8903 | A singleton ` { A } ` is n... |
| nndomog 8904 | Cardinal ordering agrees w... |
| dif1enlem 8905 | Lemma for ~ rexdif1en and ... |
| rexdif1en 8906 | If a set is equinumerous t... |
| dif1en 8907 | If a set ` A ` is equinume... |
| findcard 8908 | Schema for induction on th... |
| findcard2 8909 | Schema for induction on th... |
| findcard2s 8910 | Variation of ~ findcard2 r... |
| findcard2d 8911 | Deduction version of ~ fin... |
| nnfi 8912 | Natural numbers are finite... |
| pssnn 8913 | A proper subset of a natur... |
| ssnnfi 8914 | A subset of a natural numb... |
| ssnnfiOLD 8915 | Obsolete version of ~ ssnn... |
| 0fin 8916 | The empty set is finite. ... |
| unfi 8917 | The union of two finite se... |
| ssfi 8918 | A subset of a finite set i... |
| ssfiALT 8919 | Shorter proof of ~ ssfi us... |
| imafi 8920 | Images of finite sets are ... |
| pwfir 8921 | If the power set of a set ... |
| pwfilem 8922 | Lemma for ~ pwfi . (Contr... |
| pwfi 8923 | The power set of a finite ... |
| cnvfi 8924 | If a set is finite, its co... |
| fnfi 8925 | A version of ~ fnex for fi... |
| f1oenfi 8926 | If the domain of a one-to-... |
| f1oenfirn 8927 | If the range of a one-to-o... |
| f1domfi 8928 | If the codomain of a one-t... |
| enreffi 8929 | Equinumerosity is reflexiv... |
| ensymfib 8930 | Symmetry of equinumerosity... |
| entrfil 8931 | Transitivity of equinumero... |
| enfii 8932 | A set equinumerous to a fi... |
| enfi 8933 | Equinumerous sets have the... |
| enfiALT 8934 | Shorter proof of ~ enfi us... |
| domfi 8935 | A set dominated by a finit... |
| entrfi 8936 | Transitivity of equinumero... |
| entrfir 8937 | Transitivity of equinumero... |
| domtrfi 8938 | Transitivity of dominance ... |
| f1imaenfi 8939 | If a function is one-to-on... |
| ssdomfi 8940 | A finite set dominates its... |
| sbthfilem 8941 | Lemma for ~ sbthfi . (Con... |
| sbthfi 8942 | Schroeder-Bernstein Theore... |
| onomeneq 8943 | An ordinal number equinume... |
| onfin 8944 | An ordinal number is finit... |
| onfin2 8945 | A set is a natural number ... |
| nnfiOLD 8946 | Obsolete version of ~ nnfi... |
| nndomo 8947 | Cardinal ordering agrees w... |
| nnsdomo 8948 | Cardinal ordering agrees w... |
| sucdom 8949 | Strict dominance of a set ... |
| 0sdom1dom 8950 | Strict dominance over zero... |
| 1sdom2 8951 | Ordinal 1 is strictly domi... |
| sdom1 8952 | A set has less than one me... |
| modom 8953 | Two ways to express "at mo... |
| modom2 8954 | Two ways to express "at mo... |
| 1sdom 8955 | A set that strictly domina... |
| unxpdomlem1 8956 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 8957 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 8958 | Lemma for ~ unxpdom . (Co... |
| unxpdom 8959 | Cartesian product dominate... |
| unxpdom2 8960 | Corollary of ~ unxpdom . ... |
| sucxpdom 8961 | Cartesian product dominate... |
| pssinf 8962 | A set equinumerous to a pr... |
| fisseneq 8963 | A finite set is equal to i... |
| ominf 8964 | The set of natural numbers... |
| isinf 8965 | Any set that is not finite... |
| fineqvlem 8966 | Lemma for ~ fineqv . (Con... |
| fineqv 8967 | If the Axiom of Infinity i... |
| enfiiOLD 8968 | Obsolete version of ~ enfi... |
| pssnnOLD 8969 | Obsolete version of ~ pssn... |
| xpfir 8970 | The components of a nonemp... |
| ssfid 8971 | A subset of a finite set i... |
| infi 8972 | The intersection of two se... |
| rabfi 8973 | A restricted class built f... |
| finresfin 8974 | The restriction of a finit... |
| f1finf1o 8975 | Any injection from one fin... |
| nfielex 8976 | If a class is not finite, ... |
| en1eqsn 8977 | A set with one element is ... |
| en1eqsnbi 8978 | A set containing an elemen... |
| diffi 8979 | If ` A ` is finite, ` ( A ... |
| dif1enALT 8980 | Alternate proof of ~ dif1e... |
| enp1ilem 8981 | Lemma for uses of ~ enp1i ... |
| enp1i 8982 | Proof induction for ~ en2i... |
| en2 8983 | A set equinumerous to ordi... |
| en3 8984 | A set equinumerous to ordi... |
| en4 8985 | A set equinumerous to ordi... |
| findcard2OLD 8986 | Obsolete version of ~ find... |
| findcard3 8987 | Schema for strong inductio... |
| ac6sfi 8988 | A version of ~ ac6s for fi... |
| frfi 8989 | A partial order is well-fo... |
| fimax2g 8990 | A finite set has a maximum... |
| fimaxg 8991 | A finite set has a maximum... |
| fisupg 8992 | Lemma showing existence an... |
| wofi 8993 | A total order on a finite ... |
| ordunifi 8994 | The maximum of a finite co... |
| nnunifi 8995 | The union (supremum) of a ... |
| unblem1 8996 | Lemma for ~ unbnn . After... |
| unblem2 8997 | Lemma for ~ unbnn . The v... |
| unblem3 8998 | Lemma for ~ unbnn . The v... |
| unblem4 8999 | Lemma for ~ unbnn . The f... |
| unbnn 9000 | Any unbounded subset of na... |
| unbnn2 9001 | Version of ~ unbnn that do... |
| isfinite2 9002 | Any set strictly dominated... |
| nnsdomg 9003 | Omega strictly dominates a... |
| isfiniteg 9004 | A set is finite iff it is ... |
| infsdomnn 9005 | An infinite set strictly d... |
| infn0 9006 | An infinite set is not emp... |
| fin2inf 9007 | This (useless) theorem, wh... |
| unfilem1 9008 | Lemma for proving that the... |
| unfilem2 9009 | Lemma for proving that the... |
| unfilem3 9010 | Lemma for proving that the... |
| unfiOLD 9011 | Obsolete version of ~ unfi... |
| unfir 9012 | If a union is finite, the ... |
| unfi2 9013 | The union of two finite se... |
| difinf 9014 | An infinite set ` A ` minu... |
| xpfi 9015 | The Cartesian product of t... |
| 3xpfi 9016 | The Cartesian product of t... |
| domunfican 9017 | A finite set union cancell... |
| infcntss 9018 | Every infinite set has a d... |
| prfi 9019 | An unordered pair is finit... |
| tpfi 9020 | An unordered triple is fin... |
| fiint 9021 | Equivalent ways of stating... |
| fodomfi 9022 | An onto function implies d... |
| fodomfib 9023 | Equivalence of an onto map... |
| fofinf1o 9024 | Any surjection from one fi... |
| rneqdmfinf1o 9025 | Any function from a finite... |
| fidomdm 9026 | Any finite set dominates i... |
| dmfi 9027 | The domain of a finite set... |
| fundmfibi 9028 | A function is finite if an... |
| resfnfinfin 9029 | The restriction of a funct... |
| residfi 9030 | A restricted identity func... |
| cnvfiALT 9031 | Shorter proof of ~ cnvfi u... |
| rnfi 9032 | The range of a finite set ... |
| f1dmvrnfibi 9033 | A one-to-one function whos... |
| f1vrnfibi 9034 | A one-to-one function whic... |
| fofi 9035 | If a function has a finite... |
| f1fi 9036 | If a 1-to-1 function has a... |
| iunfi 9037 | The finite union of finite... |
| unifi 9038 | The finite union of finite... |
| unifi2 9039 | The finite union of finite... |
| infssuni 9040 | If an infinite set ` A ` i... |
| unirnffid 9041 | The union of the range of ... |
| imafiALT 9042 | Shorter proof of ~ imafi u... |
| pwfilemOLD 9043 | Obsolete version of ~ pwfi... |
| pwfiOLD 9044 | Obsolete version of ~ pwfi... |
| mapfi 9045 | Set exponentiation of fini... |
| ixpfi 9046 | A Cartesian product of fin... |
| ixpfi2 9047 | A Cartesian product of fin... |
| mptfi 9048 | A finite mapping set is fi... |
| abrexfi 9049 | An image set from a finite... |
| cnvimamptfin 9050 | A preimage of a mapping wi... |
| elfpw 9051 | Membership in a class of f... |
| unifpw 9052 | A set is the union of its ... |
| f1opwfi 9053 | A one-to-one mapping induc... |
| fissuni 9054 | A finite subset of a union... |
| fipreima 9055 | Given a finite subset ` A ... |
| finsschain 9056 | A finite subset of the uni... |
| indexfi 9057 | If for every element of a ... |
| relfsupp 9060 | The property of a function... |
| relprcnfsupp 9061 | A proper class is never fi... |
| isfsupp 9062 | The property of a class to... |
| funisfsupp 9063 | The property of a function... |
| fsuppimp 9064 | Implications of a class be... |
| fsuppimpd 9065 | A finitely supported funct... |
| fisuppfi 9066 | A function on a finite set... |
| fdmfisuppfi 9067 | The support of a function ... |
| fdmfifsupp 9068 | A function with a finite d... |
| fsuppmptdm 9069 | A mapping with a finite do... |
| fndmfisuppfi 9070 | The support of a function ... |
| fndmfifsupp 9071 | A function with a finite d... |
| suppeqfsuppbi 9072 | If two functions have the ... |
| suppssfifsupp 9073 | If the support of a functi... |
| fsuppsssupp 9074 | If the support of a functi... |
| fsuppxpfi 9075 | The cartesian product of t... |
| fczfsuppd 9076 | A constant function with v... |
| fsuppun 9077 | The union of two finitely ... |
| fsuppunfi 9078 | The union of the support o... |
| fsuppunbi 9079 | If the union of two classe... |
| 0fsupp 9080 | The empty set is a finitel... |
| snopfsupp 9081 | A singleton containing an ... |
| funsnfsupp 9082 | Finite support for a funct... |
| fsuppres 9083 | The restriction of a finit... |
| ressuppfi 9084 | If the support of the rest... |
| resfsupp 9085 | If the restriction of a fu... |
| resfifsupp 9086 | The restriction of a funct... |
| frnfsuppbi 9087 | Two ways of saying that a ... |
| fsuppmptif 9088 | A function mapping an argu... |
| sniffsupp 9089 | A function mapping all but... |
| fsuppcolem 9090 | Lemma for ~ fsuppco . For... |
| fsuppco 9091 | The composition of a 1-1 f... |
| fsuppco2 9092 | The composition of a funct... |
| fsuppcor 9093 | The composition of a funct... |
| mapfienlem1 9094 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 9095 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 9096 | Lemma 3 for ~ mapfien . (... |
| mapfien 9097 | A bijection of the base se... |
| mapfien2 9098 | Equinumerousity relation f... |
| fival 9101 | The set of all the finite ... |
| elfi 9102 | Specific properties of an ... |
| elfi2 9103 | The empty intersection nee... |
| elfir 9104 | Sufficient condition for a... |
| intrnfi 9105 | Sufficient condition for t... |
| iinfi 9106 | An indexed intersection of... |
| inelfi 9107 | The intersection of two se... |
| ssfii 9108 | Any element of a set ` A `... |
| fi0 9109 | The set of finite intersec... |
| fieq0 9110 | A set is empty iff the cla... |
| fiin 9111 | The elements of ` ( fi `` ... |
| dffi2 9112 | The set of finite intersec... |
| fiss 9113 | Subset relationship for fu... |
| inficl 9114 | A set which is closed unde... |
| fipwuni 9115 | The set of finite intersec... |
| fisn 9116 | A singleton is closed unde... |
| fiuni 9117 | The union of the finite in... |
| fipwss 9118 | If a set is a family of su... |
| elfiun 9119 | A finite intersection of e... |
| dffi3 9120 | The set of finite intersec... |
| fifo 9121 | Describe a surjection from... |
| marypha1lem 9122 | Core induction for Philip ... |
| marypha1 9123 | (Philip) Hall's marriage t... |
| marypha2lem1 9124 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 9125 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 9126 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 9127 | Lemma for ~ marypha2 . Pr... |
| marypha2 9128 | Version of ~ marypha1 usin... |
| dfsup2 9133 | Quantifier-free definition... |
| supeq1 9134 | Equality theorem for supre... |
| supeq1d 9135 | Equality deduction for sup... |
| supeq1i 9136 | Equality inference for sup... |
| supeq2 9137 | Equality theorem for supre... |
| supeq3 9138 | Equality theorem for supre... |
| supeq123d 9139 | Equality deduction for sup... |
| nfsup 9140 | Hypothesis builder for sup... |
| supmo 9141 | Any class ` B ` has at mos... |
| supexd 9142 | A supremum is a set. (Con... |
| supeu 9143 | A supremum is unique. Sim... |
| supval2 9144 | Alternate expression for t... |
| eqsup 9145 | Sufficient condition for a... |
| eqsupd 9146 | Sufficient condition for a... |
| supcl 9147 | A supremum belongs to its ... |
| supub 9148 | A supremum is an upper bou... |
| suplub 9149 | A supremum is the least up... |
| suplub2 9150 | Bidirectional form of ~ su... |
| supnub 9151 | An upper bound is not less... |
| supex 9152 | A supremum is a set. (Con... |
| sup00 9153 | The supremum under an empt... |
| sup0riota 9154 | The supremum of an empty s... |
| sup0 9155 | The supremum of an empty s... |
| supmax 9156 | The greatest element of a ... |
| fisup2g 9157 | A finite set satisfies the... |
| fisupcl 9158 | A nonempty finite set cont... |
| supgtoreq 9159 | The supremum of a finite s... |
| suppr 9160 | The supremum of a pair. (... |
| supsn 9161 | The supremum of a singleto... |
| supisolem 9162 | Lemma for ~ supiso . (Con... |
| supisoex 9163 | Lemma for ~ supiso . (Con... |
| supiso 9164 | Image of a supremum under ... |
| infeq1 9165 | Equality theorem for infim... |
| infeq1d 9166 | Equality deduction for inf... |
| infeq1i 9167 | Equality inference for inf... |
| infeq2 9168 | Equality theorem for infim... |
| infeq3 9169 | Equality theorem for infim... |
| infeq123d 9170 | Equality deduction for inf... |
| nfinf 9171 | Hypothesis builder for inf... |
| infexd 9172 | An infimum is a set. (Con... |
| eqinf 9173 | Sufficient condition for a... |
| eqinfd 9174 | Sufficient condition for a... |
| infval 9175 | Alternate expression for t... |
| infcllem 9176 | Lemma for ~ infcl , ~ infl... |
| infcl 9177 | An infimum belongs to its ... |
| inflb 9178 | An infimum is a lower boun... |
| infglb 9179 | An infimum is the greatest... |
| infglbb 9180 | Bidirectional form of ~ in... |
| infnlb 9181 | A lower bound is not great... |
| infex 9182 | An infimum is a set. (Con... |
| infmin 9183 | The smallest element of a ... |
| infmo 9184 | Any class ` B ` has at mos... |
| infeu 9185 | An infimum is unique. (Co... |
| fimin2g 9186 | A finite set has a minimum... |
| fiming 9187 | A finite set has a minimum... |
| fiinfg 9188 | Lemma showing existence an... |
| fiinf2g 9189 | A finite set satisfies the... |
| fiinfcl 9190 | A nonempty finite set cont... |
| infltoreq 9191 | The infimum of a finite se... |
| infpr 9192 | The infimum of a pair. (C... |
| infsupprpr 9193 | The infimum of a proper pa... |
| infsn 9194 | The infimum of a singleton... |
| inf00 9195 | The infimum regarding an e... |
| infempty 9196 | The infimum of an empty se... |
| infiso 9197 | Image of an infimum under ... |
| dfoi 9200 | Rewrite ~ df-oi with abbre... |
| oieq1 9201 | Equality theorem for ordin... |
| oieq2 9202 | Equality theorem for ordin... |
| nfoi 9203 | Hypothesis builder for ord... |
| ordiso2 9204 | Generalize ~ ordiso to pro... |
| ordiso 9205 | Order-isomorphic ordinal n... |
| ordtypecbv 9206 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 9207 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 9208 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 9209 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 9210 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 9211 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 9212 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 9213 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 9214 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 9215 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 9216 | Lemma for ~ ordtype . Usi... |
| oi0 9217 | Definition of the ordinal ... |
| oicl 9218 | The order type of the well... |
| oif 9219 | The order isomorphism of t... |
| oiiso2 9220 | The order isomorphism of t... |
| ordtype 9221 | For any set-like well-orde... |
| oiiniseg 9222 | ` ran F ` is an initial se... |
| ordtype2 9223 | For any set-like well-orde... |
| oiexg 9224 | The order isomorphism on a... |
| oion 9225 | The order type of the well... |
| oiiso 9226 | The order isomorphism of t... |
| oien 9227 | The order type of a well-o... |
| oieu 9228 | Uniqueness of the unique o... |
| oismo 9229 | When ` A ` is a subclass o... |
| oiid 9230 | The order type of an ordin... |
| hartogslem1 9231 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9232 | Lemma for ~ hartogs . (Co... |
| hartogs 9233 | The class of ordinals domi... |
| wofib 9234 | The only sets which are we... |
| wemaplem1 9235 | Value of the lexicographic... |
| wemaplem2 9236 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9237 | Lemma for ~ wemapso . Tra... |
| wemappo 9238 | Construct lexicographic or... |
| wemapsolem 9239 | Lemma for ~ wemapso . (Co... |
| wemapso 9240 | Construct lexicographic or... |
| wemapso2lem 9241 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9242 | An alternative to having a... |
| card2on 9243 | The alternate definition o... |
| card2inf 9244 | The alternate definition o... |
| harf 9247 | Functionality of the Harto... |
| harcl 9248 | Values of the Hartogs func... |
| harval 9249 | Function value of the Hart... |
| elharval 9250 | The Hartogs number of a se... |
| harndom 9251 | The Hartogs number of a se... |
| harword 9252 | Weak ordering property of ... |
| relwdom 9255 | Weak dominance is a relati... |
| brwdom 9256 | Property of weak dominance... |
| brwdomi 9257 | Property of weak dominance... |
| brwdomn0 9258 | Weak dominance over nonemp... |
| 0wdom 9259 | Any set weakly dominates t... |
| fowdom 9260 | An onto function implies w... |
| wdomref 9261 | Reflexivity of weak domina... |
| brwdom2 9262 | Alternate characterization... |
| domwdom 9263 | Weak dominance is implied ... |
| wdomtr 9264 | Transitivity of weak domin... |
| wdomen1 9265 | Equality-like theorem for ... |
| wdomen2 9266 | Equality-like theorem for ... |
| wdompwdom 9267 | Weak dominance strengthens... |
| canthwdom 9268 | Cantor's Theorem, stated u... |
| wdom2d 9269 | Deduce weak dominance from... |
| wdomd 9270 | Deduce weak dominance from... |
| brwdom3 9271 | Condition for weak dominan... |
| brwdom3i 9272 | Weak dominance implies exi... |
| unwdomg 9273 | Weak dominance of a (disjo... |
| xpwdomg 9274 | Weak dominance of a Cartes... |
| wdomima2g 9275 | A set is weakly dominant o... |
| wdomimag 9276 | A set is weakly dominant o... |
| unxpwdom2 9277 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9278 | If a Cartesian product is ... |
| ixpiunwdom 9279 | Describe an onto function ... |
| harwdom 9280 | The value of the Hartogs f... |
| axreg2 9282 | Axiom of Regularity expres... |
| zfregcl 9283 | The Axiom of Regularity wi... |
| zfreg 9284 | The Axiom of Regularity us... |
| elirrv 9285 | The membership relation is... |
| elirr 9286 | No class is a member of it... |
| elneq 9287 | A class is not equal to an... |
| nelaneq 9288 | A class is not an element ... |
| epinid0 9289 | The membership relation an... |
| sucprcreg 9290 | A class is equal to its su... |
| ruv 9291 | The Russell class is equal... |
| ruALT 9292 | Alternate proof of ~ ru , ... |
| zfregfr 9293 | The membership relation is... |
| en2lp 9294 | No class has 2-cycle membe... |
| elnanel 9295 | Two classes are not elemen... |
| cnvepnep 9296 | The membership (epsilon) r... |
| epnsym 9297 | The membership (epsilon) r... |
| elnotel 9298 | A class cannot be an eleme... |
| elnel 9299 | A class cannot be an eleme... |
| en3lplem1 9300 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9301 | Lemma for ~ en3lp . (Cont... |
| en3lp 9302 | No class has 3-cycle membe... |
| preleqg 9303 | Equality of two unordered ... |
| preleq 9304 | Equality of two unordered ... |
| preleqALT 9305 | Alternate proof of ~ prele... |
| opthreg 9306 | Theorem for alternate repr... |
| suc11reg 9307 | The successor operation be... |
| dford2 9308 | Assuming ~ ax-reg , an ord... |
| inf0 9309 | Existence of ` _om ` impli... |
| inf1 9310 | Variation of Axiom of Infi... |
| inf2 9311 | Variation of Axiom of Infi... |
| inf3lema 9312 | Lemma for our Axiom of Inf... |
| inf3lemb 9313 | Lemma for our Axiom of Inf... |
| inf3lemc 9314 | Lemma for our Axiom of Inf... |
| inf3lemd 9315 | Lemma for our Axiom of Inf... |
| inf3lem1 9316 | Lemma for our Axiom of Inf... |
| inf3lem2 9317 | Lemma for our Axiom of Inf... |
| inf3lem3 9318 | Lemma for our Axiom of Inf... |
| inf3lem4 9319 | Lemma for our Axiom of Inf... |
| inf3lem5 9320 | Lemma for our Axiom of Inf... |
| inf3lem6 9321 | Lemma for our Axiom of Inf... |
| inf3lem7 9322 | Lemma for our Axiom of Inf... |
| inf3 9323 | Our Axiom of Infinity ~ ax... |
| infeq5i 9324 | Half of ~ infeq5 . (Contr... |
| infeq5 9325 | The statement "there exist... |
| zfinf 9327 | Axiom of Infinity expresse... |
| axinf2 9328 | A standard version of Axio... |
| zfinf2 9330 | A standard version of the ... |
| omex 9331 | The existence of omega (th... |
| axinf 9332 | The first version of the A... |
| inf5 9333 | The statement "there exist... |
| omelon 9334 | Omega is an ordinal number... |
| dfom3 9335 | The class of natural numbe... |
| elom3 9336 | A simplification of ~ elom... |
| dfom4 9337 | A simplification of ~ df-o... |
| dfom5 9338 | ` _om ` is the smallest li... |
| oancom 9339 | Ordinal addition is not co... |
| isfinite 9340 | A set is finite iff it is ... |
| fict 9341 | A finite set is countable ... |
| nnsdom 9342 | A natural number is strict... |
| omenps 9343 | Omega is equinumerous to a... |
| omensuc 9344 | The set of natural numbers... |
| infdifsn 9345 | Removing a singleton from ... |
| infdiffi 9346 | Removing a finite set from... |
| unbnn3 9347 | Any unbounded subset of na... |
| noinfep 9348 | Using the Axiom of Regular... |
| cantnffval 9351 | The value of the Cantor no... |
| cantnfdm 9352 | The domain of the Cantor n... |
| cantnfvalf 9353 | Lemma for ~ cantnf . The ... |
| cantnfs 9354 | Elementhood in the set of ... |
| cantnfcl 9355 | Basic properties of the or... |
| cantnfval 9356 | The value of the Cantor no... |
| cantnfval2 9357 | Alternate expression for t... |
| cantnfsuc 9358 | The value of the recursive... |
| cantnfle 9359 | A lower bound on the ` CNF... |
| cantnflt 9360 | An upper bound on the part... |
| cantnflt2 9361 | An upper bound on the ` CN... |
| cantnff 9362 | The ` CNF ` function is a ... |
| cantnf0 9363 | The value of the zero func... |
| cantnfrescl 9364 | A function is finitely sup... |
| cantnfres 9365 | The ` CNF ` function respe... |
| cantnfp1lem1 9366 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9367 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9368 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9369 | If ` F ` is created by add... |
| oemapso 9370 | The relation ` T ` is a st... |
| oemapval 9371 | Value of the relation ` T ... |
| oemapvali 9372 | If ` F < G ` , then there ... |
| cantnflem1a 9373 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9374 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9375 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9376 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9377 | Lemma for ~ cantnf . This... |
| cantnflem2 9378 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9379 | Lemma for ~ cantnf . Here... |
| cantnflem4 9380 | Lemma for ~ cantnf . Comp... |
| cantnf 9381 | The Cantor Normal Form the... |
| oemapwe 9382 | The lexicographic order on... |
| cantnffval2 9383 | An alternate definition of... |
| cantnff1o 9384 | Simplify the isomorphism o... |
| wemapwe 9385 | Construct lexicographic or... |
| oef1o 9386 | A bijection of the base se... |
| cnfcomlem 9387 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9388 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9389 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9390 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9391 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9392 | Any infinite ordinal ` B `... |
| cnfcom3clem 9393 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9394 | Wrap the construction of ~... |
| dftrpred2 9397 | A definition of the transi... |
| trpredeq1 9398 | Equality theorem for trans... |
| trpredeq2 9399 | Equality theorem for trans... |
| trpredeq3 9400 | Equality theorem for trans... |
| trpredeq1d 9401 | Equality deduction for tra... |
| trpredeq2d 9402 | Equality deduction for tra... |
| trpredeq3d 9403 | Equality deduction for tra... |
| eltrpred 9404 | A class is a transitive pr... |
| trpredlem1 9405 | Technical lemma for transi... |
| trpredpred 9406 | Assuming it is a set, the ... |
| trpredss 9407 | The transitive predecessor... |
| trpredtr 9408 | Predecessors of a transiti... |
| trpredmintr 9409 | The transitive predecessor... |
| trpred0 9410 | The class of transitive pr... |
| trpredelss 9411 | Given a transitive predece... |
| dftrpred3g 9412 | The transitive predecessor... |
| dftrpred4g 9413 | Another recursive expressi... |
| trpredpo 9414 | If ` R ` partially orders ... |
| trpredrec 9415 | A transitive predecessor o... |
| trpredex 9416 | The transitive predecessor... |
| trcl 9417 | For any set ` A ` , show t... |
| tz9.1 9418 | Every set has a transitive... |
| tz9.1c 9419 | Alternate expression for t... |
| epfrs 9420 | The strong form of the Axi... |
| zfregs 9421 | The strong form of the Axi... |
| zfregs2 9422 | Alternate strong form of t... |
| setind 9423 | Set (epsilon) induction. ... |
| setind2 9424 | Set (epsilon) induction, s... |
| tcvalg 9427 | Value of the transitive cl... |
| tcid 9428 | Defining property of the t... |
| tctr 9429 | Defining property of the t... |
| tcmin 9430 | Defining property of the t... |
| tc2 9431 | A variant of the definitio... |
| tcsni 9432 | The transitive closure of ... |
| tcss 9433 | The transitive closure fun... |
| tcel 9434 | The transitive closure fun... |
| tcidm 9435 | The transitive closure fun... |
| tc0 9436 | The transitive closure of ... |
| tc00 9437 | The transitive closure is ... |
| frmin 9438 | Every (possibly proper) su... |
| frind 9439 | A subclass of a well-found... |
| frinsg 9440 | Well-Founded Induction Sch... |
| frins 9441 | Well-Founded Induction Sch... |
| frins2f 9442 | Well-Founded Induction sch... |
| frins2 9443 | Well-Founded Induction sch... |
| frins3 9444 | Well-Founded Induction sch... |
| frr3g 9445 | Functions defined by well-... |
| frrlem15 9446 | Lemma for general well-fou... |
| frrlem16 9447 | Lemma for general well-fou... |
| frr1 9448 | Law of general well-founde... |
| frr2 9449 | Law of general well-founde... |
| frr3 9450 | Law of general well-founde... |
| r1funlim 9455 | The cumulative hierarchy o... |
| r1fnon 9456 | The cumulative hierarchy o... |
| r10 9457 | Value of the cumulative hi... |
| r1sucg 9458 | Value of the cumulative hi... |
| r1suc 9459 | Value of the cumulative hi... |
| r1limg 9460 | Value of the cumulative hi... |
| r1lim 9461 | Value of the cumulative hi... |
| r1fin 9462 | The first ` _om ` levels o... |
| r1sdom 9463 | Each stage in the cumulati... |
| r111 9464 | The cumulative hierarchy i... |
| r1tr 9465 | The cumulative hierarchy o... |
| r1tr2 9466 | The union of a cumulative ... |
| r1ordg 9467 | Ordering relation for the ... |
| r1ord3g 9468 | Ordering relation for the ... |
| r1ord 9469 | Ordering relation for the ... |
| r1ord2 9470 | Ordering relation for the ... |
| r1ord3 9471 | Ordering relation for the ... |
| r1sssuc 9472 | The value of the cumulativ... |
| r1pwss 9473 | Each set of the cumulative... |
| r1sscl 9474 | Each set of the cumulative... |
| r1val1 9475 | The value of the cumulativ... |
| tz9.12lem1 9476 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9477 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9478 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9479 | A set is well-founded if a... |
| tz9.13 9480 | Every set is well-founded,... |
| tz9.13g 9481 | Every set is well-founded,... |
| rankwflemb 9482 | Two ways of saying a set i... |
| rankf 9483 | The domain and range of th... |
| rankon 9484 | The rank of a set is an or... |
| r1elwf 9485 | Any member of the cumulati... |
| rankvalb 9486 | Value of the rank function... |
| rankr1ai 9487 | One direction of ~ rankr1a... |
| rankvaln 9488 | Value of the rank function... |
| rankidb 9489 | Identity law for the rank ... |
| rankdmr1 9490 | A rank is a member of the ... |
| rankr1ag 9491 | A version of ~ rankr1a tha... |
| rankr1bg 9492 | A relationship between ran... |
| r1rankidb 9493 | Any set is a subset of the... |
| r1elssi 9494 | The range of the ` R1 ` fu... |
| r1elss 9495 | The range of the ` R1 ` fu... |
| pwwf 9496 | A power set is well-founde... |
| sswf 9497 | A subset of a well-founded... |
| snwf 9498 | A singleton is well-founde... |
| unwf 9499 | A binary union is well-fou... |
| prwf 9500 | An unordered pair is well-... |
| opwf 9501 | An ordered pair is well-fo... |
| unir1 9502 | The cumulative hierarchy o... |
| jech9.3 9503 | Every set belongs to some ... |
| rankwflem 9504 | Every set is well-founded,... |
| rankval 9505 | Value of the rank function... |
| rankvalg 9506 | Value of the rank function... |
| rankval2 9507 | Value of an alternate defi... |
| uniwf 9508 | A union is well-founded if... |
| rankr1clem 9509 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9510 | A relationship between the... |
| rankidn 9511 | A relationship between the... |
| rankpwi 9512 | The rank of a power set. ... |
| rankelb 9513 | The membership relation is... |
| wfelirr 9514 | A well-founded set is not ... |
| rankval3b 9515 | The value of the rank func... |
| ranksnb 9516 | The rank of a singleton. ... |
| rankonidlem 9517 | Lemma for ~ rankonid . (C... |
| rankonid 9518 | The rank of an ordinal num... |
| onwf 9519 | The ordinals are all well-... |
| onssr1 9520 | Initial segments of the or... |
| rankr1g 9521 | A relationship between the... |
| rankid 9522 | Identity law for the rank ... |
| rankr1 9523 | A relationship between the... |
| ssrankr1 9524 | A relationship between an ... |
| rankr1a 9525 | A relationship between ran... |
| r1val2 9526 | The value of the cumulativ... |
| r1val3 9527 | The value of the cumulativ... |
| rankel 9528 | The membership relation is... |
| rankval3 9529 | The value of the rank func... |
| bndrank 9530 | Any class whose elements h... |
| unbndrank 9531 | The elements of a proper c... |
| rankpw 9532 | The rank of a power set. ... |
| ranklim 9533 | The rank of a set belongs ... |
| r1pw 9534 | A stronger property of ` R... |
| r1pwALT 9535 | Alternate shorter proof of... |
| r1pwcl 9536 | The cumulative hierarchy o... |
| rankssb 9537 | The subset relation is inh... |
| rankss 9538 | The subset relation is inh... |
| rankunb 9539 | The rank of the union of t... |
| rankprb 9540 | The rank of an unordered p... |
| rankopb 9541 | The rank of an ordered pai... |
| rankuni2b 9542 | The value of the rank func... |
| ranksn 9543 | The rank of a singleton. ... |
| rankuni2 9544 | The rank of a union. Part... |
| rankun 9545 | The rank of the union of t... |
| rankpr 9546 | The rank of an unordered p... |
| rankop 9547 | The rank of an ordered pai... |
| r1rankid 9548 | Any set is a subset of the... |
| rankeq0b 9549 | A set is empty iff its ran... |
| rankeq0 9550 | A set is empty iff its ran... |
| rankr1id 9551 | The rank of the hierarchy ... |
| rankuni 9552 | The rank of a union. Part... |
| rankr1b 9553 | A relationship between ran... |
| ranksuc 9554 | The rank of a successor. ... |
| rankuniss 9555 | Upper bound of the rank of... |
| rankval4 9556 | The rank of a set is the s... |
| rankbnd 9557 | The rank of a set is bound... |
| rankbnd2 9558 | The rank of a set is bound... |
| rankc1 9559 | A relationship that can be... |
| rankc2 9560 | A relationship that can be... |
| rankelun 9561 | Rank membership is inherit... |
| rankelpr 9562 | Rank membership is inherit... |
| rankelop 9563 | Rank membership is inherit... |
| rankxpl 9564 | A lower bound on the rank ... |
| rankxpu 9565 | An upper bound on the rank... |
| rankfu 9566 | An upper bound on the rank... |
| rankmapu 9567 | An upper bound on the rank... |
| rankxplim 9568 | The rank of a Cartesian pr... |
| rankxplim2 9569 | If the rank of a Cartesian... |
| rankxplim3 9570 | The rank of a Cartesian pr... |
| rankxpsuc 9571 | The rank of a Cartesian pr... |
| tcwf 9572 | The transitive closure fun... |
| tcrank 9573 | This theorem expresses two... |
| scottex 9574 | Scott's trick collects all... |
| scott0 9575 | Scott's trick collects all... |
| scottexs 9576 | Theorem scheme version of ... |
| scott0s 9577 | Theorem scheme version of ... |
| cplem1 9578 | Lemma for the Collection P... |
| cplem2 9579 | Lemma for the Collection P... |
| cp 9580 | Collection Principle. Thi... |
| bnd 9581 | A very strong generalizati... |
| bnd2 9582 | A variant of the Boundedne... |
| kardex 9583 | The collection of all sets... |
| karden 9584 | If we allow the Axiom of R... |
| htalem 9585 | Lemma for defining an emul... |
| hta 9586 | A ZFC emulation of Hilbert... |
| djueq12 9593 | Equality theorem for disjo... |
| djueq1 9594 | Equality theorem for disjo... |
| djueq2 9595 | Equality theorem for disjo... |
| nfdju 9596 | Bound-variable hypothesis ... |
| djuex 9597 | The disjoint union of sets... |
| djuexb 9598 | The disjoint union of two ... |
| djulcl 9599 | Left closure of disjoint u... |
| djurcl 9600 | Right closure of disjoint ... |
| djulf1o 9601 | The left injection functio... |
| djurf1o 9602 | The right injection functi... |
| inlresf 9603 | The left injection restric... |
| inlresf1 9604 | The left injection restric... |
| inrresf 9605 | The right injection restri... |
| inrresf1 9606 | The right injection restri... |
| djuin 9607 | The images of any classes ... |
| djur 9608 | A member of a disjoint uni... |
| djuss 9609 | A disjoint union is a subc... |
| djuunxp 9610 | The union of a disjoint un... |
| djuexALT 9611 | Alternate proof of ~ djuex... |
| eldju1st 9612 | The first component of an ... |
| eldju2ndl 9613 | The second component of an... |
| eldju2ndr 9614 | The second component of an... |
| djuun 9615 | The disjoint union of two ... |
| 1stinl 9616 | The first component of the... |
| 2ndinl 9617 | The second component of th... |
| 1stinr 9618 | The first component of the... |
| 2ndinr 9619 | The second component of th... |
| updjudhf 9620 | The mapping of an element ... |
| updjudhcoinlf 9621 | The composition of the map... |
| updjudhcoinrg 9622 | The composition of the map... |
| updjud 9623 | Universal property of the ... |
| cardf2 9632 | The cardinality function i... |
| cardon 9633 | The cardinal number of a s... |
| isnum2 9634 | A way to express well-orde... |
| isnumi 9635 | A set equinumerous to an o... |
| ennum 9636 | Equinumerous sets are equi... |
| finnum 9637 | Every finite set is numera... |
| onenon 9638 | Every ordinal number is nu... |
| tskwe 9639 | A Tarski set is well-order... |
| xpnum 9640 | The cartesian product of n... |
| cardval3 9641 | An alternate definition of... |
| cardid2 9642 | Any numerable set is equin... |
| isnum3 9643 | A set is numerable iff it ... |
| oncardval 9644 | The value of the cardinal ... |
| oncardid 9645 | Any ordinal number is equi... |
| cardonle 9646 | The cardinal of an ordinal... |
| card0 9647 | The cardinality of the emp... |
| cardidm 9648 | The cardinality function i... |
| oncard 9649 | A set is a cardinal number... |
| ficardom 9650 | The cardinal number of a f... |
| ficardid 9651 | A finite set is equinumero... |
| cardnn 9652 | The cardinality of a natur... |
| cardnueq0 9653 | The empty set is the only ... |
| cardne 9654 | No member of a cardinal nu... |
| carden2a 9655 | If two sets have equal non... |
| carden2b 9656 | If two sets are equinumero... |
| card1 9657 | A set has cardinality one ... |
| cardsn 9658 | A singleton has cardinalit... |
| carddomi2 9659 | Two sets have the dominanc... |
| sdomsdomcardi 9660 | A set strictly dominates i... |
| cardlim 9661 | An infinite cardinal is a ... |
| cardsdomelir 9662 | A cardinal strictly domina... |
| cardsdomel 9663 | A cardinal strictly domina... |
| iscard 9664 | Two ways to express the pr... |
| iscard2 9665 | Two ways to express the pr... |
| carddom2 9666 | Two numerable sets have th... |
| harcard 9667 | The class of ordinal numbe... |
| cardprclem 9668 | Lemma for ~ cardprc . (Co... |
| cardprc 9669 | The class of all cardinal ... |
| carduni 9670 | The union of a set of card... |
| cardiun 9671 | The indexed union of a set... |
| cardennn 9672 | If ` A ` is equinumerous t... |
| cardsucinf 9673 | The cardinality of the suc... |
| cardsucnn 9674 | The cardinality of the suc... |
| cardom 9675 | The set of natural numbers... |
| carden2 9676 | Two numerable sets are equ... |
| cardsdom2 9677 | A numerable set is strictl... |
| domtri2 9678 | Trichotomy of dominance fo... |
| nnsdomel 9679 | Strict dominance and eleme... |
| cardval2 9680 | An alternate version of th... |
| isinffi 9681 | An infinite set contains s... |
| fidomtri 9682 | Trichotomy of dominance wi... |
| fidomtri2 9683 | Trichotomy of dominance wi... |
| harsdom 9684 | The Hartogs number of a we... |
| onsdom 9685 | Any well-orderable set is ... |
| harval2 9686 | An alternate expression fo... |
| harsucnn 9687 | The next cardinal after a ... |
| cardmin2 9688 | The smallest ordinal that ... |
| pm54.43lem 9689 | In Theorem *54.43 of [Whit... |
| pm54.43 9690 | Theorem *54.43 of [Whitehe... |
| pr2nelem 9691 | Lemma for ~ pr2ne . (Cont... |
| pr2ne 9692 | If an unordered pair has t... |
| prdom2 9693 | An unordered pair has at m... |
| en2eqpr 9694 | Building a set with two el... |
| en2eleq 9695 | Express a set of pair card... |
| en2other2 9696 | Taking the other element t... |
| dif1card 9697 | The cardinality of a nonem... |
| leweon 9698 | Lexicographical order is a... |
| r0weon 9699 | A set-like well-ordering o... |
| infxpenlem 9700 | Lemma for ~ infxpen . (Co... |
| infxpen 9701 | Every infinite ordinal is ... |
| xpomen 9702 | The Cartesian product of o... |
| xpct 9703 | The cartesian product of t... |
| infxpidm2 9704 | Every infinite well-ordera... |
| infxpenc 9705 | A canonical version of ~ i... |
| infxpenc2lem1 9706 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 9707 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 9708 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 9709 | Existence form of ~ infxpe... |
| iunmapdisj 9710 | The union ` U_ n e. C ( A ... |
| fseqenlem1 9711 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 9712 | Lemma for ~ fseqen . (Con... |
| fseqdom 9713 | One half of ~ fseqen . (C... |
| fseqen 9714 | A set that is equinumerous... |
| infpwfidom 9715 | The collection of finite s... |
| dfac8alem 9716 | Lemma for ~ dfac8a . If t... |
| dfac8a 9717 | Numeration theorem: every ... |
| dfac8b 9718 | The well-ordering theorem:... |
| dfac8clem 9719 | Lemma for ~ dfac8c . (Con... |
| dfac8c 9720 | If the union of a set is w... |
| ac10ct 9721 | A proof of the well-orderi... |
| ween 9722 | A set is numerable iff it ... |
| ac5num 9723 | A version of ~ ac5b with t... |
| ondomen 9724 | If a set is dominated by a... |
| numdom 9725 | A set dominated by a numer... |
| ssnum 9726 | A subset of a numerable se... |
| onssnum 9727 | All subsets of the ordinal... |
| indcardi 9728 | Indirect strong induction ... |
| acnrcl 9729 | Reverse closure for the ch... |
| acneq 9730 | Equality theorem for the c... |
| isacn 9731 | The property of being a ch... |
| acni 9732 | The property of being a ch... |
| acni2 9733 | The property of being a ch... |
| acni3 9734 | The property of being a ch... |
| acnlem 9735 | Construct a mapping satisf... |
| numacn 9736 | A well-orderable set has c... |
| finacn 9737 | Every set has finite choic... |
| acndom 9738 | A set with long choice seq... |
| acnnum 9739 | A set ` X ` which has choi... |
| acnen 9740 | The class of choice sets o... |
| acndom2 9741 | A set smaller than one wit... |
| acnen2 9742 | The class of sets with cho... |
| fodomacn 9743 | A version of ~ fodom that ... |
| fodomnum 9744 | A version of ~ fodom that ... |
| fonum 9745 | A surjection maps numerabl... |
| numwdom 9746 | A surjection maps numerabl... |
| fodomfi2 9747 | Onto functions define domi... |
| wdomfil 9748 | Weak dominance agrees with... |
| infpwfien 9749 | Any infinite well-orderabl... |
| inffien 9750 | The set of finite intersec... |
| wdomnumr 9751 | Weak dominance agrees with... |
| alephfnon 9752 | The aleph function is a fu... |
| aleph0 9753 | The first infinite cardina... |
| alephlim 9754 | Value of the aleph functio... |
| alephsuc 9755 | Value of the aleph functio... |
| alephon 9756 | An aleph is an ordinal num... |
| alephcard 9757 | Every aleph is a cardinal ... |
| alephnbtwn 9758 | No cardinal can be sandwic... |
| alephnbtwn2 9759 | No set has equinumerosity ... |
| alephordilem1 9760 | Lemma for ~ alephordi . (... |
| alephordi 9761 | Strict ordering property o... |
| alephord 9762 | Ordering property of the a... |
| alephord2 9763 | Ordering property of the a... |
| alephord2i 9764 | Ordering property of the a... |
| alephord3 9765 | Ordering property of the a... |
| alephsucdom 9766 | A set dominated by an alep... |
| alephsuc2 9767 | An alternate representatio... |
| alephdom 9768 | Relationship between inclu... |
| alephgeom 9769 | Every aleph is greater tha... |
| alephislim 9770 | Every aleph is a limit ord... |
| aleph11 9771 | The aleph function is one-... |
| alephf1 9772 | The aleph function is a on... |
| alephsdom 9773 | If an ordinal is smaller t... |
| alephdom2 9774 | A dominated initial ordina... |
| alephle 9775 | The argument of the aleph ... |
| cardaleph 9776 | Given any transfinite card... |
| cardalephex 9777 | Every transfinite cardinal... |
| infenaleph 9778 | An infinite numerable set ... |
| isinfcard 9779 | Two ways to express the pr... |
| iscard3 9780 | Two ways to express the pr... |
| cardnum 9781 | Two ways to express the cl... |
| alephinit 9782 | An infinite initial ordina... |
| carduniima 9783 | The union of the image of ... |
| cardinfima 9784 | If a mapping to cardinals ... |
| alephiso 9785 | Aleph is an order isomorph... |
| alephprc 9786 | The class of all transfini... |
| alephsson 9787 | The class of transfinite c... |
| unialeph 9788 | The union of the class of ... |
| alephsmo 9789 | The aleph function is stri... |
| alephf1ALT 9790 | Alternate proof of ~ aleph... |
| alephfplem1 9791 | Lemma for ~ alephfp . (Co... |
| alephfplem2 9792 | Lemma for ~ alephfp . (Co... |
| alephfplem3 9793 | Lemma for ~ alephfp . (Co... |
| alephfplem4 9794 | Lemma for ~ alephfp . (Co... |
| alephfp 9795 | The aleph function has a f... |
| alephfp2 9796 | The aleph function has at ... |
| alephval3 9797 | An alternate way to expres... |
| alephsucpw2 9798 | The power set of an aleph ... |
| mappwen 9799 | Power rule for cardinal ar... |
| finnisoeu 9800 | A finite totally ordered s... |
| iunfictbso 9801 | Countability of a countabl... |
| aceq1 9804 | Equivalence of two version... |
| aceq0 9805 | Equivalence of two version... |
| aceq2 9806 | Equivalence of two version... |
| aceq3lem 9807 | Lemma for ~ dfac3 . (Cont... |
| dfac3 9808 | Equivalence of two version... |
| dfac4 9809 | Equivalence of two version... |
| dfac5lem1 9810 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 9811 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 9812 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 9813 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 9814 | Lemma for ~ dfac5 . (Cont... |
| dfac5 9815 | Equivalence of two version... |
| dfac2a 9816 | Our Axiom of Choice (in th... |
| dfac2b 9817 | Axiom of Choice (first for... |
| dfac2 9818 | Axiom of Choice (first for... |
| dfac7 9819 | Equivalence of the Axiom o... |
| dfac0 9820 | Equivalence of two version... |
| dfac1 9821 | Equivalence of two version... |
| dfac8 9822 | A proof of the equivalency... |
| dfac9 9823 | Equivalence of the axiom o... |
| dfac10 9824 | Axiom of Choice equivalent... |
| dfac10c 9825 | Axiom of Choice equivalent... |
| dfac10b 9826 | Axiom of Choice equivalent... |
| acacni 9827 | A choice equivalent: every... |
| dfacacn 9828 | A choice equivalent: every... |
| dfac13 9829 | The axiom of choice holds ... |
| dfac12lem1 9830 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 9831 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 9832 | Lemma for ~ dfac12 . (Con... |
| dfac12r 9833 | The axiom of choice holds ... |
| dfac12k 9834 | Equivalence of ~ dfac12 an... |
| dfac12a 9835 | The axiom of choice holds ... |
| dfac12 9836 | The axiom of choice holds ... |
| kmlem1 9837 | Lemma for 5-quantifier AC ... |
| kmlem2 9838 | Lemma for 5-quantifier AC ... |
| kmlem3 9839 | Lemma for 5-quantifier AC ... |
| kmlem4 9840 | Lemma for 5-quantifier AC ... |
| kmlem5 9841 | Lemma for 5-quantifier AC ... |
| kmlem6 9842 | Lemma for 5-quantifier AC ... |
| kmlem7 9843 | Lemma for 5-quantifier AC ... |
| kmlem8 9844 | Lemma for 5-quantifier AC ... |
| kmlem9 9845 | Lemma for 5-quantifier AC ... |
| kmlem10 9846 | Lemma for 5-quantifier AC ... |
| kmlem11 9847 | Lemma for 5-quantifier AC ... |
| kmlem12 9848 | Lemma for 5-quantifier AC ... |
| kmlem13 9849 | Lemma for 5-quantifier AC ... |
| kmlem14 9850 | Lemma for 5-quantifier AC ... |
| kmlem15 9851 | Lemma for 5-quantifier AC ... |
| kmlem16 9852 | Lemma for 5-quantifier AC ... |
| dfackm 9853 | Equivalence of the Axiom o... |
| undjudom 9854 | Cardinal addition dominate... |
| endjudisj 9855 | Equinumerosity of a disjoi... |
| djuen 9856 | Disjoint unions of equinum... |
| djuenun 9857 | Disjoint union is equinume... |
| dju1en 9858 | Cardinal addition with car... |
| dju1dif 9859 | Adding and subtracting one... |
| dju1p1e2 9860 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 9861 | Alternate proof of ~ dju1p... |
| dju0en 9862 | Cardinal addition with car... |
| xp2dju 9863 | Two times a cardinal numbe... |
| djucomen 9864 | Commutative law for cardin... |
| djuassen 9865 | Associative law for cardin... |
| xpdjuen 9866 | Cardinal multiplication di... |
| mapdjuen 9867 | Sum of exponents law for c... |
| pwdjuen 9868 | Sum of exponents law for c... |
| djudom1 9869 | Ordering law for cardinal ... |
| djudom2 9870 | Ordering law for cardinal ... |
| djudoml 9871 | A set is dominated by its ... |
| djuxpdom 9872 | Cartesian product dominate... |
| djufi 9873 | The disjoint union of two ... |
| cdainflem 9874 | Any partition of omega int... |
| djuinf 9875 | A set is infinite iff the ... |
| infdju1 9876 | An infinite set is equinum... |
| pwdju1 9877 | The sum of a powerset with... |
| pwdjuidm 9878 | If the natural numbers inj... |
| djulepw 9879 | If ` A ` is idempotent und... |
| onadju 9880 | The cardinal and ordinal s... |
| cardadju 9881 | The cardinal sum is equinu... |
| djunum 9882 | The disjoint union of two ... |
| unnum 9883 | The union of two numerable... |
| nnadju 9884 | The cardinal and ordinal s... |
| nnadjuALT 9885 | Shorter proof of ~ nnadju ... |
| ficardadju 9886 | The disjoint union of fini... |
| ficardun 9887 | The cardinality of the uni... |
| ficardunOLD 9888 | Obsolete version of ~ fica... |
| ficardun2 9889 | The cardinality of the uni... |
| ficardun2OLD 9890 | Obsolete version of ~ fica... |
| pwsdompw 9891 | Lemma for ~ domtriom . Th... |
| unctb 9892 | The union of two countable... |
| infdjuabs 9893 | Absorption law for additio... |
| infunabs 9894 | An infinite set is equinum... |
| infdju 9895 | The sum of two cardinal nu... |
| infdif 9896 | The cardinality of an infi... |
| infdif2 9897 | Cardinality ordering for a... |
| infxpdom 9898 | Dominance law for multipli... |
| infxpabs 9899 | Absorption law for multipl... |
| infunsdom1 9900 | The union of two sets that... |
| infunsdom 9901 | The union of two sets that... |
| infxp 9902 | Absorption law for multipl... |
| pwdjudom 9903 | A property of dominance ov... |
| infpss 9904 | Every infinite set has an ... |
| infmap2 9905 | An exponentiation law for ... |
| ackbij2lem1 9906 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 9907 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 9908 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 9909 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 9910 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 9911 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 9912 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 9913 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 9914 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 9915 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 9916 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 9917 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 9918 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 9919 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 9920 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 9921 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 9922 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 9923 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 9924 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 9925 | The Ackermann bijection, p... |
| ackbij1b 9926 | The Ackermann bijection, p... |
| ackbij2lem2 9927 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 9928 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 9929 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 9930 | The Ackermann bijection, p... |
| r1om 9931 | The set of hereditarily fi... |
| fictb 9932 | A set is countable iff its... |
| cflem 9933 | A lemma used to simplify c... |
| cfval 9934 | Value of the cofinality fu... |
| cff 9935 | Cofinality is a function o... |
| cfub 9936 | An upper bound on cofinali... |
| cflm 9937 | Value of the cofinality fu... |
| cf0 9938 | Value of the cofinality fu... |
| cardcf 9939 | Cofinality is a cardinal n... |
| cflecard 9940 | Cofinality is bounded by t... |
| cfle 9941 | Cofinality is bounded by i... |
| cfon 9942 | The cofinality of any set ... |
| cfeq0 9943 | Only the ordinal zero has ... |
| cfsuc 9944 | Value of the cofinality fu... |
| cff1 9945 | There is always a map from... |
| cfflb 9946 | If there is a cofinal map ... |
| cfval2 9947 | Another expression for the... |
| coflim 9948 | A simpler expression for t... |
| cflim3 9949 | Another expression for the... |
| cflim2 9950 | The cofinality function is... |
| cfom 9951 | Value of the cofinality fu... |
| cfss 9952 | There is a cofinal subset ... |
| cfslb 9953 | Any cofinal subset of ` A ... |
| cfslbn 9954 | Any subset of ` A ` smalle... |
| cfslb2n 9955 | Any small collection of sm... |
| cofsmo 9956 | Any cofinal map implies th... |
| cfsmolem 9957 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 9958 | The map in ~ cff1 can be a... |
| cfcoflem 9959 | Lemma for ~ cfcof , showin... |
| coftr 9960 | If there is a cofinal map ... |
| cfcof 9961 | If there is a cofinal map ... |
| cfidm 9962 | The cofinality function is... |
| alephsing 9963 | The cofinality of a limit ... |
| sornom 9964 | The range of a single-step... |
| isfin1a 9979 | Definition of a Ia-finite ... |
| fin1ai 9980 | Property of a Ia-finite se... |
| isfin2 9981 | Definition of a II-finite ... |
| fin2i 9982 | Property of a II-finite se... |
| isfin3 9983 | Definition of a III-finite... |
| isfin4 9984 | Definition of a IV-finite ... |
| fin4i 9985 | Infer that a set is IV-inf... |
| isfin5 9986 | Definition of a V-finite s... |
| isfin6 9987 | Definition of a VI-finite ... |
| isfin7 9988 | Definition of a VII-finite... |
| sdom2en01 9989 | A set with less than two e... |
| infpssrlem1 9990 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 9991 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 9992 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 9993 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 9994 | Lemma for ~ infpssr . (Co... |
| infpssr 9995 | Dedekind infinity implies ... |
| fin4en1 9996 | Dedekind finite is a cardi... |
| ssfin4 9997 | Dedekind finite sets have ... |
| domfin4 9998 | A set dominated by a Dedek... |
| ominf4 9999 | ` _om ` is Dedekind infini... |
| infpssALT 10000 | Alternate proof of ~ infps... |
| isfin4-2 10001 | Alternate definition of IV... |
| isfin4p1 10002 | Alternate definition of IV... |
| fin23lem7 10003 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 10004 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 10005 | A II-finite set contains m... |
| isfin2-2 10006 | ` Fin2 ` expressed in term... |
| ssfin2 10007 | A subset of a II-finite se... |
| enfin2i 10008 | II-finiteness is a cardina... |
| fin23lem24 10009 | Lemma for ~ fin23 . In a ... |
| fincssdom 10010 | In a chain of finite sets,... |
| fin23lem25 10011 | Lemma for ~ fin23 . In a ... |
| fin23lem26 10012 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 10013 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 10014 | Lemma for ~ fin23 but coul... |
| fin23lem27 10015 | The mapping constructed in... |
| isfin3ds 10016 | Property of a III-finite s... |
| ssfin3ds 10017 | A subset of a III-finite s... |
| fin23lem12 10018 | The beginning of the proof... |
| fin23lem13 10019 | Lemma for ~ fin23 . Each ... |
| fin23lem14 10020 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 10021 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 10022 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 10023 | Lemma for ~ fin23 . The f... |
| fin23lem20 10024 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 10025 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 10026 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 10027 | Lemma for ~ fin23 . The r... |
| fin23lem29 10028 | Lemma for ~ fin23 . The r... |
| fin23lem30 10029 | Lemma for ~ fin23 . The r... |
| fin23lem31 10030 | Lemma for ~ fin23 . The r... |
| fin23lem32 10031 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 10032 | Lemma for ~ fin23 . Disch... |
| fin23lem34 10033 | Lemma for ~ fin23 . Estab... |
| fin23lem35 10034 | Lemma for ~ fin23 . Stric... |
| fin23lem36 10035 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 10036 | Lemma for ~ fin23 . The c... |
| fin23lem39 10037 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 10038 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 10039 | Lemma for ~ fin23 . A set... |
| isf32lem1 10040 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 10041 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 10042 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 10043 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 10044 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 10045 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 10046 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 10047 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 10048 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 10049 | Lemma for isfin3-2 . Writ... |
| isf32lem11 10050 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 10051 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 10052 | One half of ~ isfin3-2 . ... |
| isf33lem 10053 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 10054 | Weakly Dedekind-infinite s... |
| isfin3-3 10055 | Weakly Dedekind-infinite s... |
| fin33i 10056 | Inference from ~ isfin3-3 ... |
| compsscnvlem 10057 | Lemma for ~ compsscnv . (... |
| compsscnv 10058 | Complementation on a power... |
| isf34lem1 10059 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 10060 | Lemma for ~ isfin3-4 . (C... |
| compssiso 10061 | Complementation is an anti... |
| isf34lem3 10062 | Lemma for ~ isfin3-4 . (C... |
| compss 10063 | Express image under of the... |
| isf34lem4 10064 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 10065 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 10066 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 10067 | Lemma for ~ isfin3-4 . (C... |
| fin34i 10068 | Inference from ~ isfin3-4 ... |
| isfin3-4 10069 | Weakly Dedekind-infinite s... |
| fin11a 10070 | Every I-finite set is Ia-f... |
| enfin1ai 10071 | Ia-finiteness is a cardina... |
| isfin1-2 10072 | A set is finite in the usu... |
| isfin1-3 10073 | A set is I-finite iff ever... |
| isfin1-4 10074 | A set is I-finite iff ever... |
| dffin1-5 10075 | Compact quantifier-free ve... |
| fin23 10076 | Every II-finite set (every... |
| fin34 10077 | Every III-finite set is IV... |
| isfin5-2 10078 | Alternate definition of V-... |
| fin45 10079 | Every IV-finite set is V-f... |
| fin56 10080 | Every V-finite set is VI-f... |
| fin17 10081 | Every I-finite set is VII-... |
| fin67 10082 | Every VI-finite set is VII... |
| isfin7-2 10083 | A set is VII-finite iff it... |
| fin71num 10084 | A well-orderable set is VI... |
| dffin7-2 10085 | Class form of ~ isfin7-2 .... |
| dfacfin7 10086 | Axiom of Choice equivalent... |
| fin1a2lem1 10087 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 10088 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem3 10089 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 10090 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 10091 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 10092 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 10093 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 10094 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 10095 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 10096 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 10097 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 10098 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 10099 | Lemma for ~ fin1a2 . (Con... |
| fin12 10100 | Weak theorem which skips I... |
| fin1a2s 10101 | An II-infinite set can hav... |
| fin1a2 10102 | Every Ia-finite set is II-... |
| itunifval 10103 | Function value of iterated... |
| itunifn 10104 | Functionality of the itera... |
| ituni0 10105 | A zero-fold iterated union... |
| itunisuc 10106 | Successor iterated union. ... |
| itunitc1 10107 | Each union iterate is a me... |
| itunitc 10108 | The union of all union ite... |
| ituniiun 10109 | Unwrap an iterated union f... |
| hsmexlem7 10110 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 10111 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 10112 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 10113 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 10114 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 10115 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 10116 | Lemma for ~ hsmex . The c... |
| hsmexlem5 10117 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 10118 | Lemma for ~ hsmex . (Cont... |
| hsmex 10119 | The collection of heredita... |
| hsmex2 10120 | The set of hereditary size... |
| hsmex3 10121 | The set of hereditary size... |
| axcc2lem 10123 | Lemma for ~ axcc2 . (Cont... |
| axcc2 10124 | A possibly more useful ver... |
| axcc3 10125 | A possibly more useful ver... |
| axcc4 10126 | A version of ~ axcc3 that ... |
| acncc 10127 | An ~ ax-cc equivalent: eve... |
| axcc4dom 10128 | Relax the constraint on ~ ... |
| domtriomlem 10129 | Lemma for ~ domtriom . (C... |
| domtriom 10130 | Trichotomy of equinumerosi... |
| fin41 10131 | Under countable choice, th... |
| dominf 10132 | A nonempty set that is a s... |
| dcomex 10134 | The Axiom of Dependent Cho... |
| axdc2lem 10135 | Lemma for ~ axdc2 . We co... |
| axdc2 10136 | An apparent strengthening ... |
| axdc3lem 10137 | The class ` S ` of finite ... |
| axdc3lem2 10138 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 10139 | Simple substitution lemma ... |
| axdc3lem4 10140 | Lemma for ~ axdc3 . We ha... |
| axdc3 10141 | Dependent Choice. Axiom D... |
| axdc4lem 10142 | Lemma for ~ axdc4 . (Cont... |
| axdc4 10143 | A more general version of ... |
| axcclem 10144 | Lemma for ~ axcc . (Contr... |
| axcc 10145 | Although CC can be proven ... |
| zfac 10147 | Axiom of Choice expressed ... |
| ac2 10148 | Axiom of Choice equivalent... |
| ac3 10149 | Axiom of Choice using abbr... |
| axac3 10151 | This theorem asserts that ... |
| ackm 10152 | A remarkable equivalent to... |
| axac2 10153 | Derive ~ ax-ac2 from ~ ax-... |
| axac 10154 | Derive ~ ax-ac from ~ ax-a... |
| axaci 10155 | Apply a choice equivalent.... |
| cardeqv 10156 | All sets are well-orderabl... |
| numth3 10157 | All sets are well-orderabl... |
| numth2 10158 | Numeration theorem: any se... |
| numth 10159 | Numeration theorem: every ... |
| ac7 10160 | An Axiom of Choice equival... |
| ac7g 10161 | An Axiom of Choice equival... |
| ac4 10162 | Equivalent of Axiom of Cho... |
| ac4c 10163 | Equivalent of Axiom of Cho... |
| ac5 10164 | An Axiom of Choice equival... |
| ac5b 10165 | Equivalent of Axiom of Cho... |
| ac6num 10166 | A version of ~ ac6 which t... |
| ac6 10167 | Equivalent of Axiom of Cho... |
| ac6c4 10168 | Equivalent of Axiom of Cho... |
| ac6c5 10169 | Equivalent of Axiom of Cho... |
| ac9 10170 | An Axiom of Choice equival... |
| ac6s 10171 | Equivalent of Axiom of Cho... |
| ac6n 10172 | Equivalent of Axiom of Cho... |
| ac6s2 10173 | Generalization of the Axio... |
| ac6s3 10174 | Generalization of the Axio... |
| ac6sg 10175 | ~ ac6s with sethood as ant... |
| ac6sf 10176 | Version of ~ ac6 with boun... |
| ac6s4 10177 | Generalization of the Axio... |
| ac6s5 10178 | Generalization of the Axio... |
| ac8 10179 | An Axiom of Choice equival... |
| ac9s 10180 | An Axiom of Choice equival... |
| numthcor 10181 | Any set is strictly domina... |
| weth 10182 | Well-ordering theorem: any... |
| zorn2lem1 10183 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 10184 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 10185 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 10186 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 10187 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 10188 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 10189 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 10190 | Zorn's Lemma of [Monk1] p.... |
| zorng 10191 | Zorn's Lemma. If the unio... |
| zornn0g 10192 | Variant of Zorn's lemma ~ ... |
| zorn2 10193 | Zorn's Lemma of [Monk1] p.... |
| zorn 10194 | Zorn's Lemma. If the unio... |
| zornn0 10195 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 10196 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 10197 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 10198 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 10199 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 10200 | Lemma for ~ ttukey . The ... |
| ttukeylem6 10201 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 10202 | Lemma for ~ ttukey . (Con... |
| ttukey2g 10203 | The Teichmüller-Tukey... |
| ttukeyg 10204 | The Teichmüller-Tukey... |
| ttukey 10205 | The Teichmüller-Tukey... |
| axdclem 10206 | Lemma for ~ axdc . (Contr... |
| axdclem2 10207 | Lemma for ~ axdc . Using ... |
| axdc 10208 | This theorem derives ~ ax-... |
| fodomg 10209 | An onto function implies d... |
| fodom 10210 | An onto function implies d... |
| dmct 10211 | The domain of a countable ... |
| rnct 10212 | The range of a countable s... |
| fodomb 10213 | Equivalence of an onto map... |
| wdomac 10214 | When assuming AC, weak and... |
| brdom3 10215 | Equivalence to a dominance... |
| brdom5 10216 | An equivalence to a domina... |
| brdom4 10217 | An equivalence to a domina... |
| brdom7disj 10218 | An equivalence to a domina... |
| brdom6disj 10219 | An equivalence to a domina... |
| fin71ac 10220 | Once we allow AC, the "str... |
| imadomg 10221 | An image of a function und... |
| fimact 10222 | The image by a function of... |
| fnrndomg 10223 | The range of a function is... |
| fnct 10224 | If the domain of a functio... |
| mptct 10225 | A countable mapping set is... |
| iunfo 10226 | Existence of an onto funct... |
| iundom2g 10227 | An upper bound for the car... |
| iundomg 10228 | An upper bound for the car... |
| iundom 10229 | An upper bound for the car... |
| unidom 10230 | An upper bound for the car... |
| uniimadom 10231 | An upper bound for the car... |
| uniimadomf 10232 | An upper bound for the car... |
| cardval 10233 | The value of the cardinal ... |
| cardid 10234 | Any set is equinumerous to... |
| cardidg 10235 | Any set is equinumerous to... |
| cardidd 10236 | Any set is equinumerous to... |
| cardf 10237 | The cardinality function i... |
| carden 10238 | Two sets are equinumerous ... |
| cardeq0 10239 | Only the empty set has car... |
| unsnen 10240 | Equinumerosity of a set wi... |
| carddom 10241 | Two sets have the dominanc... |
| cardsdom 10242 | Two sets have the strict d... |
| domtri 10243 | Trichotomy law for dominan... |
| entric 10244 | Trichotomy of equinumerosi... |
| entri2 10245 | Trichotomy of dominance an... |
| entri3 10246 | Trichotomy of dominance. ... |
| sdomsdomcard 10247 | A set strictly dominates i... |
| canth3 10248 | Cantor's theorem in terms ... |
| infxpidm 10249 | Every infinite class is eq... |
| ondomon 10250 | The class of ordinals domi... |
| cardmin 10251 | The smallest ordinal that ... |
| ficard 10252 | A set is finite iff its ca... |
| infinf 10253 | Equivalence between two in... |
| unirnfdomd 10254 | The union of the range of ... |
| konigthlem 10255 | Lemma for ~ konigth . (Co... |
| konigth 10256 | Konig's Theorem. If ` m (... |
| alephsucpw 10257 | The power set of an aleph ... |
| aleph1 10258 | The set exponentiation of ... |
| alephval2 10259 | An alternate way to expres... |
| dominfac 10260 | A nonempty set that is a s... |
| iunctb 10261 | The countable union of cou... |
| unictb 10262 | The countable union of cou... |
| infmap 10263 | An exponentiation law for ... |
| alephadd 10264 | The sum of two alephs is t... |
| alephmul 10265 | The product of two alephs ... |
| alephexp1 10266 | An exponentiation law for ... |
| alephsuc3 10267 | An alternate representatio... |
| alephexp2 10268 | An expression equinumerous... |
| alephreg 10269 | A successor aleph is regul... |
| pwcfsdom 10270 | A corollary of Konig's The... |
| cfpwsdom 10271 | A corollary of Konig's The... |
| alephom 10272 | From ~ canth2 , we know th... |
| smobeth 10273 | The beth function is stric... |
| nd1 10274 | A lemma for proving condit... |
| nd2 10275 | A lemma for proving condit... |
| nd3 10276 | A lemma for proving condit... |
| nd4 10277 | A lemma for proving condit... |
| axextnd 10278 | A version of the Axiom of ... |
| axrepndlem1 10279 | Lemma for the Axiom of Rep... |
| axrepndlem2 10280 | Lemma for the Axiom of Rep... |
| axrepnd 10281 | A version of the Axiom of ... |
| axunndlem1 10282 | Lemma for the Axiom of Uni... |
| axunnd 10283 | A version of the Axiom of ... |
| axpowndlem1 10284 | Lemma for the Axiom of Pow... |
| axpowndlem2 10285 | Lemma for the Axiom of Pow... |
| axpowndlem3 10286 | Lemma for the Axiom of Pow... |
| axpowndlem4 10287 | Lemma for the Axiom of Pow... |
| axpownd 10288 | A version of the Axiom of ... |
| axregndlem1 10289 | Lemma for the Axiom of Reg... |
| axregndlem2 10290 | Lemma for the Axiom of Reg... |
| axregnd 10291 | A version of the Axiom of ... |
| axinfndlem1 10292 | Lemma for the Axiom of Inf... |
| axinfnd 10293 | A version of the Axiom of ... |
| axacndlem1 10294 | Lemma for the Axiom of Cho... |
| axacndlem2 10295 | Lemma for the Axiom of Cho... |
| axacndlem3 10296 | Lemma for the Axiom of Cho... |
| axacndlem4 10297 | Lemma for the Axiom of Cho... |
| axacndlem5 10298 | Lemma for the Axiom of Cho... |
| axacnd 10299 | A version of the Axiom of ... |
| zfcndext 10300 | Axiom of Extensionality ~ ... |
| zfcndrep 10301 | Axiom of Replacement ~ ax-... |
| zfcndun 10302 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10303 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10304 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10305 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10306 | Axiom of Choice ~ ax-ac , ... |
| elgch 10309 | Elementhood in the collect... |
| fingch 10310 | A finite set is a GCH-set.... |
| gchi 10311 | The only GCH-sets which ha... |
| gchen1 10312 | If ` A <_ B < ~P A ` , and... |
| gchen2 10313 | If ` A < B <_ ~P A ` , and... |
| gchor 10314 | If ` A <_ B <_ ~P A ` , an... |
| engch 10315 | The property of being a GC... |
| gchdomtri 10316 | Under certain conditions, ... |
| fpwwe2cbv 10317 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10318 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10319 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10320 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem4 10321 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10322 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10323 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10324 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem8 10325 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem9 10326 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10327 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem11 10328 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10329 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10330 | Given any function ` F ` f... |
| fpwwecbv 10331 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10332 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10333 | Given any function ` F ` f... |
| canth4 10334 | An "effective" form of Can... |
| canthnumlem 10335 | Lemma for ~ canthnum . (C... |
| canthnum 10336 | The set of well-orderable ... |
| canthwelem 10337 | Lemma for ~ canthwe . (Co... |
| canthwe 10338 | The set of well-orders of ... |
| canthp1lem1 10339 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10340 | Lemma for ~ canthp1 . (Co... |
| canthp1 10341 | A slightly stronger form o... |
| finngch 10342 | The exclusion of finite se... |
| gchdju1 10343 | An infinite GCH-set is ide... |
| gchinf 10344 | An infinite GCH-set is Ded... |
| pwfseqlem1 10345 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10346 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10347 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10348 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10349 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10350 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10351 | The powerset of a Dedekind... |
| pwxpndom2 10352 | The powerset of a Dedekind... |
| pwxpndom 10353 | The powerset of a Dedekind... |
| pwdjundom 10354 | The powerset of a Dedekind... |
| gchdjuidm 10355 | An infinite GCH-set is ide... |
| gchxpidm 10356 | An infinite GCH-set is ide... |
| gchpwdom 10357 | A relationship between dom... |
| gchaleph 10358 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10359 | If ` ( aleph `` A ) ` and ... |
| hargch 10360 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10361 | If ` ( aleph `` suc A ) ` ... |
| gch2 10362 | It is sufficient to requir... |
| gch3 10363 | An equivalent formulation ... |
| gch-kn 10364 | The equivalence of two ver... |
| gchaclem 10365 | Lemma for ~ gchac (obsolet... |
| gchhar 10366 | A "local" form of ~ gchac ... |
| gchacg 10367 | A "local" form of ~ gchac ... |
| gchac 10368 | The Generalized Continuum ... |
| elwina 10373 | Conditions of weak inacces... |
| elina 10374 | Conditions of strong inacc... |
| winaon 10375 | A weakly inaccessible card... |
| inawinalem 10376 | Lemma for ~ inawina . (Co... |
| inawina 10377 | Every strongly inaccessibl... |
| omina 10378 | ` _om ` is a strongly inac... |
| winacard 10379 | A weakly inaccessible card... |
| winainflem 10380 | A weakly inaccessible card... |
| winainf 10381 | A weakly inaccessible card... |
| winalim 10382 | A weakly inaccessible card... |
| winalim2 10383 | A nontrivial weakly inacce... |
| winafp 10384 | A nontrivial weakly inacce... |
| winafpi 10385 | This theorem, which states... |
| gchina 10386 | Assuming the GCH, weakly a... |
| iswun 10391 | Properties of a weak unive... |
| wuntr 10392 | A weak universe is transit... |
| wununi 10393 | A weak universe is closed ... |
| wunpw 10394 | A weak universe is closed ... |
| wunelss 10395 | The elements of a weak uni... |
| wunpr 10396 | A weak universe is closed ... |
| wunun 10397 | A weak universe is closed ... |
| wuntp 10398 | A weak universe is closed ... |
| wunss 10399 | A weak universe is closed ... |
| wunin 10400 | A weak universe is closed ... |
| wundif 10401 | A weak universe is closed ... |
| wunint 10402 | A weak universe is closed ... |
| wunsn 10403 | A weak universe is closed ... |
| wunsuc 10404 | A weak universe is closed ... |
| wun0 10405 | A weak universe contains t... |
| wunr1om 10406 | A weak universe is infinit... |
| wunom 10407 | A weak universe contains a... |
| wunfi 10408 | A weak universe contains a... |
| wunop 10409 | A weak universe is closed ... |
| wunot 10410 | A weak universe is closed ... |
| wunxp 10411 | A weak universe is closed ... |
| wunpm 10412 | A weak universe is closed ... |
| wunmap 10413 | A weak universe is closed ... |
| wunf 10414 | A weak universe is closed ... |
| wundm 10415 | A weak universe is closed ... |
| wunrn 10416 | A weak universe is closed ... |
| wuncnv 10417 | A weak universe is closed ... |
| wunres 10418 | A weak universe is closed ... |
| wunfv 10419 | A weak universe is closed ... |
| wunco 10420 | A weak universe is closed ... |
| wuntpos 10421 | A weak universe is closed ... |
| intwun 10422 | The intersection of a coll... |
| r1limwun 10423 | Each limit stage in the cu... |
| r1wunlim 10424 | The weak universes in the ... |
| wunex2 10425 | Construct a weak universe ... |
| wunex 10426 | Construct a weak universe ... |
| uniwun 10427 | Every set is contained in ... |
| wunex3 10428 | Construct a weak universe ... |
| wuncval 10429 | Value of the weak universe... |
| wuncid 10430 | The weak universe closure ... |
| wunccl 10431 | The weak universe closure ... |
| wuncss 10432 | The weak universe closure ... |
| wuncidm 10433 | The weak universe closure ... |
| wuncval2 10434 | Our earlier expression for... |
| eltskg 10437 | Properties of a Tarski cla... |
| eltsk2g 10438 | Properties of a Tarski cla... |
| tskpwss 10439 | First axiom of a Tarski cl... |
| tskpw 10440 | Second axiom of a Tarski c... |
| tsken 10441 | Third axiom of a Tarski cl... |
| 0tsk 10442 | The empty set is a (transi... |
| tsksdom 10443 | An element of a Tarski cla... |
| tskssel 10444 | A part of a Tarski class s... |
| tskss 10445 | The subsets of an element ... |
| tskin 10446 | The intersection of two el... |
| tsksn 10447 | A singleton of an element ... |
| tsktrss 10448 | A transitive element of a ... |
| tsksuc 10449 | If an element of a Tarski ... |
| tsk0 10450 | A nonempty Tarski class co... |
| tsk1 10451 | One is an element of a non... |
| tsk2 10452 | Two is an element of a non... |
| 2domtsk 10453 | If a Tarski class is not e... |
| tskr1om 10454 | A nonempty Tarski class is... |
| tskr1om2 10455 | A nonempty Tarski class co... |
| tskinf 10456 | A nonempty Tarski class is... |
| tskpr 10457 | If ` A ` and ` B ` are mem... |
| tskop 10458 | If ` A ` and ` B ` are mem... |
| tskxpss 10459 | A Cartesian product of two... |
| tskwe2 10460 | A Tarski class is well-ord... |
| inttsk 10461 | The intersection of a coll... |
| inar1 10462 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10463 | Alternate proof of ~ r1om ... |
| rankcf 10464 | Any set must be at least a... |
| inatsk 10465 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10466 | The set of hereditarily fi... |
| tskord 10467 | A Tarski class contains al... |
| tskcard 10468 | An even more direct relati... |
| r1tskina 10469 | There is a direct relation... |
| tskuni 10470 | The union of an element of... |
| tskwun 10471 | A nonempty transitive Tars... |
| tskint 10472 | The intersection of an ele... |
| tskun 10473 | The union of two elements ... |
| tskxp 10474 | The Cartesian product of t... |
| tskmap 10475 | Set exponentiation is an e... |
| tskurn 10476 | A transitive Tarski class ... |
| elgrug 10479 | Properties of a Grothendie... |
| grutr 10480 | A Grothendieck universe is... |
| gruelss 10481 | A Grothendieck universe is... |
| grupw 10482 | A Grothendieck universe co... |
| gruss 10483 | Any subset of an element o... |
| grupr 10484 | A Grothendieck universe co... |
| gruurn 10485 | A Grothendieck universe co... |
| gruiun 10486 | If ` B ( x ) ` is a family... |
| gruuni 10487 | A Grothendieck universe co... |
| grurn 10488 | A Grothendieck universe co... |
| gruima 10489 | A Grothendieck universe co... |
| gruel 10490 | Any element of an element ... |
| grusn 10491 | A Grothendieck universe co... |
| gruop 10492 | A Grothendieck universe co... |
| gruun 10493 | A Grothendieck universe co... |
| gruxp 10494 | A Grothendieck universe co... |
| grumap 10495 | A Grothendieck universe co... |
| gruixp 10496 | A Grothendieck universe co... |
| gruiin 10497 | A Grothendieck universe co... |
| gruf 10498 | A Grothendieck universe co... |
| gruen 10499 | A Grothendieck universe co... |
| gruwun 10500 | A nonempty Grothendieck un... |
| intgru 10501 | The intersection of a fami... |
| ingru 10502 | The intersection of a univ... |
| wfgru 10503 | The wellfounded part of a ... |
| grudomon 10504 | Each ordinal that is compa... |
| gruina 10505 | If a Grothendieck universe... |
| grur1a 10506 | A characterization of Grot... |
| grur1 10507 | A characterization of Grot... |
| grutsk1 10508 | Grothendieck universes are... |
| grutsk 10509 | Grothendieck universes are... |
| axgroth5 10511 | The Tarski-Grothendieck ax... |
| axgroth2 10512 | Alternate version of the T... |
| grothpw 10513 | Derive the Axiom of Power ... |
| grothpwex 10514 | Derive the Axiom of Power ... |
| axgroth6 10515 | The Tarski-Grothendieck ax... |
| grothomex 10516 | The Tarski-Grothendieck Ax... |
| grothac 10517 | The Tarski-Grothendieck Ax... |
| axgroth3 10518 | Alternate version of the T... |
| axgroth4 10519 | Alternate version of the T... |
| grothprimlem 10520 | Lemma for ~ grothprim . E... |
| grothprim 10521 | The Tarski-Grothendieck Ax... |
| grothtsk 10522 | The Tarski-Grothendieck Ax... |
| inaprc 10523 | An equivalent to the Tarsk... |
| tskmval 10526 | Value of our tarski map. ... |
| tskmid 10527 | The set ` A ` is an elemen... |
| tskmcl 10528 | A Tarski class that contai... |
| sstskm 10529 | Being a part of ` ( tarski... |
| eltskm 10530 | Belonging to ` ( tarskiMap... |
| elni 10563 | Membership in the class of... |
| elni2 10564 | Membership in the class of... |
| pinn 10565 | A positive integer is a na... |
| pion 10566 | A positive integer is an o... |
| piord 10567 | A positive integer is ordi... |
| niex 10568 | The class of positive inte... |
| 0npi 10569 | The empty set is not a pos... |
| 1pi 10570 | Ordinal 'one' is a positiv... |
| addpiord 10571 | Positive integer addition ... |
| mulpiord 10572 | Positive integer multiplic... |
| mulidpi 10573 | 1 is an identity element f... |
| ltpiord 10574 | Positive integer 'less tha... |
| ltsopi 10575 | Positive integer 'less tha... |
| ltrelpi 10576 | Positive integer 'less tha... |
| dmaddpi 10577 | Domain of addition on posi... |
| dmmulpi 10578 | Domain of multiplication o... |
| addclpi 10579 | Closure of addition of pos... |
| mulclpi 10580 | Closure of multiplication ... |
| addcompi 10581 | Addition of positive integ... |
| addasspi 10582 | Addition of positive integ... |
| mulcompi 10583 | Multiplication of positive... |
| mulasspi 10584 | Multiplication of positive... |
| distrpi 10585 | Multiplication of positive... |
| addcanpi 10586 | Addition cancellation law ... |
| mulcanpi 10587 | Multiplication cancellatio... |
| addnidpi 10588 | There is no identity eleme... |
| ltexpi 10589 | Ordering on positive integ... |
| ltapi 10590 | Ordering property of addit... |
| ltmpi 10591 | Ordering property of multi... |
| 1lt2pi 10592 | One is less than two (one ... |
| nlt1pi 10593 | No positive integer is les... |
| indpi 10594 | Principle of Finite Induct... |
| enqbreq 10606 | Equivalence relation for p... |
| enqbreq2 10607 | Equivalence relation for p... |
| enqer 10608 | The equivalence relation f... |
| enqex 10609 | The equivalence relation f... |
| nqex 10610 | The class of positive frac... |
| 0nnq 10611 | The empty set is not a pos... |
| elpqn 10612 | Each positive fraction is ... |
| ltrelnq 10613 | Positive fraction 'less th... |
| pinq 10614 | The representatives of pos... |
| 1nq 10615 | The positive fraction 'one... |
| nqereu 10616 | There is a unique element ... |
| nqerf 10617 | Corollary of ~ nqereu : th... |
| nqercl 10618 | Corollary of ~ nqereu : cl... |
| nqerrel 10619 | Any member of ` ( N. X. N.... |
| nqerid 10620 | Corollary of ~ nqereu : th... |
| enqeq 10621 | Corollary of ~ nqereu : if... |
| nqereq 10622 | The function ` /Q ` acts a... |
| addpipq2 10623 | Addition of positive fract... |
| addpipq 10624 | Addition of positive fract... |
| addpqnq 10625 | Addition of positive fract... |
| mulpipq2 10626 | Multiplication of positive... |
| mulpipq 10627 | Multiplication of positive... |
| mulpqnq 10628 | Multiplication of positive... |
| ordpipq 10629 | Ordering of positive fract... |
| ordpinq 10630 | Ordering of positive fract... |
| addpqf 10631 | Closure of addition on pos... |
| addclnq 10632 | Closure of addition on pos... |
| mulpqf 10633 | Closure of multiplication ... |
| mulclnq 10634 | Closure of multiplication ... |
| addnqf 10635 | Domain of addition on posi... |
| mulnqf 10636 | Domain of multiplication o... |
| addcompq 10637 | Addition of positive fract... |
| addcomnq 10638 | Addition of positive fract... |
| mulcompq 10639 | Multiplication of positive... |
| mulcomnq 10640 | Multiplication of positive... |
| adderpqlem 10641 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 10642 | Lemma for ~ mulerpq . (Co... |
| adderpq 10643 | Addition is compatible wit... |
| mulerpq 10644 | Multiplication is compatib... |
| addassnq 10645 | Addition of positive fract... |
| mulassnq 10646 | Multiplication of positive... |
| mulcanenq 10647 | Lemma for distributive law... |
| distrnq 10648 | Multiplication of positive... |
| 1nqenq 10649 | The equivalence class of r... |
| mulidnq 10650 | Multiplication identity el... |
| recmulnq 10651 | Relationship between recip... |
| recidnq 10652 | A positive fraction times ... |
| recclnq 10653 | Closure law for positive f... |
| recrecnq 10654 | Reciprocal of reciprocal o... |
| dmrecnq 10655 | Domain of reciprocal on po... |
| ltsonq 10656 | 'Less than' is a strict or... |
| lterpq 10657 | Compatibility of ordering ... |
| ltanq 10658 | Ordering property of addit... |
| ltmnq 10659 | Ordering property of multi... |
| 1lt2nq 10660 | One is less than two (one ... |
| ltaddnq 10661 | The sum of two fractions i... |
| ltexnq 10662 | Ordering on positive fract... |
| halfnq 10663 | One-half of any positive f... |
| nsmallnq 10664 | The is no smallest positiv... |
| ltbtwnnq 10665 | There exists a number betw... |
| ltrnq 10666 | Ordering property of recip... |
| archnq 10667 | For any fraction, there is... |
| npex 10673 | The class of positive real... |
| elnp 10674 | Membership in positive rea... |
| elnpi 10675 | Membership in positive rea... |
| prn0 10676 | A positive real is not emp... |
| prpssnq 10677 | A positive real is a subse... |
| elprnq 10678 | A positive real is a set o... |
| 0npr 10679 | The empty set is not a pos... |
| prcdnq 10680 | A positive real is closed ... |
| prub 10681 | A positive fraction not in... |
| prnmax 10682 | A positive real has no lar... |
| npomex 10683 | A simplifying observation,... |
| prnmadd 10684 | A positive real has no lar... |
| ltrelpr 10685 | Positive real 'less than' ... |
| genpv 10686 | Value of general operation... |
| genpelv 10687 | Membership in value of gen... |
| genpprecl 10688 | Pre-closure law for genera... |
| genpdm 10689 | Domain of general operatio... |
| genpn0 10690 | The result of an operation... |
| genpss 10691 | The result of an operation... |
| genpnnp 10692 | The result of an operation... |
| genpcd 10693 | Downward closure of an ope... |
| genpnmax 10694 | An operation on positive r... |
| genpcl 10695 | Closure of an operation on... |
| genpass 10696 | Associativity of an operat... |
| plpv 10697 | Value of addition on posit... |
| mpv 10698 | Value of multiplication on... |
| dmplp 10699 | Domain of addition on posi... |
| dmmp 10700 | Domain of multiplication o... |
| nqpr 10701 | The canonical embedding of... |
| 1pr 10702 | The positive real number '... |
| addclprlem1 10703 | Lemma to prove downward cl... |
| addclprlem2 10704 | Lemma to prove downward cl... |
| addclpr 10705 | Closure of addition on pos... |
| mulclprlem 10706 | Lemma to prove downward cl... |
| mulclpr 10707 | Closure of multiplication ... |
| addcompr 10708 | Addition of positive reals... |
| addasspr 10709 | Addition of positive reals... |
| mulcompr 10710 | Multiplication of positive... |
| mulasspr 10711 | Multiplication of positive... |
| distrlem1pr 10712 | Lemma for distributive law... |
| distrlem4pr 10713 | Lemma for distributive law... |
| distrlem5pr 10714 | Lemma for distributive law... |
| distrpr 10715 | Multiplication of positive... |
| 1idpr 10716 | 1 is an identity element f... |
| ltprord 10717 | Positive real 'less than' ... |
| psslinpr 10718 | Proper subset is a linear ... |
| ltsopr 10719 | Positive real 'less than' ... |
| prlem934 10720 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 10721 | The sum of two positive re... |
| ltaddpr2 10722 | The sum of two positive re... |
| ltexprlem1 10723 | Lemma for Proposition 9-3.... |
| ltexprlem2 10724 | Lemma for Proposition 9-3.... |
| ltexprlem3 10725 | Lemma for Proposition 9-3.... |
| ltexprlem4 10726 | Lemma for Proposition 9-3.... |
| ltexprlem5 10727 | Lemma for Proposition 9-3.... |
| ltexprlem6 10728 | Lemma for Proposition 9-3.... |
| ltexprlem7 10729 | Lemma for Proposition 9-3.... |
| ltexpri 10730 | Proposition 9-3.5(iv) of [... |
| ltaprlem 10731 | Lemma for Proposition 9-3.... |
| ltapr 10732 | Ordering property of addit... |
| addcanpr 10733 | Addition cancellation law ... |
| prlem936 10734 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 10735 | Lemma for Proposition 9-3.... |
| reclem3pr 10736 | Lemma for Proposition 9-3.... |
| reclem4pr 10737 | Lemma for Proposition 9-3.... |
| recexpr 10738 | The reciprocal of a positi... |
| suplem1pr 10739 | The union of a nonempty, b... |
| suplem2pr 10740 | The union of a set of posi... |
| supexpr 10741 | The union of a nonempty, b... |
| enrer 10750 | The equivalence relation f... |
| nrex1 10751 | The class of signed reals ... |
| enrbreq 10752 | Equivalence relation for s... |
| enreceq 10753 | Equivalence class equality... |
| enrex 10754 | The equivalence relation f... |
| ltrelsr 10755 | Signed real 'less than' is... |
| addcmpblnr 10756 | Lemma showing compatibilit... |
| mulcmpblnrlem 10757 | Lemma used in lemma showin... |
| mulcmpblnr 10758 | Lemma showing compatibilit... |
| prsrlem1 10759 | Decomposing signed reals i... |
| addsrmo 10760 | There is at most one resul... |
| mulsrmo 10761 | There is at most one resul... |
| addsrpr 10762 | Addition of signed reals i... |
| mulsrpr 10763 | Multiplication of signed r... |
| ltsrpr 10764 | Ordering of signed reals i... |
| gt0srpr 10765 | Greater than zero in terms... |
| 0nsr 10766 | The empty set is not a sig... |
| 0r 10767 | The constant ` 0R ` is a s... |
| 1sr 10768 | The constant ` 1R ` is a s... |
| m1r 10769 | The constant ` -1R ` is a ... |
| addclsr 10770 | Closure of addition on sig... |
| mulclsr 10771 | Closure of multiplication ... |
| dmaddsr 10772 | Domain of addition on sign... |
| dmmulsr 10773 | Domain of multiplication o... |
| addcomsr 10774 | Addition of signed reals i... |
| addasssr 10775 | Addition of signed reals i... |
| mulcomsr 10776 | Multiplication of signed r... |
| mulasssr 10777 | Multiplication of signed r... |
| distrsr 10778 | Multiplication of signed r... |
| m1p1sr 10779 | Minus one plus one is zero... |
| m1m1sr 10780 | Minus one times minus one ... |
| ltsosr 10781 | Signed real 'less than' is... |
| 0lt1sr 10782 | 0 is less than 1 for signe... |
| 1ne0sr 10783 | 1 and 0 are distinct for s... |
| 0idsr 10784 | The signed real number 0 i... |
| 1idsr 10785 | 1 is an identity element f... |
| 00sr 10786 | A signed real times 0 is 0... |
| ltasr 10787 | Ordering property of addit... |
| pn0sr 10788 | A signed real plus its neg... |
| negexsr 10789 | Existence of negative sign... |
| recexsrlem 10790 | The reciprocal of a positi... |
| addgt0sr 10791 | The sum of two positive si... |
| mulgt0sr 10792 | The product of two positiv... |
| sqgt0sr 10793 | The square of a nonzero si... |
| recexsr 10794 | The reciprocal of a nonzer... |
| mappsrpr 10795 | Mapping from positive sign... |
| ltpsrpr 10796 | Mapping of order from posi... |
| map2psrpr 10797 | Equivalence for positive s... |
| supsrlem 10798 | Lemma for supremum theorem... |
| supsr 10799 | A nonempty, bounded set of... |
| opelcn 10816 | Ordered pair membership in... |
| opelreal 10817 | Ordered pair membership in... |
| elreal 10818 | Membership in class of rea... |
| elreal2 10819 | Ordered pair membership in... |
| 0ncn 10820 | The empty set is not a com... |
| ltrelre 10821 | 'Less than' is a relation ... |
| addcnsr 10822 | Addition of complex number... |
| mulcnsr 10823 | Multiplication of complex ... |
| eqresr 10824 | Equality of real numbers i... |
| addresr 10825 | Addition of real numbers i... |
| mulresr 10826 | Multiplication of real num... |
| ltresr 10827 | Ordering of real subset of... |
| ltresr2 10828 | Ordering of real subset of... |
| dfcnqs 10829 | Technical trick to permit ... |
| addcnsrec 10830 | Technical trick to permit ... |
| mulcnsrec 10831 | Technical trick to permit ... |
| axaddf 10832 | Addition is an operation o... |
| axmulf 10833 | Multiplication is an opera... |
| axcnex 10834 | The complex numbers form a... |
| axresscn 10835 | The real numbers are a sub... |
| ax1cn 10836 | 1 is a complex number. Ax... |
| axicn 10837 | ` _i ` is a complex number... |
| axaddcl 10838 | Closure law for addition o... |
| axaddrcl 10839 | Closure law for addition i... |
| axmulcl 10840 | Closure law for multiplica... |
| axmulrcl 10841 | Closure law for multiplica... |
| axmulcom 10842 | Multiplication of complex ... |
| axaddass 10843 | Addition of complex number... |
| axmulass 10844 | Multiplication of complex ... |
| axdistr 10845 | Distributive law for compl... |
| axi2m1 10846 | i-squared equals -1 (expre... |
| ax1ne0 10847 | 1 and 0 are distinct. Axi... |
| ax1rid 10848 | ` 1 ` is an identity eleme... |
| axrnegex 10849 | Existence of negative of r... |
| axrrecex 10850 | Existence of reciprocal of... |
| axcnre 10851 | A complex number can be ex... |
| axpre-lttri 10852 | Ordering on reals satisfie... |
| axpre-lttrn 10853 | Ordering on reals is trans... |
| axpre-ltadd 10854 | Ordering property of addit... |
| axpre-mulgt0 10855 | The product of two positiv... |
| axpre-sup 10856 | A nonempty, bounded-above ... |
| wuncn 10857 | A weak universe containing... |
| cnex 10883 | Alias for ~ ax-cnex . See... |
| addcl 10884 | Alias for ~ ax-addcl , for... |
| readdcl 10885 | Alias for ~ ax-addrcl , fo... |
| mulcl 10886 | Alias for ~ ax-mulcl , for... |
| remulcl 10887 | Alias for ~ ax-mulrcl , fo... |
| mulcom 10888 | Alias for ~ ax-mulcom , fo... |
| addass 10889 | Alias for ~ ax-addass , fo... |
| mulass 10890 | Alias for ~ ax-mulass , fo... |
| adddi 10891 | Alias for ~ ax-distr , for... |
| recn 10892 | A real number is a complex... |
| reex 10893 | The real numbers form a se... |
| reelprrecn 10894 | Reals are a subset of the ... |
| cnelprrecn 10895 | Complex numbers are a subs... |
| elimne0 10896 | Hypothesis for weak deduct... |
| adddir 10897 | Distributive law for compl... |
| 0cn 10898 | Zero is a complex number. ... |
| 0cnd 10899 | Zero is a complex number, ... |
| c0ex 10900 | Zero is a set. (Contribut... |
| 1cnd 10901 | One is a complex number, d... |
| 1ex 10902 | One is a set. (Contribute... |
| cnre 10903 | Alias for ~ ax-cnre , for ... |
| mulid1 10904 | The number 1 is an identit... |
| mulid2 10905 | Identity law for multiplic... |
| 1re 10906 | The number 1 is real. Thi... |
| 1red 10907 | The number 1 is real, dedu... |
| 0re 10908 | The number 0 is real. Rem... |
| 0red 10909 | The number 0 is real, dedu... |
| mulid1i 10910 | Identity law for multiplic... |
| mulid2i 10911 | Identity law for multiplic... |
| addcli 10912 | Closure law for addition. ... |
| mulcli 10913 | Closure law for multiplica... |
| mulcomi 10914 | Commutative law for multip... |
| mulcomli 10915 | Commutative law for multip... |
| addassi 10916 | Associative law for additi... |
| mulassi 10917 | Associative law for multip... |
| adddii 10918 | Distributive law (left-dis... |
| adddiri 10919 | Distributive law (right-di... |
| recni 10920 | A real number is a complex... |
| readdcli 10921 | Closure law for addition o... |
| remulcli 10922 | Closure law for multiplica... |
| mulid1d 10923 | Identity law for multiplic... |
| mulid2d 10924 | Identity law for multiplic... |
| addcld 10925 | Closure law for addition. ... |
| mulcld 10926 | Closure law for multiplica... |
| mulcomd 10927 | Commutative law for multip... |
| addassd 10928 | Associative law for additi... |
| mulassd 10929 | Associative law for multip... |
| adddid 10930 | Distributive law (left-dis... |
| adddird 10931 | Distributive law (right-di... |
| adddirp1d 10932 | Distributive law, plus 1 v... |
| joinlmuladdmuld 10933 | Join AB+CB into (A+C) on L... |
| recnd 10934 | Deduction from real number... |
| readdcld 10935 | Closure law for addition o... |
| remulcld 10936 | Closure law for multiplica... |
| pnfnre 10947 | Plus infinity is not a rea... |
| pnfnre2 10948 | Plus infinity is not a rea... |
| mnfnre 10949 | Minus infinity is not a re... |
| ressxr 10950 | The standard reals are a s... |
| rexpssxrxp 10951 | The Cartesian product of s... |
| rexr 10952 | A standard real is an exte... |
| 0xr 10953 | Zero is an extended real. ... |
| renepnf 10954 | No (finite) real equals pl... |
| renemnf 10955 | No real equals minus infin... |
| rexrd 10956 | A standard real is an exte... |
| renepnfd 10957 | No (finite) real equals pl... |
| renemnfd 10958 | No real equals minus infin... |
| pnfex 10959 | Plus infinity exists. (Co... |
| pnfxr 10960 | Plus infinity belongs to t... |
| pnfnemnf 10961 | Plus and minus infinity ar... |
| mnfnepnf 10962 | Minus and plus infinity ar... |
| mnfxr 10963 | Minus infinity belongs to ... |
| rexri 10964 | A standard real is an exte... |
| 1xr 10965 | ` 1 ` is an extended real ... |
| renfdisj 10966 | The reals and the infiniti... |
| ltrelxr 10967 | "Less than" is a relation ... |
| ltrel 10968 | "Less than" is a relation.... |
| lerelxr 10969 | "Less than or equal to" is... |
| lerel 10970 | "Less than or equal to" is... |
| xrlenlt 10971 | "Less than or equal to" ex... |
| xrlenltd 10972 | "Less than or equal to" ex... |
| xrltnle 10973 | "Less than" expressed in t... |
| xrnltled 10974 | "Not less than" implies "l... |
| ssxr 10975 | The three (non-exclusive) ... |
| ltxrlt 10976 | The standard less-than ` <... |
| axlttri 10977 | Ordering on reals satisfie... |
| axlttrn 10978 | Ordering on reals is trans... |
| axltadd 10979 | Ordering property of addit... |
| axmulgt0 10980 | The product of two positiv... |
| axsup 10981 | A nonempty, bounded-above ... |
| lttr 10982 | Alias for ~ axlttrn , for ... |
| mulgt0 10983 | The product of two positiv... |
| lenlt 10984 | 'Less than or equal to' ex... |
| ltnle 10985 | 'Less than' expressed in t... |
| ltso 10986 | 'Less than' is a strict or... |
| gtso 10987 | 'Greater than' is a strict... |
| lttri2 10988 | Consequence of trichotomy.... |
| lttri3 10989 | Trichotomy law for 'less t... |
| lttri4 10990 | Trichotomy law for 'less t... |
| letri3 10991 | Trichotomy law. (Contribu... |
| leloe 10992 | 'Less than or equal to' ex... |
| eqlelt 10993 | Equality in terms of 'less... |
| ltle 10994 | 'Less than' implies 'less ... |
| leltne 10995 | 'Less than or equal to' im... |
| lelttr 10996 | Transitive law. (Contribu... |
| ltletr 10997 | Transitive law. (Contribu... |
| ltleletr 10998 | Transitive law, weaker for... |
| letr 10999 | Transitive law. (Contribu... |
| ltnr 11000 | 'Less than' is irreflexive... |
| leid 11001 | 'Less than or equal to' is... |
| ltne 11002 | 'Less than' implies not eq... |
| ltnsym 11003 | 'Less than' is not symmetr... |
| ltnsym2 11004 | 'Less than' is antisymmetr... |
| letric 11005 | Trichotomy law. (Contribu... |
| ltlen 11006 | 'Less than' expressed in t... |
| eqle 11007 | Equality implies 'less tha... |
| eqled 11008 | Equality implies 'less tha... |
| ltadd2 11009 | Addition to both sides of ... |
| ne0gt0 11010 | A nonzero nonnegative numb... |
| lecasei 11011 | Ordering elimination by ca... |
| lelttric 11012 | Trichotomy law. (Contribu... |
| ltlecasei 11013 | Ordering elimination by ca... |
| ltnri 11014 | 'Less than' is irreflexive... |
| eqlei 11015 | Equality implies 'less tha... |
| eqlei2 11016 | Equality implies 'less tha... |
| gtneii 11017 | 'Less than' implies not eq... |
| ltneii 11018 | 'Greater than' implies not... |
| lttri2i 11019 | Consequence of trichotomy.... |
| lttri3i 11020 | Consequence of trichotomy.... |
| letri3i 11021 | Consequence of trichotomy.... |
| leloei 11022 | 'Less than or equal to' in... |
| ltleni 11023 | 'Less than' expressed in t... |
| ltnsymi 11024 | 'Less than' is not symmetr... |
| lenlti 11025 | 'Less than or equal to' in... |
| ltnlei 11026 | 'Less than' in terms of 'l... |
| ltlei 11027 | 'Less than' implies 'less ... |
| ltleii 11028 | 'Less than' implies 'less ... |
| ltnei 11029 | 'Less than' implies not eq... |
| letrii 11030 | Trichotomy law for 'less t... |
| lttri 11031 | 'Less than' is transitive.... |
| lelttri 11032 | 'Less than or equal to', '... |
| ltletri 11033 | 'Less than', 'less than or... |
| letri 11034 | 'Less than or equal to' is... |
| le2tri3i 11035 | Extended trichotomy law fo... |
| ltadd2i 11036 | Addition to both sides of ... |
| mulgt0i 11037 | The product of two positiv... |
| mulgt0ii 11038 | The product of two positiv... |
| ltnrd 11039 | 'Less than' is irreflexive... |
| gtned 11040 | 'Less than' implies not eq... |
| ltned 11041 | 'Greater than' implies not... |
| ne0gt0d 11042 | A nonzero nonnegative numb... |
| lttrid 11043 | Ordering on reals satisfie... |
| lttri2d 11044 | Consequence of trichotomy.... |
| lttri3d 11045 | Consequence of trichotomy.... |
| lttri4d 11046 | Trichotomy law for 'less t... |
| letri3d 11047 | Consequence of trichotomy.... |
| leloed 11048 | 'Less than or equal to' in... |
| eqleltd 11049 | Equality in terms of 'less... |
| ltlend 11050 | 'Less than' expressed in t... |
| lenltd 11051 | 'Less than or equal to' in... |
| ltnled 11052 | 'Less than' in terms of 'l... |
| ltled 11053 | 'Less than' implies 'less ... |
| ltnsymd 11054 | 'Less than' implies 'less ... |
| nltled 11055 | 'Not less than ' implies '... |
| lensymd 11056 | 'Less than or equal to' im... |
| letrid 11057 | Trichotomy law for 'less t... |
| leltned 11058 | 'Less than or equal to' im... |
| leneltd 11059 | 'Less than or equal to' an... |
| mulgt0d 11060 | The product of two positiv... |
| ltadd2d 11061 | Addition to both sides of ... |
| letrd 11062 | Transitive law deduction f... |
| lelttrd 11063 | Transitive law deduction f... |
| ltadd2dd 11064 | Addition to both sides of ... |
| ltletrd 11065 | Transitive law deduction f... |
| lttrd 11066 | Transitive law deduction f... |
| lelttrdi 11067 | If a number is less than a... |
| dedekind 11068 | The Dedekind cut theorem. ... |
| dedekindle 11069 | The Dedekind cut theorem, ... |
| mul12 11070 | Commutative/associative la... |
| mul32 11071 | Commutative/associative la... |
| mul31 11072 | Commutative/associative la... |
| mul4 11073 | Rearrangement of 4 factors... |
| mul4r 11074 | Rearrangement of 4 factors... |
| muladd11 11075 | A simple product of sums e... |
| 1p1times 11076 | Two times a number. (Cont... |
| peano2cn 11077 | A theorem for complex numb... |
| peano2re 11078 | A theorem for reals analog... |
| readdcan 11079 | Cancellation law for addit... |
| 00id 11080 | ` 0 ` is its own additive ... |
| mul02lem1 11081 | Lemma for ~ mul02 . If an... |
| mul02lem2 11082 | Lemma for ~ mul02 . Zero ... |
| mul02 11083 | Multiplication by ` 0 ` . ... |
| mul01 11084 | Multiplication by ` 0 ` . ... |
| addid1 11085 | ` 0 ` is an additive ident... |
| cnegex 11086 | Existence of the negative ... |
| cnegex2 11087 | Existence of a left invers... |
| addid2 11088 | ` 0 ` is a left identity f... |
| addcan 11089 | Cancellation law for addit... |
| addcan2 11090 | Cancellation law for addit... |
| addcom 11091 | Addition commutes. This u... |
| addid1i 11092 | ` 0 ` is an additive ident... |
| addid2i 11093 | ` 0 ` is a left identity f... |
| mul02i 11094 | Multiplication by 0. Theo... |
| mul01i 11095 | Multiplication by ` 0 ` . ... |
| addcomi 11096 | Addition commutes. Based ... |
| addcomli 11097 | Addition commutes. (Contr... |
| addcani 11098 | Cancellation law for addit... |
| addcan2i 11099 | Cancellation law for addit... |
| mul12i 11100 | Commutative/associative la... |
| mul32i 11101 | Commutative/associative la... |
| mul4i 11102 | Rearrangement of 4 factors... |
| mul02d 11103 | Multiplication by 0. Theo... |
| mul01d 11104 | Multiplication by ` 0 ` . ... |
| addid1d 11105 | ` 0 ` is an additive ident... |
| addid2d 11106 | ` 0 ` is a left identity f... |
| addcomd 11107 | Addition commutes. Based ... |
| addcand 11108 | Cancellation law for addit... |
| addcan2d 11109 | Cancellation law for addit... |
| addcanad 11110 | Cancelling a term on the l... |
| addcan2ad 11111 | Cancelling a term on the r... |
| addneintrd 11112 | Introducing a term on the ... |
| addneintr2d 11113 | Introducing a term on the ... |
| mul12d 11114 | Commutative/associative la... |
| mul32d 11115 | Commutative/associative la... |
| mul31d 11116 | Commutative/associative la... |
| mul4d 11117 | Rearrangement of 4 factors... |
| muladd11r 11118 | A simple product of sums e... |
| comraddd 11119 | Commute RHS addition, in d... |
| ltaddneg 11120 | Adding a negative number t... |
| ltaddnegr 11121 | Adding a negative number t... |
| add12 11122 | Commutative/associative la... |
| add32 11123 | Commutative/associative la... |
| add32r 11124 | Commutative/associative la... |
| add4 11125 | Rearrangement of 4 terms i... |
| add42 11126 | Rearrangement of 4 terms i... |
| add12i 11127 | Commutative/associative la... |
| add32i 11128 | Commutative/associative la... |
| add4i 11129 | Rearrangement of 4 terms i... |
| add42i 11130 | Rearrangement of 4 terms i... |
| add12d 11131 | Commutative/associative la... |
| add32d 11132 | Commutative/associative la... |
| add4d 11133 | Rearrangement of 4 terms i... |
| add42d 11134 | Rearrangement of 4 terms i... |
| 0cnALT 11139 | Alternate proof of ~ 0cn w... |
| 0cnALT2 11140 | Alternate proof of ~ 0cnAL... |
| negeu 11141 | Existential uniqueness of ... |
| subval 11142 | Value of subtraction, whic... |
| negeq 11143 | Equality theorem for negat... |
| negeqi 11144 | Equality inference for neg... |
| negeqd 11145 | Equality deduction for neg... |
| nfnegd 11146 | Deduction version of ~ nfn... |
| nfneg 11147 | Bound-variable hypothesis ... |
| csbnegg 11148 | Move class substitution in... |
| negex 11149 | A negative is a set. (Con... |
| subcl 11150 | Closure law for subtractio... |
| negcl 11151 | Closure law for negative. ... |
| negicn 11152 | ` -u _i ` is a complex num... |
| subf 11153 | Subtraction is an operatio... |
| subadd 11154 | Relationship between subtr... |
| subadd2 11155 | Relationship between subtr... |
| subsub23 11156 | Swap subtrahend and result... |
| pncan 11157 | Cancellation law for subtr... |
| pncan2 11158 | Cancellation law for subtr... |
| pncan3 11159 | Subtraction and addition o... |
| npcan 11160 | Cancellation law for subtr... |
| addsubass 11161 | Associative-type law for a... |
| addsub 11162 | Law for addition and subtr... |
| subadd23 11163 | Commutative/associative la... |
| addsub12 11164 | Commutative/associative la... |
| 2addsub 11165 | Law for subtraction and ad... |
| addsubeq4 11166 | Relation between sums and ... |
| pncan3oi 11167 | Subtraction and addition o... |
| mvrraddi 11168 | Move the right term in a s... |
| mvlladdi 11169 | Move the left term in a su... |
| subid 11170 | Subtraction of a number fr... |
| subid1 11171 | Identity law for subtracti... |
| npncan 11172 | Cancellation law for subtr... |
| nppcan 11173 | Cancellation law for subtr... |
| nnpcan 11174 | Cancellation law for subtr... |
| nppcan3 11175 | Cancellation law for subtr... |
| subcan2 11176 | Cancellation law for subtr... |
| subeq0 11177 | If the difference between ... |
| npncan2 11178 | Cancellation law for subtr... |
| subsub2 11179 | Law for double subtraction... |
| nncan 11180 | Cancellation law for subtr... |
| subsub 11181 | Law for double subtraction... |
| nppcan2 11182 | Cancellation law for subtr... |
| subsub3 11183 | Law for double subtraction... |
| subsub4 11184 | Law for double subtraction... |
| sub32 11185 | Swap the second and third ... |
| nnncan 11186 | Cancellation law for subtr... |
| nnncan1 11187 | Cancellation law for subtr... |
| nnncan2 11188 | Cancellation law for subtr... |
| npncan3 11189 | Cancellation law for subtr... |
| pnpcan 11190 | Cancellation law for mixed... |
| pnpcan2 11191 | Cancellation law for mixed... |
| pnncan 11192 | Cancellation law for mixed... |
| ppncan 11193 | Cancellation law for mixed... |
| addsub4 11194 | Rearrangement of 4 terms i... |
| subadd4 11195 | Rearrangement of 4 terms i... |
| sub4 11196 | Rearrangement of 4 terms i... |
| neg0 11197 | Minus 0 equals 0. (Contri... |
| negid 11198 | Addition of a number and i... |
| negsub 11199 | Relationship between subtr... |
| subneg 11200 | Relationship between subtr... |
| negneg 11201 | A number is equal to the n... |
| neg11 11202 | Negative is one-to-one. (... |
| negcon1 11203 | Negative contraposition la... |
| negcon2 11204 | Negative contraposition la... |
| negeq0 11205 | A number is zero iff its n... |
| subcan 11206 | Cancellation law for subtr... |
| negsubdi 11207 | Distribution of negative o... |
| negdi 11208 | Distribution of negative o... |
| negdi2 11209 | Distribution of negative o... |
| negsubdi2 11210 | Distribution of negative o... |
| neg2sub 11211 | Relationship between subtr... |
| renegcli 11212 | Closure law for negative o... |
| resubcli 11213 | Closure law for subtractio... |
| renegcl 11214 | Closure law for negative o... |
| resubcl 11215 | Closure law for subtractio... |
| negreb 11216 | The negative of a real is ... |
| peano2cnm 11217 | "Reverse" second Peano pos... |
| peano2rem 11218 | "Reverse" second Peano pos... |
| negcli 11219 | Closure law for negative. ... |
| negidi 11220 | Addition of a number and i... |
| negnegi 11221 | A number is equal to the n... |
| subidi 11222 | Subtraction of a number fr... |
| subid1i 11223 | Identity law for subtracti... |
| negne0bi 11224 | A number is nonzero iff it... |
| negrebi 11225 | The negative of a real is ... |
| negne0i 11226 | The negative of a nonzero ... |
| subcli 11227 | Closure law for subtractio... |
| pncan3i 11228 | Subtraction and addition o... |
| negsubi 11229 | Relationship between subtr... |
| subnegi 11230 | Relationship between subtr... |
| subeq0i 11231 | If the difference between ... |
| neg11i 11232 | Negative is one-to-one. (... |
| negcon1i 11233 | Negative contraposition la... |
| negcon2i 11234 | Negative contraposition la... |
| negdii 11235 | Distribution of negative o... |
| negsubdii 11236 | Distribution of negative o... |
| negsubdi2i 11237 | Distribution of negative o... |
| subaddi 11238 | Relationship between subtr... |
| subadd2i 11239 | Relationship between subtr... |
| subaddrii 11240 | Relationship between subtr... |
| subsub23i 11241 | Swap subtrahend and result... |
| addsubassi 11242 | Associative-type law for s... |
| addsubi 11243 | Law for subtraction and ad... |
| subcani 11244 | Cancellation law for subtr... |
| subcan2i 11245 | Cancellation law for subtr... |
| pnncani 11246 | Cancellation law for mixed... |
| addsub4i 11247 | Rearrangement of 4 terms i... |
| 0reALT 11248 | Alternate proof of ~ 0re .... |
| negcld 11249 | Closure law for negative. ... |
| subidd 11250 | Subtraction of a number fr... |
| subid1d 11251 | Identity law for subtracti... |
| negidd 11252 | Addition of a number and i... |
| negnegd 11253 | A number is equal to the n... |
| negeq0d 11254 | A number is zero iff its n... |
| negne0bd 11255 | A number is nonzero iff it... |
| negcon1d 11256 | Contraposition law for una... |
| negcon1ad 11257 | Contraposition law for una... |
| neg11ad 11258 | The negatives of two compl... |
| negned 11259 | If two complex numbers are... |
| negne0d 11260 | The negative of a nonzero ... |
| negrebd 11261 | The negative of a real is ... |
| subcld 11262 | Closure law for subtractio... |
| pncand 11263 | Cancellation law for subtr... |
| pncan2d 11264 | Cancellation law for subtr... |
| pncan3d 11265 | Subtraction and addition o... |
| npcand 11266 | Cancellation law for subtr... |
| nncand 11267 | Cancellation law for subtr... |
| negsubd 11268 | Relationship between subtr... |
| subnegd 11269 | Relationship between subtr... |
| subeq0d 11270 | If the difference between ... |
| subne0d 11271 | Two unequal numbers have n... |
| subeq0ad 11272 | The difference of two comp... |
| subne0ad 11273 | If the difference of two c... |
| neg11d 11274 | If the difference between ... |
| negdid 11275 | Distribution of negative o... |
| negdi2d 11276 | Distribution of negative o... |
| negsubdid 11277 | Distribution of negative o... |
| negsubdi2d 11278 | Distribution of negative o... |
| neg2subd 11279 | Relationship between subtr... |
| subaddd 11280 | Relationship between subtr... |
| subadd2d 11281 | Relationship between subtr... |
| addsubassd 11282 | Associative-type law for s... |
| addsubd 11283 | Law for subtraction and ad... |
| subadd23d 11284 | Commutative/associative la... |
| addsub12d 11285 | Commutative/associative la... |
| npncand 11286 | Cancellation law for subtr... |
| nppcand 11287 | Cancellation law for subtr... |
| nppcan2d 11288 | Cancellation law for subtr... |
| nppcan3d 11289 | Cancellation law for subtr... |
| subsubd 11290 | Law for double subtraction... |
| subsub2d 11291 | Law for double subtraction... |
| subsub3d 11292 | Law for double subtraction... |
| subsub4d 11293 | Law for double subtraction... |
| sub32d 11294 | Swap the second and third ... |
| nnncand 11295 | Cancellation law for subtr... |
| nnncan1d 11296 | Cancellation law for subtr... |
| nnncan2d 11297 | Cancellation law for subtr... |
| npncan3d 11298 | Cancellation law for subtr... |
| pnpcand 11299 | Cancellation law for mixed... |
| pnpcan2d 11300 | Cancellation law for mixed... |
| pnncand 11301 | Cancellation law for mixed... |
| ppncand 11302 | Cancellation law for mixed... |
| subcand 11303 | Cancellation law for subtr... |
| subcan2d 11304 | Cancellation law for subtr... |
| subcanad 11305 | Cancellation law for subtr... |
| subneintrd 11306 | Introducing subtraction on... |
| subcan2ad 11307 | Cancellation law for subtr... |
| subneintr2d 11308 | Introducing subtraction on... |
| addsub4d 11309 | Rearrangement of 4 terms i... |
| subadd4d 11310 | Rearrangement of 4 terms i... |
| sub4d 11311 | Rearrangement of 4 terms i... |
| 2addsubd 11312 | Law for subtraction and ad... |
| addsubeq4d 11313 | Relation between sums and ... |
| subeqxfrd 11314 | Transfer two terms of a su... |
| mvlraddd 11315 | Move the right term in a s... |
| mvlladdd 11316 | Move the left term in a su... |
| mvrraddd 11317 | Move the right term in a s... |
| mvrladdd 11318 | Move the left term in a su... |
| assraddsubd 11319 | Associate RHS addition-sub... |
| subaddeqd 11320 | Transfer two terms of a su... |
| addlsub 11321 | Left-subtraction: Subtrac... |
| addrsub 11322 | Right-subtraction: Subtra... |
| subexsub 11323 | A subtraction law: Exchan... |
| addid0 11324 | If adding a number to a an... |
| addn0nid 11325 | Adding a nonzero number to... |
| pnpncand 11326 | Addition/subtraction cance... |
| subeqrev 11327 | Reverse the order of subtr... |
| addeq0 11328 | Two complex numbers add up... |
| pncan1 11329 | Cancellation law for addit... |
| npcan1 11330 | Cancellation law for subtr... |
| subeq0bd 11331 | If two complex numbers are... |
| renegcld 11332 | Closure law for negative o... |
| resubcld 11333 | Closure law for subtractio... |
| negn0 11334 | The image under negation o... |
| negf1o 11335 | Negation is an isomorphism... |
| kcnktkm1cn 11336 | k times k minus 1 is a com... |
| muladd 11337 | Product of two sums. (Con... |
| subdi 11338 | Distribution of multiplica... |
| subdir 11339 | Distribution of multiplica... |
| ine0 11340 | The imaginary unit ` _i ` ... |
| mulneg1 11341 | Product with negative is n... |
| mulneg2 11342 | The product with a negativ... |
| mulneg12 11343 | Swap the negative sign in ... |
| mul2neg 11344 | Product of two negatives. ... |
| submul2 11345 | Convert a subtraction to a... |
| mulm1 11346 | Product with minus one is ... |
| addneg1mul 11347 | Addition with product with... |
| mulsub 11348 | Product of two differences... |
| mulsub2 11349 | Swap the order of subtract... |
| mulm1i 11350 | Product with minus one is ... |
| mulneg1i 11351 | Product with negative is n... |
| mulneg2i 11352 | Product with negative is n... |
| mul2negi 11353 | Product of two negatives. ... |
| subdii 11354 | Distribution of multiplica... |
| subdiri 11355 | Distribution of multiplica... |
| muladdi 11356 | Product of two sums. (Con... |
| mulm1d 11357 | Product with minus one is ... |
| mulneg1d 11358 | Product with negative is n... |
| mulneg2d 11359 | Product with negative is n... |
| mul2negd 11360 | Product of two negatives. ... |
| subdid 11361 | Distribution of multiplica... |
| subdird 11362 | Distribution of multiplica... |
| muladdd 11363 | Product of two sums. (Con... |
| mulsubd 11364 | Product of two differences... |
| muls1d 11365 | Multiplication by one minu... |
| mulsubfacd 11366 | Multiplication followed by... |
| addmulsub 11367 | The product of a sum and a... |
| subaddmulsub 11368 | The difference with a prod... |
| mulsubaddmulsub 11369 | A special difference of a ... |
| gt0ne0 11370 | Positive implies nonzero. ... |
| lt0ne0 11371 | A number which is less tha... |
| ltadd1 11372 | Addition to both sides of ... |
| leadd1 11373 | Addition to both sides of ... |
| leadd2 11374 | Addition to both sides of ... |
| ltsubadd 11375 | 'Less than' relationship b... |
| ltsubadd2 11376 | 'Less than' relationship b... |
| lesubadd 11377 | 'Less than or equal to' re... |
| lesubadd2 11378 | 'Less than or equal to' re... |
| ltaddsub 11379 | 'Less than' relationship b... |
| ltaddsub2 11380 | 'Less than' relationship b... |
| leaddsub 11381 | 'Less than or equal to' re... |
| leaddsub2 11382 | 'Less than or equal to' re... |
| suble 11383 | Swap subtrahends in an ine... |
| lesub 11384 | Swap subtrahends in an ine... |
| ltsub23 11385 | 'Less than' relationship b... |
| ltsub13 11386 | 'Less than' relationship b... |
| le2add 11387 | Adding both sides of two '... |
| ltleadd 11388 | Adding both sides of two o... |
| leltadd 11389 | Adding both sides of two o... |
| lt2add 11390 | Adding both sides of two '... |
| addgt0 11391 | The sum of 2 positive numb... |
| addgegt0 11392 | The sum of nonnegative and... |
| addgtge0 11393 | The sum of nonnegative and... |
| addge0 11394 | The sum of 2 nonnegative n... |
| ltaddpos 11395 | Adding a positive number t... |
| ltaddpos2 11396 | Adding a positive number t... |
| ltsubpos 11397 | Subtracting a positive num... |
| posdif 11398 | Comparison of two numbers ... |
| lesub1 11399 | Subtraction from both side... |
| lesub2 11400 | Subtraction of both sides ... |
| ltsub1 11401 | Subtraction from both side... |
| ltsub2 11402 | Subtraction of both sides ... |
| lt2sub 11403 | Subtracting both sides of ... |
| le2sub 11404 | Subtracting both sides of ... |
| ltneg 11405 | Negative of both sides of ... |
| ltnegcon1 11406 | Contraposition of negative... |
| ltnegcon2 11407 | Contraposition of negative... |
| leneg 11408 | Negative of both sides of ... |
| lenegcon1 11409 | Contraposition of negative... |
| lenegcon2 11410 | Contraposition of negative... |
| lt0neg1 11411 | Comparison of a number and... |
| lt0neg2 11412 | Comparison of a number and... |
| le0neg1 11413 | Comparison of a number and... |
| le0neg2 11414 | Comparison of a number and... |
| addge01 11415 | A number is less than or e... |
| addge02 11416 | A number is less than or e... |
| add20 11417 | Two nonnegative numbers ar... |
| subge0 11418 | Nonnegative subtraction. ... |
| suble0 11419 | Nonpositive subtraction. ... |
| leaddle0 11420 | The sum of a real number a... |
| subge02 11421 | Nonnegative subtraction. ... |
| lesub0 11422 | Lemma to show a nonnegativ... |
| mulge0 11423 | The product of two nonnega... |
| mullt0 11424 | The product of two negativ... |
| msqgt0 11425 | A nonzero square is positi... |
| msqge0 11426 | A square is nonnegative. ... |
| 0lt1 11427 | 0 is less than 1. Theorem... |
| 0le1 11428 | 0 is less than or equal to... |
| relin01 11429 | An interval law for less t... |
| ltordlem 11430 | Lemma for ~ ltord1 . (Con... |
| ltord1 11431 | Infer an ordering relation... |
| leord1 11432 | Infer an ordering relation... |
| eqord1 11433 | A strictly increasing real... |
| ltord2 11434 | Infer an ordering relation... |
| leord2 11435 | Infer an ordering relation... |
| eqord2 11436 | A strictly decreasing real... |
| wloglei 11437 | Form of ~ wlogle where bot... |
| wlogle 11438 | If the predicate ` ch ( x ... |
| leidi 11439 | 'Less than or equal to' is... |
| gt0ne0i 11440 | Positive means nonzero (us... |
| gt0ne0ii 11441 | Positive implies nonzero. ... |
| msqgt0i 11442 | A nonzero square is positi... |
| msqge0i 11443 | A square is nonnegative. ... |
| addgt0i 11444 | Addition of 2 positive num... |
| addge0i 11445 | Addition of 2 nonnegative ... |
| addgegt0i 11446 | Addition of nonnegative an... |
| addgt0ii 11447 | Addition of 2 positive num... |
| add20i 11448 | Two nonnegative numbers ar... |
| ltnegi 11449 | Negative of both sides of ... |
| lenegi 11450 | Negative of both sides of ... |
| ltnegcon2i 11451 | Contraposition of negative... |
| mulge0i 11452 | The product of two nonnega... |
| lesub0i 11453 | Lemma to show a nonnegativ... |
| ltaddposi 11454 | Adding a positive number t... |
| posdifi 11455 | Comparison of two numbers ... |
| ltnegcon1i 11456 | Contraposition of negative... |
| lenegcon1i 11457 | Contraposition of negative... |
| subge0i 11458 | Nonnegative subtraction. ... |
| ltadd1i 11459 | Addition to both sides of ... |
| leadd1i 11460 | Addition to both sides of ... |
| leadd2i 11461 | Addition to both sides of ... |
| ltsubaddi 11462 | 'Less than' relationship b... |
| lesubaddi 11463 | 'Less than or equal to' re... |
| ltsubadd2i 11464 | 'Less than' relationship b... |
| lesubadd2i 11465 | 'Less than or equal to' re... |
| ltaddsubi 11466 | 'Less than' relationship b... |
| lt2addi 11467 | Adding both side of two in... |
| le2addi 11468 | Adding both side of two in... |
| gt0ne0d 11469 | Positive implies nonzero. ... |
| lt0ne0d 11470 | Something less than zero i... |
| leidd 11471 | 'Less than or equal to' is... |
| msqgt0d 11472 | A nonzero square is positi... |
| msqge0d 11473 | A square is nonnegative. ... |
| lt0neg1d 11474 | Comparison of a number and... |
| lt0neg2d 11475 | Comparison of a number and... |
| le0neg1d 11476 | Comparison of a number and... |
| le0neg2d 11477 | Comparison of a number and... |
| addgegt0d 11478 | Addition of nonnegative an... |
| addgtge0d 11479 | Addition of positive and n... |
| addgt0d 11480 | Addition of 2 positive num... |
| addge0d 11481 | Addition of 2 nonnegative ... |
| mulge0d 11482 | The product of two nonnega... |
| ltnegd 11483 | Negative of both sides of ... |
| lenegd 11484 | Negative of both sides of ... |
| ltnegcon1d 11485 | Contraposition of negative... |
| ltnegcon2d 11486 | Contraposition of negative... |
| lenegcon1d 11487 | Contraposition of negative... |
| lenegcon2d 11488 | Contraposition of negative... |
| ltaddposd 11489 | Adding a positive number t... |
| ltaddpos2d 11490 | Adding a positive number t... |
| ltsubposd 11491 | Subtracting a positive num... |
| posdifd 11492 | Comparison of two numbers ... |
| addge01d 11493 | A number is less than or e... |
| addge02d 11494 | A number is less than or e... |
| subge0d 11495 | Nonnegative subtraction. ... |
| suble0d 11496 | Nonpositive subtraction. ... |
| subge02d 11497 | Nonnegative subtraction. ... |
| ltadd1d 11498 | Addition to both sides of ... |
| leadd1d 11499 | Addition to both sides of ... |
| leadd2d 11500 | Addition to both sides of ... |
| ltsubaddd 11501 | 'Less than' relationship b... |
| lesubaddd 11502 | 'Less than or equal to' re... |
| ltsubadd2d 11503 | 'Less than' relationship b... |
| lesubadd2d 11504 | 'Less than or equal to' re... |
| ltaddsubd 11505 | 'Less than' relationship b... |
| ltaddsub2d 11506 | 'Less than' relationship b... |
| leaddsub2d 11507 | 'Less than or equal to' re... |
| subled 11508 | Swap subtrahends in an ine... |
| lesubd 11509 | Swap subtrahends in an ine... |
| ltsub23d 11510 | 'Less than' relationship b... |
| ltsub13d 11511 | 'Less than' relationship b... |
| lesub1d 11512 | Subtraction from both side... |
| lesub2d 11513 | Subtraction of both sides ... |
| ltsub1d 11514 | Subtraction from both side... |
| ltsub2d 11515 | Subtraction of both sides ... |
| ltadd1dd 11516 | Addition to both sides of ... |
| ltsub1dd 11517 | Subtraction from both side... |
| ltsub2dd 11518 | Subtraction of both sides ... |
| leadd1dd 11519 | Addition to both sides of ... |
| leadd2dd 11520 | Addition to both sides of ... |
| lesub1dd 11521 | Subtraction from both side... |
| lesub2dd 11522 | Subtraction of both sides ... |
| lesub3d 11523 | The result of subtracting ... |
| le2addd 11524 | Adding both side of two in... |
| le2subd 11525 | Subtracting both sides of ... |
| ltleaddd 11526 | Adding both sides of two o... |
| leltaddd 11527 | Adding both sides of two o... |
| lt2addd 11528 | Adding both side of two in... |
| lt2subd 11529 | Subtracting both sides of ... |
| possumd 11530 | Condition for a positive s... |
| sublt0d 11531 | When a subtraction gives a... |
| ltaddsublt 11532 | Addition and subtraction o... |
| 1le1 11533 | One is less than or equal ... |
| ixi 11534 | ` _i ` times itself is min... |
| recextlem1 11535 | Lemma for ~ recex . (Cont... |
| recextlem2 11536 | Lemma for ~ recex . (Cont... |
| recex 11537 | Existence of reciprocal of... |
| mulcand 11538 | Cancellation law for multi... |
| mulcan2d 11539 | Cancellation law for multi... |
| mulcanad 11540 | Cancellation of a nonzero ... |
| mulcan2ad 11541 | Cancellation of a nonzero ... |
| mulcan 11542 | Cancellation law for multi... |
| mulcan2 11543 | Cancellation law for multi... |
| mulcani 11544 | Cancellation law for multi... |
| mul0or 11545 | If a product is zero, one ... |
| mulne0b 11546 | The product of two nonzero... |
| mulne0 11547 | The product of two nonzero... |
| mulne0i 11548 | The product of two nonzero... |
| muleqadd 11549 | Property of numbers whose ... |
| receu 11550 | Existential uniqueness of ... |
| mulnzcnopr 11551 | Multiplication maps nonzer... |
| msq0i 11552 | A number is zero iff its s... |
| mul0ori 11553 | If a product is zero, one ... |
| msq0d 11554 | A number is zero iff its s... |
| mul0ord 11555 | If a product is zero, one ... |
| mulne0bd 11556 | The product of two nonzero... |
| mulne0d 11557 | The product of two nonzero... |
| mulcan1g 11558 | A generalized form of the ... |
| mulcan2g 11559 | A generalized form of the ... |
| mulne0bad 11560 | A factor of a nonzero comp... |
| mulne0bbd 11561 | A factor of a nonzero comp... |
| 1div0 11564 | You can't divide by zero, ... |
| divval 11565 | Value of division: if ` A ... |
| divmul 11566 | Relationship between divis... |
| divmul2 11567 | Relationship between divis... |
| divmul3 11568 | Relationship between divis... |
| divcl 11569 | Closure law for division. ... |
| reccl 11570 | Closure law for reciprocal... |
| divcan2 11571 | A cancellation law for div... |
| divcan1 11572 | A cancellation law for div... |
| diveq0 11573 | A ratio is zero iff the nu... |
| divne0b 11574 | The ratio of nonzero numbe... |
| divne0 11575 | The ratio of nonzero numbe... |
| recne0 11576 | The reciprocal of a nonzer... |
| recid 11577 | Multiplication of a number... |
| recid2 11578 | Multiplication of a number... |
| divrec 11579 | Relationship between divis... |
| divrec2 11580 | Relationship between divis... |
| divass 11581 | An associative law for div... |
| div23 11582 | A commutative/associative ... |
| div32 11583 | A commutative/associative ... |
| div13 11584 | A commutative/associative ... |
| div12 11585 | A commutative/associative ... |
| divmulass 11586 | An associative law for div... |
| divmulasscom 11587 | An associative/commutative... |
| divdir 11588 | Distribution of division o... |
| divcan3 11589 | A cancellation law for div... |
| divcan4 11590 | A cancellation law for div... |
| div11 11591 | One-to-one relationship fo... |
| divid 11592 | A number divided by itself... |
| div0 11593 | Division into zero is zero... |
| div1 11594 | A number divided by 1 is i... |
| 1div1e1 11595 | 1 divided by 1 is 1. (Con... |
| diveq1 11596 | Equality in terms of unit ... |
| divneg 11597 | Move negative sign inside ... |
| muldivdir 11598 | Distribution of division o... |
| divsubdir 11599 | Distribution of division o... |
| subdivcomb1 11600 | Bring a term in a subtract... |
| subdivcomb2 11601 | Bring a term in a subtract... |
| recrec 11602 | A number is equal to the r... |
| rec11 11603 | Reciprocal is one-to-one. ... |
| rec11r 11604 | Mutual reciprocals. (Cont... |
| divmuldiv 11605 | Multiplication of two rati... |
| divdivdiv 11606 | Division of two ratios. T... |
| divcan5 11607 | Cancellation of common fac... |
| divmul13 11608 | Swap the denominators in t... |
| divmul24 11609 | Swap the numerators in the... |
| divmuleq 11610 | Cross-multiply in an equal... |
| recdiv 11611 | The reciprocal of a ratio.... |
| divcan6 11612 | Cancellation of inverted f... |
| divdiv32 11613 | Swap denominators in a div... |
| divcan7 11614 | Cancel equal divisors in a... |
| dmdcan 11615 | Cancellation law for divis... |
| divdiv1 11616 | Division into a fraction. ... |
| divdiv2 11617 | Division by a fraction. (... |
| recdiv2 11618 | Division into a reciprocal... |
| ddcan 11619 | Cancellation in a double d... |
| divadddiv 11620 | Addition of two ratios. T... |
| divsubdiv 11621 | Subtraction of two ratios.... |
| conjmul 11622 | Two numbers whose reciproc... |
| rereccl 11623 | Closure law for reciprocal... |
| redivcl 11624 | Closure law for division o... |
| eqneg 11625 | A number equal to its nega... |
| eqnegd 11626 | A complex number equals it... |
| eqnegad 11627 | If a complex number equals... |
| div2neg 11628 | Quotient of two negatives.... |
| divneg2 11629 | Move negative sign inside ... |
| recclzi 11630 | Closure law for reciprocal... |
| recne0zi 11631 | The reciprocal of a nonzer... |
| recidzi 11632 | Multiplication of a number... |
| div1i 11633 | A number divided by 1 is i... |
| eqnegi 11634 | A number equal to its nega... |
| reccli 11635 | Closure law for reciprocal... |
| recidi 11636 | Multiplication of a number... |
| recreci 11637 | A number is equal to the r... |
| dividi 11638 | A number divided by itself... |
| div0i 11639 | Division into zero is zero... |
| divclzi 11640 | Closure law for division. ... |
| divcan1zi 11641 | A cancellation law for div... |
| divcan2zi 11642 | A cancellation law for div... |
| divreczi 11643 | Relationship between divis... |
| divcan3zi 11644 | A cancellation law for div... |
| divcan4zi 11645 | A cancellation law for div... |
| rec11i 11646 | Reciprocal is one-to-one. ... |
| divcli 11647 | Closure law for division. ... |
| divcan2i 11648 | A cancellation law for div... |
| divcan1i 11649 | A cancellation law for div... |
| divreci 11650 | Relationship between divis... |
| divcan3i 11651 | A cancellation law for div... |
| divcan4i 11652 | A cancellation law for div... |
| divne0i 11653 | The ratio of nonzero numbe... |
| rec11ii 11654 | Reciprocal is one-to-one. ... |
| divasszi 11655 | An associative law for div... |
| divmulzi 11656 | Relationship between divis... |
| divdirzi 11657 | Distribution of division o... |
| divdiv23zi 11658 | Swap denominators in a div... |
| divmuli 11659 | Relationship between divis... |
| divdiv32i 11660 | Swap denominators in a div... |
| divassi 11661 | An associative law for div... |
| divdiri 11662 | Distribution of division o... |
| div23i 11663 | A commutative/associative ... |
| div11i 11664 | One-to-one relationship fo... |
| divmuldivi 11665 | Multiplication of two rati... |
| divmul13i 11666 | Swap denominators of two r... |
| divadddivi 11667 | Addition of two ratios. T... |
| divdivdivi 11668 | Division of two ratios. T... |
| rerecclzi 11669 | Closure law for reciprocal... |
| rereccli 11670 | Closure law for reciprocal... |
| redivclzi 11671 | Closure law for division o... |
| redivcli 11672 | Closure law for division o... |
| div1d 11673 | A number divided by 1 is i... |
| reccld 11674 | Closure law for reciprocal... |
| recne0d 11675 | The reciprocal of a nonzer... |
| recidd 11676 | Multiplication of a number... |
| recid2d 11677 | Multiplication of a number... |
| recrecd 11678 | A number is equal to the r... |
| dividd 11679 | A number divided by itself... |
| div0d 11680 | Division into zero is zero... |
| divcld 11681 | Closure law for division. ... |
| divcan1d 11682 | A cancellation law for div... |
| divcan2d 11683 | A cancellation law for div... |
| divrecd 11684 | Relationship between divis... |
| divrec2d 11685 | Relationship between divis... |
| divcan3d 11686 | A cancellation law for div... |
| divcan4d 11687 | A cancellation law for div... |
| diveq0d 11688 | A ratio is zero iff the nu... |
| diveq1d 11689 | Equality in terms of unit ... |
| diveq1ad 11690 | The quotient of two comple... |
| diveq0ad 11691 | A fraction of complex numb... |
| divne1d 11692 | If two complex numbers are... |
| divne0bd 11693 | A ratio is zero iff the nu... |
| divnegd 11694 | Move negative sign inside ... |
| divneg2d 11695 | Move negative sign inside ... |
| div2negd 11696 | Quotient of two negatives.... |
| divne0d 11697 | The ratio of nonzero numbe... |
| recdivd 11698 | The reciprocal of a ratio.... |
| recdiv2d 11699 | Division into a reciprocal... |
| divcan6d 11700 | Cancellation of inverted f... |
| ddcand 11701 | Cancellation in a double d... |
| rec11d 11702 | Reciprocal is one-to-one. ... |
| divmuld 11703 | Relationship between divis... |
| div32d 11704 | A commutative/associative ... |
| div13d 11705 | A commutative/associative ... |
| divdiv32d 11706 | Swap denominators in a div... |
| divcan5d 11707 | Cancellation of common fac... |
| divcan5rd 11708 | Cancellation of common fac... |
| divcan7d 11709 | Cancel equal divisors in a... |
| dmdcand 11710 | Cancellation law for divis... |
| dmdcan2d 11711 | Cancellation law for divis... |
| divdiv1d 11712 | Division into a fraction. ... |
| divdiv2d 11713 | Division by a fraction. (... |
| divmul2d 11714 | Relationship between divis... |
| divmul3d 11715 | Relationship between divis... |
| divassd 11716 | An associative law for div... |
| div12d 11717 | A commutative/associative ... |
| div23d 11718 | A commutative/associative ... |
| divdird 11719 | Distribution of division o... |
| divsubdird 11720 | Distribution of division o... |
| div11d 11721 | One-to-one relationship fo... |
| divmuldivd 11722 | Multiplication of two rati... |
| divmul13d 11723 | Swap denominators of two r... |
| divmul24d 11724 | Swap the numerators in the... |
| divadddivd 11725 | Addition of two ratios. T... |
| divsubdivd 11726 | Subtraction of two ratios.... |
| divmuleqd 11727 | Cross-multiply in an equal... |
| divdivdivd 11728 | Division of two ratios. T... |
| diveq1bd 11729 | If two complex numbers are... |
| div2sub 11730 | Swap the order of subtract... |
| div2subd 11731 | Swap subtrahend and minuen... |
| rereccld 11732 | Closure law for reciprocal... |
| redivcld 11733 | Closure law for division o... |
| subrec 11734 | Subtraction of reciprocals... |
| subreci 11735 | Subtraction of reciprocals... |
| subrecd 11736 | Subtraction of reciprocals... |
| mvllmuld 11737 | Move the left term in a pr... |
| mvllmuli 11738 | Move the left term in a pr... |
| ldiv 11739 | Left-division. (Contribut... |
| rdiv 11740 | Right-division. (Contribu... |
| mdiv 11741 | A division law. (Contribu... |
| lineq 11742 | Solution of a (scalar) lin... |
| elimgt0 11743 | Hypothesis for weak deduct... |
| elimge0 11744 | Hypothesis for weak deduct... |
| ltp1 11745 | A number is less than itse... |
| lep1 11746 | A number is less than or e... |
| ltm1 11747 | A number minus 1 is less t... |
| lem1 11748 | A number minus 1 is less t... |
| letrp1 11749 | A transitive property of '... |
| p1le 11750 | A transitive property of p... |
| recgt0 11751 | The reciprocal of a positi... |
| prodgt0 11752 | Infer that a multiplicand ... |
| prodgt02 11753 | Infer that a multiplier is... |
| ltmul1a 11754 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 11755 | Multiplication of both sid... |
| ltmul2 11756 | Multiplication of both sid... |
| lemul1 11757 | Multiplication of both sid... |
| lemul2 11758 | Multiplication of both sid... |
| lemul1a 11759 | Multiplication of both sid... |
| lemul2a 11760 | Multiplication of both sid... |
| ltmul12a 11761 | Comparison of product of t... |
| lemul12b 11762 | Comparison of product of t... |
| lemul12a 11763 | Comparison of product of t... |
| mulgt1 11764 | The product of two numbers... |
| ltmulgt11 11765 | Multiplication by a number... |
| ltmulgt12 11766 | Multiplication by a number... |
| lemulge11 11767 | Multiplication by a number... |
| lemulge12 11768 | Multiplication by a number... |
| ltdiv1 11769 | Division of both sides of ... |
| lediv1 11770 | Division of both sides of ... |
| gt0div 11771 | Division of a positive num... |
| ge0div 11772 | Division of a nonnegative ... |
| divgt0 11773 | The ratio of two positive ... |
| divge0 11774 | The ratio of nonnegative a... |
| mulge0b 11775 | A condition for multiplica... |
| mulle0b 11776 | A condition for multiplica... |
| mulsuble0b 11777 | A condition for multiplica... |
| ltmuldiv 11778 | 'Less than' relationship b... |
| ltmuldiv2 11779 | 'Less than' relationship b... |
| ltdivmul 11780 | 'Less than' relationship b... |
| ledivmul 11781 | 'Less than or equal to' re... |
| ltdivmul2 11782 | 'Less than' relationship b... |
| lt2mul2div 11783 | 'Less than' relationship b... |
| ledivmul2 11784 | 'Less than or equal to' re... |
| lemuldiv 11785 | 'Less than or equal' relat... |
| lemuldiv2 11786 | 'Less than or equal' relat... |
| ltrec 11787 | The reciprocal of both sid... |
| lerec 11788 | The reciprocal of both sid... |
| lt2msq1 11789 | Lemma for ~ lt2msq . (Con... |
| lt2msq 11790 | Two nonnegative numbers co... |
| ltdiv2 11791 | Division of a positive num... |
| ltrec1 11792 | Reciprocal swap in a 'less... |
| lerec2 11793 | Reciprocal swap in a 'less... |
| ledivdiv 11794 | Invert ratios of positive ... |
| lediv2 11795 | Division of a positive num... |
| ltdiv23 11796 | Swap denominator with othe... |
| lediv23 11797 | Swap denominator with othe... |
| lediv12a 11798 | Comparison of ratio of two... |
| lediv2a 11799 | Division of both sides of ... |
| reclt1 11800 | The reciprocal of a positi... |
| recgt1 11801 | The reciprocal of a positi... |
| recgt1i 11802 | The reciprocal of a number... |
| recp1lt1 11803 | Construct a number less th... |
| recreclt 11804 | Given a positive number ` ... |
| le2msq 11805 | The square function on non... |
| msq11 11806 | The square of a nonnegativ... |
| ledivp1 11807 | "Less than or equal to" an... |
| squeeze0 11808 | If a nonnegative number is... |
| ltp1i 11809 | A number is less than itse... |
| recgt0i 11810 | The reciprocal of a positi... |
| recgt0ii 11811 | The reciprocal of a positi... |
| prodgt0i 11812 | Infer that a multiplicand ... |
| divgt0i 11813 | The ratio of two positive ... |
| divge0i 11814 | The ratio of nonnegative a... |
| ltreci 11815 | The reciprocal of both sid... |
| lereci 11816 | The reciprocal of both sid... |
| lt2msqi 11817 | The square function on non... |
| le2msqi 11818 | The square function on non... |
| msq11i 11819 | The square of a nonnegativ... |
| divgt0i2i 11820 | The ratio of two positive ... |
| ltrecii 11821 | The reciprocal of both sid... |
| divgt0ii 11822 | The ratio of two positive ... |
| ltmul1i 11823 | Multiplication of both sid... |
| ltdiv1i 11824 | Division of both sides of ... |
| ltmuldivi 11825 | 'Less than' relationship b... |
| ltmul2i 11826 | Multiplication of both sid... |
| lemul1i 11827 | Multiplication of both sid... |
| lemul2i 11828 | Multiplication of both sid... |
| ltdiv23i 11829 | Swap denominator with othe... |
| ledivp1i 11830 | "Less than or equal to" an... |
| ltdivp1i 11831 | Less-than and division rel... |
| ltdiv23ii 11832 | Swap denominator with othe... |
| ltmul1ii 11833 | Multiplication of both sid... |
| ltdiv1ii 11834 | Division of both sides of ... |
| ltp1d 11835 | A number is less than itse... |
| lep1d 11836 | A number is less than or e... |
| ltm1d 11837 | A number minus 1 is less t... |
| lem1d 11838 | A number minus 1 is less t... |
| recgt0d 11839 | The reciprocal of a positi... |
| divgt0d 11840 | The ratio of two positive ... |
| mulgt1d 11841 | The product of two numbers... |
| lemulge11d 11842 | Multiplication by a number... |
| lemulge12d 11843 | Multiplication by a number... |
| lemul1ad 11844 | Multiplication of both sid... |
| lemul2ad 11845 | Multiplication of both sid... |
| ltmul12ad 11846 | Comparison of product of t... |
| lemul12ad 11847 | Comparison of product of t... |
| lemul12bd 11848 | Comparison of product of t... |
| fimaxre 11849 | A finite set of real numbe... |
| fimaxre2 11850 | A nonempty finite set of r... |
| fimaxre3 11851 | A nonempty finite set of r... |
| fiminre 11852 | A nonempty finite set of r... |
| fiminre2 11853 | A nonempty finite set of r... |
| negfi 11854 | The negation of a finite s... |
| lbreu 11855 | If a set of reals contains... |
| lbcl 11856 | If a set of reals contains... |
| lble 11857 | If a set of reals contains... |
| lbinf 11858 | If a set of reals contains... |
| lbinfcl 11859 | If a set of reals contains... |
| lbinfle 11860 | If a set of reals contains... |
| sup2 11861 | A nonempty, bounded-above ... |
| sup3 11862 | A version of the completen... |
| infm3lem 11863 | Lemma for ~ infm3 . (Cont... |
| infm3 11864 | The completeness axiom for... |
| suprcl 11865 | Closure of supremum of a n... |
| suprub 11866 | A member of a nonempty bou... |
| suprubd 11867 | Natural deduction form of ... |
| suprcld 11868 | Natural deduction form of ... |
| suprlub 11869 | The supremum of a nonempty... |
| suprnub 11870 | An upper bound is not less... |
| suprleub 11871 | The supremum of a nonempty... |
| supaddc 11872 | The supremum function dist... |
| supadd 11873 | The supremum function dist... |
| supmul1 11874 | The supremum function dist... |
| supmullem1 11875 | Lemma for ~ supmul . (Con... |
| supmullem2 11876 | Lemma for ~ supmul . (Con... |
| supmul 11877 | The supremum function dist... |
| sup3ii 11878 | A version of the completen... |
| suprclii 11879 | Closure of supremum of a n... |
| suprubii 11880 | A member of a nonempty bou... |
| suprlubii 11881 | The supremum of a nonempty... |
| suprnubii 11882 | An upper bound is not less... |
| suprleubii 11883 | The supremum of a nonempty... |
| riotaneg 11884 | The negative of the unique... |
| negiso 11885 | Negation is an order anti-... |
| dfinfre 11886 | The infimum of a set of re... |
| infrecl 11887 | Closure of infimum of a no... |
| infrenegsup 11888 | The infimum of a set of re... |
| infregelb 11889 | Any lower bound of a nonem... |
| infrelb 11890 | If a nonempty set of real ... |
| infrefilb 11891 | The infimum of a finite se... |
| supfirege 11892 | The supremum of a finite s... |
| inelr 11893 | The imaginary unit ` _i ` ... |
| rimul 11894 | A real number times the im... |
| cru 11895 | The representation of comp... |
| crne0 11896 | The real representation of... |
| creur 11897 | The real part of a complex... |
| creui 11898 | The imaginary part of a co... |
| cju 11899 | The complex conjugate of a... |
| ofsubeq0 11900 | Function analogue of ~ sub... |
| ofnegsub 11901 | Function analogue of ~ neg... |
| ofsubge0 11902 | Function analogue of ~ sub... |
| nnexALT 11905 | Alternate proof of ~ nnex ... |
| peano5nni 11906 | Peano's inductive postulat... |
| nnssre 11907 | The positive integers are ... |
| nnsscn 11908 | The positive integers are ... |
| nnex 11909 | The set of positive intege... |
| nnre 11910 | A positive integer is a re... |
| nncn 11911 | A positive integer is a co... |
| nnrei 11912 | A positive integer is a re... |
| nncni 11913 | A positive integer is a co... |
| 1nn 11914 | Peano postulate: 1 is a po... |
| peano2nn 11915 | Peano postulate: a success... |
| dfnn2 11916 | Alternate definition of th... |
| dfnn3 11917 | Alternate definition of th... |
| nnred 11918 | A positive integer is a re... |
| nncnd 11919 | A positive integer is a co... |
| peano2nnd 11920 | Peano postulate: a success... |
| nnind 11921 | Principle of Mathematical ... |
| nnindALT 11922 | Principle of Mathematical ... |
| nnindd 11923 | Principle of Mathematical ... |
| nn1m1nn 11924 | Every positive integer is ... |
| nn1suc 11925 | If a statement holds for 1... |
| nnaddcl 11926 | Closure of addition of pos... |
| nnmulcl 11927 | Closure of multiplication ... |
| nnmulcli 11928 | Closure of multiplication ... |
| nnmtmip 11929 | "Minus times minus is plus... |
| nn2ge 11930 | There exists a positive in... |
| nnge1 11931 | A positive integer is one ... |
| nngt1ne1 11932 | A positive integer is grea... |
| nnle1eq1 11933 | A positive integer is less... |
| nngt0 11934 | A positive integer is posi... |
| nnnlt1 11935 | A positive integer is not ... |
| nnnle0 11936 | A positive integer is not ... |
| nnne0 11937 | A positive integer is nonz... |
| nnneneg 11938 | No positive integer is equ... |
| 0nnn 11939 | Zero is not a positive int... |
| 0nnnALT 11940 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 11941 | Alternate version of ~ nnn... |
| nngt0i 11942 | A positive integer is posi... |
| nnne0i 11943 | A positive integer is nonz... |
| nndivre 11944 | The quotient of a real and... |
| nnrecre 11945 | The reciprocal of a positi... |
| nnrecgt0 11946 | The reciprocal of a positi... |
| nnsub 11947 | Subtraction of positive in... |
| nnsubi 11948 | Subtraction of positive in... |
| nndiv 11949 | Two ways to express " ` A ... |
| nndivtr 11950 | Transitive property of div... |
| nnge1d 11951 | A positive integer is one ... |
| nngt0d 11952 | A positive integer is posi... |
| nnne0d 11953 | A positive integer is nonz... |
| nnrecred 11954 | The reciprocal of a positi... |
| nnaddcld 11955 | Closure of addition of pos... |
| nnmulcld 11956 | Closure of multiplication ... |
| nndivred 11957 | A positive integer is one ... |
| 0ne1 11974 | Zero is different from one... |
| 1m1e0 11975 | One minus one equals zero.... |
| 2nn 11976 | 2 is a positive integer. ... |
| 2re 11977 | The number 2 is real. (Co... |
| 2cn 11978 | The number 2 is a complex ... |
| 2cnALT 11979 | Alternate proof of ~ 2cn .... |
| 2ex 11980 | The number 2 is a set. (C... |
| 2cnd 11981 | The number 2 is a complex ... |
| 3nn 11982 | 3 is a positive integer. ... |
| 3re 11983 | The number 3 is real. (Co... |
| 3cn 11984 | The number 3 is a complex ... |
| 3ex 11985 | The number 3 is a set. (C... |
| 4nn 11986 | 4 is a positive integer. ... |
| 4re 11987 | The number 4 is real. (Co... |
| 4cn 11988 | The number 4 is a complex ... |
| 5nn 11989 | 5 is a positive integer. ... |
| 5re 11990 | The number 5 is real. (Co... |
| 5cn 11991 | The number 5 is a complex ... |
| 6nn 11992 | 6 is a positive integer. ... |
| 6re 11993 | The number 6 is real. (Co... |
| 6cn 11994 | The number 6 is a complex ... |
| 7nn 11995 | 7 is a positive integer. ... |
| 7re 11996 | The number 7 is real. (Co... |
| 7cn 11997 | The number 7 is a complex ... |
| 8nn 11998 | 8 is a positive integer. ... |
| 8re 11999 | The number 8 is real. (Co... |
| 8cn 12000 | The number 8 is a complex ... |
| 9nn 12001 | 9 is a positive integer. ... |
| 9re 12002 | The number 9 is real. (Co... |
| 9cn 12003 | The number 9 is a complex ... |
| 0le0 12004 | Zero is nonnegative. (Con... |
| 0le2 12005 | The number 0 is less than ... |
| 2pos 12006 | The number 2 is positive. ... |
| 2ne0 12007 | The number 2 is nonzero. ... |
| 3pos 12008 | The number 3 is positive. ... |
| 3ne0 12009 | The number 3 is nonzero. ... |
| 4pos 12010 | The number 4 is positive. ... |
| 4ne0 12011 | The number 4 is nonzero. ... |
| 5pos 12012 | The number 5 is positive. ... |
| 6pos 12013 | The number 6 is positive. ... |
| 7pos 12014 | The number 7 is positive. ... |
| 8pos 12015 | The number 8 is positive. ... |
| 9pos 12016 | The number 9 is positive. ... |
| neg1cn 12017 | -1 is a complex number. (... |
| neg1rr 12018 | -1 is a real number. (Con... |
| neg1ne0 12019 | -1 is nonzero. (Contribut... |
| neg1lt0 12020 | -1 is less than 0. (Contr... |
| negneg1e1 12021 | ` -u -u 1 ` is 1. (Contri... |
| 1pneg1e0 12022 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 12023 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 12024 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 12025 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 12026 | Function value at ` N + 1 ... |
| 1p0e1 12027 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 12028 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 12029 | 2 - 1 = 1. The result is ... |
| 1e2m1 12030 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 12031 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 12032 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 12033 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 12034 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 12035 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 12036 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 12037 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 12038 | Two plus two equals four. ... |
| 2times 12039 | Two times a number. (Cont... |
| times2 12040 | A number times 2. (Contri... |
| 2timesi 12041 | Two times a number. (Cont... |
| times2i 12042 | A number times 2. (Contri... |
| 2txmxeqx 12043 | Two times a complex number... |
| 2div2e1 12044 | 2 divided by 2 is 1. (Con... |
| 2p1e3 12045 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 12046 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 12047 | Alternate proof of ~ 1p2e3... |
| 3p1e4 12048 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 12049 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 12050 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 12051 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 12052 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 12053 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 12054 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 12055 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 12056 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 12057 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 12058 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 12059 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 12060 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 12061 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 12062 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 12063 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 12064 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 12065 | 1 times 1 equals 1. (Cont... |
| 2t1e2 12066 | 2 times 1 equals 2. (Cont... |
| 2t2e4 12067 | 2 times 2 equals 4. (Cont... |
| 3t1e3 12068 | 3 times 1 equals 3. (Cont... |
| 3t2e6 12069 | 3 times 2 equals 6. (Cont... |
| 3t3e9 12070 | 3 times 3 equals 9. (Cont... |
| 4t2e8 12071 | 4 times 2 equals 8. (Cont... |
| 2t0e0 12072 | 2 times 0 equals 0. (Cont... |
| 4d2e2 12073 | One half of four is two. ... |
| 1lt2 12074 | 1 is less than 2. (Contri... |
| 2lt3 12075 | 2 is less than 3. (Contri... |
| 1lt3 12076 | 1 is less than 3. (Contri... |
| 3lt4 12077 | 3 is less than 4. (Contri... |
| 2lt4 12078 | 2 is less than 4. (Contri... |
| 1lt4 12079 | 1 is less than 4. (Contri... |
| 4lt5 12080 | 4 is less than 5. (Contri... |
| 3lt5 12081 | 3 is less than 5. (Contri... |
| 2lt5 12082 | 2 is less than 5. (Contri... |
| 1lt5 12083 | 1 is less than 5. (Contri... |
| 5lt6 12084 | 5 is less than 6. (Contri... |
| 4lt6 12085 | 4 is less than 6. (Contri... |
| 3lt6 12086 | 3 is less than 6. (Contri... |
| 2lt6 12087 | 2 is less than 6. (Contri... |
| 1lt6 12088 | 1 is less than 6. (Contri... |
| 6lt7 12089 | 6 is less than 7. (Contri... |
| 5lt7 12090 | 5 is less than 7. (Contri... |
| 4lt7 12091 | 4 is less than 7. (Contri... |
| 3lt7 12092 | 3 is less than 7. (Contri... |
| 2lt7 12093 | 2 is less than 7. (Contri... |
| 1lt7 12094 | 1 is less than 7. (Contri... |
| 7lt8 12095 | 7 is less than 8. (Contri... |
| 6lt8 12096 | 6 is less than 8. (Contri... |
| 5lt8 12097 | 5 is less than 8. (Contri... |
| 4lt8 12098 | 4 is less than 8. (Contri... |
| 3lt8 12099 | 3 is less than 8. (Contri... |
| 2lt8 12100 | 2 is less than 8. (Contri... |
| 1lt8 12101 | 1 is less than 8. (Contri... |
| 8lt9 12102 | 8 is less than 9. (Contri... |
| 7lt9 12103 | 7 is less than 9. (Contri... |
| 6lt9 12104 | 6 is less than 9. (Contri... |
| 5lt9 12105 | 5 is less than 9. (Contri... |
| 4lt9 12106 | 4 is less than 9. (Contri... |
| 3lt9 12107 | 3 is less than 9. (Contri... |
| 2lt9 12108 | 2 is less than 9. (Contri... |
| 1lt9 12109 | 1 is less than 9. (Contri... |
| 0ne2 12110 | 0 is not equal to 2. (Con... |
| 1ne2 12111 | 1 is not equal to 2. (Con... |
| 1le2 12112 | 1 is less than or equal to... |
| 2cnne0 12113 | 2 is a nonzero complex num... |
| 2rene0 12114 | 2 is a nonzero real number... |
| 1le3 12115 | 1 is less than or equal to... |
| neg1mulneg1e1 12116 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 12117 | One-half is real. (Contri... |
| halfcn 12118 | One-half is a complex numb... |
| halfgt0 12119 | One-half is greater than z... |
| halfge0 12120 | One-half is not negative. ... |
| halflt1 12121 | One-half is less than one.... |
| 1mhlfehlf 12122 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 12123 | An eighth of four thirds i... |
| halfpm6th 12124 | One half plus or minus one... |
| it0e0 12125 | i times 0 equals 0. (Cont... |
| 2mulicn 12126 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 12127 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 12128 | Closure of half of a numbe... |
| rehalfcl 12129 | Real closure of half. (Co... |
| half0 12130 | Half of a number is zero i... |
| 2halves 12131 | Two halves make a whole. ... |
| halfpos2 12132 | A number is positive iff i... |
| halfpos 12133 | A positive number is great... |
| halfnneg2 12134 | A number is nonnegative if... |
| halfaddsubcl 12135 | Closure of half-sum and ha... |
| halfaddsub 12136 | Sum and difference of half... |
| subhalfhalf 12137 | Subtracting the half of a ... |
| lt2halves 12138 | A sum is less than the who... |
| addltmul 12139 | Sum is less than product f... |
| nominpos 12140 | There is no smallest posit... |
| avglt1 12141 | Ordering property for aver... |
| avglt2 12142 | Ordering property for aver... |
| avgle1 12143 | Ordering property for aver... |
| avgle2 12144 | Ordering property for aver... |
| avgle 12145 | The average of two numbers... |
| 2timesd 12146 | Two times a number. (Cont... |
| times2d 12147 | A number times 2. (Contri... |
| halfcld 12148 | Closure of half of a numbe... |
| 2halvesd 12149 | Two halves make a whole. ... |
| rehalfcld 12150 | Real closure of half. (Co... |
| lt2halvesd 12151 | A sum is less than the who... |
| rehalfcli 12152 | Half a real number is real... |
| lt2addmuld 12153 | If two real numbers are le... |
| add1p1 12154 | Adding two times 1 to a nu... |
| sub1m1 12155 | Subtracting two times 1 fr... |
| cnm2m1cnm3 12156 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 12157 | A complex number increased... |
| div4p1lem1div2 12158 | An integer greater than 5,... |
| nnunb 12159 | The set of positive intege... |
| arch 12160 | Archimedean property of re... |
| nnrecl 12161 | There exists a positive in... |
| bndndx 12162 | A bounded real sequence ` ... |
| elnn0 12165 | Nonnegative integers expre... |
| nnssnn0 12166 | Positive naturals are a su... |
| nn0ssre 12167 | Nonnegative integers are a... |
| nn0sscn 12168 | Nonnegative integers are a... |
| nn0ex 12169 | The set of nonnegative int... |
| nnnn0 12170 | A positive integer is a no... |
| nnnn0i 12171 | A positive integer is a no... |
| nn0re 12172 | A nonnegative integer is a... |
| nn0cn 12173 | A nonnegative integer is a... |
| nn0rei 12174 | A nonnegative integer is a... |
| nn0cni 12175 | A nonnegative integer is a... |
| dfn2 12176 | The set of positive intege... |
| elnnne0 12177 | The positive integer prope... |
| 0nn0 12178 | 0 is a nonnegative integer... |
| 1nn0 12179 | 1 is a nonnegative integer... |
| 2nn0 12180 | 2 is a nonnegative integer... |
| 3nn0 12181 | 3 is a nonnegative integer... |
| 4nn0 12182 | 4 is a nonnegative integer... |
| 5nn0 12183 | 5 is a nonnegative integer... |
| 6nn0 12184 | 6 is a nonnegative integer... |
| 7nn0 12185 | 7 is a nonnegative integer... |
| 8nn0 12186 | 8 is a nonnegative integer... |
| 9nn0 12187 | 9 is a nonnegative integer... |
| nn0ge0 12188 | A nonnegative integer is g... |
| nn0nlt0 12189 | A nonnegative integer is n... |
| nn0ge0i 12190 | Nonnegative integers are n... |
| nn0le0eq0 12191 | A nonnegative integer is l... |
| nn0p1gt0 12192 | A nonnegative integer incr... |
| nnnn0addcl 12193 | A positive integer plus a ... |
| nn0nnaddcl 12194 | A nonnegative integer plus... |
| 0mnnnnn0 12195 | The result of subtracting ... |
| un0addcl 12196 | If ` S ` is closed under a... |
| un0mulcl 12197 | If ` S ` is closed under m... |
| nn0addcl 12198 | Closure of addition of non... |
| nn0mulcl 12199 | Closure of multiplication ... |
| nn0addcli 12200 | Closure of addition of non... |
| nn0mulcli 12201 | Closure of multiplication ... |
| nn0p1nn 12202 | A nonnegative integer plus... |
| peano2nn0 12203 | Second Peano postulate for... |
| nnm1nn0 12204 | A positive integer minus 1... |
| elnn0nn 12205 | The nonnegative integer pr... |
| elnnnn0 12206 | The positive integer prope... |
| elnnnn0b 12207 | The positive integer prope... |
| elnnnn0c 12208 | The positive integer prope... |
| nn0addge1 12209 | A number is less than or e... |
| nn0addge2 12210 | A number is less than or e... |
| nn0addge1i 12211 | A number is less than or e... |
| nn0addge2i 12212 | A number is less than or e... |
| nn0sub 12213 | Subtraction of nonnegative... |
| ltsubnn0 12214 | Subtracting a nonnegative ... |
| nn0negleid 12215 | A nonnegative integer is g... |
| difgtsumgt 12216 | If the difference of a rea... |
| nn0le2xi 12217 | A nonnegative integer is l... |
| nn0lele2xi 12218 | 'Less than or equal to' im... |
| frnnn0supp 12219 | Two ways to write the supp... |
| frnnn0fsupp 12220 | A function on ` NN0 ` is f... |
| frnnn0suppg 12221 | Version of ~ frnnn0supp av... |
| frnnn0fsuppg 12222 | Version of ~ frnnn0fsupp a... |
| nnnn0d 12223 | A positive integer is a no... |
| nn0red 12224 | A nonnegative integer is a... |
| nn0cnd 12225 | A nonnegative integer is a... |
| nn0ge0d 12226 | A nonnegative integer is g... |
| nn0addcld 12227 | Closure of addition of non... |
| nn0mulcld 12228 | Closure of multiplication ... |
| nn0readdcl 12229 | Closure law for addition o... |
| nn0n0n1ge2 12230 | A nonnegative integer whic... |
| nn0n0n1ge2b 12231 | A nonnegative integer is n... |
| nn0ge2m1nn 12232 | If a nonnegative integer i... |
| nn0ge2m1nn0 12233 | If a nonnegative integer i... |
| nn0nndivcl 12234 | Closure law for dividing o... |
| elxnn0 12237 | An extended nonnegative in... |
| nn0ssxnn0 12238 | The standard nonnegative i... |
| nn0xnn0 12239 | A standard nonnegative int... |
| xnn0xr 12240 | An extended nonnegative in... |
| 0xnn0 12241 | Zero is an extended nonneg... |
| pnf0xnn0 12242 | Positive infinity is an ex... |
| nn0nepnf 12243 | No standard nonnegative in... |
| nn0xnn0d 12244 | A standard nonnegative int... |
| nn0nepnfd 12245 | No standard nonnegative in... |
| xnn0nemnf 12246 | No extended nonnegative in... |
| xnn0xrnemnf 12247 | The extended nonnegative i... |
| xnn0nnn0pnf 12248 | An extended nonnegative in... |
| elz 12251 | Membership in the set of i... |
| nnnegz 12252 | The negative of a positive... |
| zre 12253 | An integer is a real. (Co... |
| zcn 12254 | An integer is a complex nu... |
| zrei 12255 | An integer is a real numbe... |
| zssre 12256 | The integers are a subset ... |
| zsscn 12257 | The integers are a subset ... |
| zex 12258 | The set of integers exists... |
| elnnz 12259 | Positive integer property ... |
| 0z 12260 | Zero is an integer. (Cont... |
| 0zd 12261 | Zero is an integer, deduct... |
| elnn0z 12262 | Nonnegative integer proper... |
| elznn0nn 12263 | Integer property expressed... |
| elznn0 12264 | Integer property expressed... |
| elznn 12265 | Integer property expressed... |
| zle0orge1 12266 | There is no integer in the... |
| elz2 12267 | Membership in the set of i... |
| dfz2 12268 | Alternative definition of ... |
| zexALT 12269 | Alternate proof of ~ zex .... |
| nnssz 12270 | Positive integers are a su... |
| nn0ssz 12271 | Nonnegative integers are a... |
| nnz 12272 | A positive integer is an i... |
| nn0z 12273 | A nonnegative integer is a... |
| nnzi 12274 | A positive integer is an i... |
| nn0zi 12275 | A nonnegative integer is a... |
| elnnz1 12276 | Positive integer property ... |
| znnnlt1 12277 | An integer is not a positi... |
| nnzrab 12278 | Positive integers expresse... |
| nn0zrab 12279 | Nonnegative integers expre... |
| 1z 12280 | One is an integer. (Contr... |
| 1zzd 12281 | One is an integer, deducti... |
| 2z 12282 | 2 is an integer. (Contrib... |
| 3z 12283 | 3 is an integer. (Contrib... |
| 4z 12284 | 4 is an integer. (Contrib... |
| znegcl 12285 | Closure law for negative i... |
| neg1z 12286 | -1 is an integer. (Contri... |
| znegclb 12287 | A complex number is an int... |
| nn0negz 12288 | The negative of a nonnegat... |
| nn0negzi 12289 | The negative of a nonnegat... |
| zaddcl 12290 | Closure of addition of int... |
| peano2z 12291 | Second Peano postulate gen... |
| zsubcl 12292 | Closure of subtraction of ... |
| peano2zm 12293 | "Reverse" second Peano pos... |
| zletr 12294 | Transitive law of ordering... |
| zrevaddcl 12295 | Reverse closure law for ad... |
| znnsub 12296 | The positive difference of... |
| znn0sub 12297 | The nonnegative difference... |
| nzadd 12298 | The sum of a real number n... |
| zmulcl 12299 | Closure of multiplication ... |
| zltp1le 12300 | Integer ordering relation.... |
| zleltp1 12301 | Integer ordering relation.... |
| zlem1lt 12302 | Integer ordering relation.... |
| zltlem1 12303 | Integer ordering relation.... |
| zgt0ge1 12304 | An integer greater than ` ... |
| nnleltp1 12305 | Positive integer ordering ... |
| nnltp1le 12306 | Positive integer ordering ... |
| nnaddm1cl 12307 | Closure of addition of pos... |
| nn0ltp1le 12308 | Nonnegative integer orderi... |
| nn0leltp1 12309 | Nonnegative integer orderi... |
| nn0ltlem1 12310 | Nonnegative integer orderi... |
| nn0sub2 12311 | Subtraction of nonnegative... |
| nn0lt10b 12312 | A nonnegative integer less... |
| nn0lt2 12313 | A nonnegative integer less... |
| nn0le2is012 12314 | A nonnegative integer whic... |
| nn0lem1lt 12315 | Nonnegative integer orderi... |
| nnlem1lt 12316 | Positive integer ordering ... |
| nnltlem1 12317 | Positive integer ordering ... |
| nnm1ge0 12318 | A positive integer decreas... |
| nn0ge0div 12319 | Division of a nonnegative ... |
| zdiv 12320 | Two ways to express " ` M ... |
| zdivadd 12321 | Property of divisibility: ... |
| zdivmul 12322 | Property of divisibility: ... |
| zextle 12323 | An extensionality-like pro... |
| zextlt 12324 | An extensionality-like pro... |
| recnz 12325 | The reciprocal of a number... |
| btwnnz 12326 | A number between an intege... |
| gtndiv 12327 | A larger number does not d... |
| halfnz 12328 | One-half is not an integer... |
| 3halfnz 12329 | Three halves is not an int... |
| suprzcl 12330 | The supremum of a bounded-... |
| prime 12331 | Two ways to express " ` A ... |
| msqznn 12332 | The square of a nonzero in... |
| zneo 12333 | No even integer equals an ... |
| nneo 12334 | A positive integer is even... |
| nneoi 12335 | A positive integer is even... |
| zeo 12336 | An integer is even or odd.... |
| zeo2 12337 | An integer is even or odd ... |
| peano2uz2 12338 | Second Peano postulate for... |
| peano5uzi 12339 | Peano's inductive postulat... |
| peano5uzti 12340 | Peano's inductive postulat... |
| dfuzi 12341 | An expression for the uppe... |
| uzind 12342 | Induction on the upper int... |
| uzind2 12343 | Induction on the upper int... |
| uzind3 12344 | Induction on the upper int... |
| nn0ind 12345 | Principle of Mathematical ... |
| nn0indALT 12346 | Principle of Mathematical ... |
| nn0indd 12347 | Principle of Mathematical ... |
| fzind 12348 | Induction on the integers ... |
| fnn0ind 12349 | Induction on the integers ... |
| nn0ind-raph 12350 | Principle of Mathematical ... |
| zindd 12351 | Principle of Mathematical ... |
| btwnz 12352 | Any real number can be san... |
| nn0zd 12353 | A positive integer is an i... |
| nnzd 12354 | A nonnegative integer is a... |
| zred 12355 | An integer is a real numbe... |
| zcnd 12356 | An integer is a complex nu... |
| znegcld 12357 | Closure law for negative i... |
| peano2zd 12358 | Deduction from second Pean... |
| zaddcld 12359 | Closure of addition of int... |
| zsubcld 12360 | Closure of subtraction of ... |
| zmulcld 12361 | Closure of multiplication ... |
| znnn0nn 12362 | The negative of a negative... |
| zadd2cl 12363 | Increasing an integer by 2... |
| zriotaneg 12364 | The negative of the unique... |
| suprfinzcl 12365 | The supremum of a nonempty... |
| 9p1e10 12368 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12369 | Version of the definition ... |
| decex 12370 | A decimal number is a set.... |
| deceq1 12371 | Equality theorem for the d... |
| deceq2 12372 | Equality theorem for the d... |
| deceq1i 12373 | Equality theorem for the d... |
| deceq2i 12374 | Equality theorem for the d... |
| deceq12i 12375 | Equality theorem for the d... |
| numnncl 12376 | Closure for a numeral (wit... |
| num0u 12377 | Add a zero in the units pl... |
| num0h 12378 | Add a zero in the higher p... |
| numcl 12379 | Closure for a decimal inte... |
| numsuc 12380 | The successor of a decimal... |
| deccl 12381 | Closure for a numeral. (C... |
| 10nn 12382 | 10 is a positive integer. ... |
| 10pos 12383 | The number 10 is positive.... |
| 10nn0 12384 | 10 is a nonnegative intege... |
| 10re 12385 | The number 10 is real. (C... |
| decnncl 12386 | Closure for a numeral. (C... |
| dec0u 12387 | Add a zero in the units pl... |
| dec0h 12388 | Add a zero in the higher p... |
| numnncl2 12389 | Closure for a decimal inte... |
| decnncl2 12390 | Closure for a decimal inte... |
| numlt 12391 | Comparing two decimal inte... |
| numltc 12392 | Comparing two decimal inte... |
| le9lt10 12393 | A "decimal digit" (i.e. a ... |
| declt 12394 | Comparing two decimal inte... |
| decltc 12395 | Comparing two decimal inte... |
| declth 12396 | Comparing two decimal inte... |
| decsuc 12397 | The successor of a decimal... |
| 3declth 12398 | Comparing two decimal inte... |
| 3decltc 12399 | Comparing two decimal inte... |
| decle 12400 | Comparing two decimal inte... |
| decleh 12401 | Comparing two decimal inte... |
| declei 12402 | Comparing a digit to a dec... |
| numlti 12403 | Comparing a digit to a dec... |
| declti 12404 | Comparing a digit to a dec... |
| decltdi 12405 | Comparing a digit to a dec... |
| numsucc 12406 | The successor of a decimal... |
| decsucc 12407 | The successor of a decimal... |
| 1e0p1 12408 | The successor of zero. (C... |
| dec10p 12409 | Ten plus an integer. (Con... |
| numma 12410 | Perform a multiply-add of ... |
| nummac 12411 | Perform a multiply-add of ... |
| numma2c 12412 | Perform a multiply-add of ... |
| numadd 12413 | Add two decimal integers `... |
| numaddc 12414 | Add two decimal integers `... |
| nummul1c 12415 | The product of a decimal i... |
| nummul2c 12416 | The product of a decimal i... |
| decma 12417 | Perform a multiply-add of ... |
| decmac 12418 | Perform a multiply-add of ... |
| decma2c 12419 | Perform a multiply-add of ... |
| decadd 12420 | Add two numerals ` M ` and... |
| decaddc 12421 | Add two numerals ` M ` and... |
| decaddc2 12422 | Add two numerals ` M ` and... |
| decrmanc 12423 | Perform a multiply-add of ... |
| decrmac 12424 | Perform a multiply-add of ... |
| decaddm10 12425 | The sum of two multiples o... |
| decaddi 12426 | Add two numerals ` M ` and... |
| decaddci 12427 | Add two numerals ` M ` and... |
| decaddci2 12428 | Add two numerals ` M ` and... |
| decsubi 12429 | Difference between a numer... |
| decmul1 12430 | The product of a numeral w... |
| decmul1c 12431 | The product of a numeral w... |
| decmul2c 12432 | The product of a numeral w... |
| decmulnc 12433 | The product of a numeral w... |
| 11multnc 12434 | The product of 11 (as nume... |
| decmul10add 12435 | A multiplication of a numb... |
| 6p5lem 12436 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12437 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12438 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12439 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12440 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12441 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12442 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12443 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12444 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12445 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12446 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12447 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12448 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12449 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12450 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12451 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12452 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12453 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12454 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12455 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12456 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12457 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12458 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12459 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12460 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12461 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12462 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12463 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12464 | 4 times 3 equals 12. (Con... |
| 4t4e16 12465 | 4 times 4 equals 16. (Con... |
| 5t2e10 12466 | 5 times 2 equals 10. (Con... |
| 5t3e15 12467 | 5 times 3 equals 15. (Con... |
| 5t4e20 12468 | 5 times 4 equals 20. (Con... |
| 5t5e25 12469 | 5 times 5 equals 25. (Con... |
| 6t2e12 12470 | 6 times 2 equals 12. (Con... |
| 6t3e18 12471 | 6 times 3 equals 18. (Con... |
| 6t4e24 12472 | 6 times 4 equals 24. (Con... |
| 6t5e30 12473 | 6 times 5 equals 30. (Con... |
| 6t6e36 12474 | 6 times 6 equals 36. (Con... |
| 7t2e14 12475 | 7 times 2 equals 14. (Con... |
| 7t3e21 12476 | 7 times 3 equals 21. (Con... |
| 7t4e28 12477 | 7 times 4 equals 28. (Con... |
| 7t5e35 12478 | 7 times 5 equals 35. (Con... |
| 7t6e42 12479 | 7 times 6 equals 42. (Con... |
| 7t7e49 12480 | 7 times 7 equals 49. (Con... |
| 8t2e16 12481 | 8 times 2 equals 16. (Con... |
| 8t3e24 12482 | 8 times 3 equals 24. (Con... |
| 8t4e32 12483 | 8 times 4 equals 32. (Con... |
| 8t5e40 12484 | 8 times 5 equals 40. (Con... |
| 8t6e48 12485 | 8 times 6 equals 48. (Con... |
| 8t7e56 12486 | 8 times 7 equals 56. (Con... |
| 8t8e64 12487 | 8 times 8 equals 64. (Con... |
| 9t2e18 12488 | 9 times 2 equals 18. (Con... |
| 9t3e27 12489 | 9 times 3 equals 27. (Con... |
| 9t4e36 12490 | 9 times 4 equals 36. (Con... |
| 9t5e45 12491 | 9 times 5 equals 45. (Con... |
| 9t6e54 12492 | 9 times 6 equals 54. (Con... |
| 9t7e63 12493 | 9 times 7 equals 63. (Con... |
| 9t8e72 12494 | 9 times 8 equals 72. (Con... |
| 9t9e81 12495 | 9 times 9 equals 81. (Con... |
| 9t11e99 12496 | 9 times 11 equals 99. (Co... |
| 9lt10 12497 | 9 is less than 10. (Contr... |
| 8lt10 12498 | 8 is less than 10. (Contr... |
| 7lt10 12499 | 7 is less than 10. (Contr... |
| 6lt10 12500 | 6 is less than 10. (Contr... |
| 5lt10 12501 | 5 is less than 10. (Contr... |
| 4lt10 12502 | 4 is less than 10. (Contr... |
| 3lt10 12503 | 3 is less than 10. (Contr... |
| 2lt10 12504 | 2 is less than 10. (Contr... |
| 1lt10 12505 | 1 is less than 10. (Contr... |
| decbin0 12506 | Decompose base 4 into base... |
| decbin2 12507 | Decompose base 4 into base... |
| decbin3 12508 | Decompose base 4 into base... |
| halfthird 12509 | Half minus a third. (Cont... |
| 5recm6rec 12510 | One fifth minus one sixth.... |
| uzval 12513 | The value of the upper int... |
| uzf 12514 | The domain and range of th... |
| eluz1 12515 | Membership in the upper se... |
| eluzel2 12516 | Implication of membership ... |
| eluz2 12517 | Membership in an upper set... |
| eluzmn 12518 | Membership in an earlier u... |
| eluz1i 12519 | Membership in an upper set... |
| eluzuzle 12520 | An integer in an upper set... |
| eluzelz 12521 | A member of an upper set o... |
| eluzelre 12522 | A member of an upper set o... |
| eluzelcn 12523 | A member of an upper set o... |
| eluzle 12524 | Implication of membership ... |
| eluz 12525 | Membership in an upper set... |
| uzid 12526 | Membership of the least me... |
| uzidd 12527 | Membership of the least me... |
| uzn0 12528 | The upper integers are all... |
| uztrn 12529 | Transitive law for sets of... |
| uztrn2 12530 | Transitive law for sets of... |
| uzneg 12531 | Contraposition law for upp... |
| uzssz 12532 | An upper set of integers i... |
| uzssre 12533 | An upper set of integers i... |
| uzss 12534 | Subset relationship for tw... |
| uztric 12535 | Totality of the ordering r... |
| uz11 12536 | The upper integers functio... |
| eluzp1m1 12537 | Membership in the next upp... |
| eluzp1l 12538 | Strict ordering implied by... |
| eluzp1p1 12539 | Membership in the next upp... |
| eluzaddi 12540 | Membership in a later uppe... |
| eluzsubi 12541 | Membership in an earlier u... |
| eluzadd 12542 | Membership in a later uppe... |
| eluzsub 12543 | Membership in an earlier u... |
| subeluzsub 12544 | Membership of a difference... |
| uzm1 12545 | Choices for an element of ... |
| uznn0sub 12546 | The nonnegative difference... |
| uzin 12547 | Intersection of two upper ... |
| uzp1 12548 | Choices for an element of ... |
| nn0uz 12549 | Nonnegative integers expre... |
| nnuz 12550 | Positive integers expresse... |
| elnnuz 12551 | A positive integer express... |
| elnn0uz 12552 | A nonnegative integer expr... |
| eluz2nn 12553 | An integer greater than or... |
| eluz4eluz2 12554 | An integer greater than or... |
| eluz4nn 12555 | An integer greater than or... |
| eluzge2nn0 12556 | If an integer is greater t... |
| eluz2n0 12557 | An integer greater than or... |
| uzuzle23 12558 | An integer in the upper se... |
| eluzge3nn 12559 | If an integer is greater t... |
| uz3m2nn 12560 | An integer greater than or... |
| 1eluzge0 12561 | 1 is an integer greater th... |
| 2eluzge0 12562 | 2 is an integer greater th... |
| 2eluzge1 12563 | 2 is an integer greater th... |
| uznnssnn 12564 | The upper integers startin... |
| raluz 12565 | Restricted universal quant... |
| raluz2 12566 | Restricted universal quant... |
| rexuz 12567 | Restricted existential qua... |
| rexuz2 12568 | Restricted existential qua... |
| 2rexuz 12569 | Double existential quantif... |
| peano2uz 12570 | Second Peano postulate for... |
| peano2uzs 12571 | Second Peano postulate for... |
| peano2uzr 12572 | Reversed second Peano axio... |
| uzaddcl 12573 | Addition closure law for a... |
| nn0pzuz 12574 | The sum of a nonnegative i... |
| uzind4 12575 | Induction on the upper set... |
| uzind4ALT 12576 | Induction on the upper set... |
| uzind4s 12577 | Induction on the upper set... |
| uzind4s2 12578 | Induction on the upper set... |
| uzind4i 12579 | Induction on the upper int... |
| uzwo 12580 | Well-ordering principle: a... |
| uzwo2 12581 | Well-ordering principle: a... |
| nnwo 12582 | Well-ordering principle: a... |
| nnwof 12583 | Well-ordering principle: a... |
| nnwos 12584 | Well-ordering principle: a... |
| indstr 12585 | Strong Mathematical Induct... |
| eluznn0 12586 | Membership in a nonnegativ... |
| eluznn 12587 | Membership in a positive u... |
| eluz2b1 12588 | Two ways to say "an intege... |
| eluz2gt1 12589 | An integer greater than or... |
| eluz2b2 12590 | Two ways to say "an intege... |
| eluz2b3 12591 | Two ways to say "an intege... |
| uz2m1nn 12592 | One less than an integer g... |
| 1nuz2 12593 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12594 | A positive integer is eith... |
| uz2mulcl 12595 | Closure of multiplication ... |
| indstr2 12596 | Strong Mathematical Induct... |
| uzinfi 12597 | Extract the lower bound of... |
| nninf 12598 | The infimum of the set of ... |
| nn0inf 12599 | The infimum of the set of ... |
| infssuzle 12600 | The infimum of a subset of... |
| infssuzcl 12601 | The infimum of a subset of... |
| ublbneg 12602 | The image under negation o... |
| eqreznegel 12603 | Two ways to express the im... |
| supminf 12604 | The supremum of a bounded-... |
| lbzbi 12605 | If a set of reals is bound... |
| zsupss 12606 | Any nonempty bounded subse... |
| suprzcl2 12607 | The supremum of a bounded-... |
| suprzub 12608 | The supremum of a bounded-... |
| uzsupss 12609 | Any bounded subset of an u... |
| nn01to3 12610 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 12611 | Alternate proof of ~ nn0ge... |
| uzwo3 12612 | Well-ordering principle: a... |
| zmin 12613 | There is a unique smallest... |
| zmax 12614 | There is a unique largest ... |
| zbtwnre 12615 | There is a unique integer ... |
| rebtwnz 12616 | There is a unique greatest... |
| elq 12619 | Membership in the set of r... |
| qmulz 12620 | If ` A ` is rational, then... |
| znq 12621 | The ratio of an integer an... |
| qre 12622 | A rational number is a rea... |
| zq 12623 | An integer is a rational n... |
| qred 12624 | A rational number is a rea... |
| zssq 12625 | The integers are a subset ... |
| nn0ssq 12626 | The nonnegative integers a... |
| nnssq 12627 | The positive integers are ... |
| qssre 12628 | The rationals are a subset... |
| qsscn 12629 | The rationals are a subset... |
| qex 12630 | The set of rational number... |
| nnq 12631 | A positive integer is rati... |
| qcn 12632 | A rational number is a com... |
| qexALT 12633 | Alternate proof of ~ qex .... |
| qaddcl 12634 | Closure of addition of rat... |
| qnegcl 12635 | Closure law for the negati... |
| qmulcl 12636 | Closure of multiplication ... |
| qsubcl 12637 | Closure of subtraction of ... |
| qreccl 12638 | Closure of reciprocal of r... |
| qdivcl 12639 | Closure of division of rat... |
| qrevaddcl 12640 | Reverse closure law for ad... |
| nnrecq 12641 | The reciprocal of a positi... |
| irradd 12642 | The sum of an irrational n... |
| irrmul 12643 | The product of an irration... |
| elpq 12644 | A positive rational is the... |
| elpqb 12645 | A class is a positive rati... |
| rpnnen1lem2 12646 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 12647 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 12648 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 12649 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 12650 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 12651 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 12652 | One half of ~ rpnnen , whe... |
| reexALT 12653 | Alternate proof of ~ reex ... |
| cnref1o 12654 | There is a natural one-to-... |
| cnexALT 12655 | The set of complex numbers... |
| xrex 12656 | The set of extended reals ... |
| addex 12657 | The addition operation is ... |
| mulex 12658 | The multiplication operati... |
| elrp 12661 | Membership in the set of p... |
| elrpii 12662 | Membership in the set of p... |
| 1rp 12663 | 1 is a positive real. (Co... |
| 2rp 12664 | 2 is a positive real. (Co... |
| 3rp 12665 | 3 is a positive real. (Co... |
| rpssre 12666 | The positive reals are a s... |
| rpre 12667 | A positive real is a real.... |
| rpxr 12668 | A positive real is an exte... |
| rpcn 12669 | A positive real is a compl... |
| nnrp 12670 | A positive integer is a po... |
| rpgt0 12671 | A positive real is greater... |
| rpge0 12672 | A positive real is greater... |
| rpregt0 12673 | A positive real is a posit... |
| rprege0 12674 | A positive real is a nonne... |
| rpne0 12675 | A positive real is nonzero... |
| rprene0 12676 | A positive real is a nonze... |
| rpcnne0 12677 | A positive real is a nonze... |
| rpcndif0 12678 | A positive real number is ... |
| ralrp 12679 | Quantification over positi... |
| rexrp 12680 | Quantification over positi... |
| rpaddcl 12681 | Closure law for addition o... |
| rpmulcl 12682 | Closure law for multiplica... |
| rpmtmip 12683 | "Minus times minus is plus... |
| rpdivcl 12684 | Closure law for division o... |
| rpreccl 12685 | Closure law for reciprocat... |
| rphalfcl 12686 | Closure law for half of a ... |
| rpgecl 12687 | A number greater than or e... |
| rphalflt 12688 | Half of a positive real is... |
| rerpdivcl 12689 | Closure law for division o... |
| ge0p1rp 12690 | A nonnegative number plus ... |
| rpneg 12691 | Either a nonzero real or i... |
| negelrp 12692 | Elementhood of a negation ... |
| negelrpd 12693 | The negation of a negative... |
| 0nrp 12694 | Zero is not a positive rea... |
| ltsubrp 12695 | Subtracting a positive rea... |
| ltaddrp 12696 | Adding a positive number t... |
| difrp 12697 | Two ways to say one number... |
| elrpd 12698 | Membership in the set of p... |
| nnrpd 12699 | A positive integer is a po... |
| zgt1rpn0n1 12700 | An integer greater than 1 ... |
| rpred 12701 | A positive real is a real.... |
| rpxrd 12702 | A positive real is an exte... |
| rpcnd 12703 | A positive real is a compl... |
| rpgt0d 12704 | A positive real is greater... |
| rpge0d 12705 | A positive real is greater... |
| rpne0d 12706 | A positive real is nonzero... |
| rpregt0d 12707 | A positive real is real an... |
| rprege0d 12708 | A positive real is real an... |
| rprene0d 12709 | A positive real is a nonze... |
| rpcnne0d 12710 | A positive real is a nonze... |
| rpreccld 12711 | Closure law for reciprocat... |
| rprecred 12712 | Closure law for reciprocat... |
| rphalfcld 12713 | Closure law for half of a ... |
| reclt1d 12714 | The reciprocal of a positi... |
| recgt1d 12715 | The reciprocal of a positi... |
| rpaddcld 12716 | Closure law for addition o... |
| rpmulcld 12717 | Closure law for multiplica... |
| rpdivcld 12718 | Closure law for division o... |
| ltrecd 12719 | The reciprocal of both sid... |
| lerecd 12720 | The reciprocal of both sid... |
| ltrec1d 12721 | Reciprocal swap in a 'less... |
| lerec2d 12722 | Reciprocal swap in a 'less... |
| lediv2ad 12723 | Division of both sides of ... |
| ltdiv2d 12724 | Division of a positive num... |
| lediv2d 12725 | Division of a positive num... |
| ledivdivd 12726 | Invert ratios of positive ... |
| divge1 12727 | The ratio of a number over... |
| divlt1lt 12728 | A real number divided by a... |
| divle1le 12729 | A real number divided by a... |
| ledivge1le 12730 | If a number is less than o... |
| ge0p1rpd 12731 | A nonnegative number plus ... |
| rerpdivcld 12732 | Closure law for division o... |
| ltsubrpd 12733 | Subtracting a positive rea... |
| ltaddrpd 12734 | Adding a positive number t... |
| ltaddrp2d 12735 | Adding a positive number t... |
| ltmulgt11d 12736 | Multiplication by a number... |
| ltmulgt12d 12737 | Multiplication by a number... |
| gt0divd 12738 | Division of a positive num... |
| ge0divd 12739 | Division of a nonnegative ... |
| rpgecld 12740 | A number greater than or e... |
| divge0d 12741 | The ratio of nonnegative a... |
| ltmul1d 12742 | The ratio of nonnegative a... |
| ltmul2d 12743 | Multiplication of both sid... |
| lemul1d 12744 | Multiplication of both sid... |
| lemul2d 12745 | Multiplication of both sid... |
| ltdiv1d 12746 | Division of both sides of ... |
| lediv1d 12747 | Division of both sides of ... |
| ltmuldivd 12748 | 'Less than' relationship b... |
| ltmuldiv2d 12749 | 'Less than' relationship b... |
| lemuldivd 12750 | 'Less than or equal to' re... |
| lemuldiv2d 12751 | 'Less than or equal to' re... |
| ltdivmuld 12752 | 'Less than' relationship b... |
| ltdivmul2d 12753 | 'Less than' relationship b... |
| ledivmuld 12754 | 'Less than or equal to' re... |
| ledivmul2d 12755 | 'Less than or equal to' re... |
| ltmul1dd 12756 | The ratio of nonnegative a... |
| ltmul2dd 12757 | Multiplication of both sid... |
| ltdiv1dd 12758 | Division of both sides of ... |
| lediv1dd 12759 | Division of both sides of ... |
| lediv12ad 12760 | Comparison of ratio of two... |
| mul2lt0rlt0 12761 | If the result of a multipl... |
| mul2lt0rgt0 12762 | If the result of a multipl... |
| mul2lt0llt0 12763 | If the result of a multipl... |
| mul2lt0lgt0 12764 | If the result of a multipl... |
| mul2lt0bi 12765 | If the result of a multipl... |
| prodge0rd 12766 | Infer that a multiplicand ... |
| prodge0ld 12767 | Infer that a multiplier is... |
| ltdiv23d 12768 | Swap denominator with othe... |
| lediv23d 12769 | Swap denominator with othe... |
| lt2mul2divd 12770 | The ratio of nonnegative a... |
| nnledivrp 12771 | Division of a positive int... |
| nn0ledivnn 12772 | Division of a nonnegative ... |
| addlelt 12773 | If the sum of a real numbe... |
| ltxr 12780 | The 'less than' binary rel... |
| elxr 12781 | Membership in the set of e... |
| xrnemnf 12782 | An extended real other tha... |
| xrnepnf 12783 | An extended real other tha... |
| xrltnr 12784 | The extended real 'less th... |
| ltpnf 12785 | Any (finite) real is less ... |
| ltpnfd 12786 | Any (finite) real is less ... |
| 0ltpnf 12787 | Zero is less than plus inf... |
| mnflt 12788 | Minus infinity is less tha... |
| mnfltd 12789 | Minus infinity is less tha... |
| mnflt0 12790 | Minus infinity is less tha... |
| mnfltpnf 12791 | Minus infinity is less tha... |
| mnfltxr 12792 | Minus infinity is less tha... |
| pnfnlt 12793 | No extended real is greate... |
| nltmnf 12794 | No extended real is less t... |
| pnfge 12795 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 12796 | An extended nonnegative in... |
| 0lepnf 12797 | 0 less than or equal to po... |
| xnn0ge0 12798 | An extended nonnegative in... |
| mnfle 12799 | Minus infinity is less tha... |
| xrltnsym 12800 | Ordering on the extended r... |
| xrltnsym2 12801 | 'Less than' is antisymmetr... |
| xrlttri 12802 | Ordering on the extended r... |
| xrlttr 12803 | Ordering on the extended r... |
| xrltso 12804 | 'Less than' is a strict or... |
| xrlttri2 12805 | Trichotomy law for 'less t... |
| xrlttri3 12806 | Trichotomy law for 'less t... |
| xrleloe 12807 | 'Less than or equal' expre... |
| xrleltne 12808 | 'Less than or equal to' im... |
| xrltlen 12809 | 'Less than' expressed in t... |
| dfle2 12810 | Alternative definition of ... |
| dflt2 12811 | Alternative definition of ... |
| xrltle 12812 | 'Less than' implies 'less ... |
| xrltled 12813 | 'Less than' implies 'less ... |
| xrleid 12814 | 'Less than or equal to' is... |
| xrleidd 12815 | 'Less than or equal to' is... |
| xrletri 12816 | Trichotomy law for extende... |
| xrletri3 12817 | Trichotomy law for extende... |
| xrletrid 12818 | Trichotomy law for extende... |
| xrlelttr 12819 | Transitive law for orderin... |
| xrltletr 12820 | Transitive law for orderin... |
| xrletr 12821 | Transitive law for orderin... |
| xrlttrd 12822 | Transitive law for orderin... |
| xrlelttrd 12823 | Transitive law for orderin... |
| xrltletrd 12824 | Transitive law for orderin... |
| xrletrd 12825 | Transitive law for orderin... |
| xrltne 12826 | 'Less than' implies not eq... |
| nltpnft 12827 | An extended real is not le... |
| xgepnf 12828 | An extended real which is ... |
| ngtmnft 12829 | An extended real is not gr... |
| xlemnf 12830 | An extended real which is ... |
| xrrebnd 12831 | An extended real is real i... |
| xrre 12832 | A way of proving that an e... |
| xrre2 12833 | An extended real between t... |
| xrre3 12834 | A way of proving that an e... |
| ge0gtmnf 12835 | A nonnegative extended rea... |
| ge0nemnf 12836 | A nonnegative extended rea... |
| xrrege0 12837 | A nonnegative extended rea... |
| xrmax1 12838 | An extended real is less t... |
| xrmax2 12839 | An extended real is less t... |
| xrmin1 12840 | The minimum of two extende... |
| xrmin2 12841 | The minimum of two extende... |
| xrmaxeq 12842 | The maximum of two extende... |
| xrmineq 12843 | The minimum of two extende... |
| xrmaxlt 12844 | Two ways of saying the max... |
| xrltmin 12845 | Two ways of saying an exte... |
| xrmaxle 12846 | Two ways of saying the max... |
| xrlemin 12847 | Two ways of saying a numbe... |
| max1 12848 | A number is less than or e... |
| max1ALT 12849 | A number is less than or e... |
| max2 12850 | A number is less than or e... |
| 2resupmax 12851 | The supremum of two real n... |
| min1 12852 | The minimum of two numbers... |
| min2 12853 | The minimum of two numbers... |
| maxle 12854 | Two ways of saying the max... |
| lemin 12855 | Two ways of saying a numbe... |
| maxlt 12856 | Two ways of saying the max... |
| ltmin 12857 | Two ways of saying a numbe... |
| lemaxle 12858 | A real number which is les... |
| max0sub 12859 | Decompose a real number in... |
| ifle 12860 | An if statement transforms... |
| z2ge 12861 | There exists an integer gr... |
| qbtwnre 12862 | The rational numbers are d... |
| qbtwnxr 12863 | The rational numbers are d... |
| qsqueeze 12864 | If a nonnegative real is l... |
| qextltlem 12865 | Lemma for ~ qextlt and qex... |
| qextlt 12866 | An extensionality-like pro... |
| qextle 12867 | An extensionality-like pro... |
| xralrple 12868 | Show that ` A ` is less th... |
| alrple 12869 | Show that ` A ` is less th... |
| xnegeq 12870 | Equality of two extended n... |
| xnegex 12871 | A negative extended real e... |
| xnegpnf 12872 | Minus ` +oo ` . Remark of... |
| xnegmnf 12873 | Minus ` -oo ` . Remark of... |
| rexneg 12874 | Minus a real number. Rema... |
| xneg0 12875 | The negative of zero. (Co... |
| xnegcl 12876 | Closure of extended real n... |
| xnegneg 12877 | Extended real version of ~... |
| xneg11 12878 | Extended real version of ~... |
| xltnegi 12879 | Forward direction of ~ xlt... |
| xltneg 12880 | Extended real version of ~... |
| xleneg 12881 | Extended real version of ~... |
| xlt0neg1 12882 | Extended real version of ~... |
| xlt0neg2 12883 | Extended real version of ~... |
| xle0neg1 12884 | Extended real version of ~... |
| xle0neg2 12885 | Extended real version of ~... |
| xaddval 12886 | Value of the extended real... |
| xaddf 12887 | The extended real addition... |
| xmulval 12888 | Value of the extended real... |
| xaddpnf1 12889 | Addition of positive infin... |
| xaddpnf2 12890 | Addition of positive infin... |
| xaddmnf1 12891 | Addition of negative infin... |
| xaddmnf2 12892 | Addition of negative infin... |
| pnfaddmnf 12893 | Addition of positive and n... |
| mnfaddpnf 12894 | Addition of negative and p... |
| rexadd 12895 | The extended real addition... |
| rexsub 12896 | Extended real subtraction ... |
| rexaddd 12897 | The extended real addition... |
| xnn0xaddcl 12898 | The extended nonnegative i... |
| xaddnemnf 12899 | Closure of extended real a... |
| xaddnepnf 12900 | Closure of extended real a... |
| xnegid 12901 | Extended real version of ~... |
| xaddcl 12902 | The extended real addition... |
| xaddcom 12903 | The extended real addition... |
| xaddid1 12904 | Extended real version of ~... |
| xaddid2 12905 | Extended real version of ~... |
| xaddid1d 12906 | ` 0 ` is a right identity ... |
| xnn0lem1lt 12907 | Extended nonnegative integ... |
| xnn0lenn0nn0 12908 | An extended nonnegative in... |
| xnn0le2is012 12909 | An extended nonnegative in... |
| xnn0xadd0 12910 | The sum of two extended no... |
| xnegdi 12911 | Extended real version of ~... |
| xaddass 12912 | Associativity of extended ... |
| xaddass2 12913 | Associativity of extended ... |
| xpncan 12914 | Extended real version of ~... |
| xnpcan 12915 | Extended real version of ~... |
| xleadd1a 12916 | Extended real version of ~... |
| xleadd2a 12917 | Commuted form of ~ xleadd1... |
| xleadd1 12918 | Weakened version of ~ xlea... |
| xltadd1 12919 | Extended real version of ~... |
| xltadd2 12920 | Extended real version of ~... |
| xaddge0 12921 | The sum of nonnegative ext... |
| xle2add 12922 | Extended real version of ~... |
| xlt2add 12923 | Extended real version of ~... |
| xsubge0 12924 | Extended real version of ~... |
| xposdif 12925 | Extended real version of ~... |
| xlesubadd 12926 | Under certain conditions, ... |
| xmullem 12927 | Lemma for ~ rexmul . (Con... |
| xmullem2 12928 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 12929 | Extended real multiplicati... |
| xmul01 12930 | Extended real version of ~... |
| xmul02 12931 | Extended real version of ~... |
| xmulneg1 12932 | Extended real version of ~... |
| xmulneg2 12933 | Extended real version of ~... |
| rexmul 12934 | The extended real multipli... |
| xmulf 12935 | The extended real multipli... |
| xmulcl 12936 | Closure of extended real m... |
| xmulpnf1 12937 | Multiplication by plus inf... |
| xmulpnf2 12938 | Multiplication by plus inf... |
| xmulmnf1 12939 | Multiplication by minus in... |
| xmulmnf2 12940 | Multiplication by minus in... |
| xmulpnf1n 12941 | Multiplication by plus inf... |
| xmulid1 12942 | Extended real version of ~... |
| xmulid2 12943 | Extended real version of ~... |
| xmulm1 12944 | Extended real version of ~... |
| xmulasslem2 12945 | Lemma for ~ xmulass . (Co... |
| xmulgt0 12946 | Extended real version of ~... |
| xmulge0 12947 | Extended real version of ~... |
| xmulasslem 12948 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 12949 | Lemma for ~ xmulass . (Co... |
| xmulass 12950 | Associativity of the exten... |
| xlemul1a 12951 | Extended real version of ~... |
| xlemul2a 12952 | Extended real version of ~... |
| xlemul1 12953 | Extended real version of ~... |
| xlemul2 12954 | Extended real version of ~... |
| xltmul1 12955 | Extended real version of ~... |
| xltmul2 12956 | Extended real version of ~... |
| xadddilem 12957 | Lemma for ~ xadddi . (Con... |
| xadddi 12958 | Distributive property for ... |
| xadddir 12959 | Commuted version of ~ xadd... |
| xadddi2 12960 | The assumption that the mu... |
| xadddi2r 12961 | Commuted version of ~ xadd... |
| x2times 12962 | Extended real version of ~... |
| xnegcld 12963 | Closure of extended real n... |
| xaddcld 12964 | The extended real addition... |
| xmulcld 12965 | Closure of extended real m... |
| xadd4d 12966 | Rearrangement of 4 terms i... |
| xnn0add4d 12967 | Rearrangement of 4 terms i... |
| xrsupexmnf 12968 | Adding minus infinity to a... |
| xrinfmexpnf 12969 | Adding plus infinity to a ... |
| xrsupsslem 12970 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 12971 | Lemma for ~ xrinfmss . (C... |
| xrsupss 12972 | Any subset of extended rea... |
| xrinfmss 12973 | Any subset of extended rea... |
| xrinfmss2 12974 | Any subset of extended rea... |
| xrub 12975 | By quantifying only over r... |
| supxr 12976 | The supremum of a set of e... |
| supxr2 12977 | The supremum of a set of e... |
| supxrcl 12978 | The supremum of an arbitra... |
| supxrun 12979 | The supremum of the union ... |
| supxrmnf 12980 | Adding minus infinity to a... |
| supxrpnf 12981 | The supremum of a set of e... |
| supxrunb1 12982 | The supremum of an unbound... |
| supxrunb2 12983 | The supremum of an unbound... |
| supxrbnd1 12984 | The supremum of a bounded-... |
| supxrbnd2 12985 | The supremum of a bounded-... |
| xrsup0 12986 | The supremum of an empty s... |
| supxrub 12987 | A member of a set of exten... |
| supxrlub 12988 | The supremum of a set of e... |
| supxrleub 12989 | The supremum of a set of e... |
| supxrre 12990 | The real and extended real... |
| supxrbnd 12991 | The supremum of a bounded-... |
| supxrgtmnf 12992 | The supremum of a nonempty... |
| supxrre1 12993 | The supremum of a nonempty... |
| supxrre2 12994 | The supremum of a nonempty... |
| supxrss 12995 | Smaller sets of extended r... |
| infxrcl 12996 | The infimum of an arbitrar... |
| infxrlb 12997 | A member of a set of exten... |
| infxrgelb 12998 | The infimum of a set of ex... |
| infxrre 12999 | The real and extended real... |
| infxrmnf 13000 | The infinimum of a set of ... |
| xrinf0 13001 | The infimum of the empty s... |
| infxrss 13002 | Larger sets of extended re... |
| reltre 13003 | For all real numbers there... |
| rpltrp 13004 | For all positive real numb... |
| reltxrnmnf 13005 | For all extended real numb... |
| infmremnf 13006 | The infimum of the reals i... |
| infmrp1 13007 | The infimum of the positiv... |
| ixxval 13016 | Value of the interval func... |
| elixx1 13017 | Membership in an interval ... |
| ixxf 13018 | The set of intervals of ex... |
| ixxex 13019 | The set of intervals of ex... |
| ixxssxr 13020 | The set of intervals of ex... |
| elixx3g 13021 | Membership in a set of ope... |
| ixxssixx 13022 | An interval is a subset of... |
| ixxdisj 13023 | Split an interval into dis... |
| ixxun 13024 | Split an interval into two... |
| ixxin 13025 | Intersection of two interv... |
| ixxss1 13026 | Subset relationship for in... |
| ixxss2 13027 | Subset relationship for in... |
| ixxss12 13028 | Subset relationship for in... |
| ixxub 13029 | Extract the upper bound of... |
| ixxlb 13030 | Extract the lower bound of... |
| iooex 13031 | The set of open intervals ... |
| iooval 13032 | Value of the open interval... |
| ioo0 13033 | An empty open interval of ... |
| ioon0 13034 | An open interval of extend... |
| ndmioo 13035 | The open interval function... |
| iooid 13036 | An open interval with iden... |
| elioo3g 13037 | Membership in a set of ope... |
| elioore 13038 | A member of an open interv... |
| lbioo 13039 | An open interval does not ... |
| ubioo 13040 | An open interval does not ... |
| iooval2 13041 | Value of the open interval... |
| iooin 13042 | Intersection of two open i... |
| iooss1 13043 | Subset relationship for op... |
| iooss2 13044 | Subset relationship for op... |
| iocval 13045 | Value of the open-below, c... |
| icoval 13046 | Value of the closed-below,... |
| iccval 13047 | Value of the closed interv... |
| elioo1 13048 | Membership in an open inte... |
| elioo2 13049 | Membership in an open inte... |
| elioc1 13050 | Membership in an open-belo... |
| elico1 13051 | Membership in a closed-bel... |
| elicc1 13052 | Membership in a closed int... |
| iccid 13053 | A closed interval with ide... |
| ico0 13054 | An empty open interval of ... |
| ioc0 13055 | An empty open interval of ... |
| icc0 13056 | An empty closed interval o... |
| dfrp2 13057 | Alternate definition of th... |
| elicod 13058 | Membership in a left-close... |
| icogelb 13059 | An element of a left-close... |
| elicore 13060 | A member of a left-closed ... |
| ubioc1 13061 | The upper bound belongs to... |
| lbico1 13062 | The lower bound belongs to... |
| iccleub 13063 | An element of a closed int... |
| iccgelb 13064 | An element of a closed int... |
| elioo5 13065 | Membership in an open inte... |
| eliooxr 13066 | A nonempty open interval s... |
| eliooord 13067 | Ordering implied by a memb... |
| elioo4g 13068 | Membership in an open inte... |
| ioossre 13069 | An open interval is a set ... |
| ioosscn 13070 | An open interval is a set ... |
| elioc2 13071 | Membership in an open-belo... |
| elico2 13072 | Membership in a closed-bel... |
| elicc2 13073 | Membership in a closed rea... |
| elicc2i 13074 | Inference for membership i... |
| elicc4 13075 | Membership in a closed rea... |
| iccss 13076 | Condition for a closed int... |
| iccssioo 13077 | Condition for a closed int... |
| icossico 13078 | Condition for a closed-bel... |
| iccss2 13079 | Condition for a closed int... |
| iccssico 13080 | Condition for a closed int... |
| iccssioo2 13081 | Condition for a closed int... |
| iccssico2 13082 | Condition for a closed int... |
| ioomax 13083 | The open interval from min... |
| iccmax 13084 | The closed interval from m... |
| ioopos 13085 | The set of positive reals ... |
| ioorp 13086 | The set of positive reals ... |
| iooshf 13087 | Shift the arguments of the... |
| iocssre 13088 | A closed-above interval wi... |
| icossre 13089 | A closed-below interval wi... |
| iccssre 13090 | A closed real interval is ... |
| iccssxr 13091 | A closed interval is a set... |
| iocssxr 13092 | An open-below, closed-abov... |
| icossxr 13093 | A closed-below, open-above... |
| ioossicc 13094 | An open interval is a subs... |
| iccssred 13095 | A closed real interval is ... |
| eliccxr 13096 | A member of a closed inter... |
| icossicc 13097 | A closed-below, open-above... |
| iocssicc 13098 | A closed-above, open-below... |
| ioossico 13099 | An open interval is a subs... |
| iocssioo 13100 | Condition for a closed int... |
| icossioo 13101 | Condition for a closed int... |
| ioossioo 13102 | Condition for an open inte... |
| iccsupr 13103 | A nonempty subset of a clo... |
| elioopnf 13104 | Membership in an unbounded... |
| elioomnf 13105 | Membership in an unbounded... |
| elicopnf 13106 | Membership in a closed unb... |
| repos 13107 | Two ways of saying that a ... |
| ioof 13108 | The set of open intervals ... |
| iccf 13109 | The set of closed interval... |
| unirnioo 13110 | The union of the range of ... |
| dfioo2 13111 | Alternate definition of th... |
| ioorebas 13112 | Open intervals are element... |
| xrge0neqmnf 13113 | A nonnegative extended rea... |
| xrge0nre 13114 | An extended real which is ... |
| elrege0 13115 | The predicate "is a nonneg... |
| nn0rp0 13116 | A nonnegative integer is a... |
| rge0ssre 13117 | Nonnegative real numbers a... |
| elxrge0 13118 | Elementhood in the set of ... |
| 0e0icopnf 13119 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 13120 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 13121 | The nonnegative reals are ... |
| ge0mulcl 13122 | The nonnegative reals are ... |
| ge0xaddcl 13123 | The nonnegative reals are ... |
| ge0xmulcl 13124 | The nonnegative extended r... |
| lbicc2 13125 | The lower bound of a close... |
| ubicc2 13126 | The upper bound of a close... |
| elicc01 13127 | Membership in the closed r... |
| elunitrn 13128 | The closed unit interval i... |
| elunitcn 13129 | The closed unit interval i... |
| 0elunit 13130 | Zero is an element of the ... |
| 1elunit 13131 | One is an element of the c... |
| iooneg 13132 | Membership in a negated op... |
| iccneg 13133 | Membership in a negated cl... |
| icoshft 13134 | A shifted real is a member... |
| icoshftf1o 13135 | Shifting a closed-below, o... |
| icoun 13136 | The union of two adjacent ... |
| icodisj 13137 | Adjacent left-closed right... |
| ioounsn 13138 | The union of an open inter... |
| snunioo 13139 | The closure of one end of ... |
| snunico 13140 | The closure of the open en... |
| snunioc 13141 | The closure of the open en... |
| prunioo 13142 | The closure of an open rea... |
| ioodisj 13143 | If the upper bound of one ... |
| ioojoin 13144 | Join two open intervals to... |
| difreicc 13145 | The class difference of ` ... |
| iccsplit 13146 | Split a closed interval in... |
| iccshftr 13147 | Membership in a shifted in... |
| iccshftri 13148 | Membership in a shifted in... |
| iccshftl 13149 | Membership in a shifted in... |
| iccshftli 13150 | Membership in a shifted in... |
| iccdil 13151 | Membership in a dilated in... |
| iccdili 13152 | Membership in a dilated in... |
| icccntr 13153 | Membership in a contracted... |
| icccntri 13154 | Membership in a contracted... |
| divelunit 13155 | A condition for a ratio to... |
| lincmb01cmp 13156 | A linear combination of tw... |
| iccf1o 13157 | Describe a bijection from ... |
| iccen 13158 | Any nontrivial closed inte... |
| xov1plusxeqvd 13159 | A complex number ` X ` is ... |
| unitssre 13160 | ` ( 0 [,] 1 ) ` is a subse... |
| unitsscn 13161 | The closed unit interval i... |
| supicc 13162 | Supremum of a bounded set ... |
| supiccub 13163 | The supremum of a bounded ... |
| supicclub 13164 | The supremum of a bounded ... |
| supicclub2 13165 | The supremum of a bounded ... |
| zltaddlt1le 13166 | The sum of an integer and ... |
| xnn0xrge0 13167 | An extended nonnegative in... |
| fzval 13170 | The value of a finite set ... |
| fzval2 13171 | An alternative way of expr... |
| fzf 13172 | Establish the domain and c... |
| elfz1 13173 | Membership in a finite set... |
| elfz 13174 | Membership in a finite set... |
| elfz2 13175 | Membership in a finite set... |
| elfzd 13176 | Membership in a finite set... |
| elfz5 13177 | Membership in a finite set... |
| elfz4 13178 | Membership in a finite set... |
| elfzuzb 13179 | Membership in a finite set... |
| eluzfz 13180 | Membership in a finite set... |
| elfzuz 13181 | A member of a finite set o... |
| elfzuz3 13182 | Membership in a finite set... |
| elfzel2 13183 | Membership in a finite set... |
| elfzel1 13184 | Membership in a finite set... |
| elfzelz 13185 | A member of a finite set o... |
| elfzelzd 13186 | A member of a finite set o... |
| fzssz 13187 | A finite sequence of integ... |
| elfzle1 13188 | A member of a finite set o... |
| elfzle2 13189 | A member of a finite set o... |
| elfzuz2 13190 | Implication of membership ... |
| elfzle3 13191 | Membership in a finite set... |
| eluzfz1 13192 | Membership in a finite set... |
| eluzfz2 13193 | Membership in a finite set... |
| eluzfz2b 13194 | Membership in a finite set... |
| elfz3 13195 | Membership in a finite set... |
| elfz1eq 13196 | Membership in a finite set... |
| elfzubelfz 13197 | If there is a member in a ... |
| peano2fzr 13198 | A Peano-postulate-like the... |
| fzn0 13199 | Properties of a finite int... |
| fz0 13200 | A finite set of sequential... |
| fzn 13201 | A finite set of sequential... |
| fzen 13202 | A shifted finite set of se... |
| fz1n 13203 | A 1-based finite set of se... |
| 0nelfz1 13204 | 0 is not an element of a f... |
| 0fz1 13205 | Two ways to say a finite 1... |
| fz10 13206 | There are no integers betw... |
| uzsubsubfz 13207 | Membership of an integer g... |
| uzsubsubfz1 13208 | Membership of an integer g... |
| ige3m2fz 13209 | Membership of an integer g... |
| fzsplit2 13210 | Split a finite interval of... |
| fzsplit 13211 | Split a finite interval of... |
| fzdisj 13212 | Condition for two finite i... |
| fz01en 13213 | 0-based and 1-based finite... |
| elfznn 13214 | A member of a finite set o... |
| elfz1end 13215 | A nonempty finite range of... |
| fz1ssnn 13216 | A finite set of positive i... |
| fznn0sub 13217 | Subtraction closure for a ... |
| fzmmmeqm 13218 | Subtracting the difference... |
| fzaddel 13219 | Membership of a sum in a f... |
| fzadd2 13220 | Membership of a sum in a f... |
| fzsubel 13221 | Membership of a difference... |
| fzopth 13222 | A finite set of sequential... |
| fzass4 13223 | Two ways to express a nond... |
| fzss1 13224 | Subset relationship for fi... |
| fzss2 13225 | Subset relationship for fi... |
| fzssuz 13226 | A finite set of sequential... |
| fzsn 13227 | A finite interval of integ... |
| fzssp1 13228 | Subset relationship for fi... |
| fzssnn 13229 | Finite sets of sequential ... |
| ssfzunsnext 13230 | A subset of a finite seque... |
| ssfzunsn 13231 | A subset of a finite seque... |
| fzsuc 13232 | Join a successor to the en... |
| fzpred 13233 | Join a predecessor to the ... |
| fzpreddisj 13234 | A finite set of sequential... |
| elfzp1 13235 | Append an element to a fin... |
| fzp1ss 13236 | Subset relationship for fi... |
| fzelp1 13237 | Membership in a set of seq... |
| fzp1elp1 13238 | Add one to an element of a... |
| fznatpl1 13239 | Shift membership in a fini... |
| fzpr 13240 | A finite interval of integ... |
| fztp 13241 | A finite interval of integ... |
| fz12pr 13242 | An integer range between 1... |
| fzsuc2 13243 | Join a successor to the en... |
| fzp1disj 13244 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 13245 | Remove a successor from th... |
| fzprval 13246 | Two ways of defining the f... |
| fztpval 13247 | Two ways of defining the f... |
| fzrev 13248 | Reversal of start and end ... |
| fzrev2 13249 | Reversal of start and end ... |
| fzrev2i 13250 | Reversal of start and end ... |
| fzrev3 13251 | The "complement" of a memb... |
| fzrev3i 13252 | The "complement" of a memb... |
| fznn 13253 | Finite set of sequential i... |
| elfz1b 13254 | Membership in a 1-based fi... |
| elfz1uz 13255 | Membership in a 1-based fi... |
| elfzm11 13256 | Membership in a finite set... |
| uzsplit 13257 | Express an upper integer s... |
| uzdisj 13258 | The first ` N ` elements o... |
| fseq1p1m1 13259 | Add/remove an item to/from... |
| fseq1m1p1 13260 | Add/remove an item to/from... |
| fz1sbc 13261 | Quantification over a one-... |
| elfzp1b 13262 | An integer is a member of ... |
| elfzm1b 13263 | An integer is a member of ... |
| elfzp12 13264 | Options for membership in ... |
| fzm1 13265 | Choices for an element of ... |
| fzneuz 13266 | No finite set of sequentia... |
| fznuz 13267 | Disjointness of the upper ... |
| uznfz 13268 | Disjointness of the upper ... |
| fzp1nel 13269 | One plus the upper bound o... |
| fzrevral 13270 | Reversal of scanning order... |
| fzrevral2 13271 | Reversal of scanning order... |
| fzrevral3 13272 | Reversal of scanning order... |
| fzshftral 13273 | Shift the scanning order i... |
| ige2m1fz1 13274 | Membership of an integer g... |
| ige2m1fz 13275 | Membership in a 0-based fi... |
| elfz2nn0 13276 | Membership in a finite set... |
| fznn0 13277 | Characterization of a fini... |
| elfznn0 13278 | A member of a finite set o... |
| elfz3nn0 13279 | The upper bound of a nonem... |
| fz0ssnn0 13280 | Finite sets of sequential ... |
| fz1ssfz0 13281 | Subset relationship for fi... |
| 0elfz 13282 | 0 is an element of a finit... |
| nn0fz0 13283 | A nonnegative integer is a... |
| elfz0add 13284 | An element of a finite set... |
| fz0sn 13285 | An integer range from 0 to... |
| fz0tp 13286 | An integer range from 0 to... |
| fz0to3un2pr 13287 | An integer range from 0 to... |
| fz0to4untppr 13288 | An integer range from 0 to... |
| elfz0ubfz0 13289 | An element of a finite set... |
| elfz0fzfz0 13290 | A member of a finite set o... |
| fz0fzelfz0 13291 | If a member of a finite se... |
| fznn0sub2 13292 | Subtraction closure for a ... |
| uzsubfz0 13293 | Membership of an integer g... |
| fz0fzdiffz0 13294 | The difference of an integ... |
| elfzmlbm 13295 | Subtracting the lower boun... |
| elfzmlbp 13296 | Subtracting the lower boun... |
| fzctr 13297 | Lemma for theorems about t... |
| difelfzle 13298 | The difference of two inte... |
| difelfznle 13299 | The difference of two inte... |
| nn0split 13300 | Express the set of nonnega... |
| nn0disj 13301 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13302 | A finite set of sequential... |
| fvffz0 13303 | The function value of a fu... |
| 1fv 13304 | A function on a singleton.... |
| 4fvwrd4 13305 | The first four function va... |
| 2ffzeq 13306 | Two functions over 0-based... |
| preduz 13307 | The value of the predecess... |
| prednn 13308 | The value of the predecess... |
| prednn0 13309 | The value of the predecess... |
| predfz 13310 | Calculate the predecessor ... |
| fzof 13313 | Functionality of the half-... |
| elfzoel1 13314 | Reverse closure for half-o... |
| elfzoel2 13315 | Reverse closure for half-o... |
| elfzoelz 13316 | Reverse closure for half-o... |
| fzoval 13317 | Value of the half-open int... |
| elfzo 13318 | Membership in a half-open ... |
| elfzo2 13319 | Membership in a half-open ... |
| elfzouz 13320 | Membership in a half-open ... |
| nelfzo 13321 | An integer not being a mem... |
| fzolb 13322 | The left endpoint of a hal... |
| fzolb2 13323 | The left endpoint of a hal... |
| elfzole1 13324 | A member in a half-open in... |
| elfzolt2 13325 | A member in a half-open in... |
| elfzolt3 13326 | Membership in a half-open ... |
| elfzolt2b 13327 | A member in a half-open in... |
| elfzolt3b 13328 | Membership in a half-open ... |
| fzonel 13329 | A half-open range does not... |
| elfzouz2 13330 | The upper bound of a half-... |
| elfzofz 13331 | A half-open range is conta... |
| elfzo3 13332 | Express membership in a ha... |
| fzon0 13333 | A half-open integer interv... |
| fzossfz 13334 | A half-open range is conta... |
| fzossz 13335 | A half-open integer interv... |
| fzon 13336 | A half-open set of sequent... |
| fzo0n 13337 | A half-open range of nonne... |
| fzonlt0 13338 | A half-open integer range ... |
| fzo0 13339 | Half-open sets with equal ... |
| fzonnsub 13340 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13341 | If ` M < N ` then ` N - M ... |
| fzoss1 13342 | Subset relationship for ha... |
| fzoss2 13343 | Subset relationship for ha... |
| fzossrbm1 13344 | Subset of a half-open rang... |
| fzo0ss1 13345 | Subset relationship for ha... |
| fzossnn0 13346 | A half-open integer range ... |
| fzospliti 13347 | One direction of splitting... |
| fzosplit 13348 | Split a half-open integer ... |
| fzodisj 13349 | Abutting half-open integer... |
| fzouzsplit 13350 | Split an upper integer set... |
| fzouzdisj 13351 | A half-open integer range ... |
| fzoun 13352 | A half-open integer range ... |
| fzodisjsn 13353 | A half-open integer range ... |
| prinfzo0 13354 | The intersection of a half... |
| lbfzo0 13355 | An integer is strictly gre... |
| elfzo0 13356 | Membership in a half-open ... |
| elfzo0z 13357 | Membership in a half-open ... |
| nn0p1elfzo 13358 | A nonnegative integer incr... |
| elfzo0le 13359 | A member in a half-open ra... |
| elfzonn0 13360 | A member of a half-open ra... |
| fzonmapblen 13361 | The result of subtracting ... |
| fzofzim 13362 | If a nonnegative integer i... |
| fz1fzo0m1 13363 | Translation of one between... |
| fzossnn 13364 | Half-open integer ranges s... |
| elfzo1 13365 | Membership in a half-open ... |
| fzo1fzo0n0 13366 | An integer between 1 and a... |
| fzo0n0 13367 | A half-open integer range ... |
| fzoaddel 13368 | Translate membership in a ... |
| fzo0addel 13369 | Translate membership in a ... |
| fzo0addelr 13370 | Translate membership in a ... |
| fzoaddel2 13371 | Translate membership in a ... |
| elfzoext 13372 | Membership of an integer i... |
| elincfzoext 13373 | Membership of an increased... |
| fzosubel 13374 | Translate membership in a ... |
| fzosubel2 13375 | Membership in a translated... |
| fzosubel3 13376 | Membership in a translated... |
| eluzgtdifelfzo 13377 | Membership of the differen... |
| ige2m2fzo 13378 | Membership of an integer g... |
| fzocatel 13379 | Translate membership in a ... |
| ubmelfzo 13380 | If an integer in a 1-based... |
| elfzodifsumelfzo 13381 | If an integer is in a half... |
| elfzom1elp1fzo 13382 | Membership of an integer i... |
| elfzom1elfzo 13383 | Membership in a half-open ... |
| fzval3 13384 | Expressing a closed intege... |
| fz0add1fz1 13385 | Translate membership in a ... |
| fzosn 13386 | Expressing a singleton as ... |
| elfzomin 13387 | Membership of an integer i... |
| zpnn0elfzo 13388 | Membership of an integer i... |
| zpnn0elfzo1 13389 | Membership of an integer i... |
| fzosplitsnm1 13390 | Removing a singleton from ... |
| elfzonlteqm1 13391 | If an element of a half-op... |
| fzonn0p1 13392 | A nonnegative integer is e... |
| fzossfzop1 13393 | A half-open range of nonne... |
| fzonn0p1p1 13394 | If a nonnegative integer i... |
| elfzom1p1elfzo 13395 | Increasing an element of a... |
| fzo0ssnn0 13396 | Half-open integer ranges s... |
| fzo01 13397 | Expressing the singleton o... |
| fzo12sn 13398 | A 1-based half-open intege... |
| fzo13pr 13399 | A 1-based half-open intege... |
| fzo0to2pr 13400 | A half-open integer range ... |
| fzo0to3tp 13401 | A half-open integer range ... |
| fzo0to42pr 13402 | A half-open integer range ... |
| fzo1to4tp 13403 | A half-open integer range ... |
| fzo0sn0fzo1 13404 | A half-open range of nonne... |
| elfzo0l 13405 | A member of a half-open ra... |
| fzoend 13406 | The endpoint of a half-ope... |
| fzo0end 13407 | The endpoint of a zero-bas... |
| ssfzo12 13408 | Subset relationship for ha... |
| ssfzoulel 13409 | If a half-open integer ran... |
| ssfzo12bi 13410 | Subset relationship for ha... |
| ubmelm1fzo 13411 | The result of subtracting ... |
| fzofzp1 13412 | If a point is in a half-op... |
| fzofzp1b 13413 | If a point is in a half-op... |
| elfzom1b 13414 | An integer is a member of ... |
| elfzom1elp1fzo1 13415 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13416 | Membership of a positive i... |
| elfzonelfzo 13417 | If an element of a half-op... |
| fzonfzoufzol 13418 | If an element of a half-op... |
| elfzomelpfzo 13419 | An integer increased by an... |
| elfznelfzo 13420 | A value in a finite set of... |
| elfznelfzob 13421 | A value in a finite set of... |
| peano2fzor 13422 | A Peano-postulate-like the... |
| fzosplitsn 13423 | Extending a half-open rang... |
| fzosplitpr 13424 | Extending a half-open inte... |
| fzosplitprm1 13425 | Extending a half-open inte... |
| fzosplitsni 13426 | Membership in a half-open ... |
| fzisfzounsn 13427 | A finite interval of integ... |
| elfzr 13428 | A member of a finite inter... |
| elfzlmr 13429 | A member of a finite inter... |
| elfz0lmr 13430 | A member of a finite inter... |
| fzostep1 13431 | Two possibilities for a nu... |
| fzoshftral 13432 | Shift the scanning order i... |
| fzind2 13433 | Induction on the integers ... |
| fvinim0ffz 13434 | The function values for th... |
| injresinjlem 13435 | Lemma for ~ injresinj . (... |
| injresinj 13436 | A function whose restricti... |
| subfzo0 13437 | The difference between two... |
| flval 13442 | Value of the floor (greate... |
| flcl 13443 | The floor (greatest intege... |
| reflcl 13444 | The floor (greatest intege... |
| fllelt 13445 | A basic property of the fl... |
| flcld 13446 | The floor (greatest intege... |
| flle 13447 | A basic property of the fl... |
| flltp1 13448 | A basic property of the fl... |
| fllep1 13449 | A basic property of the fl... |
| fraclt1 13450 | The fractional part of a r... |
| fracle1 13451 | The fractional part of a r... |
| fracge0 13452 | The fractional part of a r... |
| flge 13453 | The floor function value i... |
| fllt 13454 | The floor function value i... |
| flflp1 13455 | Move floor function betwee... |
| flid 13456 | An integer is its own floo... |
| flidm 13457 | The floor function is idem... |
| flidz 13458 | A real number equals its f... |
| flltnz 13459 | The floor of a non-integer... |
| flwordi 13460 | Ordering relation for the ... |
| flword2 13461 | Ordering relation for the ... |
| flval2 13462 | An alternate way to define... |
| flval3 13463 | An alternate way to define... |
| flbi 13464 | A condition equivalent to ... |
| flbi2 13465 | A condition equivalent to ... |
| adddivflid 13466 | The floor of a sum of an i... |
| ico01fl0 13467 | The floor of a real number... |
| flge0nn0 13468 | The floor of a number grea... |
| flge1nn 13469 | The floor of a number grea... |
| fldivnn0 13470 | The floor function of a di... |
| refldivcl 13471 | The floor function of a di... |
| divfl0 13472 | The floor of a fraction is... |
| fladdz 13473 | An integer can be moved in... |
| flzadd 13474 | An integer can be moved in... |
| flmulnn0 13475 | Move a nonnegative integer... |
| btwnzge0 13476 | A real bounded between an ... |
| 2tnp1ge0ge0 13477 | Two times an integer plus ... |
| flhalf 13478 | Ordering relation for the ... |
| fldivle 13479 | The floor function of a di... |
| fldivnn0le 13480 | The floor function of a di... |
| flltdivnn0lt 13481 | The floor function of a di... |
| ltdifltdiv 13482 | If the dividend of a divis... |
| fldiv4p1lem1div2 13483 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13484 | The floor of an integer gr... |
| fldiv4lem1div2 13485 | The floor of a positive in... |
| ceilval 13486 | The value of the ceiling f... |
| dfceil2 13487 | Alternative definition of ... |
| ceilval2 13488 | The value of the ceiling f... |
| ceicl 13489 | The ceiling function retur... |
| ceilcl 13490 | Closure of the ceiling fun... |
| ceilcld 13491 | Closure of the ceiling fun... |
| ceige 13492 | The ceiling of a real numb... |
| ceilge 13493 | The ceiling of a real numb... |
| ceilged 13494 | The ceiling of a real numb... |
| ceim1l 13495 | One less than the ceiling ... |
| ceilm1lt 13496 | One less than the ceiling ... |
| ceile 13497 | The ceiling of a real numb... |
| ceille 13498 | The ceiling of a real numb... |
| ceilid 13499 | An integer is its own ceil... |
| ceilidz 13500 | A real number equals its c... |
| flleceil 13501 | The floor of a real number... |
| fleqceilz 13502 | A real number is an intege... |
| quoremz 13503 | Quotient and remainder of ... |
| quoremnn0 13504 | Quotient and remainder of ... |
| quoremnn0ALT 13505 | Alternate proof of ~ quore... |
| intfrac2 13506 | Decompose a real into inte... |
| intfracq 13507 | Decompose a rational numbe... |
| fldiv 13508 | Cancellation of the embedd... |
| fldiv2 13509 | Cancellation of an embedde... |
| fznnfl 13510 | Finite set of sequential i... |
| uzsup 13511 | An upper set of integers i... |
| ioopnfsup 13512 | An upper set of reals is u... |
| icopnfsup 13513 | An upper set of reals is u... |
| rpsup 13514 | The positive reals are unb... |
| resup 13515 | The real numbers are unbou... |
| xrsup 13516 | The extended real numbers ... |
| modval 13519 | The value of the modulo op... |
| modvalr 13520 | The value of the modulo op... |
| modcl 13521 | Closure law for the modulo... |
| flpmodeq 13522 | Partition of a division in... |
| modcld 13523 | Closure law for the modulo... |
| mod0 13524 | ` A mod B ` is zero iff ` ... |
| mulmod0 13525 | The product of an integer ... |
| negmod0 13526 | ` A ` is divisible by ` B ... |
| modge0 13527 | The modulo operation is no... |
| modlt 13528 | The modulo operation is le... |
| modelico 13529 | Modular reduction produces... |
| moddiffl 13530 | Value of the modulo operat... |
| moddifz 13531 | The modulo operation diffe... |
| modfrac 13532 | The fractional part of a n... |
| flmod 13533 | The floor function express... |
| intfrac 13534 | Break a number into its in... |
| zmod10 13535 | An integer modulo 1 is 0. ... |
| zmod1congr 13536 | Two arbitrary integers are... |
| modmulnn 13537 | Move a positive integer in... |
| modvalp1 13538 | The value of the modulo op... |
| zmodcl 13539 | Closure law for the modulo... |
| zmodcld 13540 | Closure law for the modulo... |
| zmodfz 13541 | An integer mod ` B ` lies ... |
| zmodfzo 13542 | An integer mod ` B ` lies ... |
| zmodfzp1 13543 | An integer mod ` B ` lies ... |
| modid 13544 | Identity law for modulo. ... |
| modid0 13545 | A positive real number mod... |
| modid2 13546 | Identity law for modulo. ... |
| zmodid2 13547 | Identity law for modulo re... |
| zmodidfzo 13548 | Identity law for modulo re... |
| zmodidfzoimp 13549 | Identity law for modulo re... |
| 0mod 13550 | Special case: 0 modulo a p... |
| 1mod 13551 | Special case: 1 modulo a r... |
| modabs 13552 | Absorption law for modulo.... |
| modabs2 13553 | Absorption law for modulo.... |
| modcyc 13554 | The modulo operation is pe... |
| modcyc2 13555 | The modulo operation is pe... |
| modadd1 13556 | Addition property of the m... |
| modaddabs 13557 | Absorption law for modulo.... |
| modaddmod 13558 | The sum of a real number m... |
| muladdmodid 13559 | The sum of a positive real... |
| mulp1mod1 13560 | The product of an integer ... |
| modmuladd 13561 | Decomposition of an intege... |
| modmuladdim 13562 | Implication of a decomposi... |
| modmuladdnn0 13563 | Implication of a decomposi... |
| negmod 13564 | The negation of a number m... |
| m1modnnsub1 13565 | Minus one modulo a positiv... |
| m1modge3gt1 13566 | Minus one modulo an intege... |
| addmodid 13567 | The sum of a positive inte... |
| addmodidr 13568 | The sum of a positive inte... |
| modadd2mod 13569 | The sum of a real number m... |
| modm1p1mod0 13570 | If a real number modulo a ... |
| modltm1p1mod 13571 | If a real number modulo a ... |
| modmul1 13572 | Multiplication property of... |
| modmul12d 13573 | Multiplication property of... |
| modnegd 13574 | Negation property of the m... |
| modadd12d 13575 | Additive property of the m... |
| modsub12d 13576 | Subtraction property of th... |
| modsubmod 13577 | The difference of a real n... |
| modsubmodmod 13578 | The difference of a real n... |
| 2txmodxeq0 13579 | Two times a positive real ... |
| 2submod 13580 | If a real number is betwee... |
| modifeq2int 13581 | If a nonnegative integer i... |
| modaddmodup 13582 | The sum of an integer modu... |
| modaddmodlo 13583 | The sum of an integer modu... |
| modmulmod 13584 | The product of a real numb... |
| modmulmodr 13585 | The product of an integer ... |
| modaddmulmod 13586 | The sum of a real number a... |
| moddi 13587 | Distribute multiplication ... |
| modsubdir 13588 | Distribute the modulo oper... |
| modeqmodmin 13589 | A real number equals the d... |
| modirr 13590 | A number modulo an irratio... |
| modfzo0difsn 13591 | For a number within a half... |
| modsumfzodifsn 13592 | The sum of a number within... |
| modlteq 13593 | Two nonnegative integers l... |
| addmodlteq 13594 | Two nonnegative integers l... |
| om2uz0i 13595 | The mapping ` G ` is a one... |
| om2uzsuci 13596 | The value of ` G ` (see ~ ... |
| om2uzuzi 13597 | The value ` G ` (see ~ om2... |
| om2uzlti 13598 | Less-than relation for ` G... |
| om2uzlt2i 13599 | The mapping ` G ` (see ~ o... |
| om2uzrani 13600 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 13601 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 13602 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 13603 | An alternative definition ... |
| om2uzrdg 13604 | A helper lemma for the val... |
| uzrdglem 13605 | A helper lemma for the val... |
| uzrdgfni 13606 | The recursive definition g... |
| uzrdg0i 13607 | Initial value of a recursi... |
| uzrdgsuci 13608 | Successor value of a recur... |
| ltweuz 13609 | ` < ` is a well-founded re... |
| ltwenn 13610 | Less than well-orders the ... |
| ltwefz 13611 | Less than well-orders a se... |
| uzenom 13612 | An upper integer set is de... |
| uzinf 13613 | An upper integer set is in... |
| nnnfi 13614 | The set of positive intege... |
| uzrdgxfr 13615 | Transfer the value of the ... |
| fzennn 13616 | The cardinality of a finit... |
| fzen2 13617 | The cardinality of a finit... |
| cardfz 13618 | The cardinality of a finit... |
| hashgf1o 13619 | ` G ` maps ` _om ` one-to-... |
| fzfi 13620 | A finite interval of integ... |
| fzfid 13621 | Commonly used special case... |
| fzofi 13622 | Half-open integer sets are... |
| fsequb 13623 | The values of a finite rea... |
| fsequb2 13624 | The values of a finite rea... |
| fseqsupcl 13625 | The values of a finite rea... |
| fseqsupubi 13626 | The values of a finite rea... |
| nn0ennn 13627 | The nonnegative integers a... |
| nnenom 13628 | The set of positive intege... |
| nnct 13629 | ` NN ` is countable. (Con... |
| uzindi 13630 | Indirect strong induction ... |
| axdc4uzlem 13631 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 13632 | A version of ~ axdc4 that ... |
| ssnn0fi 13633 | A subset of the nonnegativ... |
| rabssnn0fi 13634 | A subset of the nonnegativ... |
| uzsinds 13635 | Strong (or "total") induct... |
| nnsinds 13636 | Strong (or "total") induct... |
| nn0sinds 13637 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 13638 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 13639 | If all functions of a fini... |
| fsuppmapnn0fiubex 13640 | If all functions of a fini... |
| fsuppmapnn0fiub0 13641 | If all functions of a fini... |
| suppssfz 13642 | Condition for a function o... |
| fsuppmapnn0ub 13643 | If a function over the non... |
| fsuppmapnn0fz 13644 | If a function over the non... |
| mptnn0fsupp 13645 | A mapping from the nonnega... |
| mptnn0fsuppd 13646 | A mapping from the nonnega... |
| mptnn0fsuppr 13647 | A finitely supported mappi... |
| f13idfv 13648 | A one-to-one function with... |
| seqex 13651 | Existence of the sequence ... |
| seqeq1 13652 | Equality theorem for the s... |
| seqeq2 13653 | Equality theorem for the s... |
| seqeq3 13654 | Equality theorem for the s... |
| seqeq1d 13655 | Equality deduction for the... |
| seqeq2d 13656 | Equality deduction for the... |
| seqeq3d 13657 | Equality deduction for the... |
| seqeq123d 13658 | Equality deduction for the... |
| nfseq 13659 | Hypothesis builder for the... |
| seqval 13660 | Value of the sequence buil... |
| seqfn 13661 | The sequence builder funct... |
| seq1 13662 | Value of the sequence buil... |
| seq1i 13663 | Value of the sequence buil... |
| seqp1 13664 | Value of the sequence buil... |
| seqexw 13665 | Weak version of ~ seqex th... |
| seqp1d 13666 | Value of the sequence buil... |
| seqp1iOLD 13667 | Obsolete version of ~ seqp... |
| seqm1 13668 | Value of the sequence buil... |
| seqcl2 13669 | Closure properties of the ... |
| seqf2 13670 | Range of the recursive seq... |
| seqcl 13671 | Closure properties of the ... |
| seqf 13672 | Range of the recursive seq... |
| seqfveq2 13673 | Equality of sequences. (C... |
| seqfeq2 13674 | Equality of sequences. (C... |
| seqfveq 13675 | Equality of sequences. (C... |
| seqfeq 13676 | Equality of sequences. (C... |
| seqshft2 13677 | Shifting the index set of ... |
| seqres 13678 | Restricting its characteri... |
| serf 13679 | An infinite series of comp... |
| serfre 13680 | An infinite series of real... |
| monoord 13681 | Ordering relation for a mo... |
| monoord2 13682 | Ordering relation for a mo... |
| sermono 13683 | The partial sums in an inf... |
| seqsplit 13684 | Split a sequence into two ... |
| seq1p 13685 | Removing the first term fr... |
| seqcaopr3 13686 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 13687 | The sum of two infinite se... |
| seqcaopr 13688 | The sum of two infinite se... |
| seqf1olem2a 13689 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 13690 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 13691 | Lemma for ~ seqf1o . (Con... |
| seqf1o 13692 | Rearrange a sum via an arb... |
| seradd 13693 | The sum of two infinite se... |
| sersub 13694 | The difference of two infi... |
| seqid3 13695 | A sequence that consists e... |
| seqid 13696 | Discarding the first few t... |
| seqid2 13697 | The last few partial sums ... |
| seqhomo 13698 | Apply a homomorphism to a ... |
| seqz 13699 | If the operation ` .+ ` ha... |
| seqfeq4 13700 | Equality of series under d... |
| seqfeq3 13701 | Equality of series under d... |
| seqdistr 13702 | The distributive property ... |
| ser0 13703 | The value of the partial s... |
| ser0f 13704 | A zero-valued infinite ser... |
| serge0 13705 | A finite sum of nonnegativ... |
| serle 13706 | Comparison of partial sums... |
| ser1const 13707 | Value of the partial serie... |
| seqof 13708 | Distribute function operat... |
| seqof2 13709 | Distribute function operat... |
| expval 13712 | Value of exponentiation to... |
| expnnval 13713 | Value of exponentiation to... |
| exp0 13714 | Value of a complex number ... |
| 0exp0e1 13715 | The zeroth power of zero e... |
| exp1 13716 | Value of a complex number ... |
| expp1 13717 | Value of a complex number ... |
| expneg 13718 | Value of a complex number ... |
| expneg2 13719 | Value of a complex number ... |
| expn1 13720 | A number to the negative o... |
| expcllem 13721 | Lemma for proving nonnegat... |
| expcl2lem 13722 | Lemma for proving integer ... |
| nnexpcl 13723 | Closure of exponentiation ... |
| nn0expcl 13724 | Closure of exponentiation ... |
| zexpcl 13725 | Closure of exponentiation ... |
| qexpcl 13726 | Closure of exponentiation ... |
| reexpcl 13727 | Closure of exponentiation ... |
| expcl 13728 | Closure law for nonnegativ... |
| rpexpcl 13729 | Closure law for exponentia... |
| reexpclz 13730 | Closure of exponentiation ... |
| qexpclz 13731 | Closure of exponentiation ... |
| m1expcl2 13732 | Closure of exponentiation ... |
| m1expcl 13733 | Closure of exponentiation ... |
| expclzlem 13734 | Closure law for integer ex... |
| expclz 13735 | Closure law for integer ex... |
| zexpcld 13736 | Closure of exponentiation ... |
| nn0expcli 13737 | Closure of exponentiation ... |
| nn0sqcl 13738 | The square of a nonnegativ... |
| expm1t 13739 | Exponentiation in terms of... |
| 1exp 13740 | Value of one raised to a n... |
| expeq0 13741 | Positive integer exponenti... |
| expne0 13742 | Positive integer exponenti... |
| expne0i 13743 | Nonnegative integer expone... |
| expgt0 13744 | A positive real raised to ... |
| expnegz 13745 | Value of a complex number ... |
| 0exp 13746 | Value of zero raised to a ... |
| expge0 13747 | A nonnegative real raised ... |
| expge1 13748 | A real greater than or equ... |
| expgt1 13749 | A real greater than 1 rais... |
| mulexp 13750 | Nonnegative integer expone... |
| mulexpz 13751 | Integer exponentiation of ... |
| exprec 13752 | Integer exponentiation of ... |
| expadd 13753 | Sum of exponents law for n... |
| expaddzlem 13754 | Lemma for ~ expaddz . (Co... |
| expaddz 13755 | Sum of exponents law for i... |
| expmul 13756 | Product of exponents law f... |
| expmulz 13757 | Product of exponents law f... |
| m1expeven 13758 | Exponentiation of negative... |
| expsub 13759 | Exponent subtraction law f... |
| expp1z 13760 | Value of a nonzero complex... |
| expm1 13761 | Value of a complex number ... |
| expdiv 13762 | Nonnegative integer expone... |
| sqval 13763 | Value of the square of a c... |
| sqneg 13764 | The square of the negative... |
| sqsubswap 13765 | Swap the order of subtract... |
| sqcl 13766 | Closure of square. (Contr... |
| sqmul 13767 | Distribution of square ove... |
| sqeq0 13768 | A number is zero iff its s... |
| sqdiv 13769 | Distribution of square ove... |
| sqdivid 13770 | The square of a nonzero nu... |
| sqne0 13771 | A number is nonzero iff it... |
| resqcl 13772 | Closure of the square of a... |
| sqgt0 13773 | The square of a nonzero re... |
| sqn0rp 13774 | The square of a nonzero re... |
| nnsqcl 13775 | The naturals are closed un... |
| zsqcl 13776 | Integers are closed under ... |
| qsqcl 13777 | The square of a rational i... |
| sq11 13778 | The square function is one... |
| nn0sq11 13779 | The square function is one... |
| lt2sq 13780 | The square function on non... |
| le2sq 13781 | The square function on non... |
| le2sq2 13782 | The square of a 'less than... |
| sqge0 13783 | A square of a real is nonn... |
| zsqcl2 13784 | The square of an integer i... |
| 0expd 13785 | Value of zero raised to a ... |
| exp0d 13786 | Value of a complex number ... |
| exp1d 13787 | Value of a complex number ... |
| expeq0d 13788 | Positive integer exponenti... |
| sqvald 13789 | Value of square. Inferenc... |
| sqcld 13790 | Closure of square. (Contr... |
| sqeq0d 13791 | A number is zero iff its s... |
| expcld 13792 | Closure law for nonnegativ... |
| expp1d 13793 | Value of a complex number ... |
| expaddd 13794 | Sum of exponents law for n... |
| expmuld 13795 | Product of exponents law f... |
| sqrecd 13796 | Square of reciprocal. (Co... |
| expclzd 13797 | Closure law for integer ex... |
| expne0d 13798 | Nonnegative integer expone... |
| expnegd 13799 | Value of a complex number ... |
| exprecd 13800 | Nonnegative integer expone... |
| expp1zd 13801 | Value of a nonzero complex... |
| expm1d 13802 | Value of a complex number ... |
| expsubd 13803 | Exponent subtraction law f... |
| sqmuld 13804 | Distribution of square ove... |
| sqdivd 13805 | Distribution of square ove... |
| expdivd 13806 | Nonnegative integer expone... |
| mulexpd 13807 | Positive integer exponenti... |
| znsqcld 13808 | The square of a nonzero in... |
| reexpcld 13809 | Closure of exponentiation ... |
| expge0d 13810 | A nonnegative real raised ... |
| expge1d 13811 | A real greater than or equ... |
| ltexp2a 13812 | Ordering relationship for ... |
| expmordi 13813 | Base ordering relationship... |
| rpexpmord 13814 | Base ordering relationship... |
| expcan 13815 | Cancellation law for expon... |
| ltexp2 13816 | Ordering law for exponenti... |
| leexp2 13817 | Ordering law for exponenti... |
| leexp2a 13818 | Weak ordering relationship... |
| ltexp2r 13819 | The power of a positive nu... |
| leexp2r 13820 | Weak ordering relationship... |
| leexp1a 13821 | Weak base ordering relatio... |
| exple1 13822 | A real between 0 and 1 inc... |
| expubnd 13823 | An upper bound on ` A ^ N ... |
| sumsqeq0 13824 | Two real numbers are equal... |
| sqvali 13825 | Value of square. Inferenc... |
| sqcli 13826 | Closure of square. (Contr... |
| sqeq0i 13827 | A number is zero iff its s... |
| sqrecii 13828 | Square of reciprocal. (Co... |
| sqmuli 13829 | Distribution of square ove... |
| sqdivi 13830 | Distribution of square ove... |
| resqcli 13831 | Closure of square in reals... |
| sqgt0i 13832 | The square of a nonzero re... |
| sqge0i 13833 | A square of a real is nonn... |
| lt2sqi 13834 | The square function on non... |
| le2sqi 13835 | The square function on non... |
| sq11i 13836 | The square function is one... |
| sq0 13837 | The square of 0 is 0. (Co... |
| sq0i 13838 | If a number is zero, its s... |
| sq0id 13839 | If a number is zero, its s... |
| sq1 13840 | The square of 1 is 1. (Co... |
| neg1sqe1 13841 | ` -u 1 ` squared is 1. (C... |
| sq2 13842 | The square of 2 is 4. (Co... |
| sq3 13843 | The square of 3 is 9. (Co... |
| sq4e2t8 13844 | The square of 4 is 2 times... |
| cu2 13845 | The cube of 2 is 8. (Cont... |
| irec 13846 | The reciprocal of ` _i ` .... |
| i2 13847 | ` _i ` squared. (Contribu... |
| i3 13848 | ` _i ` cubed. (Contribute... |
| i4 13849 | ` _i ` to the fourth power... |
| nnlesq 13850 | A positive integer is less... |
| iexpcyc 13851 | Taking ` _i ` to the ` K `... |
| expnass 13852 | A counterexample showing t... |
| sqlecan 13853 | Cancel one factor of a squ... |
| subsq 13854 | Factor the difference of t... |
| subsq2 13855 | Express the difference of ... |
| binom2i 13856 | The square of a binomial. ... |
| subsqi 13857 | Factor the difference of t... |
| sqeqori 13858 | The squares of two complex... |
| subsq0i 13859 | The two solutions to the d... |
| sqeqor 13860 | The squares of two complex... |
| binom2 13861 | The square of a binomial. ... |
| binom21 13862 | Special case of ~ binom2 w... |
| binom2sub 13863 | Expand the square of a sub... |
| binom2sub1 13864 | Special case of ~ binom2su... |
| binom2subi 13865 | Expand the square of a sub... |
| mulbinom2 13866 | The square of a binomial w... |
| binom3 13867 | The cube of a binomial. (... |
| sq01 13868 | If a complex number equals... |
| zesq 13869 | An integer is even iff its... |
| nnesq 13870 | A positive integer is even... |
| crreczi 13871 | Reciprocal of a complex nu... |
| bernneq 13872 | Bernoulli's inequality, du... |
| bernneq2 13873 | Variation of Bernoulli's i... |
| bernneq3 13874 | A corollary of ~ bernneq .... |
| expnbnd 13875 | Exponentiation with a base... |
| expnlbnd 13876 | The reciprocal of exponent... |
| expnlbnd2 13877 | The reciprocal of exponent... |
| expmulnbnd 13878 | Exponentiation with a base... |
| digit2 13879 | Two ways to express the ` ... |
| digit1 13880 | Two ways to express the ` ... |
| modexp 13881 | Exponentiation property of... |
| discr1 13882 | A nonnegative quadratic fo... |
| discr 13883 | If a quadratic polynomial ... |
| expnngt1 13884 | If an integer power with a... |
| expnngt1b 13885 | An integer power with an i... |
| sqoddm1div8 13886 | A squared odd number minus... |
| nnsqcld 13887 | The naturals are closed un... |
| nnexpcld 13888 | Closure of exponentiation ... |
| nn0expcld 13889 | Closure of exponentiation ... |
| rpexpcld 13890 | Closure law for exponentia... |
| ltexp2rd 13891 | The power of a positive nu... |
| reexpclzd 13892 | Closure of exponentiation ... |
| resqcld 13893 | Closure of square in reals... |
| sqge0d 13894 | A square of a real is nonn... |
| sqgt0d 13895 | The square of a nonzero re... |
| ltexp2d 13896 | Ordering relationship for ... |
| leexp2d 13897 | Ordering law for exponenti... |
| expcand 13898 | Ordering relationship for ... |
| leexp2ad 13899 | Ordering relationship for ... |
| leexp2rd 13900 | Ordering relationship for ... |
| lt2sqd 13901 | The square function on non... |
| le2sqd 13902 | The square function on non... |
| sq11d 13903 | The square function is one... |
| mulsubdivbinom2 13904 | The square of a binomial w... |
| muldivbinom2 13905 | The square of a binomial w... |
| sq10 13906 | The square of 10 is 100. ... |
| sq10e99m1 13907 | The square of 10 is 99 plu... |
| 3dec 13908 | A "decimal constructor" wh... |
| nn0le2msqi 13909 | The square function on non... |
| nn0opthlem1 13910 | A rather pretty lemma for ... |
| nn0opthlem2 13911 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 13912 | An ordered pair theorem fo... |
| nn0opth2i 13913 | An ordered pair theorem fo... |
| nn0opth2 13914 | An ordered pair theorem fo... |
| facnn 13917 | Value of the factorial fun... |
| fac0 13918 | The factorial of 0. (Cont... |
| fac1 13919 | The factorial of 1. (Cont... |
| facp1 13920 | The factorial of a success... |
| fac2 13921 | The factorial of 2. (Cont... |
| fac3 13922 | The factorial of 3. (Cont... |
| fac4 13923 | The factorial of 4. (Cont... |
| facnn2 13924 | Value of the factorial fun... |
| faccl 13925 | Closure of the factorial f... |
| faccld 13926 | Closure of the factorial f... |
| facmapnn 13927 | The factorial function res... |
| facne0 13928 | The factorial function is ... |
| facdiv 13929 | A positive integer divides... |
| facndiv 13930 | No positive integer (great... |
| facwordi 13931 | Ordering property of facto... |
| faclbnd 13932 | A lower bound for the fact... |
| faclbnd2 13933 | A lower bound for the fact... |
| faclbnd3 13934 | A lower bound for the fact... |
| faclbnd4lem1 13935 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 13936 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 13937 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 13938 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 13939 | Variant of ~ faclbnd5 prov... |
| faclbnd5 13940 | The factorial function gro... |
| faclbnd6 13941 | Geometric lower bound for ... |
| facubnd 13942 | An upper bound for the fac... |
| facavg 13943 | The product of two factori... |
| bcval 13946 | Value of the binomial coef... |
| bcval2 13947 | Value of the binomial coef... |
| bcval3 13948 | Value of the binomial coef... |
| bcval4 13949 | Value of the binomial coef... |
| bcrpcl 13950 | Closure of the binomial co... |
| bccmpl 13951 | "Complementing" its second... |
| bcn0 13952 | ` N ` choose 0 is 1. Rema... |
| bc0k 13953 | The binomial coefficient "... |
| bcnn 13954 | ` N ` choose ` N ` is 1. ... |
| bcn1 13955 | Binomial coefficient: ` N ... |
| bcnp1n 13956 | Binomial coefficient: ` N ... |
| bcm1k 13957 | The proportion of one bino... |
| bcp1n 13958 | The proportion of one bino... |
| bcp1nk 13959 | The proportion of one bino... |
| bcval5 13960 | Write out the top and bott... |
| bcn2 13961 | Binomial coefficient: ` N ... |
| bcp1m1 13962 | Compute the binomial coeff... |
| bcpasc 13963 | Pascal's rule for the bino... |
| bccl 13964 | A binomial coefficient, in... |
| bccl2 13965 | A binomial coefficient, in... |
| bcn2m1 13966 | Compute the binomial coeff... |
| bcn2p1 13967 | Compute the binomial coeff... |
| permnn 13968 | The number of permutations... |
| bcnm1 13969 | The binomial coefficent of... |
| 4bc3eq4 13970 | The value of four choose t... |
| 4bc2eq6 13971 | The value of four choose t... |
| hashkf 13974 | The finite part of the siz... |
| hashgval 13975 | The value of the ` # ` fun... |
| hashginv 13976 | The converse of ` G ` maps... |
| hashinf 13977 | The value of the ` # ` fun... |
| hashbnd 13978 | If ` A ` has size bounded ... |
| hashfxnn0 13979 | The size function is a fun... |
| hashf 13980 | The size function maps all... |
| hashxnn0 13981 | The value of the hash func... |
| hashresfn 13982 | Restriction of the domain ... |
| dmhashres 13983 | Restriction of the domain ... |
| hashnn0pnf 13984 | The value of the hash func... |
| hashnnn0genn0 13985 | If the size of a set is no... |
| hashnemnf 13986 | The size of a set is never... |
| hashv01gt1 13987 | The size of a set is eithe... |
| hashfz1 13988 | The set ` ( 1 ... N ) ` ha... |
| hashen 13989 | Two finite sets have the s... |
| hasheni 13990 | Equinumerous sets have the... |
| hasheqf1o 13991 | The size of two finite set... |
| fiinfnf1o 13992 | There is no bijection betw... |
| focdmex 13993 | The codomain of an onto fu... |
| hasheqf1oi 13994 | The size of two sets is eq... |
| hashf1rn 13995 | The size of a finite set w... |
| hasheqf1od 13996 | The size of two sets is eq... |
| fz1eqb 13997 | Two possibly-empty 1-based... |
| hashcard 13998 | The size function of the c... |
| hashcl 13999 | Closure of the ` # ` funct... |
| hashxrcl 14000 | Extended real closure of t... |
| hashclb 14001 | Reverse closure of the ` #... |
| nfile 14002 | The size of any infinite s... |
| hashvnfin 14003 | A set of finite size is a ... |
| hashnfinnn0 14004 | The size of an infinite se... |
| isfinite4 14005 | A finite set is equinumero... |
| hasheq0 14006 | Two ways of saying a finit... |
| hashneq0 14007 | Two ways of saying a set i... |
| hashgt0n0 14008 | If the size of a set is gr... |
| hashnncl 14009 | Positive natural closure o... |
| hash0 14010 | The empty set has size zer... |
| hashelne0d 14011 | A set with an element has ... |
| hashsng 14012 | The size of a singleton. ... |
| hashen1 14013 | A set has size 1 if and on... |
| hash1elsn 14014 | A set of size 1 with a kno... |
| hashrabrsn 14015 | The size of a restricted c... |
| hashrabsn01 14016 | The size of a restricted c... |
| hashrabsn1 14017 | If the size of a restricte... |
| hashfn 14018 | A function is equinumerous... |
| fseq1hash 14019 | The value of the size func... |
| hashgadd 14020 | ` G ` maps ordinal additio... |
| hashgval2 14021 | A short expression for the... |
| hashdom 14022 | Dominance relation for the... |
| hashdomi 14023 | Non-strict order relation ... |
| hashsdom 14024 | Strict dominance relation ... |
| hashun 14025 | The size of the union of d... |
| hashun2 14026 | The size of the union of f... |
| hashun3 14027 | The size of the union of f... |
| hashinfxadd 14028 | The extended real addition... |
| hashunx 14029 | The size of the union of d... |
| hashge0 14030 | The cardinality of a set i... |
| hashgt0 14031 | The cardinality of a nonem... |
| hashge1 14032 | The cardinality of a nonem... |
| 1elfz0hash 14033 | 1 is an element of the fin... |
| hashnn0n0nn 14034 | If a nonnegative integer i... |
| hashunsng 14035 | The size of the union of a... |
| hashunsngx 14036 | The size of the union of a... |
| hashunsnggt 14037 | The size of a set is great... |
| hashprg 14038 | The size of an unordered p... |
| elprchashprn2 14039 | If one element of an unord... |
| hashprb 14040 | The size of an unordered p... |
| hashprdifel 14041 | The elements of an unorder... |
| prhash2ex 14042 | There is (at least) one se... |
| hashle00 14043 | If the size of a set is le... |
| hashgt0elex 14044 | If the size of a set is gr... |
| hashgt0elexb 14045 | The size of a set is great... |
| hashp1i 14046 | Size of a finite ordinal. ... |
| hash1 14047 | Size of a finite ordinal. ... |
| hash2 14048 | Size of a finite ordinal. ... |
| hash3 14049 | Size of a finite ordinal. ... |
| hash4 14050 | Size of a finite ordinal. ... |
| pr0hash2ex 14051 | There is (at least) one se... |
| hashss 14052 | The size of a subset is le... |
| prsshashgt1 14053 | The size of a superset of ... |
| hashin 14054 | The size of the intersecti... |
| hashssdif 14055 | The size of the difference... |
| hashdif 14056 | The size of the difference... |
| hashdifsn 14057 | The size of the difference... |
| hashdifpr 14058 | The size of the difference... |
| hashsn01 14059 | The size of a singleton is... |
| hashsnle1 14060 | The size of a singleton is... |
| hashsnlei 14061 | Get an upper bound on a co... |
| hash1snb 14062 | The size of a set is 1 if ... |
| euhash1 14063 | The size of a set is 1 in ... |
| hash1n0 14064 | If the size of a set is 1 ... |
| hashgt12el 14065 | In a set with more than on... |
| hashgt12el2 14066 | In a set with more than on... |
| hashgt23el 14067 | A set with more than two e... |
| hashunlei 14068 | Get an upper bound on a co... |
| hashsslei 14069 | Get an upper bound on a co... |
| hashfz 14070 | Value of the numeric cardi... |
| fzsdom2 14071 | Condition for finite range... |
| hashfzo 14072 | Cardinality of a half-open... |
| hashfzo0 14073 | Cardinality of a half-open... |
| hashfzp1 14074 | Value of the numeric cardi... |
| hashfz0 14075 | Value of the numeric cardi... |
| hashxplem 14076 | Lemma for ~ hashxp . (Con... |
| hashxp 14077 | The size of the Cartesian ... |
| hashmap 14078 | The size of the set expone... |
| hashpw 14079 | The size of the power set ... |
| hashfun 14080 | A finite set is a function... |
| hashres 14081 | The number of elements of ... |
| hashreshashfun 14082 | The number of elements of ... |
| hashimarn 14083 | The size of the image of a... |
| hashimarni 14084 | If the size of the image o... |
| resunimafz0 14085 | TODO-AV: Revise using ` F... |
| fnfz0hash 14086 | The size of a function on ... |
| ffz0hash 14087 | The size of a function on ... |
| fnfz0hashnn0 14088 | The size of a function on ... |
| ffzo0hash 14089 | The size of a function on ... |
| fnfzo0hash 14090 | The size of a function on ... |
| fnfzo0hashnn0 14091 | The value of the size func... |
| hashbclem 14092 | Lemma for ~ hashbc : induc... |
| hashbc 14093 | The binomial coefficient c... |
| hashfacen 14094 | The number of bijections b... |
| hashfacenOLD 14095 | Obsolete version of ~ hash... |
| hashf1lem1 14096 | Lemma for ~ hashf1 . (Con... |
| hashf1lem1OLD 14097 | Obsolete version of ~ hash... |
| hashf1lem2 14098 | Lemma for ~ hashf1 . (Con... |
| hashf1 14099 | The permutation number ` |... |
| hashfac 14100 | A factorial counts the num... |
| leiso 14101 | Two ways to write a strict... |
| leisorel 14102 | Version of ~ isorel for st... |
| fz1isolem 14103 | Lemma for ~ fz1iso . (Con... |
| fz1iso 14104 | Any finite ordered set has... |
| ishashinf 14105 | Any set that is not finite... |
| seqcoll 14106 | The function ` F ` contain... |
| seqcoll2 14107 | The function ` F ` contain... |
| phphashd 14108 | Corollary of the Pigeonhol... |
| phphashrd 14109 | Corollary of the Pigeonhol... |
| hashprlei 14110 | An unordered pair has at m... |
| hash2pr 14111 | A set of size two is an un... |
| hash2prde 14112 | A set of size two is an un... |
| hash2exprb 14113 | A set of size two is an un... |
| hash2prb 14114 | A set of size two is a pro... |
| prprrab 14115 | The set of proper pairs of... |
| nehash2 14116 | The cardinality of a set w... |
| hash2prd 14117 | A set of size two is an un... |
| hash2pwpr 14118 | If the size of a subset of... |
| hashle2pr 14119 | A nonempty set of size les... |
| hashle2prv 14120 | A nonempty subset of a pow... |
| pr2pwpr 14121 | The set of subsets of a pa... |
| hashge2el2dif 14122 | A set with size at least 2... |
| hashge2el2difr 14123 | A set with at least 2 diff... |
| hashge2el2difb 14124 | A set has size at least 2 ... |
| hashdmpropge2 14125 | The size of the domain of ... |
| hashtplei 14126 | An unordered triple has at... |
| hashtpg 14127 | The size of an unordered t... |
| hashge3el3dif 14128 | A set with size at least 3... |
| elss2prb 14129 | An element of the set of s... |
| hash2sspr 14130 | A subset of size two is an... |
| exprelprel 14131 | If there is an element of ... |
| hash3tr 14132 | A set of size three is an ... |
| hash1to3 14133 | If the size of a set is be... |
| fundmge2nop0 14134 | A function with a domain c... |
| fundmge2nop 14135 | A function with a domain c... |
| fun2dmnop0 14136 | A function with a domain c... |
| fun2dmnop 14137 | A function with a domain c... |
| hashdifsnp1 14138 | If the size of a set is a ... |
| fi1uzind 14139 | Properties of an ordered p... |
| brfi1uzind 14140 | Properties of a binary rel... |
| brfi1ind 14141 | Properties of a binary rel... |
| brfi1indALT 14142 | Alternate proof of ~ brfi1... |
| opfi1uzind 14143 | Properties of an ordered p... |
| opfi1ind 14144 | Properties of an ordered p... |
| iswrd 14147 | Property of being a word o... |
| wrdval 14148 | Value of the set of words ... |
| iswrdi 14149 | A zero-based sequence is a... |
| wrdf 14150 | A word is a zero-based seq... |
| iswrdb 14151 | A word over an alphabet is... |
| wrddm 14152 | The indices of a word (i.e... |
| sswrd 14153 | The set of words respects ... |
| snopiswrd 14154 | A singleton of an ordered ... |
| wrdexg 14155 | The set of words over a se... |
| wrdexb 14156 | The set of words over a se... |
| wrdexi 14157 | The set of words over a se... |
| wrdsymbcl 14158 | A symbol within a word ove... |
| wrdfn 14159 | A word is a function with ... |
| wrdv 14160 | A word over an alphabet is... |
| wrdlndm 14161 | The length of a word is no... |
| iswrdsymb 14162 | An arbitrary word is a wor... |
| wrdfin 14163 | A word is a finite set. (... |
| lencl 14164 | The length of a word is a ... |
| lennncl 14165 | The length of a nonempty w... |
| wrdffz 14166 | A word is a function from ... |
| wrdeq 14167 | Equality theorem for the s... |
| wrdeqi 14168 | Equality theorem for the s... |
| iswrddm0 14169 | A function with empty doma... |
| wrd0 14170 | The empty set is a word (t... |
| 0wrd0 14171 | The empty word is the only... |
| ffz0iswrd 14172 | A sequence with zero-based... |
| wrdsymb 14173 | A word is a word over the ... |
| nfwrd 14174 | Hypothesis builder for ` W... |
| csbwrdg 14175 | Class substitution for the... |
| wrdnval 14176 | Words of a fixed length ar... |
| wrdmap 14177 | Words as a mapping. (Cont... |
| hashwrdn 14178 | If there is only a finite ... |
| wrdnfi 14179 | If there is only a finite ... |
| wrdsymb0 14180 | A symbol at a position "ou... |
| wrdlenge1n0 14181 | A word with length at leas... |
| len0nnbi 14182 | The length of a word is a ... |
| wrdlenge2n0 14183 | A word with length at leas... |
| wrdsymb1 14184 | The first symbol of a none... |
| wrdlen1 14185 | A word of length 1 starts ... |
| fstwrdne 14186 | The first symbol of a none... |
| fstwrdne0 14187 | The first symbol of a none... |
| eqwrd 14188 | Two words are equal iff th... |
| elovmpowrd 14189 | Implications for the value... |
| elovmptnn0wrd 14190 | Implications for the value... |
| wrdred1 14191 | A word truncated by a symb... |
| wrdred1hash 14192 | The length of a word trunc... |
| lsw 14195 | Extract the last symbol of... |
| lsw0 14196 | The last symbol of an empt... |
| lsw0g 14197 | The last symbol of an empt... |
| lsw1 14198 | The last symbol of a word ... |
| lswcl 14199 | Closure of the last symbol... |
| lswlgt0cl 14200 | The last symbol of a nonem... |
| ccatfn 14203 | The concatenation operator... |
| ccatfval 14204 | Value of the concatenation... |
| ccatcl 14205 | The concatenation of two w... |
| ccatlen 14206 | The length of a concatenat... |
| ccatlenOLD 14207 | Obsolete version of ~ ccat... |
| ccat0 14208 | The concatenation of two w... |
| ccatval1 14209 | Value of a symbol in the l... |
| ccatval1OLD 14210 | Obsolete version of ~ ccat... |
| ccatval2 14211 | Value of a symbol in the r... |
| ccatval3 14212 | Value of a symbol in the r... |
| elfzelfzccat 14213 | An element of a finite set... |
| ccatvalfn 14214 | The concatenation of two w... |
| ccatsymb 14215 | The symbol at a given posi... |
| ccatfv0 14216 | The first symbol of a conc... |
| ccatval1lsw 14217 | The last symbol of the lef... |
| ccatval21sw 14218 | The first symbol of the ri... |
| ccatlid 14219 | Concatenation of a word by... |
| ccatrid 14220 | Concatenation of a word by... |
| ccatass 14221 | Associative law for concat... |
| ccatrn 14222 | The range of a concatenate... |
| ccatidid 14223 | Concatenation of the empty... |
| lswccatn0lsw 14224 | The last symbol of a word ... |
| lswccat0lsw 14225 | The last symbol of a word ... |
| ccatalpha 14226 | A concatenation of two arb... |
| ccatrcl1 14227 | Reverse closure of a conca... |
| ids1 14230 | Identity function protecti... |
| s1val 14231 | Value of a singleton word.... |
| s1rn 14232 | The range of a singleton w... |
| s1eq 14233 | Equality theorem for a sin... |
| s1eqd 14234 | Equality theorem for a sin... |
| s1cl 14235 | A singleton word is a word... |
| s1cld 14236 | A singleton word is a word... |
| s1prc 14237 | Value of a singleton word ... |
| s1cli 14238 | A singleton word is a word... |
| s1len 14239 | Length of a singleton word... |
| s1nz 14240 | A singleton word is not th... |
| s1dm 14241 | The domain of a singleton ... |
| s1dmALT 14242 | Alternate version of ~ s1d... |
| s1fv 14243 | Sole symbol of a singleton... |
| lsws1 14244 | The last symbol of a singl... |
| eqs1 14245 | A word of length 1 is a si... |
| wrdl1exs1 14246 | A word of length 1 is a si... |
| wrdl1s1 14247 | A word of length 1 is a si... |
| s111 14248 | The singleton word functio... |
| ccatws1cl 14249 | The concatenation of a wor... |
| ccatws1clv 14250 | The concatenation of a wor... |
| ccat2s1cl 14251 | The concatenation of two s... |
| ccats1alpha 14252 | A concatenation of a word ... |
| ccatws1len 14253 | The length of the concaten... |
| ccatws1lenp1b 14254 | The length of a word is ` ... |
| wrdlenccats1lenm1 14255 | The length of a word is th... |
| ccat2s1len 14256 | The length of the concaten... |
| ccat2s1lenOLD 14257 | Obsolete version of ~ ccat... |
| ccatw2s1cl 14258 | The concatenation of a wor... |
| ccatw2s1len 14259 | The length of the concaten... |
| ccats1val1 14260 | Value of a symbol in the l... |
| ccats1val1OLD 14261 | Obsolete version of ~ ccat... |
| ccats1val2 14262 | Value of the symbol concat... |
| ccat1st1st 14263 | The first symbol of a word... |
| ccat2s1p1 14264 | Extract the first of two c... |
| ccat2s1p2 14265 | Extract the second of two ... |
| ccat2s1p1OLD 14266 | Obsolete version of ~ ccat... |
| ccat2s1p2OLD 14267 | Obsolete version of ~ ccat... |
| ccatw2s1ass 14268 | Associative law for a conc... |
| ccatw2s1assOLD 14269 | Obsolete version of ~ ccat... |
| ccatws1n0 14270 | The concatenation of a wor... |
| ccatws1ls 14271 | The last symbol of the con... |
| lswccats1 14272 | The last symbol of a word ... |
| lswccats1fst 14273 | The last symbol of a nonem... |
| ccatw2s1p1 14274 | Extract the symbol of the ... |
| ccatw2s1p1OLD 14275 | Obsolete version of ~ ccat... |
| ccatw2s1p2 14276 | Extract the second of two ... |
| ccat2s1fvw 14277 | Extract a symbol of a word... |
| ccat2s1fvwOLD 14278 | Obsolete version of ~ ccat... |
| ccat2s1fst 14279 | The first symbol of the co... |
| ccat2s1fstOLD 14280 | Obsolete version of ~ ccat... |
| swrdnznd 14283 | The value of a subword ope... |
| swrdval 14284 | Value of a subword. (Cont... |
| swrd00 14285 | A zero length substring. ... |
| swrdcl 14286 | Closure of the subword ext... |
| swrdval2 14287 | Value of the subword extra... |
| swrdlen 14288 | Length of an extracted sub... |
| swrdfv 14289 | A symbol in an extracted s... |
| swrdfv0 14290 | The first symbol in an ext... |
| swrdf 14291 | A subword of a word is a f... |
| swrdvalfn 14292 | Value of the subword extra... |
| swrdrn 14293 | The range of a subword of ... |
| swrdlend 14294 | The value of the subword e... |
| swrdnd 14295 | The value of the subword e... |
| swrdnd2 14296 | Value of the subword extra... |
| swrdnnn0nd 14297 | The value of a subword ope... |
| swrdnd0 14298 | The value of a subword ope... |
| swrd0 14299 | A subword of an empty set ... |
| swrdrlen 14300 | Length of a right-anchored... |
| swrdlen2 14301 | Length of an extracted sub... |
| swrdfv2 14302 | A symbol in an extracted s... |
| swrdwrdsymb 14303 | A subword is a word over t... |
| swrdsb0eq 14304 | Two subwords with the same... |
| swrdsbslen 14305 | Two subwords with the same... |
| swrdspsleq 14306 | Two words have a common su... |
| swrds1 14307 | Extract a single symbol fr... |
| swrdlsw 14308 | Extract the last single sy... |
| ccatswrd 14309 | Joining two adjacent subwo... |
| swrdccat2 14310 | Recover the right half of ... |
| pfxnndmnd 14313 | The value of a prefix oper... |
| pfxval 14314 | Value of a prefix operatio... |
| pfx00 14315 | The zero length prefix is ... |
| pfx0 14316 | A prefix of an empty set i... |
| pfxval0 14317 | Value of a prefix operatio... |
| pfxcl 14318 | Closure of the prefix extr... |
| pfxmpt 14319 | Value of the prefix extrac... |
| pfxres 14320 | Value of the subword extra... |
| pfxf 14321 | A prefix of a word is a fu... |
| pfxfn 14322 | Value of the prefix extrac... |
| pfxfv 14323 | A symbol in a prefix of a ... |
| pfxlen 14324 | Length of a prefix. (Cont... |
| pfxid 14325 | A word is a prefix of itse... |
| pfxrn 14326 | The range of a prefix of a... |
| pfxn0 14327 | A prefix consisting of at ... |
| pfxnd 14328 | The value of a prefix oper... |
| pfxnd0 14329 | The value of a prefix oper... |
| pfxwrdsymb 14330 | A prefix of a word is a wo... |
| addlenrevpfx 14331 | The sum of the lengths of ... |
| addlenpfx 14332 | The sum of the lengths of ... |
| pfxfv0 14333 | The first symbol of a pref... |
| pfxtrcfv 14334 | A symbol in a word truncat... |
| pfxtrcfv0 14335 | The first symbol in a word... |
| pfxfvlsw 14336 | The last symbol in a nonem... |
| pfxeq 14337 | The prefixes of two words ... |
| pfxtrcfvl 14338 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14339 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14340 | Two (nonempty) words are e... |
| disjwrdpfx 14341 | Sets of words are disjoint... |
| ccatpfx 14342 | Concatenating a prefix wit... |
| pfxccat1 14343 | Recover the left half of a... |
| pfx1 14344 | The prefix of length one o... |
| swrdswrdlem 14345 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14346 | A subword of a subword is ... |
| pfxswrd 14347 | A prefix of a subword is a... |
| swrdpfx 14348 | A subword of a prefix is a... |
| pfxpfx 14349 | A prefix of a prefix is a ... |
| pfxpfxid 14350 | A prefix of a prefix with ... |
| pfxcctswrd 14351 | The concatenation of the p... |
| lenpfxcctswrd 14352 | The length of the concaten... |
| lenrevpfxcctswrd 14353 | The length of the concaten... |
| pfxlswccat 14354 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14355 | The last symbol of a word ... |
| ccats1pfxeqrex 14356 | There exists a symbol such... |
| ccatopth 14357 | An ~ opth -like theorem fo... |
| ccatopth2 14358 | An ~ opth -like theorem fo... |
| ccatlcan 14359 | Concatenation of words is ... |
| ccatrcan 14360 | Concatenation of words is ... |
| wrdeqs1cat 14361 | Decompose a nonempty word ... |
| cats1un 14362 | Express a word with an ext... |
| wrdind 14363 | Perform induction over the... |
| wrd2ind 14364 | Perform induction over the... |
| swrdccatfn 14365 | The subword of a concatena... |
| swrdccatin1 14366 | The subword of a concatena... |
| pfxccatin12lem4 14367 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14368 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14369 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14370 | The subword of a concatena... |
| pfxccatin12lem2c 14371 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14372 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14373 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14374 | The subword of a concatena... |
| pfxccat3 14375 | The subword of a concatena... |
| swrdccat 14376 | The subword of a concatena... |
| pfxccatpfx1 14377 | A prefix of a concatenatio... |
| pfxccatpfx2 14378 | A prefix of a concatenatio... |
| pfxccat3a 14379 | A prefix of a concatenatio... |
| swrdccat3blem 14380 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14381 | A suffix of a concatenatio... |
| pfxccatid 14382 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14383 | A word is a prefix of a wo... |
| swrdccatin1d 14384 | The subword of a concatena... |
| swrdccatin2d 14385 | The subword of a concatena... |
| pfxccatin12d 14386 | The subword of a concatena... |
| reuccatpfxs1lem 14387 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14388 | There is a unique word hav... |
| reuccatpfxs1v 14389 | There is a unique word hav... |
| splval 14392 | Value of the substring rep... |
| splcl 14393 | Closure of the substring r... |
| splid 14394 | Splicing a subword for the... |
| spllen 14395 | The length of a splice. (... |
| splfv1 14396 | Symbols to the left of a s... |
| splfv2a 14397 | Symbols within the replace... |
| splval2 14398 | Value of a splice, assumin... |
| revval 14401 | Value of the word reversin... |
| revcl 14402 | The reverse of a word is a... |
| revlen 14403 | The reverse of a word has ... |
| revfv 14404 | Reverse of a word at a poi... |
| rev0 14405 | The empty word is its own ... |
| revs1 14406 | Singleton words are their ... |
| revccat 14407 | Antiautomorphic property o... |
| revrev 14408 | Reversal is an involution ... |
| reps 14411 | Construct a function mappi... |
| repsundef 14412 | A function mapping a half-... |
| repsconst 14413 | Construct a function mappi... |
| repsf 14414 | The constructed function m... |
| repswsymb 14415 | The symbols of a "repeated... |
| repsw 14416 | A function mapping a half-... |
| repswlen 14417 | The length of a "repeated ... |
| repsw0 14418 | The "repeated symbol word"... |
| repsdf2 14419 | Alternative definition of ... |
| repswsymball 14420 | All the symbols of a "repe... |
| repswsymballbi 14421 | A word is a "repeated symb... |
| repswfsts 14422 | The first symbol of a none... |
| repswlsw 14423 | The last symbol of a nonem... |
| repsw1 14424 | The "repeated symbol word"... |
| repswswrd 14425 | A subword of a "repeated s... |
| repswpfx 14426 | A prefix of a repeated sym... |
| repswccat 14427 | The concatenation of two "... |
| repswrevw 14428 | The reverse of a "repeated... |
| cshfn 14431 | Perform a cyclical shift f... |
| cshword 14432 | Perform a cyclical shift f... |
| cshnz 14433 | A cyclical shift is the em... |
| 0csh0 14434 | Cyclically shifting an emp... |
| cshw0 14435 | A word cyclically shifted ... |
| cshwmodn 14436 | Cyclically shifting a word... |
| cshwsublen 14437 | Cyclically shifting a word... |
| cshwn 14438 | A word cyclically shifted ... |
| cshwcl 14439 | A cyclically shifted word ... |
| cshwlen 14440 | The length of a cyclically... |
| cshwf 14441 | A cyclically shifted word ... |
| cshwfn 14442 | A cyclically shifted word ... |
| cshwrn 14443 | The range of a cyclically ... |
| cshwidxmod 14444 | The symbol at a given inde... |
| cshwidxmodr 14445 | The symbol at a given inde... |
| cshwidx0mod 14446 | The symbol at index 0 of a... |
| cshwidx0 14447 | The symbol at index 0 of a... |
| cshwidxm1 14448 | The symbol at index ((n-N)... |
| cshwidxm 14449 | The symbol at index (n-N) ... |
| cshwidxn 14450 | The symbol at index (n-1) ... |
| cshf1 14451 | Cyclically shifting a word... |
| cshinj 14452 | If a word is injectiv (reg... |
| repswcshw 14453 | A cyclically shifted "repe... |
| 2cshw 14454 | Cyclically shifting a word... |
| 2cshwid 14455 | Cyclically shifting a word... |
| lswcshw 14456 | The last symbol of a word ... |
| 2cshwcom 14457 | Cyclically shifting a word... |
| cshwleneq 14458 | If the results of cyclical... |
| 3cshw 14459 | Cyclically shifting a word... |
| cshweqdif2 14460 | If cyclically shifting two... |
| cshweqdifid 14461 | If cyclically shifting a w... |
| cshweqrep 14462 | If cyclically shifting a w... |
| cshw1 14463 | If cyclically shifting a w... |
| cshw1repsw 14464 | If cyclically shifting a w... |
| cshwsexa 14465 | The class of (different!) ... |
| 2cshwcshw 14466 | If a word is a cyclically ... |
| scshwfzeqfzo 14467 | For a nonempty word the se... |
| cshwcshid 14468 | A cyclically shifted word ... |
| cshwcsh2id 14469 | A cyclically shifted word ... |
| cshimadifsn 14470 | The image of a cyclically ... |
| cshimadifsn0 14471 | The image of a cyclically ... |
| wrdco 14472 | Mapping a word by a functi... |
| lenco 14473 | Length of a mapped word is... |
| s1co 14474 | Mapping of a singleton wor... |
| revco 14475 | Mapping of words (i.e., a ... |
| ccatco 14476 | Mapping of words commutes ... |
| cshco 14477 | Mapping of words commutes ... |
| swrdco 14478 | Mapping of words commutes ... |
| pfxco 14479 | Mapping of words commutes ... |
| lswco 14480 | Mapping of (nonempty) word... |
| repsco 14481 | Mapping of words commutes ... |
| cats1cld 14496 | Closure of concatenation w... |
| cats1co 14497 | Closure of concatenation w... |
| cats1cli 14498 | Closure of concatenation w... |
| cats1fvn 14499 | The last symbol of a conca... |
| cats1fv 14500 | A symbol other than the la... |
| cats1len 14501 | The length of concatenatio... |
| cats1cat 14502 | Closure of concatenation w... |
| cats2cat 14503 | Closure of concatenation o... |
| s2eqd 14504 | Equality theorem for a dou... |
| s3eqd 14505 | Equality theorem for a len... |
| s4eqd 14506 | Equality theorem for a len... |
| s5eqd 14507 | Equality theorem for a len... |
| s6eqd 14508 | Equality theorem for a len... |
| s7eqd 14509 | Equality theorem for a len... |
| s8eqd 14510 | Equality theorem for a len... |
| s3eq2 14511 | Equality theorem for a len... |
| s2cld 14512 | A doubleton word is a word... |
| s3cld 14513 | A length 3 string is a wor... |
| s4cld 14514 | A length 4 string is a wor... |
| s5cld 14515 | A length 5 string is a wor... |
| s6cld 14516 | A length 6 string is a wor... |
| s7cld 14517 | A length 7 string is a wor... |
| s8cld 14518 | A length 7 string is a wor... |
| s2cl 14519 | A doubleton word is a word... |
| s3cl 14520 | A length 3 string is a wor... |
| s2cli 14521 | A doubleton word is a word... |
| s3cli 14522 | A length 3 string is a wor... |
| s4cli 14523 | A length 4 string is a wor... |
| s5cli 14524 | A length 5 string is a wor... |
| s6cli 14525 | A length 6 string is a wor... |
| s7cli 14526 | A length 7 string is a wor... |
| s8cli 14527 | A length 8 string is a wor... |
| s2fv0 14528 | Extract the first symbol f... |
| s2fv1 14529 | Extract the second symbol ... |
| s2len 14530 | The length of a doubleton ... |
| s2dm 14531 | The domain of a doubleton ... |
| s3fv0 14532 | Extract the first symbol f... |
| s3fv1 14533 | Extract the second symbol ... |
| s3fv2 14534 | Extract the third symbol f... |
| s3len 14535 | The length of a length 3 s... |
| s4fv0 14536 | Extract the first symbol f... |
| s4fv1 14537 | Extract the second symbol ... |
| s4fv2 14538 | Extract the third symbol f... |
| s4fv3 14539 | Extract the fourth symbol ... |
| s4len 14540 | The length of a length 4 s... |
| s5len 14541 | The length of a length 5 s... |
| s6len 14542 | The length of a length 6 s... |
| s7len 14543 | The length of a length 7 s... |
| s8len 14544 | The length of a length 8 s... |
| lsws2 14545 | The last symbol of a doubl... |
| lsws3 14546 | The last symbol of a 3 let... |
| lsws4 14547 | The last symbol of a 4 let... |
| s2prop 14548 | A length 2 word is an unor... |
| s2dmALT 14549 | Alternate version of ~ s2d... |
| s3tpop 14550 | A length 3 word is an unor... |
| s4prop 14551 | A length 4 word is a union... |
| s3fn 14552 | A length 3 word is a funct... |
| funcnvs1 14553 | The converse of a singleto... |
| funcnvs2 14554 | The converse of a length 2... |
| funcnvs3 14555 | The converse of a length 3... |
| funcnvs4 14556 | The converse of a length 4... |
| s2f1o 14557 | A length 2 word with mutua... |
| f1oun2prg 14558 | A union of unordered pairs... |
| s4f1o 14559 | A length 4 word with mutua... |
| s4dom 14560 | The domain of a length 4 w... |
| s2co 14561 | Mapping a doubleton word b... |
| s3co 14562 | Mapping a length 3 string ... |
| s0s1 14563 | Concatenation of fixed len... |
| s1s2 14564 | Concatenation of fixed len... |
| s1s3 14565 | Concatenation of fixed len... |
| s1s4 14566 | Concatenation of fixed len... |
| s1s5 14567 | Concatenation of fixed len... |
| s1s6 14568 | Concatenation of fixed len... |
| s1s7 14569 | Concatenation of fixed len... |
| s2s2 14570 | Concatenation of fixed len... |
| s4s2 14571 | Concatenation of fixed len... |
| s4s3 14572 | Concatenation of fixed len... |
| s4s4 14573 | Concatenation of fixed len... |
| s3s4 14574 | Concatenation of fixed len... |
| s2s5 14575 | Concatenation of fixed len... |
| s5s2 14576 | Concatenation of fixed len... |
| s2eq2s1eq 14577 | Two length 2 words are equ... |
| s2eq2seq 14578 | Two length 2 words are equ... |
| s3eqs2s1eq 14579 | Two length 3 words are equ... |
| s3eq3seq 14580 | Two length 3 words are equ... |
| swrds2 14581 | Extract two adjacent symbo... |
| swrds2m 14582 | Extract two adjacent symbo... |
| wrdlen2i 14583 | Implications of a word of ... |
| wrd2pr2op 14584 | A word of length two repre... |
| wrdlen2 14585 | A word of length two. (Co... |
| wrdlen2s2 14586 | A word of length two as do... |
| wrdl2exs2 14587 | A word of length two is a ... |
| pfx2 14588 | A prefix of length two. (... |
| wrd3tpop 14589 | A word of length three rep... |
| wrdlen3s3 14590 | A word of length three as ... |
| repsw2 14591 | The "repeated symbol word"... |
| repsw3 14592 | The "repeated symbol word"... |
| swrd2lsw 14593 | Extract the last two symbo... |
| 2swrd2eqwrdeq 14594 | Two words of length at lea... |
| ccatw2s1ccatws2 14595 | The concatenation of a wor... |
| ccatw2s1ccatws2OLD 14596 | Obsolete version of ~ ccat... |
| ccat2s1fvwALT 14597 | Alternate proof of ~ ccat2... |
| ccat2s1fvwALTOLD 14598 | Obsolete version of ~ ccat... |
| wwlktovf 14599 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 14600 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 14601 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 14602 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 14603 | There is a bijection betwe... |
| eqwrds3 14604 | A word is equal with a len... |
| wrdl3s3 14605 | A word of length 3 is a le... |
| s3sndisj 14606 | The singletons consisting ... |
| s3iunsndisj 14607 | The union of singletons co... |
| ofccat 14608 | Letterwise operations on w... |
| ofs1 14609 | Letterwise operations on a... |
| ofs2 14610 | Letterwise operations on a... |
| coss12d 14611 | Subset deduction for compo... |
| trrelssd 14612 | The composition of subclas... |
| xpcogend 14613 | The most interesting case ... |
| xpcoidgend 14614 | If two classes are not dis... |
| cotr2g 14615 | Two ways of saying that th... |
| cotr2 14616 | Two ways of saying a relat... |
| cotr3 14617 | Two ways of saying a relat... |
| coemptyd 14618 | Deduction about compositio... |
| xptrrel 14619 | The cross product is alway... |
| 0trrel 14620 | The empty class is a trans... |
| cleq1lem 14621 | Equality implies bijection... |
| cleq1 14622 | Equality of relations impl... |
| clsslem 14623 | The closure of a subclass ... |
| trcleq1 14628 | Equality of relations impl... |
| trclsslem 14629 | The transitive closure (as... |
| trcleq2lem 14630 | Equality implies bijection... |
| cvbtrcl 14631 | Change of bound variable i... |
| trcleq12lem 14632 | Equality implies bijection... |
| trclexlem 14633 | Existence of relation impl... |
| trclublem 14634 | If a relation exists then ... |
| trclubi 14635 | The Cartesian product of t... |
| trclubgi 14636 | The union with the Cartesi... |
| trclub 14637 | The Cartesian product of t... |
| trclubg 14638 | The union with the Cartesi... |
| trclfv 14639 | The transitive closure of ... |
| brintclab 14640 | Two ways to express a bina... |
| brtrclfv 14641 | Two ways of expressing the... |
| brcnvtrclfv 14642 | Two ways of expressing the... |
| brtrclfvcnv 14643 | Two ways of expressing the... |
| brcnvtrclfvcnv 14644 | Two ways of expressing the... |
| trclfvss 14645 | The transitive closure (as... |
| trclfvub 14646 | The transitive closure of ... |
| trclfvlb 14647 | The transitive closure of ... |
| trclfvcotr 14648 | The transitive closure of ... |
| trclfvlb2 14649 | The transitive closure of ... |
| trclfvlb3 14650 | The transitive closure of ... |
| cotrtrclfv 14651 | The transitive closure of ... |
| trclidm 14652 | The transitive closure of ... |
| trclun 14653 | Transitive closure of a un... |
| trclfvg 14654 | The value of the transitiv... |
| trclfvcotrg 14655 | The value of the transitiv... |
| reltrclfv 14656 | The transitive closure of ... |
| dmtrclfv 14657 | The domain of the transiti... |
| reldmrelexp 14660 | The domain of the repeated... |
| relexp0g 14661 | A relation composed zero t... |
| relexp0 14662 | A relation composed zero t... |
| relexp0d 14663 | A relation composed zero t... |
| relexpsucnnr 14664 | A reduction for relation e... |
| relexp1g 14665 | A relation composed once i... |
| dfid5 14666 | Identity relation is equal... |
| dfid6 14667 | Identity relation expresse... |
| relexp1d 14668 | A relation composed once i... |
| relexpsucnnl 14669 | A reduction for relation e... |
| relexpsucl 14670 | A reduction for relation e... |
| relexpsucr 14671 | A reduction for relation e... |
| relexpsucrd 14672 | A reduction for relation e... |
| relexpsucld 14673 | A reduction for relation e... |
| relexpcnv 14674 | Commutation of converse an... |
| relexpcnvd 14675 | Commutation of converse an... |
| relexp0rel 14676 | The exponentiation of a cl... |
| relexprelg 14677 | The exponentiation of a cl... |
| relexprel 14678 | The exponentiation of a re... |
| relexpreld 14679 | The exponentiation of a re... |
| relexpnndm 14680 | The domain of an exponenti... |
| relexpdmg 14681 | The domain of an exponenti... |
| relexpdm 14682 | The domain of an exponenti... |
| relexpdmd 14683 | The domain of an exponenti... |
| relexpnnrn 14684 | The range of an exponentia... |
| relexprng 14685 | The range of an exponentia... |
| relexprn 14686 | The range of an exponentia... |
| relexprnd 14687 | The range of an exponentia... |
| relexpfld 14688 | The field of an exponentia... |
| relexpfldd 14689 | The field of an exponentia... |
| relexpaddnn 14690 | Relation composition becom... |
| relexpuzrel 14691 | The exponentiation of a cl... |
| relexpaddg 14692 | Relation composition becom... |
| relexpaddd 14693 | Relation composition becom... |
| rtrclreclem1 14696 | The reflexive, transitive ... |
| dfrtrclrec2 14697 | If two elements are connec... |
| rtrclreclem2 14698 | The reflexive, transitive ... |
| rtrclreclem3 14699 | The reflexive, transitive ... |
| rtrclreclem4 14700 | The reflexive, transitive ... |
| dfrtrcl2 14701 | The two definitions ` t* `... |
| relexpindlem 14702 | Principle of transitive in... |
| relexpind 14703 | Principle of transitive in... |
| rtrclind 14704 | Principle of transitive in... |
| shftlem 14707 | Two ways to write a shifte... |
| shftuz 14708 | A shift of the upper integ... |
| shftfval 14709 | The value of the sequence ... |
| shftdm 14710 | Domain of a relation shift... |
| shftfib 14711 | Value of a fiber of the re... |
| shftfn 14712 | Functionality and domain o... |
| shftval 14713 | Value of a sequence shifte... |
| shftval2 14714 | Value of a sequence shifte... |
| shftval3 14715 | Value of a sequence shifte... |
| shftval4 14716 | Value of a sequence shifte... |
| shftval5 14717 | Value of a shifted sequenc... |
| shftf 14718 | Functionality of a shifted... |
| 2shfti 14719 | Composite shift operations... |
| shftidt2 14720 | Identity law for the shift... |
| shftidt 14721 | Identity law for the shift... |
| shftcan1 14722 | Cancellation law for the s... |
| shftcan2 14723 | Cancellation law for the s... |
| seqshft 14724 | Shifting the index set of ... |
| sgnval 14727 | Value of the signum functi... |
| sgn0 14728 | The signum of 0 is 0. (Co... |
| sgnp 14729 | The signum of a positive e... |
| sgnrrp 14730 | The signum of a positive r... |
| sgn1 14731 | The signum of 1 is 1. (Co... |
| sgnpnf 14732 | The signum of ` +oo ` is 1... |
| sgnn 14733 | The signum of a negative e... |
| sgnmnf 14734 | The signum of ` -oo ` is -... |
| cjval 14741 | The value of the conjugate... |
| cjth 14742 | The defining property of t... |
| cjf 14743 | Domain and codomain of the... |
| cjcl 14744 | The conjugate of a complex... |
| reval 14745 | The value of the real part... |
| imval 14746 | The value of the imaginary... |
| imre 14747 | The imaginary part of a co... |
| reim 14748 | The real part of a complex... |
| recl 14749 | The real part of a complex... |
| imcl 14750 | The imaginary part of a co... |
| ref 14751 | Domain and codomain of the... |
| imf 14752 | Domain and codomain of the... |
| crre 14753 | The real part of a complex... |
| crim 14754 | The real part of a complex... |
| replim 14755 | Reconstruct a complex numb... |
| remim 14756 | Value of the conjugate of ... |
| reim0 14757 | The imaginary part of a re... |
| reim0b 14758 | A number is real iff its i... |
| rereb 14759 | A number is real iff it eq... |
| mulre 14760 | A product with a nonzero r... |
| rere 14761 | A real number equals its r... |
| cjreb 14762 | A number is real iff it eq... |
| recj 14763 | Real part of a complex con... |
| reneg 14764 | Real part of negative. (C... |
| readd 14765 | Real part distributes over... |
| resub 14766 | Real part distributes over... |
| remullem 14767 | Lemma for ~ remul , ~ immu... |
| remul 14768 | Real part of a product. (... |
| remul2 14769 | Real part of a product. (... |
| rediv 14770 | Real part of a division. ... |
| imcj 14771 | Imaginary part of a comple... |
| imneg 14772 | The imaginary part of a ne... |
| imadd 14773 | Imaginary part distributes... |
| imsub 14774 | Imaginary part distributes... |
| immul 14775 | Imaginary part of a produc... |
| immul2 14776 | Imaginary part of a produc... |
| imdiv 14777 | Imaginary part of a divisi... |
| cjre 14778 | A real number equals its c... |
| cjcj 14779 | The conjugate of the conju... |
| cjadd 14780 | Complex conjugate distribu... |
| cjmul 14781 | Complex conjugate distribu... |
| ipcnval 14782 | Standard inner product on ... |
| cjmulrcl 14783 | A complex number times its... |
| cjmulval 14784 | A complex number times its... |
| cjmulge0 14785 | A complex number times its... |
| cjneg 14786 | Complex conjugate of negat... |
| addcj 14787 | A number plus its conjugat... |
| cjsub 14788 | Complex conjugate distribu... |
| cjexp 14789 | Complex conjugate of posit... |
| imval2 14790 | The imaginary part of a nu... |
| re0 14791 | The real part of zero. (C... |
| im0 14792 | The imaginary part of zero... |
| re1 14793 | The real part of one. (Co... |
| im1 14794 | The imaginary part of one.... |
| rei 14795 | The real part of ` _i ` . ... |
| imi 14796 | The imaginary part of ` _i... |
| cj0 14797 | The conjugate of zero. (C... |
| cji 14798 | The complex conjugate of t... |
| cjreim 14799 | The conjugate of a represe... |
| cjreim2 14800 | The conjugate of the repre... |
| cj11 14801 | Complex conjugate is a one... |
| cjne0 14802 | A number is nonzero iff it... |
| cjdiv 14803 | Complex conjugate distribu... |
| cnrecnv 14804 | The inverse to the canonic... |
| sqeqd 14805 | A deduction for showing tw... |
| recli 14806 | The real part of a complex... |
| imcli 14807 | The imaginary part of a co... |
| cjcli 14808 | Closure law for complex co... |
| replimi 14809 | Construct a complex number... |
| cjcji 14810 | The conjugate of the conju... |
| reim0bi 14811 | A number is real iff its i... |
| rerebi 14812 | A real number equals its r... |
| cjrebi 14813 | A number is real iff it eq... |
| recji 14814 | Real part of a complex con... |
| imcji 14815 | Imaginary part of a comple... |
| cjmulrcli 14816 | A complex number times its... |
| cjmulvali 14817 | A complex number times its... |
| cjmulge0i 14818 | A complex number times its... |
| renegi 14819 | Real part of negative. (C... |
| imnegi 14820 | Imaginary part of negative... |
| cjnegi 14821 | Complex conjugate of negat... |
| addcji 14822 | A number plus its conjugat... |
| readdi 14823 | Real part distributes over... |
| imaddi 14824 | Imaginary part distributes... |
| remuli 14825 | Real part of a product. (... |
| immuli 14826 | Imaginary part of a produc... |
| cjaddi 14827 | Complex conjugate distribu... |
| cjmuli 14828 | Complex conjugate distribu... |
| ipcni 14829 | Standard inner product on ... |
| cjdivi 14830 | Complex conjugate distribu... |
| crrei 14831 | The real part of a complex... |
| crimi 14832 | The imaginary part of a co... |
| recld 14833 | The real part of a complex... |
| imcld 14834 | The imaginary part of a co... |
| cjcld 14835 | Closure law for complex co... |
| replimd 14836 | Construct a complex number... |
| remimd 14837 | Value of the conjugate of ... |
| cjcjd 14838 | The conjugate of the conju... |
| reim0bd 14839 | A number is real iff its i... |
| rerebd 14840 | A real number equals its r... |
| cjrebd 14841 | A number is real iff it eq... |
| cjne0d 14842 | A number is nonzero iff it... |
| recjd 14843 | Real part of a complex con... |
| imcjd 14844 | Imaginary part of a comple... |
| cjmulrcld 14845 | A complex number times its... |
| cjmulvald 14846 | A complex number times its... |
| cjmulge0d 14847 | A complex number times its... |
| renegd 14848 | Real part of negative. (C... |
| imnegd 14849 | Imaginary part of negative... |
| cjnegd 14850 | Complex conjugate of negat... |
| addcjd 14851 | A number plus its conjugat... |
| cjexpd 14852 | Complex conjugate of posit... |
| readdd 14853 | Real part distributes over... |
| imaddd 14854 | Imaginary part distributes... |
| resubd 14855 | Real part distributes over... |
| imsubd 14856 | Imaginary part distributes... |
| remuld 14857 | Real part of a product. (... |
| immuld 14858 | Imaginary part of a produc... |
| cjaddd 14859 | Complex conjugate distribu... |
| cjmuld 14860 | Complex conjugate distribu... |
| ipcnd 14861 | Standard inner product on ... |
| cjdivd 14862 | Complex conjugate distribu... |
| rered 14863 | A real number equals its r... |
| reim0d 14864 | The imaginary part of a re... |
| cjred 14865 | A real number equals its c... |
| remul2d 14866 | Real part of a product. (... |
| immul2d 14867 | Imaginary part of a produc... |
| redivd 14868 | Real part of a division. ... |
| imdivd 14869 | Imaginary part of a divisi... |
| crred 14870 | The real part of a complex... |
| crimd 14871 | The imaginary part of a co... |
| sqrtval 14876 | Value of square root funct... |
| absval 14877 | The absolute value (modulu... |
| rennim 14878 | A real number does not lie... |
| cnpart 14879 | The specification of restr... |
| sqr0lem 14880 | Square root of zero. (Con... |
| sqrt0 14881 | Square root of zero. (Con... |
| sqrlem1 14882 | Lemma for ~ 01sqrex . (Co... |
| sqrlem2 14883 | Lemma for ~ 01sqrex . (Co... |
| sqrlem3 14884 | Lemma for ~ 01sqrex . (Co... |
| sqrlem4 14885 | Lemma for ~ 01sqrex . (Co... |
| sqrlem5 14886 | Lemma for ~ 01sqrex . (Co... |
| sqrlem6 14887 | Lemma for ~ 01sqrex . (Co... |
| sqrlem7 14888 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 14889 | Existence of a square root... |
| resqrex 14890 | Existence of a square root... |
| sqrmo 14891 | Uniqueness for the square ... |
| resqreu 14892 | Existence and uniqueness f... |
| resqrtcl 14893 | Closure of the square root... |
| resqrtthlem 14894 | Lemma for ~ resqrtth . (C... |
| resqrtth 14895 | Square root theorem over t... |
| remsqsqrt 14896 | Square of square root. (C... |
| sqrtge0 14897 | The square root function i... |
| sqrtgt0 14898 | The square root function i... |
| sqrtmul 14899 | Square root distributes ov... |
| sqrtle 14900 | Square root is monotonic. ... |
| sqrtlt 14901 | Square root is strictly mo... |
| sqrt11 14902 | The square root function i... |
| sqrt00 14903 | A square root is zero iff ... |
| rpsqrtcl 14904 | The square root of a posit... |
| sqrtdiv 14905 | Square root distributes ov... |
| sqrtneglem 14906 | The square root of a negat... |
| sqrtneg 14907 | The square root of a negat... |
| sqrtsq2 14908 | Relationship between squar... |
| sqrtsq 14909 | Square root of square. (C... |
| sqrtmsq 14910 | Square root of square. (C... |
| sqrt1 14911 | The square root of 1 is 1.... |
| sqrt4 14912 | The square root of 4 is 2.... |
| sqrt9 14913 | The square root of 9 is 3.... |
| sqrt2gt1lt2 14914 | The square root of 2 is bo... |
| sqrtm1 14915 | The imaginary unit is the ... |
| nn0sqeq1 14916 | A natural number with squa... |
| absneg 14917 | Absolute value of the oppo... |
| abscl 14918 | Real closure of absolute v... |
| abscj 14919 | The absolute value of a nu... |
| absvalsq 14920 | Square of value of absolut... |
| absvalsq2 14921 | Square of value of absolut... |
| sqabsadd 14922 | Square of absolute value o... |
| sqabssub 14923 | Square of absolute value o... |
| absval2 14924 | Value of absolute value fu... |
| abs0 14925 | The absolute value of 0. ... |
| absi 14926 | The absolute value of the ... |
| absge0 14927 | Absolute value is nonnegat... |
| absrpcl 14928 | The absolute value of a no... |
| abs00 14929 | The absolute value of a nu... |
| abs00ad 14930 | A complex number is zero i... |
| abs00bd 14931 | If a complex number is zer... |
| absreimsq 14932 | Square of the absolute val... |
| absreim 14933 | Absolute value of a number... |
| absmul 14934 | Absolute value distributes... |
| absdiv 14935 | Absolute value distributes... |
| absid 14936 | A nonnegative number is it... |
| abs1 14937 | The absolute value of one ... |
| absnid 14938 | A negative number is the n... |
| leabs 14939 | A real number is less than... |
| absor 14940 | The absolute value of a re... |
| absre 14941 | Absolute value of a real n... |
| absresq 14942 | Square of the absolute val... |
| absmod0 14943 | ` A ` is divisible by ` B ... |
| absexp 14944 | Absolute value of positive... |
| absexpz 14945 | Absolute value of integer ... |
| abssq 14946 | Square can be moved in and... |
| sqabs 14947 | The squares of two reals a... |
| absrele 14948 | The absolute value of a co... |
| absimle 14949 | The absolute value of a co... |
| max0add 14950 | The sum of the positive an... |
| absz 14951 | A real number is an intege... |
| nn0abscl 14952 | The absolute value of an i... |
| zabscl 14953 | The absolute value of an i... |
| abslt 14954 | Absolute value and 'less t... |
| absle 14955 | Absolute value and 'less t... |
| abssubne0 14956 | If the absolute value of a... |
| absdiflt 14957 | The absolute value of a di... |
| absdifle 14958 | The absolute value of a di... |
| elicc4abs 14959 | Membership in a symmetric ... |
| lenegsq 14960 | Comparison to a nonnegativ... |
| releabs 14961 | The real part of a number ... |
| recval 14962 | Reciprocal expressed with ... |
| absidm 14963 | The absolute value functio... |
| absgt0 14964 | The absolute value of a no... |
| nnabscl 14965 | The absolute value of a no... |
| abssub 14966 | Swapping order of subtract... |
| abssubge0 14967 | Absolute value of a nonneg... |
| abssuble0 14968 | Absolute value of a nonpos... |
| absmax 14969 | The maximum of two numbers... |
| abstri 14970 | Triangle inequality for ab... |
| abs3dif 14971 | Absolute value of differen... |
| abs2dif 14972 | Difference of absolute val... |
| abs2dif2 14973 | Difference of absolute val... |
| abs2difabs 14974 | Absolute value of differen... |
| abs1m 14975 | For any complex number, th... |
| recan 14976 | Cancellation law involving... |
| absf 14977 | Mapping domain and codomai... |
| abs3lem 14978 | Lemma involving absolute v... |
| abslem2 14979 | Lemma involving absolute v... |
| rddif 14980 | The difference between a r... |
| absrdbnd 14981 | Bound on the absolute valu... |
| fzomaxdiflem 14982 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 14983 | A bound on the separation ... |
| uzin2 14984 | The upper integers are clo... |
| rexanuz 14985 | Combine two different uppe... |
| rexanre 14986 | Combine two different uppe... |
| rexfiuz 14987 | Combine finitely many diff... |
| rexuz3 14988 | Restrict the base of the u... |
| rexanuz2 14989 | Combine two different uppe... |
| r19.29uz 14990 | A version of ~ 19.29 for u... |
| r19.2uz 14991 | A version of ~ r19.2z for ... |
| rexuzre 14992 | Convert an upper real quan... |
| rexico 14993 | Restrict the base of an up... |
| cau3lem 14994 | Lemma for ~ cau3 . (Contr... |
| cau3 14995 | Convert between three-quan... |
| cau4 14996 | Change the base of a Cauch... |
| caubnd2 14997 | A Cauchy sequence of compl... |
| caubnd 14998 | A Cauchy sequence of compl... |
| sqreulem 14999 | Lemma for ~ sqreu : write ... |
| sqreu 15000 | Existence and uniqueness f... |
| sqrtcl 15001 | Closure of the square root... |
| sqrtthlem 15002 | Lemma for ~ sqrtth . (Con... |
| sqrtf 15003 | Mapping domain and codomai... |
| sqrtth 15004 | Square root theorem over t... |
| sqrtrege0 15005 | The square root function m... |
| eqsqrtor 15006 | Solve an equation containi... |
| eqsqrtd 15007 | A deduction for showing th... |
| eqsqrt2d 15008 | A deduction for showing th... |
| amgm2 15009 | Arithmetic-geometric mean ... |
| sqrtthi 15010 | Square root theorem. Theo... |
| sqrtcli 15011 | The square root of a nonne... |
| sqrtgt0i 15012 | The square root of a posit... |
| sqrtmsqi 15013 | Square root of square. (C... |
| sqrtsqi 15014 | Square root of square. (C... |
| sqsqrti 15015 | Square of square root. (C... |
| sqrtge0i 15016 | The square root of a nonne... |
| absidi 15017 | A nonnegative number is it... |
| absnidi 15018 | A negative number is the n... |
| leabsi 15019 | A real number is less than... |
| absori 15020 | The absolute value of a re... |
| absrei 15021 | Absolute value of a real n... |
| sqrtpclii 15022 | The square root of a posit... |
| sqrtgt0ii 15023 | The square root of a posit... |
| sqrt11i 15024 | The square root function i... |
| sqrtmuli 15025 | Square root distributes ov... |
| sqrtmulii 15026 | Square root distributes ov... |
| sqrtmsq2i 15027 | Relationship between squar... |
| sqrtlei 15028 | Square root is monotonic. ... |
| sqrtlti 15029 | Square root is strictly mo... |
| abslti 15030 | Absolute value and 'less t... |
| abslei 15031 | Absolute value and 'less t... |
| cnsqrt00 15032 | A square root of a complex... |
| absvalsqi 15033 | Square of value of absolut... |
| absvalsq2i 15034 | Square of value of absolut... |
| abscli 15035 | Real closure of absolute v... |
| absge0i 15036 | Absolute value is nonnegat... |
| absval2i 15037 | Value of absolute value fu... |
| abs00i 15038 | The absolute value of a nu... |
| absgt0i 15039 | The absolute value of a no... |
| absnegi 15040 | Absolute value of negative... |
| abscji 15041 | The absolute value of a nu... |
| releabsi 15042 | The real part of a number ... |
| abssubi 15043 | Swapping order of subtract... |
| absmuli 15044 | Absolute value distributes... |
| sqabsaddi 15045 | Square of absolute value o... |
| sqabssubi 15046 | Square of absolute value o... |
| absdivzi 15047 | Absolute value distributes... |
| abstrii 15048 | Triangle inequality for ab... |
| abs3difi 15049 | Absolute value of differen... |
| abs3lemi 15050 | Lemma involving absolute v... |
| rpsqrtcld 15051 | The square root of a posit... |
| sqrtgt0d 15052 | The square root of a posit... |
| absnidd 15053 | A negative number is the n... |
| leabsd 15054 | A real number is less than... |
| absord 15055 | The absolute value of a re... |
| absred 15056 | Absolute value of a real n... |
| resqrtcld 15057 | The square root of a nonne... |
| sqrtmsqd 15058 | Square root of square. (C... |
| sqrtsqd 15059 | Square root of square. (C... |
| sqrtge0d 15060 | The square root of a nonne... |
| sqrtnegd 15061 | The square root of a negat... |
| absidd 15062 | A nonnegative number is it... |
| sqrtdivd 15063 | Square root distributes ov... |
| sqrtmuld 15064 | Square root distributes ov... |
| sqrtsq2d 15065 | Relationship between squar... |
| sqrtled 15066 | Square root is monotonic. ... |
| sqrtltd 15067 | Square root is strictly mo... |
| sqr11d 15068 | The square root function i... |
| absltd 15069 | Absolute value and 'less t... |
| absled 15070 | Absolute value and 'less t... |
| abssubge0d 15071 | Absolute value of a nonneg... |
| abssuble0d 15072 | Absolute value of a nonpos... |
| absdifltd 15073 | The absolute value of a di... |
| absdifled 15074 | The absolute value of a di... |
| icodiamlt 15075 | Two elements in a half-ope... |
| abscld 15076 | Real closure of absolute v... |
| sqrtcld 15077 | Closure of the square root... |
| sqrtrege0d 15078 | The real part of the squar... |
| sqsqrtd 15079 | Square root theorem. Theo... |
| msqsqrtd 15080 | Square root theorem. Theo... |
| sqr00d 15081 | A square root is zero iff ... |
| absvalsqd 15082 | Square of value of absolut... |
| absvalsq2d 15083 | Square of value of absolut... |
| absge0d 15084 | Absolute value is nonnegat... |
| absval2d 15085 | Value of absolute value fu... |
| abs00d 15086 | The absolute value of a nu... |
| absne0d 15087 | The absolute value of a nu... |
| absrpcld 15088 | The absolute value of a no... |
| absnegd 15089 | Absolute value of negative... |
| abscjd 15090 | The absolute value of a nu... |
| releabsd 15091 | The real part of a number ... |
| absexpd 15092 | Absolute value of positive... |
| abssubd 15093 | Swapping order of subtract... |
| absmuld 15094 | Absolute value distributes... |
| absdivd 15095 | Absolute value distributes... |
| abstrid 15096 | Triangle inequality for ab... |
| abs2difd 15097 | Difference of absolute val... |
| abs2dif2d 15098 | Difference of absolute val... |
| abs2difabsd 15099 | Absolute value of differen... |
| abs3difd 15100 | Absolute value of differen... |
| abs3lemd 15101 | Lemma involving absolute v... |
| reusq0 15102 | A complex number is the sq... |
| bhmafibid1cn 15103 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 15104 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 15105 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 15106 | The Brahmagupta-Fibonacci ... |
| limsupgord 15109 | Ordering property of the s... |
| limsupcl 15110 | Closure of the superior li... |
| limsupval 15111 | The superior limit of an i... |
| limsupgf 15112 | Closure of the superior li... |
| limsupgval 15113 | Value of the superior limi... |
| limsupgle 15114 | The defining property of t... |
| limsuple 15115 | The defining property of t... |
| limsuplt 15116 | The defining property of t... |
| limsupval2 15117 | The superior limit, relati... |
| limsupgre 15118 | If a sequence of real numb... |
| limsupbnd1 15119 | If a sequence is eventuall... |
| limsupbnd2 15120 | If a sequence is eventuall... |
| climrel 15129 | The limit relation is a re... |
| rlimrel 15130 | The limit relation is a re... |
| clim 15131 | Express the predicate: Th... |
| rlim 15132 | Express the predicate: Th... |
| rlim2 15133 | Rewrite ~ rlim for a mappi... |
| rlim2lt 15134 | Use strictly less-than in ... |
| rlim3 15135 | Restrict the range of the ... |
| climcl 15136 | Closure of the limit of a ... |
| rlimpm 15137 | Closure of a function with... |
| rlimf 15138 | Closure of a function with... |
| rlimss 15139 | Domain closure of a functi... |
| rlimcl 15140 | Closure of the limit of a ... |
| clim2 15141 | Express the predicate: Th... |
| clim2c 15142 | Express the predicate ` F ... |
| clim0 15143 | Express the predicate ` F ... |
| clim0c 15144 | Express the predicate ` F ... |
| rlim0 15145 | Express the predicate ` B ... |
| rlim0lt 15146 | Use strictly less-than in ... |
| climi 15147 | Convergence of a sequence ... |
| climi2 15148 | Convergence of a sequence ... |
| climi0 15149 | Convergence of a sequence ... |
| rlimi 15150 | Convergence at infinity of... |
| rlimi2 15151 | Convergence at infinity of... |
| ello1 15152 | Elementhood in the set of ... |
| ello12 15153 | Elementhood in the set of ... |
| ello12r 15154 | Sufficient condition for e... |
| lo1f 15155 | An eventually upper bounde... |
| lo1dm 15156 | An eventually upper bounde... |
| lo1bdd 15157 | The defining property of a... |
| ello1mpt 15158 | Elementhood in the set of ... |
| ello1mpt2 15159 | Elementhood in the set of ... |
| ello1d 15160 | Sufficient condition for e... |
| lo1bdd2 15161 | If an eventually bounded f... |
| lo1bddrp 15162 | Refine ~ o1bdd2 to give a ... |
| elo1 15163 | Elementhood in the set of ... |
| elo12 15164 | Elementhood in the set of ... |
| elo12r 15165 | Sufficient condition for e... |
| o1f 15166 | An eventually bounded func... |
| o1dm 15167 | An eventually bounded func... |
| o1bdd 15168 | The defining property of a... |
| lo1o1 15169 | A function is eventually b... |
| lo1o12 15170 | A function is eventually b... |
| elo1mpt 15171 | Elementhood in the set of ... |
| elo1mpt2 15172 | Elementhood in the set of ... |
| elo1d 15173 | Sufficient condition for e... |
| o1lo1 15174 | A real function is eventua... |
| o1lo12 15175 | A lower bounded real funct... |
| o1lo1d 15176 | A real eventually bounded ... |
| icco1 15177 | Derive eventual boundednes... |
| o1bdd2 15178 | If an eventually bounded f... |
| o1bddrp 15179 | Refine ~ o1bdd2 to give a ... |
| climconst 15180 | An (eventually) constant s... |
| rlimconst 15181 | A constant sequence conver... |
| rlimclim1 15182 | Forward direction of ~ rli... |
| rlimclim 15183 | A sequence on an upper int... |
| climrlim2 15184 | Produce a real limit from ... |
| climconst2 15185 | A constant sequence conver... |
| climz 15186 | The zero sequence converge... |
| rlimuni 15187 | A real function whose doma... |
| rlimdm 15188 | Two ways to express that a... |
| climuni 15189 | An infinite sequence of co... |
| fclim 15190 | The limit relation is func... |
| climdm 15191 | Two ways to express that a... |
| climeu 15192 | An infinite sequence of co... |
| climreu 15193 | An infinite sequence of co... |
| climmo 15194 | An infinite sequence of co... |
| rlimres 15195 | The restriction of a funct... |
| lo1res 15196 | The restriction of an even... |
| o1res 15197 | The restriction of an even... |
| rlimres2 15198 | The restriction of a funct... |
| lo1res2 15199 | The restriction of a funct... |
| o1res2 15200 | The restriction of a funct... |
| lo1resb 15201 | The restriction of a funct... |
| rlimresb 15202 | The restriction of a funct... |
| o1resb 15203 | The restriction of a funct... |
| climeq 15204 | Two functions that are eve... |
| lo1eq 15205 | Two functions that are eve... |
| rlimeq 15206 | Two functions that are eve... |
| o1eq 15207 | Two functions that are eve... |
| climmpt 15208 | Exhibit a function ` G ` w... |
| 2clim 15209 | If two sequences converge ... |
| climmpt2 15210 | Relate an integer limit on... |
| climshftlem 15211 | A shifted function converg... |
| climres 15212 | A function restricted to u... |
| climshft 15213 | A shifted function converg... |
| serclim0 15214 | The zero series converges ... |
| rlimcld2 15215 | If ` D ` is a closed set i... |
| rlimrege0 15216 | The limit of a sequence of... |
| rlimrecl 15217 | The limit of a real sequen... |
| rlimge0 15218 | The limit of a sequence of... |
| climshft2 15219 | A shifted function converg... |
| climrecl 15220 | The limit of a convergent ... |
| climge0 15221 | A nonnegative sequence con... |
| climabs0 15222 | Convergence to zero of the... |
| o1co 15223 | Sufficient condition for t... |
| o1compt 15224 | Sufficient condition for t... |
| rlimcn1 15225 | Image of a limit under a c... |
| rlimcn1b 15226 | Image of a limit under a c... |
| rlimcn3 15227 | Image of a limit under a c... |
| rlimcn2 15228 | Image of a limit under a c... |
| climcn1 15229 | Image of a limit under a c... |
| climcn2 15230 | Image of a limit under a c... |
| addcn2 15231 | Complex number addition is... |
| subcn2 15232 | Complex number subtraction... |
| mulcn2 15233 | Complex number multiplicat... |
| reccn2 15234 | The reciprocal function is... |
| cn1lem 15235 | A sufficient condition for... |
| abscn2 15236 | The absolute value functio... |
| cjcn2 15237 | The complex conjugate func... |
| recn2 15238 | The real part function is ... |
| imcn2 15239 | The imaginary part functio... |
| climcn1lem 15240 | The limit of a continuous ... |
| climabs 15241 | Limit of the absolute valu... |
| climcj 15242 | Limit of the complex conju... |
| climre 15243 | Limit of the real part of ... |
| climim 15244 | Limit of the imaginary par... |
| rlimmptrcl 15245 | Reverse closure for a real... |
| rlimabs 15246 | Limit of the absolute valu... |
| rlimcj 15247 | Limit of the complex conju... |
| rlimre 15248 | Limit of the real part of ... |
| rlimim 15249 | Limit of the imaginary par... |
| o1of2 15250 | Show that a binary operati... |
| o1add 15251 | The sum of two eventually ... |
| o1mul 15252 | The product of two eventua... |
| o1sub 15253 | The difference of two even... |
| rlimo1 15254 | Any function with a finite... |
| rlimdmo1 15255 | A convergent function is e... |
| o1rlimmul 15256 | The product of an eventual... |
| o1const 15257 | A constant function is eve... |
| lo1const 15258 | A constant function is eve... |
| lo1mptrcl 15259 | Reverse closure for an eve... |
| o1mptrcl 15260 | Reverse closure for an eve... |
| o1add2 15261 | The sum of two eventually ... |
| o1mul2 15262 | The product of two eventua... |
| o1sub2 15263 | The product of two eventua... |
| lo1add 15264 | The sum of two eventually ... |
| lo1mul 15265 | The product of an eventual... |
| lo1mul2 15266 | The product of an eventual... |
| o1dif 15267 | If the difference of two f... |
| lo1sub 15268 | The difference of an event... |
| climadd 15269 | Limit of the sum of two co... |
| climmul 15270 | Limit of the product of tw... |
| climsub 15271 | Limit of the difference of... |
| climaddc1 15272 | Limit of a constant ` C ` ... |
| climaddc2 15273 | Limit of a constant ` C ` ... |
| climmulc2 15274 | Limit of a sequence multip... |
| climsubc1 15275 | Limit of a constant ` C ` ... |
| climsubc2 15276 | Limit of a constant ` C ` ... |
| climle 15277 | Comparison of the limits o... |
| climsqz 15278 | Convergence of a sequence ... |
| climsqz2 15279 | Convergence of a sequence ... |
| rlimadd 15280 | Limit of the sum of two co... |
| rlimaddOLD 15281 | Obsolete version of ~ rlim... |
| rlimsub 15282 | Limit of the difference of... |
| rlimmul 15283 | Limit of the product of tw... |
| rlimmulOLD 15284 | Obsolete version of ~ rlim... |
| rlimdiv 15285 | Limit of the quotient of t... |
| rlimneg 15286 | Limit of the negative of a... |
| rlimle 15287 | Comparison of the limits o... |
| rlimsqzlem 15288 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15289 | Convergence of a sequence ... |
| rlimsqz2 15290 | Convergence of a sequence ... |
| lo1le 15291 | Transfer eventual upper bo... |
| o1le 15292 | Transfer eventual boundedn... |
| rlimno1 15293 | A function whose inverse c... |
| clim2ser 15294 | The limit of an infinite s... |
| clim2ser2 15295 | The limit of an infinite s... |
| iserex 15296 | An infinite series converg... |
| isermulc2 15297 | Multiplication of an infin... |
| climlec2 15298 | Comparison of a constant t... |
| iserle 15299 | Comparison of the limits o... |
| iserge0 15300 | The limit of an infinite s... |
| climub 15301 | The limit of a monotonic s... |
| climserle 15302 | The partial sums of a conv... |
| isershft 15303 | Index shift of the limit o... |
| isercolllem1 15304 | Lemma for ~ isercoll . (C... |
| isercolllem2 15305 | Lemma for ~ isercoll . (C... |
| isercolllem3 15306 | Lemma for ~ isercoll . (C... |
| isercoll 15307 | Rearrange an infinite seri... |
| isercoll2 15308 | Generalize ~ isercoll so t... |
| climsup 15309 | A bounded monotonic sequen... |
| climcau 15310 | A converging sequence of c... |
| climbdd 15311 | A converging sequence of c... |
| caucvgrlem 15312 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15313 | A Cauchy sequence of real ... |
| caucvgrlem2 15314 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15315 | A Cauchy sequence of compl... |
| caurcvg 15316 | A Cauchy sequence of real ... |
| caurcvg2 15317 | A Cauchy sequence of real ... |
| caucvg 15318 | A Cauchy sequence of compl... |
| caucvgb 15319 | A function is convergent i... |
| serf0 15320 | If an infinite series conv... |
| iseraltlem1 15321 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15322 | Lemma for ~ iseralt . The... |
| iseraltlem3 15323 | Lemma for ~ iseralt . Fro... |
| iseralt 15324 | The alternating series tes... |
| sumex 15327 | A sum is a set. (Contribu... |
| sumeq1 15328 | Equality theorem for a sum... |
| nfsum1 15329 | Bound-variable hypothesis ... |
| nfsum 15330 | Bound-variable hypothesis ... |
| nfsumOLD 15331 | Obsolete version of ~ nfsu... |
| sumeq2w 15332 | Equality theorem for sum, ... |
| sumeq2ii 15333 | Equality theorem for sum, ... |
| sumeq2 15334 | Equality theorem for sum. ... |
| cbvsum 15335 | Change bound variable in a... |
| cbvsumv 15336 | Change bound variable in a... |
| cbvsumi 15337 | Change bound variable in a... |
| sumeq1i 15338 | Equality inference for sum... |
| sumeq2i 15339 | Equality inference for sum... |
| sumeq12i 15340 | Equality inference for sum... |
| sumeq1d 15341 | Equality deduction for sum... |
| sumeq2d 15342 | Equality deduction for sum... |
| sumeq2dv 15343 | Equality deduction for sum... |
| sumeq2sdv 15344 | Equality deduction for sum... |
| 2sumeq2dv 15345 | Equality deduction for dou... |
| sumeq12dv 15346 | Equality deduction for sum... |
| sumeq12rdv 15347 | Equality deduction for sum... |
| sum2id 15348 | The second class argument ... |
| sumfc 15349 | A lemma to facilitate conv... |
| fz1f1o 15350 | A lemma for working with f... |
| sumrblem 15351 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15352 | The sequence of partial su... |
| sumrb 15353 | Rebase the starting point ... |
| summolem3 15354 | Lemma for ~ summo . (Cont... |
| summolem2a 15355 | Lemma for ~ summo . (Cont... |
| summolem2 15356 | Lemma for ~ summo . (Cont... |
| summo 15357 | A sum has at most one limi... |
| zsum 15358 | Series sum with index set ... |
| isum 15359 | Series sum with an upper i... |
| fsum 15360 | The value of a sum over a ... |
| sum0 15361 | Any sum over the empty set... |
| sumz 15362 | Any sum of zero over a sum... |
| fsumf1o 15363 | Re-index a finite sum usin... |
| sumss 15364 | Change the index set to a ... |
| fsumss 15365 | Change the index set to a ... |
| sumss2 15366 | Change the index set of a ... |
| fsumcvg2 15367 | The sequence of partial su... |
| fsumsers 15368 | Special case of series sum... |
| fsumcvg3 15369 | A finite sum is convergent... |
| fsumser 15370 | A finite sum expressed in ... |
| fsumcl2lem 15371 | - Lemma for finite sum clo... |
| fsumcllem 15372 | - Lemma for finite sum clo... |
| fsumcl 15373 | Closure of a finite sum of... |
| fsumrecl 15374 | Closure of a finite sum of... |
| fsumzcl 15375 | Closure of a finite sum of... |
| fsumnn0cl 15376 | Closure of a finite sum of... |
| fsumrpcl 15377 | Closure of a finite sum of... |
| fsumclf 15378 | Closure of a finite sum of... |
| fsumzcl2 15379 | A finite sum with integer ... |
| fsumadd 15380 | The sum of two finite sums... |
| fsumsplit 15381 | Split a sum into two parts... |
| fsumsplitf 15382 | Split a sum into two parts... |
| sumsnf 15383 | A sum of a singleton is th... |
| fsumsplitsn 15384 | Separate out a term in a f... |
| fsumsplit1 15385 | Separate out a term in a f... |
| sumsn 15386 | A sum of a singleton is th... |
| fsum1 15387 | The finite sum of ` A ( k ... |
| sumpr 15388 | A sum over a pair is the s... |
| sumtp 15389 | A sum over a triple is the... |
| sumsns 15390 | A sum of a singleton is th... |
| fsumm1 15391 | Separate out the last term... |
| fzosump1 15392 | Separate out the last term... |
| fsum1p 15393 | Separate out the first ter... |
| fsummsnunz 15394 | A finite sum all of whose ... |
| fsumsplitsnun 15395 | Separate out a term in a f... |
| fsump1 15396 | The addition of the next t... |
| isumclim 15397 | An infinite sum equals the... |
| isumclim2 15398 | A converging series conver... |
| isumclim3 15399 | The sequence of partial fi... |
| sumnul 15400 | The sum of a non-convergen... |
| isumcl 15401 | The sum of a converging in... |
| isummulc2 15402 | An infinite sum multiplied... |
| isummulc1 15403 | An infinite sum multiplied... |
| isumdivc 15404 | An infinite sum divided by... |
| isumrecl 15405 | The sum of a converging in... |
| isumge0 15406 | An infinite sum of nonnega... |
| isumadd 15407 | Addition of infinite sums.... |
| sumsplit 15408 | Split a sum into two parts... |
| fsump1i 15409 | Optimized version of ~ fsu... |
| fsum2dlem 15410 | Lemma for ~ fsum2d - induc... |
| fsum2d 15411 | Write a double sum as a su... |
| fsumxp 15412 | Combine two sums into a si... |
| fsumcnv 15413 | Transform a region of summ... |
| fsumcom2 15414 | Interchange order of summa... |
| fsumcom 15415 | Interchange order of summa... |
| fsum0diaglem 15416 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15417 | Two ways to express "the s... |
| mptfzshft 15418 | 1-1 onto function in maps-... |
| fsumrev 15419 | Reversal of a finite sum. ... |
| fsumshft 15420 | Index shift of a finite su... |
| fsumshftm 15421 | Negative index shift of a ... |
| fsumrev2 15422 | Reversal of a finite sum. ... |
| fsum0diag2 15423 | Two ways to express "the s... |
| fsummulc2 15424 | A finite sum multiplied by... |
| fsummulc1 15425 | A finite sum multiplied by... |
| fsumdivc 15426 | A finite sum divided by a ... |
| fsumneg 15427 | Negation of a finite sum. ... |
| fsumsub 15428 | Split a finite sum over a ... |
| fsum2mul 15429 | Separate the nested sum of... |
| fsumconst 15430 | The sum of constant terms ... |
| fsumdifsnconst 15431 | The sum of constant terms ... |
| modfsummodslem1 15432 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15433 | Induction step for ~ modfs... |
| modfsummod 15434 | A finite sum modulo a posi... |
| fsumge0 15435 | If all of the terms of a f... |
| fsumless 15436 | A shorter sum of nonnegati... |
| fsumge1 15437 | A sum of nonnegative numbe... |
| fsum00 15438 | A sum of nonnegative numbe... |
| fsumle 15439 | If all of the terms of fin... |
| fsumlt 15440 | If every term in one finit... |
| fsumabs 15441 | Generalized triangle inequ... |
| telfsumo 15442 | Sum of a telescoping serie... |
| telfsumo2 15443 | Sum of a telescoping serie... |
| telfsum 15444 | Sum of a telescoping serie... |
| telfsum2 15445 | Sum of a telescoping serie... |
| fsumparts 15446 | Summation by parts. (Cont... |
| fsumrelem 15447 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15448 | The real part of a sum. (... |
| fsumim 15449 | The imaginary part of a su... |
| fsumcj 15450 | The complex conjugate of a... |
| fsumrlim 15451 | Limit of a finite sum of c... |
| fsumo1 15452 | The finite sum of eventual... |
| o1fsum 15453 | If ` A ( k ) ` is O(1), th... |
| seqabs 15454 | Generalized triangle inequ... |
| iserabs 15455 | Generalized triangle inequ... |
| cvgcmp 15456 | A comparison test for conv... |
| cvgcmpub 15457 | An upper bound for the lim... |
| cvgcmpce 15458 | A comparison test for conv... |
| abscvgcvg 15459 | An absolutely convergent s... |
| climfsum 15460 | Limit of a finite sum of c... |
| fsumiun 15461 | Sum over a disjoint indexe... |
| hashiun 15462 | The cardinality of a disjo... |
| hash2iun 15463 | The cardinality of a neste... |
| hash2iun1dif1 15464 | The cardinality of a neste... |
| hashrabrex 15465 | The number of elements in ... |
| hashuni 15466 | The cardinality of a disjo... |
| qshash 15467 | The cardinality of a set w... |
| ackbijnn 15468 | Translate the Ackermann bi... |
| binomlem 15469 | Lemma for ~ binom (binomia... |
| binom 15470 | The binomial theorem: ` ( ... |
| binom1p 15471 | Special case of the binomi... |
| binom11 15472 | Special case of the binomi... |
| binom1dif 15473 | A summation for the differ... |
| bcxmaslem1 15474 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15475 | Parallel summation (Christ... |
| incexclem 15476 | Lemma for ~ incexc . (Con... |
| incexc 15477 | The inclusion/exclusion pr... |
| incexc2 15478 | The inclusion/exclusion pr... |
| isumshft 15479 | Index shift of an infinite... |
| isumsplit 15480 | Split off the first ` N ` ... |
| isum1p 15481 | The infinite sum of a conv... |
| isumnn0nn 15482 | Sum from 0 to infinity in ... |
| isumrpcl 15483 | The infinite sum of positi... |
| isumle 15484 | Comparison of two infinite... |
| isumless 15485 | A finite sum of nonnegativ... |
| isumsup2 15486 | An infinite sum of nonnega... |
| isumsup 15487 | An infinite sum of nonnega... |
| isumltss 15488 | A partial sum of a series ... |
| climcndslem1 15489 | Lemma for ~ climcnds : bou... |
| climcndslem2 15490 | Lemma for ~ climcnds : bou... |
| climcnds 15491 | The Cauchy condensation te... |
| divrcnv 15492 | The sequence of reciprocal... |
| divcnv 15493 | The sequence of reciprocal... |
| flo1 15494 | The floor function satisfi... |
| divcnvshft 15495 | Limit of a ratio function.... |
| supcvg 15496 | Extract a sequence ` f ` i... |
| infcvgaux1i 15497 | Auxiliary theorem for appl... |
| infcvgaux2i 15498 | Auxiliary theorem for appl... |
| harmonic 15499 | The harmonic series ` H ` ... |
| arisum 15500 | Arithmetic series sum of t... |
| arisum2 15501 | Arithmetic series sum of t... |
| trireciplem 15502 | Lemma for ~ trirecip . Sh... |
| trirecip 15503 | The sum of the reciprocals... |
| expcnv 15504 | A sequence of powers of a ... |
| explecnv 15505 | A sequence of terms conver... |
| geoserg 15506 | The value of the finite ge... |
| geoser 15507 | The value of the finite ge... |
| pwdif 15508 | The difference of two numb... |
| pwm1geoser 15509 | The n-th power of a number... |
| geolim 15510 | The partial sums in the in... |
| geolim2 15511 | The partial sums in the ge... |
| georeclim 15512 | The limit of a geometric s... |
| geo2sum 15513 | The value of the finite ge... |
| geo2sum2 15514 | The value of the finite ge... |
| geo2lim 15515 | The value of the infinite ... |
| geomulcvg 15516 | The geometric series conve... |
| geoisum 15517 | The infinite sum of ` 1 + ... |
| geoisumr 15518 | The infinite sum of recipr... |
| geoisum1 15519 | The infinite sum of ` A ^ ... |
| geoisum1c 15520 | The infinite sum of ` A x.... |
| 0.999... 15521 | The recurring decimal 0.99... |
| geoihalfsum 15522 | Prove that the infinite ge... |
| cvgrat 15523 | Ratio test for convergence... |
| mertenslem1 15524 | Lemma for ~ mertens . (Co... |
| mertenslem2 15525 | Lemma for ~ mertens . (Co... |
| mertens 15526 | Mertens' theorem. If ` A ... |
| prodf 15527 | An infinite product of com... |
| clim2prod 15528 | The limit of an infinite p... |
| clim2div 15529 | The limit of an infinite p... |
| prodfmul 15530 | The product of two infinit... |
| prodf1 15531 | The value of the partial p... |
| prodf1f 15532 | A one-valued infinite prod... |
| prodfclim1 15533 | The constant one product c... |
| prodfn0 15534 | No term of a nonzero infin... |
| prodfrec 15535 | The reciprocal of an infin... |
| prodfdiv 15536 | The quotient of two infini... |
| ntrivcvg 15537 | A non-trivially converging... |
| ntrivcvgn0 15538 | A product that converges t... |
| ntrivcvgfvn0 15539 | Any value of a product seq... |
| ntrivcvgtail 15540 | A tail of a non-trivially ... |
| ntrivcvgmullem 15541 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15542 | The product of two non-tri... |
| prodex 15545 | A product is a set. (Cont... |
| prodeq1f 15546 | Equality theorem for a pro... |
| prodeq1 15547 | Equality theorem for a pro... |
| nfcprod1 15548 | Bound-variable hypothesis ... |
| nfcprod 15549 | Bound-variable hypothesis ... |
| prodeq2w 15550 | Equality theorem for produ... |
| prodeq2ii 15551 | Equality theorem for produ... |
| prodeq2 15552 | Equality theorem for produ... |
| cbvprod 15553 | Change bound variable in a... |
| cbvprodv 15554 | Change bound variable in a... |
| cbvprodi 15555 | Change bound variable in a... |
| prodeq1i 15556 | Equality inference for pro... |
| prodeq2i 15557 | Equality inference for pro... |
| prodeq12i 15558 | Equality inference for pro... |
| prodeq1d 15559 | Equality deduction for pro... |
| prodeq2d 15560 | Equality deduction for pro... |
| prodeq2dv 15561 | Equality deduction for pro... |
| prodeq2sdv 15562 | Equality deduction for pro... |
| 2cprodeq2dv 15563 | Equality deduction for dou... |
| prodeq12dv 15564 | Equality deduction for pro... |
| prodeq12rdv 15565 | Equality deduction for pro... |
| prod2id 15566 | The second class argument ... |
| prodrblem 15567 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15568 | The sequence of partial pr... |
| prodrblem2 15569 | Lemma for ~ prodrb . (Con... |
| prodrb 15570 | Rebase the starting point ... |
| prodmolem3 15571 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15572 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15573 | Lemma for ~ prodmo . (Con... |
| prodmo 15574 | A product has at most one ... |
| zprod 15575 | Series product with index ... |
| iprod 15576 | Series product with an upp... |
| zprodn0 15577 | Nonzero series product wit... |
| iprodn0 15578 | Nonzero series product wit... |
| fprod 15579 | The value of a product ove... |
| fprodntriv 15580 | A non-triviality lemma for... |
| prod0 15581 | A product over the empty s... |
| prod1 15582 | Any product of one over a ... |
| prodfc 15583 | A lemma to facilitate conv... |
| fprodf1o 15584 | Re-index a finite product ... |
| prodss 15585 | Change the index set to a ... |
| fprodss 15586 | Change the index set to a ... |
| fprodser 15587 | A finite product expressed... |
| fprodcl2lem 15588 | Finite product closure lem... |
| fprodcllem 15589 | Finite product closure lem... |
| fprodcl 15590 | Closure of a finite produc... |
| fprodrecl 15591 | Closure of a finite produc... |
| fprodzcl 15592 | Closure of a finite produc... |
| fprodnncl 15593 | Closure of a finite produc... |
| fprodrpcl 15594 | Closure of a finite produc... |
| fprodnn0cl 15595 | Closure of a finite produc... |
| fprodcllemf 15596 | Finite product closure lem... |
| fprodreclf 15597 | Closure of a finite produc... |
| fprodmul 15598 | The product of two finite ... |
| fproddiv 15599 | The quotient of two finite... |
| prodsn 15600 | A product of a singleton i... |
| fprod1 15601 | A finite product of only o... |
| prodsnf 15602 | A product of a singleton i... |
| climprod1 15603 | The limit of a product ove... |
| fprodsplit 15604 | Split a finite product int... |
| fprodm1 15605 | Separate out the last term... |
| fprod1p 15606 | Separate out the first ter... |
| fprodp1 15607 | Multiply in the last term ... |
| fprodm1s 15608 | Separate out the last term... |
| fprodp1s 15609 | Multiply in the last term ... |
| prodsns 15610 | A product of the singleton... |
| fprodfac 15611 | Factorial using product no... |
| fprodabs 15612 | The absolute value of a fi... |
| fprodeq0 15613 | Any finite product contain... |
| fprodshft 15614 | Shift the index of a finit... |
| fprodrev 15615 | Reversal of a finite produ... |
| fprodconst 15616 | The product of constant te... |
| fprodn0 15617 | A finite product of nonzer... |
| fprod2dlem 15618 | Lemma for ~ fprod2d - indu... |
| fprod2d 15619 | Write a double product as ... |
| fprodxp 15620 | Combine two products into ... |
| fprodcnv 15621 | Transform a product region... |
| fprodcom2 15622 | Interchange order of multi... |
| fprodcom 15623 | Interchange product order.... |
| fprod0diag 15624 | Two ways to express "the p... |
| fproddivf 15625 | The quotient of two finite... |
| fprodsplitf 15626 | Split a finite product int... |
| fprodsplitsn 15627 | Separate out a term in a f... |
| fprodsplit1f 15628 | Separate out a term in a f... |
| fprodn0f 15629 | A finite product of nonzer... |
| fprodclf 15630 | Closure of a finite produc... |
| fprodge0 15631 | If all the terms of a fini... |
| fprodeq0g 15632 | Any finite product contain... |
| fprodge1 15633 | If all of the terms of a f... |
| fprodle 15634 | If all the terms of two fi... |
| fprodmodd 15635 | If all factors of two fini... |
| iprodclim 15636 | An infinite product equals... |
| iprodclim2 15637 | A converging product conve... |
| iprodclim3 15638 | The sequence of partial fi... |
| iprodcl 15639 | The product of a non-trivi... |
| iprodrecl 15640 | The product of a non-trivi... |
| iprodmul 15641 | Multiplication of infinite... |
| risefacval 15646 | The value of the rising fa... |
| fallfacval 15647 | The value of the falling f... |
| risefacval2 15648 | One-based value of rising ... |
| fallfacval2 15649 | One-based value of falling... |
| fallfacval3 15650 | A product representation o... |
| risefaccllem 15651 | Lemma for rising factorial... |
| fallfaccllem 15652 | Lemma for falling factoria... |
| risefaccl 15653 | Closure law for rising fac... |
| fallfaccl 15654 | Closure law for falling fa... |
| rerisefaccl 15655 | Closure law for rising fac... |
| refallfaccl 15656 | Closure law for falling fa... |
| nnrisefaccl 15657 | Closure law for rising fac... |
| zrisefaccl 15658 | Closure law for rising fac... |
| zfallfaccl 15659 | Closure law for falling fa... |
| nn0risefaccl 15660 | Closure law for rising fac... |
| rprisefaccl 15661 | Closure law for rising fac... |
| risefallfac 15662 | A relationship between ris... |
| fallrisefac 15663 | A relationship between fal... |
| risefall0lem 15664 | Lemma for ~ risefac0 and ~... |
| risefac0 15665 | The value of the rising fa... |
| fallfac0 15666 | The value of the falling f... |
| risefacp1 15667 | The value of the rising fa... |
| fallfacp1 15668 | The value of the falling f... |
| risefacp1d 15669 | The value of the rising fa... |
| fallfacp1d 15670 | The value of the falling f... |
| risefac1 15671 | The value of rising factor... |
| fallfac1 15672 | The value of falling facto... |
| risefacfac 15673 | Relate rising factorial to... |
| fallfacfwd 15674 | The forward difference of ... |
| 0fallfac 15675 | The value of the zero fall... |
| 0risefac 15676 | The value of the zero risi... |
| binomfallfaclem1 15677 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 15678 | Lemma for ~ binomfallfac .... |
| binomfallfac 15679 | A version of the binomial ... |
| binomrisefac 15680 | A version of the binomial ... |
| fallfacval4 15681 | Represent the falling fact... |
| bcfallfac 15682 | Binomial coefficient in te... |
| fallfacfac 15683 | Relate falling factorial t... |
| bpolylem 15686 | Lemma for ~ bpolyval . (C... |
| bpolyval 15687 | The value of the Bernoulli... |
| bpoly0 15688 | The value of the Bernoulli... |
| bpoly1 15689 | The value of the Bernoulli... |
| bpolycl 15690 | Closure law for Bernoulli ... |
| bpolysum 15691 | A sum for Bernoulli polyno... |
| bpolydiflem 15692 | Lemma for ~ bpolydif . (C... |
| bpolydif 15693 | Calculate the difference b... |
| fsumkthpow 15694 | A closed-form expression f... |
| bpoly2 15695 | The Bernoulli polynomials ... |
| bpoly3 15696 | The Bernoulli polynomials ... |
| bpoly4 15697 | The Bernoulli polynomials ... |
| fsumcube 15698 | Express the sum of cubes i... |
| eftcl 15711 | Closure of a term in the s... |
| reeftcl 15712 | The terms of the series ex... |
| eftabs 15713 | The absolute value of a te... |
| eftval 15714 | The value of a term in the... |
| efcllem 15715 | Lemma for ~ efcl . The se... |
| ef0lem 15716 | The series defining the ex... |
| efval 15717 | Value of the exponential f... |
| esum 15718 | Value of Euler's constant ... |
| eff 15719 | Domain and codomain of the... |
| efcl 15720 | Closure law for the expone... |
| efval2 15721 | Value of the exponential f... |
| efcvg 15722 | The series that defines th... |
| efcvgfsum 15723 | Exponential function conve... |
| reefcl 15724 | The exponential function i... |
| reefcld 15725 | The exponential function i... |
| ere 15726 | Euler's constant ` _e ` = ... |
| ege2le3 15727 | Lemma for ~ egt2lt3 . (Co... |
| ef0 15728 | Value of the exponential f... |
| efcj 15729 | The exponential of a compl... |
| efaddlem 15730 | Lemma for ~ efadd (exponen... |
| efadd 15731 | Sum of exponents law for e... |
| fprodefsum 15732 | Move the exponential funct... |
| efcan 15733 | Cancellation law for expon... |
| efne0 15734 | The exponential of a compl... |
| efneg 15735 | The exponential of the opp... |
| eff2 15736 | The exponential function m... |
| efsub 15737 | Difference of exponents la... |
| efexp 15738 | The exponential of an inte... |
| efzval 15739 | Value of the exponential f... |
| efgt0 15740 | The exponential of a real ... |
| rpefcl 15741 | The exponential of a real ... |
| rpefcld 15742 | The exponential of a real ... |
| eftlcvg 15743 | The tail series of the exp... |
| eftlcl 15744 | Closure of the sum of an i... |
| reeftlcl 15745 | Closure of the sum of an i... |
| eftlub 15746 | An upper bound on the abso... |
| efsep 15747 | Separate out the next term... |
| effsumlt 15748 | The partial sums of the se... |
| eft0val 15749 | The value of the first ter... |
| ef4p 15750 | Separate out the first fou... |
| efgt1p2 15751 | The exponential of a posit... |
| efgt1p 15752 | The exponential of a posit... |
| efgt1 15753 | The exponential of a posit... |
| eflt 15754 | The exponential function o... |
| efle 15755 | The exponential function o... |
| reef11 15756 | The exponential function o... |
| reeff1 15757 | The exponential function m... |
| eflegeo 15758 | The exponential function o... |
| sinval 15759 | Value of the sine function... |
| cosval 15760 | Value of the cosine functi... |
| sinf 15761 | Domain and codomain of the... |
| cosf 15762 | Domain and codomain of the... |
| sincl 15763 | Closure of the sine functi... |
| coscl 15764 | Closure of the cosine func... |
| tanval 15765 | Value of the tangent funct... |
| tancl 15766 | The closure of the tangent... |
| sincld 15767 | Closure of the sine functi... |
| coscld 15768 | Closure of the cosine func... |
| tancld 15769 | Closure of the tangent fun... |
| tanval2 15770 | Express the tangent functi... |
| tanval3 15771 | Express the tangent functi... |
| resinval 15772 | The sine of a real number ... |
| recosval 15773 | The cosine of a real numbe... |
| efi4p 15774 | Separate out the first fou... |
| resin4p 15775 | Separate out the first fou... |
| recos4p 15776 | Separate out the first fou... |
| resincl 15777 | The sine of a real number ... |
| recoscl 15778 | The cosine of a real numbe... |
| retancl 15779 | The closure of the tangent... |
| resincld 15780 | Closure of the sine functi... |
| recoscld 15781 | Closure of the cosine func... |
| retancld 15782 | Closure of the tangent fun... |
| sinneg 15783 | The sine of a negative is ... |
| cosneg 15784 | The cosines of a number an... |
| tanneg 15785 | The tangent of a negative ... |
| sin0 15786 | Value of the sine function... |
| cos0 15787 | Value of the cosine functi... |
| tan0 15788 | The value of the tangent f... |
| efival 15789 | The exponential function i... |
| efmival 15790 | The exponential function i... |
| sinhval 15791 | Value of the hyperbolic si... |
| coshval 15792 | Value of the hyperbolic co... |
| resinhcl 15793 | The hyperbolic sine of a r... |
| rpcoshcl 15794 | The hyperbolic cosine of a... |
| recoshcl 15795 | The hyperbolic cosine of a... |
| retanhcl 15796 | The hyperbolic tangent of ... |
| tanhlt1 15797 | The hyperbolic tangent of ... |
| tanhbnd 15798 | The hyperbolic tangent of ... |
| efeul 15799 | Eulerian representation of... |
| efieq 15800 | The exponentials of two im... |
| sinadd 15801 | Addition formula for sine.... |
| cosadd 15802 | Addition formula for cosin... |
| tanaddlem 15803 | A useful intermediate step... |
| tanadd 15804 | Addition formula for tange... |
| sinsub 15805 | Sine of difference. (Cont... |
| cossub 15806 | Cosine of difference. (Co... |
| addsin 15807 | Sum of sines. (Contribute... |
| subsin 15808 | Difference of sines. (Con... |
| sinmul 15809 | Product of sines can be re... |
| cosmul 15810 | Product of cosines can be ... |
| addcos 15811 | Sum of cosines. (Contribu... |
| subcos 15812 | Difference of cosines. (C... |
| sincossq 15813 | Sine squared plus cosine s... |
| sin2t 15814 | Double-angle formula for s... |
| cos2t 15815 | Double-angle formula for c... |
| cos2tsin 15816 | Double-angle formula for c... |
| sinbnd 15817 | The sine of a real number ... |
| cosbnd 15818 | The cosine of a real numbe... |
| sinbnd2 15819 | The sine of a real number ... |
| cosbnd2 15820 | The cosine of a real numbe... |
| ef01bndlem 15821 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 15822 | Bounds on the sine of a po... |
| cos01bnd 15823 | Bounds on the cosine of a ... |
| cos1bnd 15824 | Bounds on the cosine of 1.... |
| cos2bnd 15825 | Bounds on the cosine of 2.... |
| sinltx 15826 | The sine of a positive rea... |
| sin01gt0 15827 | The sine of a positive rea... |
| cos01gt0 15828 | The cosine of a positive r... |
| sin02gt0 15829 | The sine of a positive rea... |
| sincos1sgn 15830 | The signs of the sine and ... |
| sincos2sgn 15831 | The signs of the sine and ... |
| sin4lt0 15832 | The sine of 4 is negative.... |
| absefi 15833 | The absolute value of the ... |
| absef 15834 | The absolute value of the ... |
| absefib 15835 | A complex number is real i... |
| efieq1re 15836 | A number whose imaginary e... |
| demoivre 15837 | De Moivre's Formula. Proo... |
| demoivreALT 15838 | Alternate proof of ~ demoi... |
| eirrlem 15841 | Lemma for ~ eirr . (Contr... |
| eirr 15842 | ` _e ` is irrational. (Co... |
| egt2lt3 15843 | Euler's constant ` _e ` = ... |
| epos 15844 | Euler's constant ` _e ` is... |
| epr 15845 | Euler's constant ` _e ` is... |
| ene0 15846 | ` _e ` is not 0. (Contrib... |
| ene1 15847 | ` _e ` is not 1. (Contrib... |
| xpnnen 15848 | The Cartesian product of t... |
| znnen 15849 | The set of integers and th... |
| qnnen 15850 | The rational numbers are c... |
| rpnnen2lem1 15851 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 15852 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 15853 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 15854 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 15855 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 15856 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 15857 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 15858 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 15859 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 15860 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 15861 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 15862 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 15863 | The other half of ~ rpnnen... |
| rpnnen 15864 | The cardinality of the con... |
| rexpen 15865 | The real numbers are equin... |
| cpnnen 15866 | The complex numbers are eq... |
| rucALT 15867 | Alternate proof of ~ ruc .... |
| ruclem1 15868 | Lemma for ~ ruc (the reals... |
| ruclem2 15869 | Lemma for ~ ruc . Orderin... |
| ruclem3 15870 | Lemma for ~ ruc . The con... |
| ruclem4 15871 | Lemma for ~ ruc . Initial... |
| ruclem6 15872 | Lemma for ~ ruc . Domain ... |
| ruclem7 15873 | Lemma for ~ ruc . Success... |
| ruclem8 15874 | Lemma for ~ ruc . The int... |
| ruclem9 15875 | Lemma for ~ ruc . The fir... |
| ruclem10 15876 | Lemma for ~ ruc . Every f... |
| ruclem11 15877 | Lemma for ~ ruc . Closure... |
| ruclem12 15878 | Lemma for ~ ruc . The sup... |
| ruclem13 15879 | Lemma for ~ ruc . There i... |
| ruc 15880 | The set of positive intege... |
| resdomq 15881 | The set of rationals is st... |
| aleph1re 15882 | There are at least aleph-o... |
| aleph1irr 15883 | There are at least aleph-o... |
| cnso 15884 | The complex numbers can be... |
| sqrt2irrlem 15885 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 15886 | The square root of 2 is ir... |
| sqrt2re 15887 | The square root of 2 exist... |
| sqrt2irr0 15888 | The square root of 2 is an... |
| nthruc 15889 | The sequence ` NN ` , ` ZZ... |
| nthruz 15890 | The sequence ` NN ` , ` NN... |
| divides 15893 | Define the divides relatio... |
| dvdsval2 15894 | One nonzero integer divide... |
| dvdsval3 15895 | One nonzero integer divide... |
| dvdszrcl 15896 | Reverse closure for the di... |
| dvdsmod0 15897 | If a positive integer divi... |
| p1modz1 15898 | If a number greater than 1... |
| dvdsmodexp 15899 | If a positive integer divi... |
| nndivdvds 15900 | Strong form of ~ dvdsval2 ... |
| nndivides 15901 | Definition of the divides ... |
| moddvds 15902 | Two ways to say ` A == B `... |
| modm1div 15903 | An integer greater than on... |
| dvds0lem 15904 | A lemma to assist theorems... |
| dvds1lem 15905 | A lemma to assist theorems... |
| dvds2lem 15906 | A lemma to assist theorems... |
| iddvds 15907 | An integer divides itself.... |
| 1dvds 15908 | 1 divides any integer. Th... |
| dvds0 15909 | Any integer divides 0. Th... |
| negdvdsb 15910 | An integer divides another... |
| dvdsnegb 15911 | An integer divides another... |
| absdvdsb 15912 | An integer divides another... |
| dvdsabsb 15913 | An integer divides another... |
| 0dvds 15914 | Only 0 is divisible by 0. ... |
| dvdsmul1 15915 | An integer divides a multi... |
| dvdsmul2 15916 | An integer divides a multi... |
| iddvdsexp 15917 | An integer divides a posit... |
| muldvds1 15918 | If a product divides an in... |
| muldvds2 15919 | If a product divides an in... |
| dvdscmul 15920 | Multiplication by a consta... |
| dvdsmulc 15921 | Multiplication by a consta... |
| dvdscmulr 15922 | Cancellation law for the d... |
| dvdsmulcr 15923 | Cancellation law for the d... |
| summodnegmod 15924 | The sum of two integers mo... |
| modmulconst 15925 | Constant multiplication in... |
| dvds2ln 15926 | If an integer divides each... |
| dvds2add 15927 | If an integer divides each... |
| dvds2sub 15928 | If an integer divides each... |
| dvds2addd 15929 | Deduction form of ~ dvds2a... |
| dvds2subd 15930 | Deduction form of ~ dvds2s... |
| dvdstr 15931 | The divides relation is tr... |
| dvdstrd 15932 | The divides relation is tr... |
| dvdsmultr1 15933 | If an integer divides anot... |
| dvdsmultr1d 15934 | Deduction form of ~ dvdsmu... |
| dvdsmultr2 15935 | If an integer divides anot... |
| dvdsmultr2d 15936 | Deduction form of ~ dvdsmu... |
| ordvdsmul 15937 | If an integer divides eith... |
| dvdssub2 15938 | If an integer divides a di... |
| dvdsadd 15939 | An integer divides another... |
| dvdsaddr 15940 | An integer divides another... |
| dvdssub 15941 | An integer divides another... |
| dvdssubr 15942 | An integer divides another... |
| dvdsadd2b 15943 | Adding a multiple of the b... |
| dvdsaddre2b 15944 | Adding a multiple of the b... |
| fsumdvds 15945 | If every term in a sum is ... |
| dvdslelem 15946 | Lemma for ~ dvdsle . (Con... |
| dvdsle 15947 | The divisors of a positive... |
| dvdsleabs 15948 | The divisors of a nonzero ... |
| dvdsleabs2 15949 | Transfer divisibility to a... |
| dvdsabseq 15950 | If two integers divide eac... |
| dvdseq 15951 | If two nonnegative integer... |
| divconjdvds 15952 | If a nonzero integer ` M `... |
| dvdsdivcl 15953 | The complement of a diviso... |
| dvdsflip 15954 | An involution of the divis... |
| dvdsssfz1 15955 | The set of divisors of a n... |
| dvds1 15956 | The only nonnegative integ... |
| alzdvds 15957 | Only 0 is divisible by all... |
| dvdsext 15958 | Poset extensionality for d... |
| fzm1ndvds 15959 | No number between ` 1 ` an... |
| fzo0dvdseq 15960 | Zero is the only one of th... |
| fzocongeq 15961 | Two different elements of ... |
| addmodlteqALT 15962 | Two nonnegative integers l... |
| dvdsfac 15963 | A positive integer divides... |
| dvdsexp2im 15964 | If an integer divides anot... |
| dvdsexp 15965 | A power divides a power wi... |
| dvdsmod 15966 | Any number ` K ` whose mod... |
| mulmoddvds 15967 | If an integer is divisible... |
| 3dvds 15968 | A rule for divisibility by... |
| 3dvdsdec 15969 | A decimal number is divisi... |
| 3dvds2dec 15970 | A decimal number is divisi... |
| fprodfvdvdsd 15971 | A finite product of intege... |
| fproddvdsd 15972 | A finite product of intege... |
| evenelz 15973 | An even number is an integ... |
| zeo3 15974 | An integer is even or odd.... |
| zeo4 15975 | An integer is even or odd ... |
| zeneo 15976 | No even integer equals an ... |
| odd2np1lem 15977 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 15978 | An integer is odd iff it i... |
| even2n 15979 | An integer is even iff it ... |
| oddm1even 15980 | An integer is odd iff its ... |
| oddp1even 15981 | An integer is odd iff its ... |
| oexpneg 15982 | The exponential of the neg... |
| mod2eq0even 15983 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 15984 | An integer is 1 modulo 2 i... |
| oddnn02np1 15985 | A nonnegative integer is o... |
| oddge22np1 15986 | An integer greater than on... |
| evennn02n 15987 | A nonnegative integer is e... |
| evennn2n 15988 | A positive integer is even... |
| 2tp1odd 15989 | A number which is twice an... |
| mulsucdiv2z 15990 | An integer multiplied with... |
| sqoddm1div8z 15991 | A squared odd number minus... |
| 2teven 15992 | A number which is twice an... |
| zeo5 15993 | An integer is either even ... |
| evend2 15994 | An integer is even iff its... |
| oddp1d2 15995 | An integer is odd iff its ... |
| zob 15996 | Alternate characterization... |
| oddm1d2 15997 | An integer is odd iff its ... |
| ltoddhalfle 15998 | An integer is less than ha... |
| halfleoddlt 15999 | An integer is greater than... |
| opoe 16000 | The sum of two odds is eve... |
| omoe 16001 | The difference of two odds... |
| opeo 16002 | The sum of an odd and an e... |
| omeo 16003 | The difference of an odd a... |
| z0even 16004 | 2 divides 0. That means 0... |
| n2dvds1 16005 | 2 does not divide 1. That... |
| n2dvdsm1 16006 | 2 does not divide -1. Tha... |
| z2even 16007 | 2 divides 2. That means 2... |
| n2dvds3 16008 | 2 does not divide 3. That... |
| z4even 16009 | 2 divides 4. That means 4... |
| 4dvdseven 16010 | An integer which is divisi... |
| m1expe 16011 | Exponentiation of -1 by an... |
| m1expo 16012 | Exponentiation of -1 by an... |
| m1exp1 16013 | Exponentiation of negative... |
| nn0enne 16014 | A positive integer is an e... |
| nn0ehalf 16015 | The half of an even nonneg... |
| nnehalf 16016 | The half of an even positi... |
| nn0onn 16017 | An odd nonnegative integer... |
| nn0o1gt2 16018 | An odd nonnegative integer... |
| nno 16019 | An alternate characterizat... |
| nn0o 16020 | An alternate characterizat... |
| nn0ob 16021 | Alternate characterization... |
| nn0oddm1d2 16022 | A positive integer is odd ... |
| nnoddm1d2 16023 | A positive integer is odd ... |
| sumeven 16024 | If every term in a sum is ... |
| sumodd 16025 | If every term in a sum is ... |
| evensumodd 16026 | If every term in a sum wit... |
| oddsumodd 16027 | If every term in a sum wit... |
| pwp1fsum 16028 | The n-th power of a number... |
| oddpwp1fsum 16029 | An odd power of a number i... |
| divalglem0 16030 | Lemma for ~ divalg . (Con... |
| divalglem1 16031 | Lemma for ~ divalg . (Con... |
| divalglem2 16032 | Lemma for ~ divalg . (Con... |
| divalglem4 16033 | Lemma for ~ divalg . (Con... |
| divalglem5 16034 | Lemma for ~ divalg . (Con... |
| divalglem6 16035 | Lemma for ~ divalg . (Con... |
| divalglem7 16036 | Lemma for ~ divalg . (Con... |
| divalglem8 16037 | Lemma for ~ divalg . (Con... |
| divalglem9 16038 | Lemma for ~ divalg . (Con... |
| divalglem10 16039 | Lemma for ~ divalg . (Con... |
| divalg 16040 | The division algorithm (th... |
| divalgb 16041 | Express the division algor... |
| divalg2 16042 | The division algorithm (th... |
| divalgmod 16043 | The result of the ` mod ` ... |
| divalgmodcl 16044 | The result of the ` mod ` ... |
| modremain 16045 | The result of the modulo o... |
| ndvdssub 16046 | Corollary of the division ... |
| ndvdsadd 16047 | Corollary of the division ... |
| ndvdsp1 16048 | Special case of ~ ndvdsadd... |
| ndvdsi 16049 | A quick test for non-divis... |
| flodddiv4 16050 | The floor of an odd intege... |
| fldivndvdslt 16051 | The floor of an integer di... |
| flodddiv4lt 16052 | The floor of an odd number... |
| flodddiv4t2lthalf 16053 | The floor of an odd number... |
| bitsfval 16058 | Expand the definition of t... |
| bitsval 16059 | Expand the definition of t... |
| bitsval2 16060 | Expand the definition of t... |
| bitsss 16061 | The set of bits of an inte... |
| bitsf 16062 | The ` bits ` function is a... |
| bits0 16063 | Value of the zeroth bit. ... |
| bits0e 16064 | The zeroth bit of an even ... |
| bits0o 16065 | The zeroth bit of an odd n... |
| bitsp1 16066 | The ` M + 1 ` -th bit of `... |
| bitsp1e 16067 | The ` M + 1 ` -th bit of `... |
| bitsp1o 16068 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 16069 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 16070 | The bits of a number are a... |
| bitsmod 16071 | Truncating the bit sequenc... |
| bitsfi 16072 | Every number is associated... |
| bitscmp 16073 | The bit complement of ` N ... |
| 0bits 16074 | The bits of zero. (Contri... |
| m1bits 16075 | The bits of negative one. ... |
| bitsinv1lem 16076 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 16077 | There is an explicit inver... |
| bitsinv2 16078 | There is an explicit inver... |
| bitsf1ocnv 16079 | The ` bits ` function rest... |
| bitsf1o 16080 | The ` bits ` function rest... |
| bitsf1 16081 | The ` bits ` function is a... |
| 2ebits 16082 | The bits of a power of two... |
| bitsinv 16083 | The inverse of the ` bits ... |
| bitsinvp1 16084 | Recursive definition of th... |
| sadadd2lem2 16085 | The core of the proof of ~... |
| sadfval 16087 | Define the addition of two... |
| sadcf 16088 | The carry sequence is a se... |
| sadc0 16089 | The initial element of the... |
| sadcp1 16090 | The carry sequence (which ... |
| sadval 16091 | The full adder sequence is... |
| sadcaddlem 16092 | Lemma for ~ sadcadd . (Co... |
| sadcadd 16093 | Non-recursive definition o... |
| sadadd2lem 16094 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 16095 | Sum of initial segments of... |
| sadadd3 16096 | Sum of initial segments of... |
| sadcl 16097 | The sum of two sequences i... |
| sadcom 16098 | The adder sequence functio... |
| saddisjlem 16099 | Lemma for ~ sadadd . (Con... |
| saddisj 16100 | The sum of disjoint sequen... |
| sadaddlem 16101 | Lemma for ~ sadadd . (Con... |
| sadadd 16102 | For sequences that corresp... |
| sadid1 16103 | The adder sequence functio... |
| sadid2 16104 | The adder sequence functio... |
| sadasslem 16105 | Lemma for ~ sadass . (Con... |
| sadass 16106 | Sequence addition is assoc... |
| sadeq 16107 | Any element of a sequence ... |
| bitsres 16108 | Restrict the bits of a num... |
| bitsuz 16109 | The bits of a number are a... |
| bitsshft 16110 | Shifting a bit sequence to... |
| smufval 16112 | The multiplication of two ... |
| smupf 16113 | The sequence of partial su... |
| smup0 16114 | The initial element of the... |
| smupp1 16115 | The initial element of the... |
| smuval 16116 | Define the addition of two... |
| smuval2 16117 | The partial sum sequence s... |
| smupvallem 16118 | If ` A ` only has elements... |
| smucl 16119 | The product of two sequenc... |
| smu01lem 16120 | Lemma for ~ smu01 and ~ sm... |
| smu01 16121 | Multiplication of a sequen... |
| smu02 16122 | Multiplication of a sequen... |
| smupval 16123 | Rewrite the elements of th... |
| smup1 16124 | Rewrite ~ smupp1 using onl... |
| smueqlem 16125 | Any element of a sequence ... |
| smueq 16126 | Any element of a sequence ... |
| smumullem 16127 | Lemma for ~ smumul . (Con... |
| smumul 16128 | For sequences that corresp... |
| gcdval 16131 | The value of the ` gcd ` o... |
| gcd0val 16132 | The value, by convention, ... |
| gcdn0val 16133 | The value of the ` gcd ` o... |
| gcdcllem1 16134 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 16135 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 16136 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 16137 | Closure of the ` gcd ` ope... |
| gcddvds 16138 | The gcd of two integers di... |
| dvdslegcd 16139 | An integer which divides b... |
| nndvdslegcd 16140 | A positive integer which d... |
| gcdcl 16141 | Closure of the ` gcd ` ope... |
| gcdnncl 16142 | Closure of the ` gcd ` ope... |
| gcdcld 16143 | Closure of the ` gcd ` ope... |
| gcd2n0cl 16144 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 16145 | An integer is the product ... |
| divgcdz 16146 | An integer divided by the ... |
| gcdf 16147 | Domain and codomain of the... |
| gcdcom 16148 | The ` gcd ` operator is co... |
| gcdcomd 16149 | The ` gcd ` operator is co... |
| divgcdnn 16150 | A positive integer divided... |
| divgcdnnr 16151 | A positive integer divided... |
| gcdeq0 16152 | The gcd of two integers is... |
| gcdn0gt0 16153 | The gcd of two integers is... |
| gcd0id 16154 | The gcd of 0 and an intege... |
| gcdid0 16155 | The gcd of an integer and ... |
| nn0gcdid0 16156 | The gcd of a nonnegative i... |
| gcdneg 16157 | Negating one operand of th... |
| neggcd 16158 | Negating one operand of th... |
| gcdaddmlem 16159 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 16160 | Adding a multiple of one o... |
| gcdadd 16161 | The GCD of two numbers is ... |
| gcdid 16162 | The gcd of a number and it... |
| gcd1 16163 | The gcd of a number with 1... |
| gcdabs1 16164 | ` gcd ` of the absolute va... |
| gcdabs2 16165 | ` gcd ` of the absolute va... |
| gcdabs 16166 | The gcd of two integers is... |
| gcdabsOLD 16167 | Obsolete version of ~ gcda... |
| modgcd 16168 | The gcd remains unchanged ... |
| 1gcd 16169 | The GCD of one and an inte... |
| gcdmultipled 16170 | The greatest common diviso... |
| gcdmultiplez 16171 | The GCD of a multiple of a... |
| gcdmultiple 16172 | The GCD of a multiple of a... |
| dvdsgcdidd 16173 | The greatest common diviso... |
| 6gcd4e2 16174 | The greatest common diviso... |
| bezoutlem1 16175 | Lemma for ~ bezout . (Con... |
| bezoutlem2 16176 | Lemma for ~ bezout . (Con... |
| bezoutlem3 16177 | Lemma for ~ bezout . (Con... |
| bezoutlem4 16178 | Lemma for ~ bezout . (Con... |
| bezout 16179 | Bézout's identity: ... |
| dvdsgcd 16180 | An integer which divides e... |
| dvdsgcdb 16181 | Biconditional form of ~ dv... |
| dfgcd2 16182 | Alternate definition of th... |
| gcdass 16183 | Associative law for ` gcd ... |
| mulgcd 16184 | Distribute multiplication ... |
| absmulgcd 16185 | Distribute absolute value ... |
| mulgcdr 16186 | Reverse distribution law f... |
| gcddiv 16187 | Division law for GCD. (Con... |
| gcdmultipleOLD 16188 | Obsolete proof of ~ gcdmul... |
| gcdmultiplezOLD 16189 | Obsolete proof of ~ gcdmul... |
| gcdzeq 16190 | A positive integer ` A ` i... |
| gcdeq 16191 | ` A ` is equal to its gcd ... |
| dvdssqim 16192 | Unidirectional form of ~ d... |
| dvdsmulgcd 16193 | A divisibility equivalent ... |
| rpmulgcd 16194 | If ` K ` and ` M ` are rel... |
| rplpwr 16195 | If ` A ` and ` B ` are rel... |
| rprpwr 16196 | If ` A ` and ` B ` are rel... |
| rppwr 16197 | If ` A ` and ` B ` are rel... |
| sqgcd 16198 | Square distributes over gc... |
| dvdssqlem 16199 | Lemma for ~ dvdssq . (Con... |
| dvdssq 16200 | Two numbers are divisible ... |
| bezoutr 16201 | Partial converse to ~ bezo... |
| bezoutr1 16202 | Converse of ~ bezout for w... |
| nn0seqcvgd 16203 | A strictly-decreasing nonn... |
| seq1st 16204 | A sequence whose iteration... |
| algr0 16205 | The value of the algorithm... |
| algrf 16206 | An algorithm is a step fun... |
| algrp1 16207 | The value of the algorithm... |
| alginv 16208 | If ` I ` is an invariant o... |
| algcvg 16209 | One way to prove that an a... |
| algcvgblem 16210 | Lemma for ~ algcvgb . (Co... |
| algcvgb 16211 | Two ways of expressing tha... |
| algcvga 16212 | The countdown function ` C... |
| algfx 16213 | If ` F ` reaches a fixed p... |
| eucalgval2 16214 | The value of the step func... |
| eucalgval 16215 | Euclid's Algorithm ~ eucal... |
| eucalgf 16216 | Domain and codomain of the... |
| eucalginv 16217 | The invariant of the step ... |
| eucalglt 16218 | The second member of the s... |
| eucalgcvga 16219 | Once Euclid's Algorithm ha... |
| eucalg 16220 | Euclid's Algorithm compute... |
| lcmval 16225 | Value of the ` lcm ` opera... |
| lcmcom 16226 | The ` lcm ` operator is co... |
| lcm0val 16227 | The value, by convention, ... |
| lcmn0val 16228 | The value of the ` lcm ` o... |
| lcmcllem 16229 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 16230 | Closure of the ` lcm ` ope... |
| dvdslcm 16231 | The lcm of two integers is... |
| lcmledvds 16232 | A positive integer which b... |
| lcmeq0 16233 | The lcm of two integers is... |
| lcmcl 16234 | Closure of the ` lcm ` ope... |
| gcddvdslcm 16235 | The greatest common diviso... |
| lcmneg 16236 | Negating one operand of th... |
| neglcm 16237 | Negating one operand of th... |
| lcmabs 16238 | The lcm of two integers is... |
| lcmgcdlem 16239 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 16240 | The product of two numbers... |
| lcmdvds 16241 | The lcm of two integers di... |
| lcmid 16242 | The lcm of an integer and ... |
| lcm1 16243 | The lcm of an integer and ... |
| lcmgcdnn 16244 | The product of two positiv... |
| lcmgcdeq 16245 | Two integers' absolute val... |
| lcmdvdsb 16246 | Biconditional form of ~ lc... |
| lcmass 16247 | Associative law for ` lcm ... |
| 3lcm2e6woprm 16248 | The least common multiple ... |
| 6lcm4e12 16249 | The least common multiple ... |
| absproddvds 16250 | The absolute value of the ... |
| absprodnn 16251 | The absolute value of the ... |
| fissn0dvds 16252 | For each finite subset of ... |
| fissn0dvdsn0 16253 | For each finite subset of ... |
| lcmfval 16254 | Value of the ` _lcm ` func... |
| lcmf0val 16255 | The value, by convention, ... |
| lcmfn0val 16256 | The value of the ` _lcm ` ... |
| lcmfnnval 16257 | The value of the ` _lcm ` ... |
| lcmfcllem 16258 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 16259 | Closure of the ` _lcm ` fu... |
| lcmfpr 16260 | The value of the ` _lcm ` ... |
| lcmfcl 16261 | Closure of the ` _lcm ` fu... |
| lcmfnncl 16262 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 16263 | The least common multiple ... |
| dvdslcmf 16264 | The least common multiple ... |
| lcmfledvds 16265 | A positive integer which i... |
| lcmf 16266 | Characterization of the le... |
| lcmf0 16267 | The least common multiple ... |
| lcmfsn 16268 | The least common multiple ... |
| lcmftp 16269 | The least common multiple ... |
| lcmfunsnlem1 16270 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 16271 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 16272 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 16273 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 16274 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 16275 | The least common multiple ... |
| lcmfdvdsb 16276 | Biconditional form of ~ lc... |
| lcmfunsn 16277 | The ` _lcm ` function for ... |
| lcmfun 16278 | The ` _lcm ` function for ... |
| lcmfass 16279 | Associative law for the ` ... |
| lcmf2a3a4e12 16280 | The least common multiple ... |
| lcmflefac 16281 | The least common multiple ... |
| coprmgcdb 16282 | Two positive integers are ... |
| ncoprmgcdne1b 16283 | Two positive integers are ... |
| ncoprmgcdgt1b 16284 | Two positive integers are ... |
| coprmdvds1 16285 | If two positive integers a... |
| coprmdvds 16286 | Euclid's Lemma (see ProofW... |
| coprmdvds2 16287 | If an integer is divisible... |
| mulgcddvds 16288 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 16289 | If ` M ` is relatively pri... |
| qredeq 16290 | Two equal reduced fraction... |
| qredeu 16291 | Every rational number has ... |
| rpmul 16292 | If ` K ` is relatively pri... |
| rpdvds 16293 | If ` K ` is relatively pri... |
| coprmprod 16294 | The product of the element... |
| coprmproddvdslem 16295 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16296 | If a positive integer is d... |
| congr 16297 | Definition of congruence b... |
| divgcdcoprm0 16298 | Integers divided by gcd ar... |
| divgcdcoprmex 16299 | Integers divided by gcd ar... |
| cncongr1 16300 | One direction of the bicon... |
| cncongr2 16301 | The other direction of the... |
| cncongr 16302 | Cancellability of Congruen... |
| cncongrcoprm 16303 | Corollary 1 of Cancellabil... |
| isprm 16306 | The predicate "is a prime ... |
| prmnn 16307 | A prime number is a positi... |
| prmz 16308 | A prime number is an integ... |
| prmssnn 16309 | The prime numbers are a su... |
| prmex 16310 | The set of prime numbers e... |
| 0nprm 16311 | 0 is not a prime number. ... |
| 1nprm 16312 | 1 is not a prime number. ... |
| 1idssfct 16313 | The positive divisors of a... |
| isprm2lem 16314 | Lemma for ~ isprm2 . (Con... |
| isprm2 16315 | The predicate "is a prime ... |
| isprm3 16316 | The predicate "is a prime ... |
| isprm4 16317 | The predicate "is a prime ... |
| prmind2 16318 | A variation on ~ prmind as... |
| prmind 16319 | Perform induction over the... |
| dvdsprime 16320 | If ` M ` divides a prime, ... |
| nprm 16321 | A product of two integers ... |
| nprmi 16322 | An inference for composite... |
| dvdsnprmd 16323 | If a number is divisible b... |
| prm2orodd 16324 | A prime number is either 2... |
| 2prm 16325 | 2 is a prime number. (Con... |
| 2mulprm 16326 | A multiple of two is prime... |
| 3prm 16327 | 3 is a prime number. (Con... |
| 4nprm 16328 | 4 is not a prime number. ... |
| prmuz2 16329 | A prime number is an integ... |
| prmgt1 16330 | A prime number is an integ... |
| prmm2nn0 16331 | Subtracting 2 from a prime... |
| oddprmgt2 16332 | An odd prime is greater th... |
| oddprmge3 16333 | An odd prime is greater th... |
| ge2nprmge4 16334 | A composite integer greate... |
| sqnprm 16335 | A square is never prime. ... |
| dvdsprm 16336 | An integer greater than or... |
| exprmfct 16337 | Every integer greater than... |
| prmdvdsfz 16338 | Each integer greater than ... |
| nprmdvds1 16339 | No prime number divides 1.... |
| isprm5 16340 | One need only check prime ... |
| isprm7 16341 | One need only check prime ... |
| maxprmfct 16342 | The set of prime factors o... |
| divgcdodd 16343 | Either ` A / ( A gcd B ) `... |
| coprm 16344 | A prime number either divi... |
| prmrp 16345 | Unequal prime numbers are ... |
| euclemma 16346 | Euclid's lemma. A prime n... |
| isprm6 16347 | A number is prime iff it s... |
| prmdvdsexp 16348 | A prime divides a positive... |
| prmdvdsexpb 16349 | A prime divides a positive... |
| prmdvdsexpr 16350 | If a prime divides a nonne... |
| prmdvdssq 16351 | Condition for a prime divi... |
| prmdvdssqOLD 16352 | Obsolete version of ~ prmd... |
| prmexpb 16353 | Two positive prime powers ... |
| prmfac1 16354 | The factorial of a number ... |
| rpexp 16355 | If two numbers ` A ` and `... |
| rpexp1i 16356 | Relative primality passes ... |
| rpexp12i 16357 | Relative primality passes ... |
| prmndvdsfaclt 16358 | A prime number does not di... |
| prmdvdsncoprmbd 16359 | Two positive integers are ... |
| ncoprmlnprm 16360 | If two positive integers a... |
| cncongrprm 16361 | Corollary 2 of Cancellabil... |
| isevengcd2 16362 | The predicate "is an even ... |
| isoddgcd1 16363 | The predicate "is an odd n... |
| 3lcm2e6 16364 | The least common multiple ... |
| qnumval 16369 | Value of the canonical num... |
| qdenval 16370 | Value of the canonical den... |
| qnumdencl 16371 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16372 | The canonical numerator of... |
| qdencl 16373 | The canonical denominator ... |
| fnum 16374 | Canonical numerator define... |
| fden 16375 | Canonical denominator defi... |
| qnumdenbi 16376 | Two numbers are the canoni... |
| qnumdencoprm 16377 | The canonical representati... |
| qeqnumdivden 16378 | Recover a rational number ... |
| qmuldeneqnum 16379 | Multiplying a rational by ... |
| divnumden 16380 | Calculate the reduced form... |
| divdenle 16381 | Reducing a quotient never ... |
| qnumgt0 16382 | A rational is positive iff... |
| qgt0numnn 16383 | A rational is positive iff... |
| nn0gcdsq 16384 | Squaring commutes with GCD... |
| zgcdsq 16385 | ~ nn0gcdsq extended to int... |
| numdensq 16386 | Squaring a rational square... |
| numsq 16387 | Square commutes with canon... |
| densq 16388 | Square commutes with canon... |
| qden1elz 16389 | A rational is an integer i... |
| zsqrtelqelz 16390 | If an integer has a ration... |
| nonsq 16391 | Any integer strictly betwe... |
| phival 16396 | Value of the Euler ` phi `... |
| phicl2 16397 | Bounds and closure for the... |
| phicl 16398 | Closure for the value of t... |
| phibndlem 16399 | Lemma for ~ phibnd . (Con... |
| phibnd 16400 | A slightly tighter bound o... |
| phicld 16401 | Closure for the value of t... |
| phi1 16402 | Value of the Euler ` phi `... |
| dfphi2 16403 | Alternate definition of th... |
| hashdvds 16404 | The number of numbers in a... |
| phiprmpw 16405 | Value of the Euler ` phi `... |
| phiprm 16406 | Value of the Euler ` phi `... |
| crth 16407 | The Chinese Remainder Theo... |
| phimullem 16408 | Lemma for ~ phimul . (Con... |
| phimul 16409 | The Euler ` phi ` function... |
| eulerthlem1 16410 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16411 | Lemma for ~ eulerth . (Co... |
| eulerth 16412 | Euler's theorem, a general... |
| fermltl 16413 | Fermat's little theorem. ... |
| prmdiv 16414 | Show an explicit expressio... |
| prmdiveq 16415 | The modular inverse of ` A... |
| prmdivdiv 16416 | The (modular) inverse of t... |
| hashgcdlem 16417 | A correspondence between e... |
| hashgcdeq 16418 | Number of initial positive... |
| phisum 16419 | The divisor sum identity o... |
| odzval 16420 | Value of the order functio... |
| odzcllem 16421 | - Lemma for ~ odzcl , show... |
| odzcl 16422 | The order of a group eleme... |
| odzid 16423 | Any element raised to the ... |
| odzdvds 16424 | The only powers of ` A ` t... |
| odzphi 16425 | The order of any group ele... |
| modprm1div 16426 | A prime number divides an ... |
| m1dvdsndvds 16427 | If an integer minus 1 is d... |
| modprminv 16428 | Show an explicit expressio... |
| modprminveq 16429 | The modular inverse of ` A... |
| vfermltl 16430 | Variant of Fermat's little... |
| vfermltlALT 16431 | Alternate proof of ~ vferm... |
| powm2modprm 16432 | If an integer minus 1 is d... |
| reumodprminv 16433 | For any prime number and f... |
| modprm0 16434 | For two positive integers ... |
| nnnn0modprm0 16435 | For a positive integer and... |
| modprmn0modprm0 16436 | For an integer not being 0... |
| coprimeprodsq 16437 | If three numbers are copri... |
| coprimeprodsq2 16438 | If three numbers are copri... |
| oddprm 16439 | A prime not equal to ` 2 `... |
| nnoddn2prm 16440 | A prime not equal to ` 2 `... |
| oddn2prm 16441 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16442 | A number is a prime number... |
| prm23lt5 16443 | A prime less than 5 is eit... |
| prm23ge5 16444 | A prime is either 2 or 3 o... |
| pythagtriplem1 16445 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16446 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16447 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16448 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16449 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16450 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16451 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16452 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16453 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16454 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16455 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16456 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16457 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16458 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16459 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16460 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16461 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16462 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16463 | Parameterize the Pythagore... |
| iserodd 16464 | Collect the odd terms in a... |
| pclem 16467 | - Lemma for the prime powe... |
| pcprecl 16468 | Closure of the prime power... |
| pcprendvds 16469 | Non-divisibility property ... |
| pcprendvds2 16470 | Non-divisibility property ... |
| pcpre1 16471 | Value of the prime power p... |
| pcpremul 16472 | Multiplicative property of... |
| pcval 16473 | The value of the prime pow... |
| pceulem 16474 | Lemma for ~ pceu . (Contr... |
| pceu 16475 | Uniqueness for the prime p... |
| pczpre 16476 | Connect the prime count pr... |
| pczcl 16477 | Closure of the prime power... |
| pccl 16478 | Closure of the prime power... |
| pccld 16479 | Closure of the prime power... |
| pcmul 16480 | Multiplication property of... |
| pcdiv 16481 | Division property of the p... |
| pcqmul 16482 | Multiplication property of... |
| pc0 16483 | The value of the prime pow... |
| pc1 16484 | Value of the prime count f... |
| pcqcl 16485 | Closure of the general pri... |
| pcqdiv 16486 | Division property of the p... |
| pcrec 16487 | Prime power of a reciproca... |
| pcexp 16488 | Prime power of an exponent... |
| pcxnn0cl 16489 | Extended nonnegative integ... |
| pcxcl 16490 | Extended real closure of t... |
| pcge0 16491 | The prime count of an inte... |
| pczdvds 16492 | Defining property of the p... |
| pcdvds 16493 | Defining property of the p... |
| pczndvds 16494 | Defining property of the p... |
| pcndvds 16495 | Defining property of the p... |
| pczndvds2 16496 | The remainder after dividi... |
| pcndvds2 16497 | The remainder after dividi... |
| pcdvdsb 16498 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16499 | There are a positive numbe... |
| pceq0 16500 | There are zero powers of a... |
| pcidlem 16501 | The prime count of a prime... |
| pcid 16502 | The prime count of a prime... |
| pcneg 16503 | The prime count of a negat... |
| pcabs 16504 | The prime count of an abso... |
| pcdvdstr 16505 | The prime count increases ... |
| pcgcd1 16506 | The prime count of a GCD i... |
| pcgcd 16507 | The prime count of a GCD i... |
| pc2dvds 16508 | A characterization of divi... |
| pc11 16509 | The prime count function, ... |
| pcz 16510 | The prime count function c... |
| pcprmpw2 16511 | Self-referential expressio... |
| pcprmpw 16512 | Self-referential expressio... |
| dvdsprmpweq 16513 | If a positive integer divi... |
| dvdsprmpweqnn 16514 | If an integer greater than... |
| dvdsprmpweqle 16515 | If a positive integer divi... |
| difsqpwdvds 16516 | If the difference of two s... |
| pcaddlem 16517 | Lemma for ~ pcadd . The o... |
| pcadd 16518 | An inequality for the prim... |
| pcadd2 16519 | The inequality of ~ pcadd ... |
| pcmptcl 16520 | Closure for the prime powe... |
| pcmpt 16521 | Construct a function with ... |
| pcmpt2 16522 | Dividing two prime count m... |
| pcmptdvds 16523 | The partial products of th... |
| pcprod 16524 | The product of the primes ... |
| sumhash 16525 | The sum of 1 over a set is... |
| fldivp1 16526 | The difference between the... |
| pcfaclem 16527 | Lemma for ~ pcfac . (Cont... |
| pcfac 16528 | Calculate the prime count ... |
| pcbc 16529 | Calculate the prime count ... |
| qexpz 16530 | If a power of a rational n... |
| expnprm 16531 | A second or higher power o... |
| oddprmdvds 16532 | Every positive integer whi... |
| prmpwdvds 16533 | A relation involving divis... |
| pockthlem 16534 | Lemma for ~ pockthg . (Co... |
| pockthg 16535 | The generalized Pocklingto... |
| pockthi 16536 | Pocklington's theorem, whi... |
| unbenlem 16537 | Lemma for ~ unben . (Cont... |
| unben 16538 | An unbounded set of positi... |
| infpnlem1 16539 | Lemma for ~ infpn . The s... |
| infpnlem2 16540 | Lemma for ~ infpn . For a... |
| infpn 16541 | There exist infinitely man... |
| infpn2 16542 | There exist infinitely man... |
| prmunb 16543 | The primes are unbounded. ... |
| prminf 16544 | There are an infinite numb... |
| prmreclem1 16545 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16546 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16547 | Lemma for ~ prmrec . The ... |
| prmreclem4 16548 | Lemma for ~ prmrec . Show... |
| prmreclem5 16549 | Lemma for ~ prmrec . Here... |
| prmreclem6 16550 | Lemma for ~ prmrec . If t... |
| prmrec 16551 | The sum of the reciprocals... |
| 1arithlem1 16552 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16553 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16554 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16555 | Lemma for ~ 1arith . (Con... |
| 1arith 16556 | Fundamental theorem of ari... |
| 1arith2 16557 | Fundamental theorem of ari... |
| elgz 16560 | Elementhood in the gaussia... |
| gzcn 16561 | A gaussian integer is a co... |
| zgz 16562 | An integer is a gaussian i... |
| igz 16563 | ` _i ` is a gaussian integ... |
| gznegcl 16564 | The gaussian integers are ... |
| gzcjcl 16565 | The gaussian integers are ... |
| gzaddcl 16566 | The gaussian integers are ... |
| gzmulcl 16567 | The gaussian integers are ... |
| gzreim 16568 | Construct a gaussian integ... |
| gzsubcl 16569 | The gaussian integers are ... |
| gzabssqcl 16570 | The squared norm of a gaus... |
| 4sqlem5 16571 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16572 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16573 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16574 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16575 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16576 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16577 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16578 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16579 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 16580 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 16581 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 16582 | Lemma for ~ mul4sq : algeb... |
| mul4sq 16583 | Euler's four-square identi... |
| 4sqlem11 16584 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 16585 | Lemma for ~ 4sq . For any... |
| 4sqlem13 16586 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 16587 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 16588 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 16589 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 16590 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 16591 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 16592 | Lemma for ~ 4sq . The pro... |
| 4sq 16593 | Lagrange's four-square the... |
| vdwapfval 16600 | Define the arithmetic prog... |
| vdwapf 16601 | The arithmetic progression... |
| vdwapval 16602 | Value of the arithmetic pr... |
| vdwapun 16603 | Remove the first element o... |
| vdwapid1 16604 | The first element of an ar... |
| vdwap0 16605 | Value of a length-1 arithm... |
| vdwap1 16606 | Value of a length-1 arithm... |
| vdwmc 16607 | The predicate " The ` <. R... |
| vdwmc2 16608 | Expand out the definition ... |
| vdwpc 16609 | The predicate " The colori... |
| vdwlem1 16610 | Lemma for ~ vdw . (Contri... |
| vdwlem2 16611 | Lemma for ~ vdw . (Contri... |
| vdwlem3 16612 | Lemma for ~ vdw . (Contri... |
| vdwlem4 16613 | Lemma for ~ vdw . (Contri... |
| vdwlem5 16614 | Lemma for ~ vdw . (Contri... |
| vdwlem6 16615 | Lemma for ~ vdw . (Contri... |
| vdwlem7 16616 | Lemma for ~ vdw . (Contri... |
| vdwlem8 16617 | Lemma for ~ vdw . (Contri... |
| vdwlem9 16618 | Lemma for ~ vdw . (Contri... |
| vdwlem10 16619 | Lemma for ~ vdw . Set up ... |
| vdwlem11 16620 | Lemma for ~ vdw . (Contri... |
| vdwlem12 16621 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 16622 | Lemma for ~ vdw . Main in... |
| vdw 16623 | Van der Waerden's theorem.... |
| vdwnnlem1 16624 | Corollary of ~ vdw , and l... |
| vdwnnlem2 16625 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 16626 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 16627 | Van der Waerden's theorem,... |
| ramtlecl 16629 | The set ` T ` of numbers w... |
| hashbcval 16631 | Value of the "binomial set... |
| hashbccl 16632 | The binomial set is a fini... |
| hashbcss 16633 | Subset relation for the bi... |
| hashbc0 16634 | The set of subsets of size... |
| hashbc2 16635 | The size of the binomial s... |
| 0hashbc 16636 | There are no subsets of th... |
| ramval 16637 | The value of the Ramsey nu... |
| ramcl2lem 16638 | Lemma for extended real cl... |
| ramtcl 16639 | The Ramsey number has the ... |
| ramtcl2 16640 | The Ramsey number is an in... |
| ramtub 16641 | The Ramsey number is a low... |
| ramub 16642 | The Ramsey number is a low... |
| ramub2 16643 | It is sufficient to check ... |
| rami 16644 | The defining property of a... |
| ramcl2 16645 | The Ramsey number is eithe... |
| ramxrcl 16646 | The Ramsey number is an ex... |
| ramubcl 16647 | If the Ramsey number is up... |
| ramlb 16648 | Establish a lower bound on... |
| 0ram 16649 | The Ramsey number when ` M... |
| 0ram2 16650 | The Ramsey number when ` M... |
| ram0 16651 | The Ramsey number when ` R... |
| 0ramcl 16652 | Lemma for ~ ramcl : Exist... |
| ramz2 16653 | The Ramsey number when ` F... |
| ramz 16654 | The Ramsey number when ` F... |
| ramub1lem1 16655 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 16656 | Lemma for ~ ramub1 . (Con... |
| ramub1 16657 | Inductive step for Ramsey'... |
| ramcl 16658 | Ramsey's theorem: the Rams... |
| ramsey 16659 | Ramsey's theorem with the ... |
| prmoval 16662 | Value of the primorial fun... |
| prmocl 16663 | Closure of the primorial f... |
| prmone0 16664 | The primorial function is ... |
| prmo0 16665 | The primorial of 0. (Cont... |
| prmo1 16666 | The primorial of 1. (Cont... |
| prmop1 16667 | The primorial of a success... |
| prmonn2 16668 | Value of the primorial fun... |
| prmo2 16669 | The primorial of 2. (Cont... |
| prmo3 16670 | The primorial of 3. (Cont... |
| prmdvdsprmo 16671 | The primorial of a number ... |
| prmdvdsprmop 16672 | The primorial of a number ... |
| fvprmselelfz 16673 | The value of the prime sel... |
| fvprmselgcd1 16674 | The greatest common diviso... |
| prmolefac 16675 | The primorial of a positiv... |
| prmodvdslcmf 16676 | The primorial of a nonnega... |
| prmolelcmf 16677 | The primorial of a positiv... |
| prmgaplem1 16678 | Lemma for ~ prmgap : The ... |
| prmgaplem2 16679 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 16680 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 16681 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 16682 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 16683 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 16684 | Lemma for ~ prmgap : for e... |
| prmgaplem6 16685 | Lemma for ~ prmgap : for e... |
| prmgaplem7 16686 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 16687 | Lemma for ~ prmgap . (Con... |
| prmgap 16688 | The prime gap theorem: for... |
| prmgaplcm 16689 | Alternate proof of ~ prmga... |
| prmgapprmolem 16690 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 16691 | Alternate proof of ~ prmga... |
| dec2dvds 16692 | Divisibility by two is obv... |
| dec5dvds 16693 | Divisibility by five is ob... |
| dec5dvds2 16694 | Divisibility by five is ob... |
| dec5nprm 16695 | Divisibility by five is ob... |
| dec2nprm 16696 | Divisibility by two is obv... |
| modxai 16697 | Add exponents in a power m... |
| mod2xi 16698 | Double exponents in a powe... |
| modxp1i 16699 | Add one to an exponent in ... |
| mod2xnegi 16700 | Version of ~ mod2xi with a... |
| modsubi 16701 | Subtract from within a mod... |
| gcdi 16702 | Calculate a GCD via Euclid... |
| gcdmodi 16703 | Calculate a GCD via Euclid... |
| decexp2 16704 | Calculate a power of two. ... |
| numexp0 16705 | Calculate an integer power... |
| numexp1 16706 | Calculate an integer power... |
| numexpp1 16707 | Calculate an integer power... |
| numexp2x 16708 | Double an integer power. ... |
| decsplit0b 16709 | Split a decimal number int... |
| decsplit0 16710 | Split a decimal number int... |
| decsplit1 16711 | Split a decimal number int... |
| decsplit 16712 | Split a decimal number int... |
| karatsuba 16713 | The Karatsuba multiplicati... |
| 2exp4 16714 | Two to the fourth power is... |
| 2exp5 16715 | Two to the fifth power is ... |
| 2exp6 16716 | Two to the sixth power is ... |
| 2exp7 16717 | Two to the seventh power i... |
| 2exp8 16718 | Two to the eighth power is... |
| 2exp11 16719 | Two to the eleventh power ... |
| 2exp16 16720 | Two to the sixteenth power... |
| 3exp3 16721 | Three to the third power i... |
| 2expltfac 16722 | The factorial grows faster... |
| cshwsidrepsw 16723 | If cyclically shifting a w... |
| cshwsidrepswmod0 16724 | If cyclically shifting a w... |
| cshwshashlem1 16725 | If cyclically shifting a w... |
| cshwshashlem2 16726 | If cyclically shifting a w... |
| cshwshashlem3 16727 | If cyclically shifting a w... |
| cshwsdisj 16728 | The singletons resulting b... |
| cshwsiun 16729 | The set of (different!) wo... |
| cshwsex 16730 | The class of (different!) ... |
| cshws0 16731 | The size of the set of (di... |
| cshwrepswhash1 16732 | The size of the set of (di... |
| cshwshashnsame 16733 | If a word (not consisting ... |
| cshwshash 16734 | If a word has a length bei... |
| prmlem0 16735 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 16736 | A quick proof skeleton to ... |
| prmlem1 16737 | A quick proof skeleton to ... |
| 5prm 16738 | 5 is a prime number. (Con... |
| 6nprm 16739 | 6 is not a prime number. ... |
| 7prm 16740 | 7 is a prime number. (Con... |
| 8nprm 16741 | 8 is not a prime number. ... |
| 9nprm 16742 | 9 is not a prime number. ... |
| 10nprm 16743 | 10 is not a prime number. ... |
| 11prm 16744 | 11 is a prime number. (Co... |
| 13prm 16745 | 13 is a prime number. (Co... |
| 17prm 16746 | 17 is a prime number. (Co... |
| 19prm 16747 | 19 is a prime number. (Co... |
| 23prm 16748 | 23 is a prime number. (Co... |
| prmlem2 16749 | Our last proving session g... |
| 37prm 16750 | 37 is a prime number. (Co... |
| 43prm 16751 | 43 is a prime number. (Co... |
| 83prm 16752 | 83 is a prime number. (Co... |
| 139prm 16753 | 139 is a prime number. (C... |
| 163prm 16754 | 163 is a prime number. (C... |
| 317prm 16755 | 317 is a prime number. (C... |
| 631prm 16756 | 631 is a prime number. (C... |
| prmo4 16757 | The primorial of 4. (Cont... |
| prmo5 16758 | The primorial of 5. (Cont... |
| prmo6 16759 | The primorial of 6. (Cont... |
| 1259lem1 16760 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 16761 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 16762 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 16763 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 16764 | Lemma for ~ 1259prm . Cal... |
| 1259prm 16765 | 1259 is a prime number. (... |
| 2503lem1 16766 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 16767 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 16768 | Lemma for ~ 2503prm . Cal... |
| 2503prm 16769 | 2503 is a prime number. (... |
| 4001lem1 16770 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 16771 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 16772 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 16773 | Lemma for ~ 4001prm . Cal... |
| 4001prm 16774 | 4001 is a prime number. (... |
| brstruct 16777 | The structure relation is ... |
| isstruct2 16778 | The property of being a st... |
| structex 16779 | A structure is a set. (Co... |
| structn0fun 16780 | A structure without the em... |
| isstruct 16781 | The property of being a st... |
| structcnvcnv 16782 | Two ways to express the re... |
| structfung 16783 | The converse of the conver... |
| structfun 16784 | Convert between two kinds ... |
| structfn 16785 | Convert between two kinds ... |
| strleun 16786 | Combine two structures int... |
| strle1 16787 | Make a structure from a si... |
| strle2 16788 | Make a structure from a pa... |
| strle3 16789 | Make a structure from a tr... |
| sbcie2s 16790 | A special version of class... |
| sbcie3s 16791 | A special version of class... |
| reldmsets 16794 | The structure override ope... |
| setsvalg 16795 | Value of the structure rep... |
| setsval 16796 | Value of the structure rep... |
| fvsetsid 16797 | The value of the structure... |
| fsets 16798 | The structure replacement ... |
| setsdm 16799 | The domain of a structure ... |
| setsfun 16800 | A structure with replaceme... |
| setsfun0 16801 | A structure with replaceme... |
| setsn0fun 16802 | The value of the structure... |
| setsstruct2 16803 | An extensible structure wi... |
| setsexstruct2 16804 | An extensible structure wi... |
| setsstruct 16805 | An extensible structure wi... |
| wunsets 16806 | Closure of structure repla... |
| setsres 16807 | The structure replacement ... |
| setsabs 16808 | Replacing the same compone... |
| setscom 16809 | Component-setting is commu... |
| sloteq 16812 | Equality theorem for the `... |
| slotfn 16813 | A slot is a function on se... |
| strfvnd 16814 | Deduction version of ~ str... |
| strfvn 16815 | Value of a structure compo... |
| strfvss 16816 | A structure component extr... |
| wunstr 16817 | Closure of a structure ind... |
| str0 16818 | All components of the empt... |
| strfvi 16819 | Structure slot extractors ... |
| fveqprc 16820 | Lemma for showing the equa... |
| oveqprc 16821 | Lemma for showing the equa... |
| wunndx 16824 | Closure of the index extra... |
| ndxarg 16825 | Get the numeric argument f... |
| ndxid 16826 | A structure component extr... |
| strndxid 16827 | The value of a structure c... |
| setsidvald 16828 | Value of the structure rep... |
| setsidvaldOLD 16829 | Obsolete version of ~ sets... |
| strfvd 16830 | Deduction version of ~ str... |
| strfv2d 16831 | Deduction version of ~ str... |
| strfv2 16832 | A variation on ~ strfv to ... |
| strfv 16833 | Extract a structure compon... |
| strfv3 16834 | Variant on ~ strfv for lar... |
| strssd 16835 | Deduction version of ~ str... |
| strss 16836 | Propagate component extrac... |
| setsid 16837 | Value of the structure rep... |
| setsnid 16838 | Value of the structure rep... |
| setsnidOLD 16839 | Obsolete proof of ~ setsni... |
| baseval 16842 | Value of the base set extr... |
| baseid 16843 | Utility theorem: index-ind... |
| basfn 16844 | The base set extractor is ... |
| base0 16845 | The base set of the empty ... |
| elbasfv 16846 | Utility theorem: reverse c... |
| elbasov 16847 | Utility theorem: reverse c... |
| strov2rcl 16848 | Partial reverse closure fo... |
| basendx 16849 | Index value of the base se... |
| basendxnn 16850 | The index value of the bas... |
| basendxnnOLD 16851 | Obsolete proof of ~ basend... |
| basndxelwund 16852 | The index of the base set ... |
| basprssdmsets 16853 | The pair of the base index... |
| opelstrbas 16854 | The base set of a structur... |
| 1strstr 16855 | A constructed one-slot str... |
| 1strbas 16856 | The base set of a construc... |
| 1strwunbndx 16857 | A constructed one-slot str... |
| 1strwun 16858 | A constructed one-slot str... |
| 1strwunOLD 16859 | Obsolete version of ~ 1str... |
| 2strstr 16860 | A constructed two-slot str... |
| 2strbas 16861 | The base set of a construc... |
| 2strop 16862 | The other slot of a constr... |
| 2strstr1 16863 | A constructed two-slot str... |
| 2strstr1OLD 16864 | Obsolete version of ~ 2str... |
| 2strbas1 16865 | The base set of a construc... |
| 2strop1 16866 | The other slot of a constr... |
| reldmress 16869 | The structure restriction ... |
| ressval 16870 | Value of structure restric... |
| ressid2 16871 | General behavior of trivia... |
| ressval2 16872 | Value of nontrivial struct... |
| ressbas 16873 | Base set of a structure re... |
| ressbasOLD 16874 | Obsolete proof of ~ ressba... |
| ressbas2 16875 | Base set of a structure re... |
| ressbasss 16876 | The base set of a restrict... |
| resseqnbas 16877 | The components of an exten... |
| resslemOLD 16878 | Obsolete version of ~ ress... |
| ress0 16879 | All restrictions of the nu... |
| ressid 16880 | Behavior of trivial restri... |
| ressinbas 16881 | Restriction only cares abo... |
| ressval3d 16882 | Value of structure restric... |
| ressval3dOLD 16883 | Obsolete version of ~ ress... |
| ressress 16884 | Restriction composition la... |
| ressabs 16885 | Restriction absorption law... |
| wunress 16886 | Closure of structure restr... |
| wunressOLD 16887 | Obsolete proof of ~ wunres... |
| plusgndx 16914 | Index value of the ~ df-pl... |
| plusgid 16915 | Utility theorem: index-ind... |
| plusgndxnn 16916 | The index of the slot for ... |
| basendxltplusgndx 16917 | The index of the slot for ... |
| basendxnplusgndx 16918 | The slot for the base set ... |
| basendxnplusgndxOLD 16919 | Obsolete version of ~ base... |
| grpstr 16920 | A constructed group is a s... |
| grpstrndx 16921 | A constructed group is a s... |
| grpbase 16922 | The base set of a construc... |
| grpbaseOLD 16923 | Obsolete version of ~ grpb... |
| grpplusg 16924 | The operation of a constru... |
| grpplusgOLD 16925 | Obsolete version of ~ grpp... |
| ressplusg 16926 | ` +g ` is unaffected by re... |
| grpbasex 16927 | The base of an explicitly ... |
| grpplusgx 16928 | The operation of an explic... |
| mulrndx 16929 | Index value of the ~ df-mu... |
| mulrid 16930 | Utility theorem: index-ind... |
| basendxnmulrndx 16931 | The slot for the base set ... |
| basendxnmulrndxOLD 16932 | Obsolete proof of ~ basend... |
| plusgndxnmulrndx 16933 | The slot for the group (ad... |
| rngstr 16934 | A constructed ring is a st... |
| rngbase 16935 | The base set of a construc... |
| rngplusg 16936 | The additive operation of ... |
| rngmulr 16937 | The multiplicative operati... |
| starvndx 16938 | Index value of the ~ df-st... |
| starvid 16939 | Utility theorem: index-ind... |
| starvndxnbasendx 16940 | The slot for the involutio... |
| starvndxnplusgndx 16941 | The slot for the involutio... |
| starvndxnmulrndx 16942 | The slot for the involutio... |
| ressmulr 16943 | ` .r ` is unaffected by re... |
| ressstarv 16944 | ` *r ` is unaffected by re... |
| srngstr 16945 | A constructed star ring is... |
| srngbase 16946 | The base set of a construc... |
| srngplusg 16947 | The addition operation of ... |
| srngmulr 16948 | The multiplication operati... |
| srnginvl 16949 | The involution function of... |
| scandx 16950 | Index value of the ~ df-sc... |
| scaid 16951 | Utility theorem: index-ind... |
| scandxnbasendx 16952 | The slot for the scalar is... |
| scandxnplusgndx 16953 | The slot for the scalar fi... |
| scandxnmulrndx 16954 | The slot for the scalar fi... |
| vscandx 16955 | Index value of the ~ df-vs... |
| vscaid 16956 | Utility theorem: index-ind... |
| vscandxnbasendx 16957 | The slot for the scalar pr... |
| vscandxnplusgndx 16958 | The slot for the scalar pr... |
| vscandxnmulrndx 16959 | The slot for the scalar pr... |
| vscandxnscandx 16960 | The slot for the scalar pr... |
| lmodstr 16961 | A constructed left module ... |
| lmodbase 16962 | The base set of a construc... |
| lmodplusg 16963 | The additive operation of ... |
| lmodsca 16964 | The set of scalars of a co... |
| lmodvsca 16965 | The scalar product operati... |
| ipndx 16966 | Index value of the ~ df-ip... |
| ipid 16967 | Utility theorem: index-ind... |
| ipndxnbasendx 16968 | The slot for the inner pro... |
| ipndxnplusgndx 16969 | The slot for the inner pro... |
| ipndxnmulrndx 16970 | The slot for the inner pro... |
| ipsstr 16971 | Lemma to shorten proofs of... |
| ipsbase 16972 | The base set of a construc... |
| ipsaddg 16973 | The additive operation of ... |
| ipsmulr 16974 | The multiplicative operati... |
| ipssca 16975 | The set of scalars of a co... |
| ipsvsca 16976 | The scalar product operati... |
| ipsip 16977 | The multiplicative operati... |
| resssca 16978 | ` Scalar ` is unaffected b... |
| ressvsca 16979 | ` .s ` is unaffected by re... |
| ressip 16980 | The inner product is unaff... |
| phlstr 16981 | A constructed pre-Hilbert ... |
| phlbase 16982 | The base set of a construc... |
| phlplusg 16983 | The additive operation of ... |
| phlsca 16984 | The ring of scalars of a c... |
| phlvsca 16985 | The scalar product operati... |
| phlip 16986 | The inner product (Hermiti... |
| tsetndx 16987 | Index value of the ~ df-ts... |
| tsetid 16988 | Utility theorem: index-ind... |
| tsetndxnn 16989 | The index of the slot for ... |
| basendxlttsetndx 16990 | The index of the slot for ... |
| tsetndxnbasendx 16991 | The slot for the topology ... |
| tsetndxnplusgndx 16992 | The slot for the topology ... |
| tsetndxnmulrndx 16993 | The slot for the topology ... |
| slotstnscsi 16994 | The slots ` Scalar ` , ` .... |
| topgrpstr 16995 | A constructed topological ... |
| topgrpbas 16996 | The base set of a construc... |
| topgrpplusg 16997 | The additive operation of ... |
| topgrptset 16998 | The topology of a construc... |
| resstset 16999 | ` TopSet ` is unaffected b... |
| plendx 17000 | Index value of the ~ df-pl... |
| pleid 17001 | Utility theorem: self-refe... |
| plendxnn 17002 | The index value of the ord... |
| basendxltplendx 17003 | The index value of the ` B... |
| plendxnbasendx 17004 | The slot for the order is ... |
| plendxnplusgndx 17005 | The slot for the "less tha... |
| plendxnmulrndx 17006 | The slot for the "less tha... |
| plendxnscandx 17007 | The slot for the "less tha... |
| plendxnvscandx 17008 | The slot for the "less tha... |
| otpsstr 17009 | Functionality of a topolog... |
| otpsbas 17010 | The base set of a topologi... |
| otpstset 17011 | The open sets of a topolog... |
| otpsle 17012 | The order of a topological... |
| ressle 17013 | ` le ` is unaffected by re... |
| ocndx 17014 | Index value of the ~ df-oc... |
| ocid 17015 | Utility theorem: index-ind... |
| dsndx 17016 | Index value of the ~ df-ds... |
| dsid 17017 | Utility theorem: index-ind... |
| dsndxnn 17018 | The index of the slot for ... |
| basendxltdsndx 17019 | The index of the slot for ... |
| dsndxnbasendx 17020 | The slot for the distance ... |
| dsndxnplusgndx 17021 | The slot for the distance ... |
| dsndxnmulrndx 17022 | The slot for the distance ... |
| slotsdnscsi 17023 | The slots ` Scalar ` , ` .... |
| dsndxntsetndx 17024 | The slot for the distance ... |
| unifndx 17025 | Index value of the ~ df-un... |
| unifid 17026 | Utility theorem: index-ind... |
| unifndxnn 17027 | The index of the slot for ... |
| basendxltunifndx 17028 | The index of the slot for ... |
| unifndxnbasendx 17029 | The slot for the uniform s... |
| unifndxntsetndx 17030 | The slot for the uniform s... |
| ressunif 17031 | ` UnifSet ` is unaffected ... |
| odrngstr 17032 | Functionality of an ordere... |
| odrngbas 17033 | The base set of an ordered... |
| odrngplusg 17034 | The addition operation of ... |
| odrngmulr 17035 | The multiplication operati... |
| odrngtset 17036 | The open sets of an ordere... |
| odrngle 17037 | The order of an ordered me... |
| odrngds 17038 | The metric of an ordered m... |
| ressds 17039 | ` dist ` is unaffected by ... |
| homndx 17040 | Index value of the ~ df-ho... |
| homid 17041 | Utility theorem: index-ind... |
| ccondx 17042 | Index value of the ~ df-cc... |
| ccoid 17043 | Utility theorem: index-ind... |
| slotsbhcdif 17044 | The slots ` Base ` , ` Hom... |
| slotsbhcdifOLD 17045 | Obsolete proof of ~ slotsb... |
| resshom 17046 | ` Hom ` is unaffected by r... |
| ressco 17047 | ` comp ` is unaffected by ... |
| restfn 17052 | The subspace topology oper... |
| topnfn 17053 | The topology extractor fun... |
| restval 17054 | The subspace topology indu... |
| elrest 17055 | The predicate "is an open ... |
| elrestr 17056 | Sufficient condition for b... |
| 0rest 17057 | Value of the structure res... |
| restid2 17058 | The subspace topology over... |
| restsspw 17059 | The subspace topology is a... |
| firest 17060 | The finite intersections o... |
| restid 17061 | The subspace topology of t... |
| topnval 17062 | Value of the topology extr... |
| topnid 17063 | Value of the topology extr... |
| topnpropd 17064 | The topology extractor fun... |
| reldmprds 17076 | The structure product is a... |
| prdsbasex 17078 | Lemma for structure produc... |
| imasvalstr 17079 | An image structure value i... |
| prdsvalstr 17080 | Structure product value is... |
| prdsbaslem 17081 | Lemma for ~ prdsbas and si... |
| prdsvallem 17082 | Lemma for ~ prdsval . (Co... |
| prdsval 17083 | Value of the structure pro... |
| prdssca 17084 | Scalar ring of a structure... |
| prdsbas 17085 | Base set of a structure pr... |
| prdsplusg 17086 | Addition in a structure pr... |
| prdsmulr 17087 | Multiplication in a struct... |
| prdsvsca 17088 | Scalar multiplication in a... |
| prdsip 17089 | Inner product in a structu... |
| prdsle 17090 | Structure product weak ord... |
| prdsless 17091 | Closure of the order relat... |
| prdsds 17092 | Structure product distance... |
| prdsdsfn 17093 | Structure product distance... |
| prdstset 17094 | Structure product topology... |
| prdshom 17095 | Structure product hom-sets... |
| prdsco 17096 | Structure product composit... |
| prdsbas2 17097 | The base set of a structur... |
| prdsbasmpt 17098 | A constructed tuple is a p... |
| prdsbasfn 17099 | Points in the structure pr... |
| prdsbasprj 17100 | Each point in a structure ... |
| prdsplusgval 17101 | Value of a componentwise s... |
| prdsplusgfval 17102 | Value of a structure produ... |
| prdsmulrval 17103 | Value of a componentwise r... |
| prdsmulrfval 17104 | Value of a structure produ... |
| prdsleval 17105 | Value of the product order... |
| prdsdsval 17106 | Value of the metric in a s... |
| prdsvscaval 17107 | Scalar multiplication in a... |
| prdsvscafval 17108 | Scalar multiplication of a... |
| prdsbas3 17109 | The base set of an indexed... |
| prdsbasmpt2 17110 | A constructed tuple is a p... |
| prdsbascl 17111 | An element of the base has... |
| prdsdsval2 17112 | Value of the metric in a s... |
| prdsdsval3 17113 | Value of the metric in a s... |
| pwsval 17114 | Value of a structure power... |
| pwsbas 17115 | Base set of a structure po... |
| pwselbasb 17116 | Membership in the base set... |
| pwselbas 17117 | An element of a structure ... |
| pwsplusgval 17118 | Value of addition in a str... |
| pwsmulrval 17119 | Value of multiplication in... |
| pwsle 17120 | Ordering in a structure po... |
| pwsleval 17121 | Ordering in a structure po... |
| pwsvscafval 17122 | Scalar multiplication in a... |
| pwsvscaval 17123 | Scalar multiplication of a... |
| pwssca 17124 | The ring of scalars of a s... |
| pwsdiagel 17125 | Membership of diagonal ele... |
| pwssnf1o 17126 | Triviality of singleton po... |
| imasval 17139 | Value of an image structur... |
| imasbas 17140 | The base set of an image s... |
| imasds 17141 | The distance function of a... |
| imasdsfn 17142 | The distance function is a... |
| imasdsval 17143 | The distance function of a... |
| imasdsval2 17144 | The distance function of a... |
| imasplusg 17145 | The group operation in an ... |
| imasmulr 17146 | The ring multiplication in... |
| imassca 17147 | The scalar field of an ima... |
| imasvsca 17148 | The scalar multiplication ... |
| imasip 17149 | The inner product of an im... |
| imastset 17150 | The topology of an image s... |
| imasle 17151 | The ordering of an image s... |
| f1ocpbllem 17152 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 17153 | An injection is compatible... |
| f1ovscpbl 17154 | An injection is compatible... |
| f1olecpbl 17155 | An injection is compatible... |
| imasaddfnlem 17156 | The image structure operat... |
| imasaddvallem 17157 | The operation of an image ... |
| imasaddflem 17158 | The image set operations a... |
| imasaddfn 17159 | The image structure's grou... |
| imasaddval 17160 | The value of an image stru... |
| imasaddf 17161 | The image structure's grou... |
| imasmulfn 17162 | The image structure's ring... |
| imasmulval 17163 | The value of an image stru... |
| imasmulf 17164 | The image structure's ring... |
| imasvscafn 17165 | The image structure's scal... |
| imasvscaval 17166 | The value of an image stru... |
| imasvscaf 17167 | The image structure's scal... |
| imasless 17168 | The order relation defined... |
| imasleval 17169 | The value of the image str... |
| qusval 17170 | Value of a quotient struct... |
| quslem 17171 | The function in ~ qusval i... |
| qusin 17172 | Restrict the equivalence r... |
| qusbas 17173 | Base set of a quotient str... |
| quss 17174 | The scalar field of a quot... |
| divsfval 17175 | Value of the function in ~... |
| ercpbllem 17176 | Lemma for ~ ercpbl . (Con... |
| ercpbl 17177 | Translate the function com... |
| erlecpbl 17178 | Translate the relation com... |
| qusaddvallem 17179 | Value of an operation defi... |
| qusaddflem 17180 | The operation of a quotien... |
| qusaddval 17181 | The base set of an image s... |
| qusaddf 17182 | The base set of an image s... |
| qusmulval 17183 | The base set of an image s... |
| qusmulf 17184 | The base set of an image s... |
| fnpr2o 17185 | Function with a domain of ... |
| fnpr2ob 17186 | Biconditional version of ~... |
| fvpr0o 17187 | The value of a function wi... |
| fvpr1o 17188 | The value of a function wi... |
| fvprif 17189 | The value of the pair func... |
| xpsfrnel 17190 | Elementhood in the target ... |
| xpsfeq 17191 | A function on ` 2o ` is de... |
| xpsfrnel2 17192 | Elementhood in the target ... |
| xpscf 17193 | Equivalent condition for t... |
| xpsfval 17194 | The value of the function ... |
| xpsff1o 17195 | The function appearing in ... |
| xpsfrn 17196 | A short expression for the... |
| xpsff1o2 17197 | The function appearing in ... |
| xpsval 17198 | Value of the binary struct... |
| xpsrnbas 17199 | The indexed structure prod... |
| xpsbas 17200 | The base set of the binary... |
| xpsaddlem 17201 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 17202 | Value of the addition oper... |
| xpsmul 17203 | Value of the multiplicatio... |
| xpssca 17204 | Value of the scalar field ... |
| xpsvsca 17205 | Value of the scalar multip... |
| xpsless 17206 | Closure of the ordering in... |
| xpsle 17207 | Value of the ordering in a... |
| ismre 17216 | Property of being a Moore ... |
| fnmre 17217 | The Moore collection gener... |
| mresspw 17218 | A Moore collection is a su... |
| mress 17219 | A Moore-closed subset is a... |
| mre1cl 17220 | In any Moore collection th... |
| mreintcl 17221 | A nonempty collection of c... |
| mreiincl 17222 | A nonempty indexed interse... |
| mrerintcl 17223 | The relative intersection ... |
| mreriincl 17224 | The relative intersection ... |
| mreincl 17225 | Two closed sets have a clo... |
| mreuni 17226 | Since the entire base set ... |
| mreunirn 17227 | Two ways to express the no... |
| ismred 17228 | Properties that determine ... |
| ismred2 17229 | Properties that determine ... |
| mremre 17230 | The Moore collections of s... |
| submre 17231 | The subcollection of a clo... |
| mrcflem 17232 | The domain and range of th... |
| fnmrc 17233 | Moore-closure is a well-be... |
| mrcfval 17234 | Value of the function expr... |
| mrcf 17235 | The Moore closure is a fun... |
| mrcval 17236 | Evaluation of the Moore cl... |
| mrccl 17237 | The Moore closure of a set... |
| mrcsncl 17238 | The Moore closure of a sin... |
| mrcid 17239 | The closure of a closed se... |
| mrcssv 17240 | The closure of a set is a ... |
| mrcidb 17241 | A set is closed iff it is ... |
| mrcss 17242 | Closure preserves subset o... |
| mrcssid 17243 | The closure of a set is a ... |
| mrcidb2 17244 | A set is closed iff it con... |
| mrcidm 17245 | The closure operation is i... |
| mrcsscl 17246 | The closure is the minimal... |
| mrcuni 17247 | Idempotence of closure und... |
| mrcun 17248 | Idempotence of closure und... |
| mrcssvd 17249 | The Moore closure of a set... |
| mrcssd 17250 | Moore closure preserves su... |
| mrcssidd 17251 | A set is contained in its ... |
| mrcidmd 17252 | Moore closure is idempoten... |
| mressmrcd 17253 | In a Moore system, if a se... |
| submrc 17254 | In a closure system which ... |
| mrieqvlemd 17255 | In a Moore system, if ` Y ... |
| mrisval 17256 | Value of the set of indepe... |
| ismri 17257 | Criterion for a set to be ... |
| ismri2 17258 | Criterion for a subset of ... |
| ismri2d 17259 | Criterion for a subset of ... |
| ismri2dd 17260 | Definition of independence... |
| mriss 17261 | An independent set of a Mo... |
| mrissd 17262 | An independent set of a Mo... |
| ismri2dad 17263 | Consequence of a set in a ... |
| mrieqvd 17264 | In a Moore system, a set i... |
| mrieqv2d 17265 | In a Moore system, a set i... |
| mrissmrcd 17266 | In a Moore system, if an i... |
| mrissmrid 17267 | In a Moore system, subsets... |
| mreexd 17268 | In a Moore system, the clo... |
| mreexmrid 17269 | In a Moore system whose cl... |
| mreexexlemd 17270 | This lemma is used to gene... |
| mreexexlem2d 17271 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 17272 | Base case of the induction... |
| mreexexlem4d 17273 | Induction step of the indu... |
| mreexexd 17274 | Exchange-type theorem. In... |
| mreexdomd 17275 | In a Moore system whose cl... |
| mreexfidimd 17276 | In a Moore system whose cl... |
| isacs 17277 | A set is an algebraic clos... |
| acsmre 17278 | Algebraic closure systems ... |
| isacs2 17279 | In the definition of an al... |
| acsfiel 17280 | A set is closed in an alge... |
| acsfiel2 17281 | A set is closed in an alge... |
| acsmred 17282 | An algebraic closure syste... |
| isacs1i 17283 | A closure system determine... |
| mreacs 17284 | Algebraicity is a composab... |
| acsfn 17285 | Algebraicity of a conditio... |
| acsfn0 17286 | Algebraicity of a point cl... |
| acsfn1 17287 | Algebraicity of a one-argu... |
| acsfn1c 17288 | Algebraicity of a one-argu... |
| acsfn2 17289 | Algebraicity of a two-argu... |
| iscat 17298 | The predicate "is a catego... |
| iscatd 17299 | Properties that determine ... |
| catidex 17300 | Each object in a category ... |
| catideu 17301 | Each object in a category ... |
| cidfval 17302 | Each object in a category ... |
| cidval 17303 | Each object in a category ... |
| cidffn 17304 | The identity arrow constru... |
| cidfn 17305 | The identity arrow operato... |
| catidd 17306 | Deduce the identity arrow ... |
| iscatd2 17307 | Version of ~ iscatd with a... |
| catidcl 17308 | Each object in a category ... |
| catlid 17309 | Left identity property of ... |
| catrid 17310 | Right identity property of... |
| catcocl 17311 | Closure of a composition a... |
| catass 17312 | Associativity of compositi... |
| catcone0 17313 | Composition of non-empty h... |
| 0catg 17314 | Any structure with an empt... |
| 0cat 17315 | The empty set is a categor... |
| homffval 17316 | Value of the functionalize... |
| fnhomeqhomf 17317 | If the Hom-set operation i... |
| homfval 17318 | Value of the functionalize... |
| homffn 17319 | The functionalized Hom-set... |
| homfeq 17320 | Condition for two categori... |
| homfeqd 17321 | If two structures have the... |
| homfeqbas 17322 | Deduce equality of base se... |
| homfeqval 17323 | Value of the functionalize... |
| comfffval 17324 | Value of the functionalize... |
| comffval 17325 | Value of the functionalize... |
| comfval 17326 | Value of the functionalize... |
| comfffval2 17327 | Value of the functionalize... |
| comffval2 17328 | Value of the functionalize... |
| comfval2 17329 | Value of the functionalize... |
| comfffn 17330 | The functionalized composi... |
| comffn 17331 | The functionalized composi... |
| comfeq 17332 | Condition for two categori... |
| comfeqd 17333 | Condition for two categori... |
| comfeqval 17334 | Equality of two compositio... |
| catpropd 17335 | Two structures with the sa... |
| cidpropd 17336 | Two structures with the sa... |
| oppcval 17339 | Value of the opposite cate... |
| oppchomfval 17340 | Hom-sets of the opposite c... |
| oppchomfvalOLD 17341 | Obsolete proof of ~ oppcho... |
| oppchom 17342 | Hom-sets of the opposite c... |
| oppccofval 17343 | Composition in the opposit... |
| oppcco 17344 | Composition in the opposit... |
| oppcbas 17345 | Base set of an opposite ca... |
| oppcbasOLD 17346 | Obsolete version of ~ oppc... |
| oppccatid 17347 | Lemma for ~ oppccat . (Co... |
| oppchomf 17348 | Hom-sets of the opposite c... |
| oppcid 17349 | Identity function of an op... |
| oppccat 17350 | An opposite category is a ... |
| 2oppcbas 17351 | The double opposite catego... |
| 2oppchomf 17352 | The double opposite catego... |
| 2oppccomf 17353 | The double opposite catego... |
| oppchomfpropd 17354 | If two categories have the... |
| oppccomfpropd 17355 | If two categories have the... |
| oppccatf 17356 | ` oppCat ` restricted to `... |
| monfval 17361 | Definition of a monomorphi... |
| ismon 17362 | Definition of a monomorphi... |
| ismon2 17363 | Write out the monomorphism... |
| monhom 17364 | A monomorphism is a morphi... |
| moni 17365 | Property of a monomorphism... |
| monpropd 17366 | If two categories have the... |
| oppcmon 17367 | A monomorphism in the oppo... |
| oppcepi 17368 | An epimorphism in the oppo... |
| isepi 17369 | Definition of an epimorphi... |
| isepi2 17370 | Write out the epimorphism ... |
| epihom 17371 | An epimorphism is a morphi... |
| epii 17372 | Property of an epimorphism... |
| sectffval 17379 | Value of the section opera... |
| sectfval 17380 | Value of the section relat... |
| sectss 17381 | The section relation is a ... |
| issect 17382 | The property " ` F ` is a ... |
| issect2 17383 | Property of being a sectio... |
| sectcan 17384 | If ` G ` is a section of `... |
| sectco 17385 | Composition of two section... |
| isofval 17386 | Function value of the func... |
| invffval 17387 | Value of the inverse relat... |
| invfval 17388 | Value of the inverse relat... |
| isinv 17389 | Value of the inverse relat... |
| invss 17390 | The inverse relation is a ... |
| invsym 17391 | The inverse relation is sy... |
| invsym2 17392 | The inverse relation is sy... |
| invfun 17393 | The inverse relation is a ... |
| isoval 17394 | The isomorphisms are the d... |
| inviso1 17395 | If ` G ` is an inverse to ... |
| inviso2 17396 | If ` G ` is an inverse to ... |
| invf 17397 | The inverse relation is a ... |
| invf1o 17398 | The inverse relation is a ... |
| invinv 17399 | The inverse of the inverse... |
| invco 17400 | The composition of two iso... |
| dfiso2 17401 | Alternate definition of an... |
| dfiso3 17402 | Alternate definition of an... |
| inveq 17403 | If there are two inverses ... |
| isofn 17404 | The function value of the ... |
| isohom 17405 | An isomorphism is a homomo... |
| isoco 17406 | The composition of two iso... |
| oppcsect 17407 | A section in the opposite ... |
| oppcsect2 17408 | A section in the opposite ... |
| oppcinv 17409 | An inverse in the opposite... |
| oppciso 17410 | An isomorphism in the oppo... |
| sectmon 17411 | If ` F ` is a section of `... |
| monsect 17412 | If ` F ` is a monomorphism... |
| sectepi 17413 | If ` F ` is a section of `... |
| episect 17414 | If ` F ` is an epimorphism... |
| sectid 17415 | The identity is a section ... |
| invid 17416 | The inverse of the identit... |
| idiso 17417 | The identity is an isomorp... |
| idinv 17418 | The inverse of the identit... |
| invisoinvl 17419 | The inverse of an isomorph... |
| invisoinvr 17420 | The inverse of an isomorph... |
| invcoisoid 17421 | The inverse of an isomorph... |
| isocoinvid 17422 | The inverse of an isomorph... |
| rcaninv 17423 | Right cancellation of an i... |
| cicfval 17426 | The set of isomorphic obje... |
| brcic 17427 | The relation "is isomorphi... |
| cic 17428 | Objects ` X ` and ` Y ` in... |
| brcici 17429 | Prove that two objects are... |
| cicref 17430 | Isomorphism is reflexive. ... |
| ciclcl 17431 | Isomorphism implies the le... |
| cicrcl 17432 | Isomorphism implies the ri... |
| cicsym 17433 | Isomorphism is symmetric. ... |
| cictr 17434 | Isomorphism is transitive.... |
| cicer 17435 | Isomorphism is an equivale... |
| sscrel 17442 | The subcategory subset rel... |
| brssc 17443 | The subcategory subset rel... |
| sscpwex 17444 | An analogue of ~ pwex for ... |
| subcrcl 17445 | Reverse closure for the su... |
| sscfn1 17446 | The subcategory subset rel... |
| sscfn2 17447 | The subcategory subset rel... |
| ssclem 17448 | Lemma for ~ ssc1 and simil... |
| isssc 17449 | Value of the subcategory s... |
| ssc1 17450 | Infer subset relation on o... |
| ssc2 17451 | Infer subset relation on m... |
| sscres 17452 | Any function restricted to... |
| sscid 17453 | The subcategory subset rel... |
| ssctr 17454 | The subcategory subset rel... |
| ssceq 17455 | The subcategory subset rel... |
| rescval 17456 | Value of the category rest... |
| rescval2 17457 | Value of the category rest... |
| rescbas 17458 | Base set of the category r... |
| rescbasOLD 17459 | Obsolete version of ~ resc... |
| reschom 17460 | Hom-sets of the category r... |
| reschomf 17461 | Hom-sets of the category r... |
| rescco 17462 | Composition in the categor... |
| resccoOLD 17463 | Obsolete proof of ~ rescco... |
| rescabs 17464 | Restriction absorption law... |
| rescabs2 17465 | Restriction absorption law... |
| issubc 17466 | Elementhood in the set of ... |
| issubc2 17467 | Elementhood in the set of ... |
| 0ssc 17468 | For any category ` C ` , t... |
| 0subcat 17469 | For any category ` C ` , t... |
| catsubcat 17470 | For any category ` C ` , `... |
| subcssc 17471 | An element in the set of s... |
| subcfn 17472 | An element in the set of s... |
| subcss1 17473 | The objects of a subcatego... |
| subcss2 17474 | The morphisms of a subcate... |
| subcidcl 17475 | The identity of the origin... |
| subccocl 17476 | A subcategory is closed un... |
| subccatid 17477 | A subcategory is a categor... |
| subcid 17478 | The identity in a subcateg... |
| subccat 17479 | A subcategory is a categor... |
| issubc3 17480 | Alternate definition of a ... |
| fullsubc 17481 | The full subcategory gener... |
| fullresc 17482 | The category formed by str... |
| resscat 17483 | A category restricted to a... |
| subsubc 17484 | A subcategory of a subcate... |
| relfunc 17493 | The set of functors is a r... |
| funcrcl 17494 | Reverse closure for a func... |
| isfunc 17495 | Value of the set of functo... |
| isfuncd 17496 | Deduce that an operation i... |
| funcf1 17497 | The object part of a funct... |
| funcixp 17498 | The morphism part of a fun... |
| funcf2 17499 | The morphism part of a fun... |
| funcfn2 17500 | The morphism part of a fun... |
| funcid 17501 | A functor maps each identi... |
| funcco 17502 | A functor maps composition... |
| funcsect 17503 | The image of a section und... |
| funcinv 17504 | The image of an inverse un... |
| funciso 17505 | The image of an isomorphis... |
| funcoppc 17506 | A functor on categories yi... |
| idfuval 17507 | Value of the identity func... |
| idfu2nd 17508 | Value of the morphism part... |
| idfu2 17509 | Value of the morphism part... |
| idfu1st 17510 | Value of the object part o... |
| idfu1 17511 | Value of the object part o... |
| idfucl 17512 | The identity functor is a ... |
| cofuval 17513 | Value of the composition o... |
| cofu1st 17514 | Value of the object part o... |
| cofu1 17515 | Value of the object part o... |
| cofu2nd 17516 | Value of the morphism part... |
| cofu2 17517 | Value of the morphism part... |
| cofuval2 17518 | Value of the composition o... |
| cofucl 17519 | The composition of two fun... |
| cofuass 17520 | Functor composition is ass... |
| cofulid 17521 | The identity functor is a ... |
| cofurid 17522 | The identity functor is a ... |
| resfval 17523 | Value of the functor restr... |
| resfval2 17524 | Value of the functor restr... |
| resf1st 17525 | Value of the functor restr... |
| resf2nd 17526 | Value of the functor restr... |
| funcres 17527 | A functor restricted to a ... |
| funcres2b 17528 | Condition for a functor to... |
| funcres2 17529 | A functor into a restricte... |
| wunfunc 17530 | A weak universe is closed ... |
| wunfuncOLD 17531 | Obsolete proof of ~ wunfun... |
| funcpropd 17532 | If two categories have the... |
| funcres2c 17533 | Condition for a functor to... |
| fullfunc 17538 | A full functor is a functo... |
| fthfunc 17539 | A faithful functor is a fu... |
| relfull 17540 | The set of full functors i... |
| relfth 17541 | The set of faithful functo... |
| isfull 17542 | Value of the set of full f... |
| isfull2 17543 | Equivalent condition for a... |
| fullfo 17544 | The morphism map of a full... |
| fulli 17545 | The morphism map of a full... |
| isfth 17546 | Value of the set of faithf... |
| isfth2 17547 | Equivalent condition for a... |
| isffth2 17548 | A fully faithful functor i... |
| fthf1 17549 | The morphism map of a fait... |
| fthi 17550 | The morphism map of a fait... |
| ffthf1o 17551 | The morphism map of a full... |
| fullpropd 17552 | If two categories have the... |
| fthpropd 17553 | If two categories have the... |
| fulloppc 17554 | The opposite functor of a ... |
| fthoppc 17555 | The opposite functor of a ... |
| ffthoppc 17556 | The opposite functor of a ... |
| fthsect 17557 | A faithful functor reflect... |
| fthinv 17558 | A faithful functor reflect... |
| fthmon 17559 | A faithful functor reflect... |
| fthepi 17560 | A faithful functor reflect... |
| ffthiso 17561 | A fully faithful functor r... |
| fthres2b 17562 | Condition for a faithful f... |
| fthres2c 17563 | Condition for a faithful f... |
| fthres2 17564 | A faithful functor into a ... |
| idffth 17565 | The identity functor is a ... |
| cofull 17566 | The composition of two ful... |
| cofth 17567 | The composition of two fai... |
| coffth 17568 | The composition of two ful... |
| rescfth 17569 | The inclusion functor from... |
| ressffth 17570 | The inclusion functor from... |
| fullres2c 17571 | Condition for a full funct... |
| ffthres2c 17572 | Condition for a fully fait... |
| fnfuc 17577 | The ` FuncCat ` operation ... |
| natfval 17578 | Value of the function givi... |
| isnat 17579 | Property of being a natura... |
| isnat2 17580 | Property of being a natura... |
| natffn 17581 | The natural transformation... |
| natrcl 17582 | Reverse closure for a natu... |
| nat1st2nd 17583 | Rewrite the natural transf... |
| natixp 17584 | A natural transformation i... |
| natcl 17585 | A component of a natural t... |
| natfn 17586 | A natural transformation i... |
| nati 17587 | Naturality property of a n... |
| wunnat 17588 | A weak universe is closed ... |
| wunnatOLD 17589 | Obsolete proof of ~ wunnat... |
| catstr 17590 | A category structure is a ... |
| fucval 17591 | Value of the functor categ... |
| fuccofval 17592 | Value of the functor categ... |
| fucbas 17593 | The objects of the functor... |
| fuchom 17594 | The morphisms in the funct... |
| fuchomOLD 17595 | Obsolete proof of ~ fuchom... |
| fucco 17596 | Value of the composition o... |
| fuccoval 17597 | Value of the functor categ... |
| fuccocl 17598 | The composition of two nat... |
| fucidcl 17599 | The identity natural trans... |
| fuclid 17600 | Left identity of natural t... |
| fucrid 17601 | Right identity of natural ... |
| fucass 17602 | Associativity of natural t... |
| fuccatid 17603 | The functor category is a ... |
| fuccat 17604 | The functor category is a ... |
| fucid 17605 | The identity morphism in t... |
| fucsect 17606 | Two natural transformation... |
| fucinv 17607 | Two natural transformation... |
| invfuc 17608 | If ` V ( x ) ` is an inver... |
| fuciso 17609 | A natural transformation i... |
| natpropd 17610 | If two categories have the... |
| fucpropd 17611 | If two categories have the... |
| initofn 17618 | ` InitO ` is a function on... |
| termofn 17619 | ` TermO ` is a function on... |
| zeroofn 17620 | ` ZeroO ` is a function on... |
| initorcl 17621 | Reverse closure for an ini... |
| termorcl 17622 | Reverse closure for a term... |
| zeroorcl 17623 | Reverse closure for a zero... |
| initoval 17624 | The value of the initial o... |
| termoval 17625 | The value of the terminal ... |
| zerooval 17626 | The value of the zero obje... |
| isinito 17627 | The predicate "is an initi... |
| istermo 17628 | The predicate "is a termin... |
| iszeroo 17629 | The predicate "is a zero o... |
| isinitoi 17630 | Implication of a class bei... |
| istermoi 17631 | Implication of a class bei... |
| initoid 17632 | For an initial object, the... |
| termoid 17633 | For a terminal object, the... |
| dfinito2 17634 | An initial object is a ter... |
| dftermo2 17635 | A terminal object is an in... |
| dfinito3 17636 | An alternate definition of... |
| dftermo3 17637 | An alternate definition of... |
| initoo 17638 | An initial object is an ob... |
| termoo 17639 | A terminal object is an ob... |
| iszeroi 17640 | Implication of a class bei... |
| 2initoinv 17641 | Morphisms between two init... |
| initoeu1 17642 | Initial objects are essent... |
| initoeu1w 17643 | Initial objects are essent... |
| initoeu2lem0 17644 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 17645 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 17646 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 17647 | Initial objects are essent... |
| 2termoinv 17648 | Morphisms between two term... |
| termoeu1 17649 | Terminal objects are essen... |
| termoeu1w 17650 | Terminal objects are essen... |
| homarcl 17659 | Reverse closure for an arr... |
| homafval 17660 | Value of the disjointified... |
| homaf 17661 | Functionality of the disjo... |
| homaval 17662 | Value of the disjointified... |
| elhoma 17663 | Value of the disjointified... |
| elhomai 17664 | Produce an arrow from a mo... |
| elhomai2 17665 | Produce an arrow from a mo... |
| homarcl2 17666 | Reverse closure for the do... |
| homarel 17667 | An arrow is an ordered pai... |
| homa1 17668 | The first component of an ... |
| homahom2 17669 | The second component of an... |
| homahom 17670 | The second component of an... |
| homadm 17671 | The domain of an arrow wit... |
| homacd 17672 | The codomain of an arrow w... |
| homadmcd 17673 | Decompose an arrow into do... |
| arwval 17674 | The set of arrows is the u... |
| arwrcl 17675 | The first component of an ... |
| arwhoma 17676 | An arrow is contained in t... |
| homarw 17677 | A hom-set is a subset of t... |
| arwdm 17678 | The domain of an arrow is ... |
| arwcd 17679 | The codomain of an arrow i... |
| dmaf 17680 | The domain function is a f... |
| cdaf 17681 | The codomain function is a... |
| arwhom 17682 | The second component of an... |
| arwdmcd 17683 | Decompose an arrow into do... |
| idafval 17688 | Value of the identity arro... |
| idaval 17689 | Value of the identity arro... |
| ida2 17690 | Morphism part of the ident... |
| idahom 17691 | Domain and codomain of the... |
| idadm 17692 | Domain of the identity arr... |
| idacd 17693 | Codomain of the identity a... |
| idaf 17694 | The identity arrow functio... |
| coafval 17695 | The value of the compositi... |
| eldmcoa 17696 | A pair ` <. G , F >. ` is ... |
| dmcoass 17697 | The domain of composition ... |
| homdmcoa 17698 | If ` F : X --> Y ` and ` G... |
| coaval 17699 | Value of composition for c... |
| coa2 17700 | The morphism part of arrow... |
| coahom 17701 | The composition of two com... |
| coapm 17702 | Composition of arrows is a... |
| arwlid 17703 | Left identity of a categor... |
| arwrid 17704 | Right identity of a catego... |
| arwass 17705 | Associativity of compositi... |
| setcval 17708 | Value of the category of s... |
| setcbas 17709 | Set of objects of the cate... |
| setchomfval 17710 | Set of arrows of the categ... |
| setchom 17711 | Set of arrows of the categ... |
| elsetchom 17712 | A morphism of sets is a fu... |
| setccofval 17713 | Composition in the categor... |
| setcco 17714 | Composition in the categor... |
| setccatid 17715 | Lemma for ~ setccat . (Co... |
| setccat 17716 | The category of sets is a ... |
| setcid 17717 | The identity arrow in the ... |
| setcmon 17718 | A monomorphism of sets is ... |
| setcepi 17719 | An epimorphism of sets is ... |
| setcsect 17720 | A section in the category ... |
| setcinv 17721 | An inverse in the category... |
| setciso 17722 | An isomorphism in the cate... |
| resssetc 17723 | The restriction of the cat... |
| funcsetcres2 17724 | A functor into a smaller c... |
| setc2obas 17725 | ` (/) ` and ` 1o ` are dis... |
| setc2ohom 17726 | ` ( SetCat `` 2o ) ` is a ... |
| cat1lem 17727 | The category of sets in a ... |
| cat1 17728 | The definition of category... |
| catcval 17731 | Value of the category of c... |
| catcbas 17732 | Set of objects of the cate... |
| catchomfval 17733 | Set of arrows of the categ... |
| catchom 17734 | Set of arrows of the categ... |
| catccofval 17735 | Composition in the categor... |
| catcco 17736 | Composition in the categor... |
| catccatid 17737 | Lemma for ~ catccat . (Co... |
| catcid 17738 | The identity arrow in the ... |
| catccat 17739 | The category of categories... |
| resscatc 17740 | The restriction of the cat... |
| catcisolem 17741 | Lemma for ~ catciso . (Co... |
| catciso 17742 | A functor is an isomorphis... |
| catcbascl 17743 | An element of the base set... |
| catcslotelcl 17744 | A slot entry of an element... |
| catcbaselcl 17745 | The base set of an element... |
| catchomcl 17746 | The Hom-set of an element ... |
| catcccocl 17747 | The composition operation ... |
| catcoppccl 17748 | The category of categories... |
| catcoppcclOLD 17749 | Obsolete proof of ~ catcop... |
| catcfuccl 17750 | The category of categories... |
| catcfucclOLD 17751 | Obsolete proof of ~ catcfu... |
| fncnvimaeqv 17752 | The inverse images of the ... |
| bascnvimaeqv 17753 | The inverse image of the u... |
| estrcval 17756 | Value of the category of e... |
| estrcbas 17757 | Set of objects of the cate... |
| estrchomfval 17758 | Set of morphisms ("arrows"... |
| estrchom 17759 | The morphisms between exte... |
| elestrchom 17760 | A morphism between extensi... |
| estrccofval 17761 | Composition in the categor... |
| estrcco 17762 | Composition in the categor... |
| estrcbasbas 17763 | An element of the base set... |
| estrccatid 17764 | Lemma for ~ estrccat . (C... |
| estrccat 17765 | The category of extensible... |
| estrcid 17766 | The identity arrow in the ... |
| estrchomfn 17767 | The Hom-set operation in t... |
| estrchomfeqhom 17768 | The functionalized Hom-set... |
| estrreslem1 17769 | Lemma 1 for ~ estrres . (... |
| estrreslem1OLD 17770 | Obsolete version of ~ estr... |
| estrreslem2 17771 | Lemma 2 for ~ estrres . (... |
| estrres 17772 | Any restriction of a categ... |
| funcestrcsetclem1 17773 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 17774 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 17775 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 17776 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 17777 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 17778 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 17779 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 17780 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 17781 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 17782 | The "natural forgetful fun... |
| fthestrcsetc 17783 | The "natural forgetful fun... |
| fullestrcsetc 17784 | The "natural forgetful fun... |
| equivestrcsetc 17785 | The "natural forgetful fun... |
| setc1strwun 17786 | A constructed one-slot str... |
| funcsetcestrclem1 17787 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 17788 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 17789 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 17790 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 17791 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 17792 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 17793 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 17794 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 17795 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 17796 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 17797 | The "embedding functor" fr... |
| fthsetcestrc 17798 | The "embedding functor" fr... |
| fullsetcestrc 17799 | The "embedding functor" fr... |
| embedsetcestrc 17800 | The "embedding functor" fr... |
| fnxpc 17809 | The binary product of cate... |
| xpcval 17810 | Value of the binary produc... |
| xpcbas 17811 | Set of objects of the bina... |
| xpchomfval 17812 | Set of morphisms of the bi... |
| xpchom 17813 | Set of morphisms of the bi... |
| relxpchom 17814 | A hom-set in the binary pr... |
| xpccofval 17815 | Value of composition in th... |
| xpcco 17816 | Value of composition in th... |
| xpcco1st 17817 | Value of composition in th... |
| xpcco2nd 17818 | Value of composition in th... |
| xpchom2 17819 | Value of the set of morphi... |
| xpcco2 17820 | Value of composition in th... |
| xpccatid 17821 | The product of two categor... |
| xpcid 17822 | The identity morphism in t... |
| xpccat 17823 | The product of two categor... |
| 1stfval 17824 | Value of the first project... |
| 1stf1 17825 | Value of the first project... |
| 1stf2 17826 | Value of the first project... |
| 2ndfval 17827 | Value of the first project... |
| 2ndf1 17828 | Value of the first project... |
| 2ndf2 17829 | Value of the first project... |
| 1stfcl 17830 | The first projection funct... |
| 2ndfcl 17831 | The second projection func... |
| prfval 17832 | Value of the pairing funct... |
| prf1 17833 | Value of the pairing funct... |
| prf2fval 17834 | Value of the pairing funct... |
| prf2 17835 | Value of the pairing funct... |
| prfcl 17836 | The pairing of functors ` ... |
| prf1st 17837 | Cancellation of pairing wi... |
| prf2nd 17838 | Cancellation of pairing wi... |
| 1st2ndprf 17839 | Break a functor into a pro... |
| catcxpccl 17840 | The category of categories... |
| catcxpcclOLD 17841 | Obsolete proof of ~ catcxp... |
| xpcpropd 17842 | If two categories have the... |
| evlfval 17851 | Value of the evaluation fu... |
| evlf2 17852 | Value of the evaluation fu... |
| evlf2val 17853 | Value of the evaluation na... |
| evlf1 17854 | Value of the evaluation fu... |
| evlfcllem 17855 | Lemma for ~ evlfcl . (Con... |
| evlfcl 17856 | The evaluation functor is ... |
| curfval 17857 | Value of the curry functor... |
| curf1fval 17858 | Value of the object part o... |
| curf1 17859 | Value of the object part o... |
| curf11 17860 | Value of the double evalua... |
| curf12 17861 | The partially evaluated cu... |
| curf1cl 17862 | The partially evaluated cu... |
| curf2 17863 | Value of the curry functor... |
| curf2val 17864 | Value of a component of th... |
| curf2cl 17865 | The curry functor at a mor... |
| curfcl 17866 | The curry functor of a fun... |
| curfpropd 17867 | If two categories have the... |
| uncfval 17868 | Value of the uncurry funct... |
| uncfcl 17869 | The uncurry operation take... |
| uncf1 17870 | Value of the uncurry funct... |
| uncf2 17871 | Value of the uncurry funct... |
| curfuncf 17872 | Cancellation of curry with... |
| uncfcurf 17873 | Cancellation of uncurry wi... |
| diagval 17874 | Define the diagonal functo... |
| diagcl 17875 | The diagonal functor is a ... |
| diag1cl 17876 | The constant functor of ` ... |
| diag11 17877 | Value of the constant func... |
| diag12 17878 | Value of the constant func... |
| diag2 17879 | Value of the diagonal func... |
| diag2cl 17880 | The diagonal functor at a ... |
| curf2ndf 17881 | As shown in ~ diagval , th... |
| hofval 17886 | Value of the Hom functor, ... |
| hof1fval 17887 | The object part of the Hom... |
| hof1 17888 | The object part of the Hom... |
| hof2fval 17889 | The morphism part of the H... |
| hof2val 17890 | The morphism part of the H... |
| hof2 17891 | The morphism part of the H... |
| hofcllem 17892 | Lemma for ~ hofcl . (Cont... |
| hofcl 17893 | Closure of the Hom functor... |
| oppchofcl 17894 | Closure of the opposite Ho... |
| yonval 17895 | Value of the Yoneda embedd... |
| yoncl 17896 | The Yoneda embedding is a ... |
| yon1cl 17897 | The Yoneda embedding at an... |
| yon11 17898 | Value of the Yoneda embedd... |
| yon12 17899 | Value of the Yoneda embedd... |
| yon2 17900 | Value of the Yoneda embedd... |
| hofpropd 17901 | If two categories have the... |
| yonpropd 17902 | If two categories have the... |
| oppcyon 17903 | Value of the opposite Yone... |
| oyoncl 17904 | The opposite Yoneda embedd... |
| oyon1cl 17905 | The opposite Yoneda embedd... |
| yonedalem1 17906 | Lemma for ~ yoneda . (Con... |
| yonedalem21 17907 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 17908 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 17909 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 17910 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 17911 | Lemma for ~ yoneda . (Con... |
| yonedalem22 17912 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 17913 | Lemma for ~ yoneda . (Con... |
| yonedalem3 17914 | Lemma for ~ yoneda . (Con... |
| yonedainv 17915 | The Yoneda Lemma with expl... |
| yonffthlem 17916 | Lemma for ~ yonffth . (Co... |
| yoneda 17917 | The Yoneda Lemma. There i... |
| yonffth 17918 | The Yoneda Lemma. The Yon... |
| yoniso 17919 | If the codomain is recover... |
| oduval 17922 | Value of an order dual str... |
| oduleval 17923 | Value of the less-equal re... |
| oduleg 17924 | Truth of the less-equal re... |
| odubas 17925 | Base set of an order dual ... |
| isprs 17930 | Property of being a preord... |
| prslem 17931 | Lemma for ~ prsref and ~ p... |
| prsref 17932 | "Less than or equal to" is... |
| prstr 17933 | "Less than or equal to" is... |
| isdrs 17934 | Property of being a direct... |
| drsdir 17935 | Direction of a directed se... |
| drsprs 17936 | A directed set is a proset... |
| drsbn0 17937 | The base of a directed set... |
| drsdirfi 17938 | Any _finite_ number of ele... |
| isdrs2 17939 | Directed sets may be defin... |
| ispos 17947 | The predicate "is a poset"... |
| ispos2 17948 | A poset is an antisymmetri... |
| posprs 17949 | A poset is a proset. (Con... |
| posi 17950 | Lemma for poset properties... |
| posref 17951 | A poset ordering is reflex... |
| posasymb 17952 | A poset ordering is asymme... |
| postr 17953 | A poset ordering is transi... |
| 0pos 17954 | Technical lemma to simplif... |
| 0posOLD 17955 | Obsolete proof of ~ 0pos a... |
| isposd 17956 | Properties that determine ... |
| isposi 17957 | Properties that determine ... |
| isposix 17958 | Properties that determine ... |
| isposixOLD 17959 | Obsolete proof of ~ isposi... |
| pospropd 17960 | Posethood is determined on... |
| odupos 17961 | Being a poset is a self-du... |
| oduposb 17962 | Being a poset is a self-du... |
| pltfval 17964 | Value of the less-than rel... |
| pltval 17965 | Less-than relation. ( ~ d... |
| pltle 17966 | "Less than" implies "less ... |
| pltne 17967 | The "less than" relation i... |
| pltirr 17968 | The "less than" relation i... |
| pleval2i 17969 | One direction of ~ pleval2... |
| pleval2 17970 | "Less than or equal to" in... |
| pltnle 17971 | "Less than" implies not co... |
| pltval3 17972 | Alternate expression for t... |
| pltnlt 17973 | The less-than relation imp... |
| pltn2lp 17974 | The less-than relation has... |
| plttr 17975 | The less-than relation is ... |
| pltletr 17976 | Transitive law for chained... |
| plelttr 17977 | Transitive law for chained... |
| pospo 17978 | Write a poset structure in... |
| lubfval 17983 | Value of the least upper b... |
| lubdm 17984 | Domain of the least upper ... |
| lubfun 17985 | The LUB is a function. (C... |
| lubeldm 17986 | Member of the domain of th... |
| lubelss 17987 | A member of the domain of ... |
| lubeu 17988 | Unique existence proper of... |
| lubval 17989 | Value of the least upper b... |
| lubcl 17990 | The least upper bound func... |
| lubprop 17991 | Properties of greatest low... |
| luble 17992 | The greatest lower bound i... |
| lublecllem 17993 | Lemma for ~ lublecl and ~ ... |
| lublecl 17994 | The set of all elements le... |
| lubid 17995 | The LUB of elements less t... |
| glbfval 17996 | Value of the greatest lowe... |
| glbdm 17997 | Domain of the greatest low... |
| glbfun 17998 | The GLB is a function. (C... |
| glbeldm 17999 | Member of the domain of th... |
| glbelss 18000 | A member of the domain of ... |
| glbeu 18001 | Unique existence proper of... |
| glbval 18002 | Value of the greatest lowe... |
| glbcl 18003 | The least upper bound func... |
| glbprop 18004 | Properties of greatest low... |
| glble 18005 | The greatest lower bound i... |
| joinfval 18006 | Value of join function for... |
| joinfval2 18007 | Value of join function for... |
| joindm 18008 | Domain of join function fo... |
| joindef 18009 | Two ways to say that a joi... |
| joinval 18010 | Join value. Since both si... |
| joincl 18011 | Closure of join of element... |
| joindmss 18012 | Subset property of domain ... |
| joinval2lem 18013 | Lemma for ~ joinval2 and ~... |
| joinval2 18014 | Value of join for a poset ... |
| joineu 18015 | Uniqueness of join of elem... |
| joinlem 18016 | Lemma for join properties.... |
| lejoin1 18017 | A join's first argument is... |
| lejoin2 18018 | A join's second argument i... |
| joinle 18019 | A join is less than or equ... |
| meetfval 18020 | Value of meet function for... |
| meetfval2 18021 | Value of meet function for... |
| meetdm 18022 | Domain of meet function fo... |
| meetdef 18023 | Two ways to say that a mee... |
| meetval 18024 | Meet value. Since both si... |
| meetcl 18025 | Closure of meet of element... |
| meetdmss 18026 | Subset property of domain ... |
| meetval2lem 18027 | Lemma for ~ meetval2 and ~... |
| meetval2 18028 | Value of meet for a poset ... |
| meeteu 18029 | Uniqueness of meet of elem... |
| meetlem 18030 | Lemma for meet properties.... |
| lemeet1 18031 | A meet's first argument is... |
| lemeet2 18032 | A meet's second argument i... |
| meetle 18033 | A meet is less than or equ... |
| joincomALT 18034 | The join of a poset is com... |
| joincom 18035 | The join of a poset is com... |
| meetcomALT 18036 | The meet of a poset is com... |
| meetcom 18037 | The meet of a poset is com... |
| join0 18038 | Lemma for ~ odumeet . (Co... |
| meet0 18039 | Lemma for ~ odujoin . (Co... |
| odulub 18040 | Least upper bounds in a du... |
| odujoin 18041 | Joins in a dual order are ... |
| oduglb 18042 | Greatest lower bounds in a... |
| odumeet 18043 | Meets in a dual order are ... |
| poslubmo 18044 | Least upper bounds in a po... |
| posglbmo 18045 | Greatest lower bounds in a... |
| poslubd 18046 | Properties which determine... |
| poslubdg 18047 | Properties which determine... |
| posglbdg 18048 | Properties which determine... |
| istos 18051 | The predicate "is a toset"... |
| tosso 18052 | Write the totally ordered ... |
| tospos 18053 | A Toset is a Poset. (Cont... |
| tleile 18054 | In a Toset, any two elemen... |
| tltnle 18055 | In a Toset, "less than" is... |
| p0val 18060 | Value of poset zero. (Con... |
| p1val 18061 | Value of poset zero. (Con... |
| p0le 18062 | Any element is less than o... |
| ple1 18063 | Any element is less than o... |
| islat 18066 | The predicate "is a lattic... |
| odulatb 18067 | Being a lattice is self-du... |
| odulat 18068 | Being a lattice is self-du... |
| latcl2 18069 | The join and meet of any t... |
| latlem 18070 | Lemma for lattice properti... |
| latpos 18071 | A lattice is a poset. (Co... |
| latjcl 18072 | Closure of join operation ... |
| latmcl 18073 | Closure of meet operation ... |
| latref 18074 | A lattice ordering is refl... |
| latasymb 18075 | A lattice ordering is asym... |
| latasym 18076 | A lattice ordering is asym... |
| lattr 18077 | A lattice ordering is tran... |
| latasymd 18078 | Deduce equality from latti... |
| lattrd 18079 | A lattice ordering is tran... |
| latjcom 18080 | The join of a lattice comm... |
| latlej1 18081 | A join's first argument is... |
| latlej2 18082 | A join's second argument i... |
| latjle12 18083 | A join is less than or equ... |
| latleeqj1 18084 | "Less than or equal to" in... |
| latleeqj2 18085 | "Less than or equal to" in... |
| latjlej1 18086 | Add join to both sides of ... |
| latjlej2 18087 | Add join to both sides of ... |
| latjlej12 18088 | Add join to both sides of ... |
| latnlej 18089 | An idiom to express that a... |
| latnlej1l 18090 | An idiom to express that a... |
| latnlej1r 18091 | An idiom to express that a... |
| latnlej2 18092 | An idiom to express that a... |
| latnlej2l 18093 | An idiom to express that a... |
| latnlej2r 18094 | An idiom to express that a... |
| latjidm 18095 | Lattice join is idempotent... |
| latmcom 18096 | The join of a lattice comm... |
| latmle1 18097 | A meet is less than or equ... |
| latmle2 18098 | A meet is less than or equ... |
| latlem12 18099 | An element is less than or... |
| latleeqm1 18100 | "Less than or equal to" in... |
| latleeqm2 18101 | "Less than or equal to" in... |
| latmlem1 18102 | Add meet to both sides of ... |
| latmlem2 18103 | Add meet to both sides of ... |
| latmlem12 18104 | Add join to both sides of ... |
| latnlemlt 18105 | Negation of "less than or ... |
| latnle 18106 | Equivalent expressions for... |
| latmidm 18107 | Lattice meet is idempotent... |
| latabs1 18108 | Lattice absorption law. F... |
| latabs2 18109 | Lattice absorption law. F... |
| latledi 18110 | An ortholattice is distrib... |
| latmlej11 18111 | Ordering of a meet and joi... |
| latmlej12 18112 | Ordering of a meet and joi... |
| latmlej21 18113 | Ordering of a meet and joi... |
| latmlej22 18114 | Ordering of a meet and joi... |
| lubsn 18115 | The least upper bound of a... |
| latjass 18116 | Lattice join is associativ... |
| latj12 18117 | Swap 1st and 2nd members o... |
| latj32 18118 | Swap 2nd and 3rd members o... |
| latj13 18119 | Swap 1st and 3rd members o... |
| latj31 18120 | Swap 2nd and 3rd members o... |
| latjrot 18121 | Rotate lattice join of 3 c... |
| latj4 18122 | Rearrangement of lattice j... |
| latj4rot 18123 | Rotate lattice join of 4 c... |
| latjjdi 18124 | Lattice join distributes o... |
| latjjdir 18125 | Lattice join distributes o... |
| mod1ile 18126 | The weak direction of the ... |
| mod2ile 18127 | The weak direction of the ... |
| latmass 18128 | Lattice meet is associativ... |
| latdisdlem 18129 | Lemma for ~ latdisd . (Co... |
| latdisd 18130 | In a lattice, joins distri... |
| isclat 18133 | The predicate "is a comple... |
| clatpos 18134 | A complete lattice is a po... |
| clatlem 18135 | Lemma for properties of a ... |
| clatlubcl 18136 | Any subset of the base set... |
| clatlubcl2 18137 | Any subset of the base set... |
| clatglbcl 18138 | Any subset of the base set... |
| clatglbcl2 18139 | Any subset of the base set... |
| oduclatb 18140 | Being a complete lattice i... |
| clatl 18141 | A complete lattice is a la... |
| isglbd 18142 | Properties that determine ... |
| lublem 18143 | Lemma for the least upper ... |
| lubub 18144 | The LUB of a complete latt... |
| lubl 18145 | The LUB of a complete latt... |
| lubss 18146 | Subset law for least upper... |
| lubel 18147 | An element of a set is les... |
| lubun 18148 | The LUB of a union. (Cont... |
| clatglb 18149 | Properties of greatest low... |
| clatglble 18150 | The greatest lower bound i... |
| clatleglb 18151 | Two ways of expressing "le... |
| clatglbss 18152 | Subset law for greatest lo... |
| isdlat 18155 | Property of being a distri... |
| dlatmjdi 18156 | In a distributive lattice,... |
| dlatl 18157 | A distributive lattice is ... |
| odudlatb 18158 | The dual of a distributive... |
| dlatjmdi 18159 | In a distributive lattice,... |
| ipostr 18162 | The structure of ~ df-ipo ... |
| ipoval 18163 | Value of the inclusion pos... |
| ipobas 18164 | Base set of the inclusion ... |
| ipolerval 18165 | Relation of the inclusion ... |
| ipotset 18166 | Topology of the inclusion ... |
| ipole 18167 | Weak order condition of th... |
| ipolt 18168 | Strict order condition of ... |
| ipopos 18169 | The inclusion poset on a f... |
| isipodrs 18170 | Condition for a family of ... |
| ipodrscl 18171 | Direction by inclusion as ... |
| ipodrsfi 18172 | Finite upper bound propert... |
| fpwipodrs 18173 | The finite subsets of any ... |
| ipodrsima 18174 | The monotone image of a di... |
| isacs3lem 18175 | An algebraic closure syste... |
| acsdrsel 18176 | An algebraic closure syste... |
| isacs4lem 18177 | In a closure system in whi... |
| isacs5lem 18178 | If closure commutes with d... |
| acsdrscl 18179 | In an algebraic closure sy... |
| acsficl 18180 | A closure in an algebraic ... |
| isacs5 18181 | A closure system is algebr... |
| isacs4 18182 | A closure system is algebr... |
| isacs3 18183 | A closure system is algebr... |
| acsficld 18184 | In an algebraic closure sy... |
| acsficl2d 18185 | In an algebraic closure sy... |
| acsfiindd 18186 | In an algebraic closure sy... |
| acsmapd 18187 | In an algebraic closure sy... |
| acsmap2d 18188 | In an algebraic closure sy... |
| acsinfd 18189 | In an algebraic closure sy... |
| acsdomd 18190 | In an algebraic closure sy... |
| acsinfdimd 18191 | In an algebraic closure sy... |
| acsexdimd 18192 | In an algebraic closure sy... |
| mrelatglb 18193 | Greatest lower bounds in a... |
| mrelatglb0 18194 | The empty intersection in ... |
| mrelatlub 18195 | Least upper bounds in a Mo... |
| mreclatBAD 18196 | A Moore space is a complet... |
| isps 18201 | The predicate "is a poset"... |
| psrel 18202 | A poset is a relation. (C... |
| psref2 18203 | A poset is antisymmetric a... |
| pstr2 18204 | A poset is transitive. (C... |
| pslem 18205 | Lemma for ~ psref and othe... |
| psdmrn 18206 | The domain and range of a ... |
| psref 18207 | A poset is reflexive. (Co... |
| psrn 18208 | The range of a poset equal... |
| psasym 18209 | A poset is antisymmetric. ... |
| pstr 18210 | A poset is transitive. (C... |
| cnvps 18211 | The converse of a poset is... |
| cnvpsb 18212 | The converse of a poset is... |
| psss 18213 | Any subset of a partially ... |
| psssdm2 18214 | Field of a subposet. (Con... |
| psssdm 18215 | Field of a subposet. (Con... |
| istsr 18216 | The predicate is a toset. ... |
| istsr2 18217 | The predicate is a toset. ... |
| tsrlin 18218 | A toset is a linear order.... |
| tsrlemax 18219 | Two ways of saying a numbe... |
| tsrps 18220 | A toset is a poset. (Cont... |
| cnvtsr 18221 | The converse of a toset is... |
| tsrss 18222 | Any subset of a totally or... |
| ledm 18223 | The domain of ` <_ ` is ` ... |
| lern 18224 | The range of ` <_ ` is ` R... |
| lefld 18225 | The field of the 'less or ... |
| letsr 18226 | The "less than or equal to... |
| isdir 18231 | A condition for a relation... |
| reldir 18232 | A direction is a relation.... |
| dirdm 18233 | A direction's domain is eq... |
| dirref 18234 | A direction is reflexive. ... |
| dirtr 18235 | A direction is transitive.... |
| dirge 18236 | For any two elements of a ... |
| tsrdir 18237 | A totally ordered set is a... |
| ismgm 18242 | The predicate "is a magma"... |
| ismgmn0 18243 | The predicate "is a magma"... |
| mgmcl 18244 | Closure of the operation o... |
| isnmgm 18245 | A condition for a structur... |
| mgmsscl 18246 | If the base set of a magma... |
| plusffval 18247 | The group addition operati... |
| plusfval 18248 | The group addition operati... |
| plusfeq 18249 | If the addition operation ... |
| plusffn 18250 | The group addition operati... |
| mgmplusf 18251 | The group addition functio... |
| issstrmgm 18252 | Characterize a substructur... |
| intopsn 18253 | The internal operation for... |
| mgmb1mgm1 18254 | The only magma with a base... |
| mgm0 18255 | Any set with an empty base... |
| mgm0b 18256 | The structure with an empt... |
| mgm1 18257 | The structure with one ele... |
| opifismgm 18258 | A structure with a group a... |
| mgmidmo 18259 | A two-sided identity eleme... |
| grpidval 18260 | The value of the identity ... |
| grpidpropd 18261 | If two structures have the... |
| fn0g 18262 | The group zero extractor i... |
| 0g0 18263 | The identity element funct... |
| ismgmid 18264 | The identity element of a ... |
| mgmidcl 18265 | The identity element of a ... |
| mgmlrid 18266 | The identity element of a ... |
| ismgmid2 18267 | Show that a given element ... |
| lidrideqd 18268 | If there is a left and rig... |
| lidrididd 18269 | If there is a left and rig... |
| grpidd 18270 | Deduce the identity elemen... |
| mgmidsssn0 18271 | Property of the set of ide... |
| grprinvlem 18272 | Lemma for ~ grprinvd . (C... |
| grprinvd 18273 | Deduce right inverse from ... |
| grpridd 18274 | Deduce right identity from... |
| gsumvalx 18275 | Expand out the substitutio... |
| gsumval 18276 | Expand out the substitutio... |
| gsumpropd 18277 | The group sum depends only... |
| gsumpropd2lem 18278 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 18279 | A stronger version of ~ gs... |
| gsummgmpropd 18280 | A stronger version of ~ gs... |
| gsumress 18281 | The group sum in a substru... |
| gsumval1 18282 | Value of the group sum ope... |
| gsum0 18283 | Value of the empty group s... |
| gsumval2a 18284 | Value of the group sum ope... |
| gsumval2 18285 | Value of the group sum ope... |
| gsumsplit1r 18286 | Splitting off the rightmos... |
| gsumprval 18287 | Value of the group sum ope... |
| gsumpr12val 18288 | Value of the group sum ope... |
| issgrp 18291 | The predicate "is a semigr... |
| issgrpv 18292 | The predicate "is a semigr... |
| issgrpn0 18293 | The predicate "is a semigr... |
| isnsgrp 18294 | A condition for a structur... |
| sgrpmgm 18295 | A semigroup is a magma. (... |
| sgrpass 18296 | A semigroup operation is a... |
| sgrp0 18297 | Any set with an empty base... |
| sgrp0b 18298 | The structure with an empt... |
| sgrp1 18299 | The structure with one ele... |
| ismnddef 18302 | The predicate "is a monoid... |
| ismnd 18303 | The predicate "is a monoid... |
| isnmnd 18304 | A condition for a structur... |
| sgrpidmnd 18305 | A semigroup with an identi... |
| mndsgrp 18306 | A monoid is a semigroup. ... |
| mndmgm 18307 | A monoid is a magma. (Con... |
| mndcl 18308 | Closure of the operation o... |
| mndass 18309 | A monoid operation is asso... |
| mndid 18310 | A monoid has a two-sided i... |
| mndideu 18311 | The two-sided identity ele... |
| mnd32g 18312 | Commutative/associative la... |
| mnd12g 18313 | Commutative/associative la... |
| mnd4g 18314 | Commutative/associative la... |
| mndidcl 18315 | The identity element of a ... |
| mndbn0 18316 | The base set of a monoid i... |
| hashfinmndnn 18317 | A finite monoid has positi... |
| mndplusf 18318 | The group addition operati... |
| mndlrid 18319 | A monoid's identity elemen... |
| mndlid 18320 | The identity element of a ... |
| mndrid 18321 | The identity element of a ... |
| ismndd 18322 | Deduce a monoid from its p... |
| mndpfo 18323 | The addition operation of ... |
| mndfo 18324 | The addition operation of ... |
| mndpropd 18325 | If two structures have the... |
| mndprop 18326 | If two structures have the... |
| issubmnd 18327 | Characterize a submonoid b... |
| ress0g 18328 | ` 0g ` is unaffected by re... |
| submnd0 18329 | The zero of a submonoid is... |
| mndinvmod 18330 | Uniqueness of an inverse e... |
| prdsplusgcl 18331 | Structure product pointwis... |
| prdsidlem 18332 | Characterization of identi... |
| prdsmndd 18333 | The product of a family of... |
| prds0g 18334 | Zero in a product of monoi... |
| pwsmnd 18335 | The structure power of a m... |
| pws0g 18336 | Zero in a structure power ... |
| imasmnd2 18337 | The image structure of a m... |
| imasmnd 18338 | The image structure of a m... |
| imasmndf1 18339 | The image of a monoid unde... |
| xpsmnd 18340 | The binary product of mono... |
| mnd1 18341 | The (smallest) structure r... |
| mnd1id 18342 | The singleton element of a... |
| ismhm 18347 | Property of a monoid homom... |
| mhmrcl1 18348 | Reverse closure of a monoi... |
| mhmrcl2 18349 | Reverse closure of a monoi... |
| mhmf 18350 | A monoid homomorphism is a... |
| mhmpropd 18351 | Monoid homomorphism depend... |
| mhmlin 18352 | A monoid homomorphism comm... |
| mhm0 18353 | A monoid homomorphism pres... |
| idmhm 18354 | The identity homomorphism ... |
| mhmf1o 18355 | A monoid homomorphism is b... |
| submrcl 18356 | Reverse closure for submon... |
| issubm 18357 | Expand definition of a sub... |
| issubm2 18358 | Submonoids are subsets tha... |
| issubmndb 18359 | The submonoid predicate. ... |
| issubmd 18360 | Deduction for proving a su... |
| mndissubm 18361 | If the base set of a monoi... |
| resmndismnd 18362 | If the base set of a monoi... |
| submss 18363 | Submonoids are subsets of ... |
| submid 18364 | Every monoid is trivially ... |
| subm0cl 18365 | Submonoids contain zero. ... |
| submcl 18366 | Submonoids are closed unde... |
| submmnd 18367 | Submonoids are themselves ... |
| submbas 18368 | The base set of a submonoi... |
| subm0 18369 | Submonoids have the same i... |
| subsubm 18370 | A submonoid of a submonoid... |
| 0subm 18371 | The zero submonoid of an a... |
| insubm 18372 | The intersection of two su... |
| 0mhm 18373 | The constant zero linear f... |
| resmhm 18374 | Restriction of a monoid ho... |
| resmhm2 18375 | One direction of ~ resmhm2... |
| resmhm2b 18376 | Restriction of the codomai... |
| mhmco 18377 | The composition of monoid ... |
| mhmima 18378 | The homomorphic image of a... |
| mhmeql 18379 | The equalizer of two monoi... |
| submacs 18380 | Submonoids are an algebrai... |
| mndind 18381 | Induction in a monoid. In... |
| prdspjmhm 18382 | A projection from a produc... |
| pwspjmhm 18383 | A projection from a struct... |
| pwsdiagmhm 18384 | Diagonal monoid homomorphi... |
| pwsco1mhm 18385 | Right composition with a f... |
| pwsco2mhm 18386 | Left composition with a mo... |
| gsumvallem2 18387 | Lemma for properties of th... |
| gsumsubm 18388 | Evaluate a group sum in a ... |
| gsumz 18389 | Value of a group sum over ... |
| gsumwsubmcl 18390 | Closure of the composite i... |
| gsumws1 18391 | A singleton composite reco... |
| gsumwcl 18392 | Closure of the composite o... |
| gsumsgrpccat 18393 | Homomorphic property of no... |
| gsumccatOLD 18394 | Obsolete version of ~ gsum... |
| gsumccat 18395 | Homomorphic property of co... |
| gsumws2 18396 | Valuation of a pair in a m... |
| gsumccatsn 18397 | Homomorphic property of co... |
| gsumspl 18398 | The primary purpose of the... |
| gsumwmhm 18399 | Behavior of homomorphisms ... |
| gsumwspan 18400 | The submonoid generated by... |
| frmdval 18405 | Value of the free monoid c... |
| frmdbas 18406 | The base set of a free mon... |
| frmdelbas 18407 | An element of the base set... |
| frmdplusg 18408 | The monoid operation of a ... |
| frmdadd 18409 | Value of the monoid operat... |
| vrmdfval 18410 | The canonical injection fr... |
| vrmdval 18411 | The value of the generatin... |
| vrmdf 18412 | The mapping from the index... |
| frmdmnd 18413 | A free monoid is a monoid.... |
| frmd0 18414 | The identity of the free m... |
| frmdsssubm 18415 | The set of words taking va... |
| frmdgsum 18416 | Any word in a free monoid ... |
| frmdss2 18417 | A subset of generators is ... |
| frmdup1 18418 | Any assignment of the gene... |
| frmdup2 18419 | The evaluation map has the... |
| frmdup3lem 18420 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18421 | Universal property of the ... |
| efmnd 18424 | The monoid of endofunction... |
| efmndbas 18425 | The base set of the monoid... |
| efmndbasabf 18426 | The base set of the monoid... |
| elefmndbas 18427 | Two ways of saying a funct... |
| elefmndbas2 18428 | Two ways of saying a funct... |
| efmndbasf 18429 | Elements in the monoid of ... |
| efmndhash 18430 | The monoid of endofunction... |
| efmndbasfi 18431 | The monoid of endofunction... |
| efmndfv 18432 | The function value of an e... |
| efmndtset 18433 | The topology of the monoid... |
| efmndplusg 18434 | The group operation of a m... |
| efmndov 18435 | The value of the group ope... |
| efmndcl 18436 | The group operation of the... |
| efmndtopn 18437 | The topology of the monoid... |
| symggrplem 18438 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18439 | The monoid of endofunction... |
| efmndsgrp 18440 | The monoid of endofunction... |
| ielefmnd 18441 | The identity function rest... |
| efmndid 18442 | The identity function rest... |
| efmndmnd 18443 | The monoid of endofunction... |
| efmnd0nmnd 18444 | Even the monoid of endofun... |
| efmndbas0 18445 | The base set of the monoid... |
| efmnd1hash 18446 | The monoid of endofunction... |
| efmnd1bas 18447 | The monoid of endofunction... |
| efmnd2hash 18448 | The monoid of endofunction... |
| submefmnd 18449 | If the base set of a monoi... |
| sursubmefmnd 18450 | The set of surjective endo... |
| injsubmefmnd 18451 | The set of injective endof... |
| idressubmefmnd 18452 | The singleton containing o... |
| idresefmnd 18453 | The structure with the sin... |
| smndex1ibas 18454 | The modulo function ` I ` ... |
| smndex1iidm 18455 | The modulo function ` I ` ... |
| smndex1gbas 18456 | The constant functions ` (... |
| smndex1gid 18457 | The composition of a const... |
| smndex1igid 18458 | The composition of the mod... |
| smndex1basss 18459 | The modulo function ` I ` ... |
| smndex1bas 18460 | The base set of the monoid... |
| smndex1mgm 18461 | The monoid of endofunction... |
| smndex1sgrp 18462 | The monoid of endofunction... |
| smndex1mndlem 18463 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18464 | The monoid of endofunction... |
| smndex1id 18465 | The modulo function ` I ` ... |
| smndex1n0mnd 18466 | The identity of the monoid... |
| nsmndex1 18467 | The base set ` B ` of the ... |
| smndex2dbas 18468 | The doubling function ` D ... |
| smndex2dnrinv 18469 | The doubling function ` D ... |
| smndex2hbas 18470 | The halving functions ` H ... |
| smndex2dlinvh 18471 | The halving functions ` H ... |
| mgm2nsgrplem1 18472 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18473 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18474 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18475 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18476 | A small magma (with two el... |
| sgrp2nmndlem1 18477 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18478 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18479 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18480 | A small semigroup (with tw... |
| sgrp2rid2ex 18481 | A small semigroup (with tw... |
| sgrp2nmndlem4 18482 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18483 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18484 | A small semigroup (with tw... |
| mgmnsgrpex 18485 | There is a magma which is ... |
| sgrpnmndex 18486 | There is a semigroup which... |
| sgrpssmgm 18487 | The class of all semigroup... |
| mndsssgrp 18488 | The class of all monoids i... |
| pwmndgplus 18489 | The operation of the monoi... |
| pwmndid 18490 | The identity of the monoid... |
| pwmnd 18491 | The power set of a class `... |
| isgrp 18498 | The predicate "is a group"... |
| grpmnd 18499 | A group is a monoid. (Con... |
| grpcl 18500 | Closure of the operation o... |
| grpass 18501 | A group operation is assoc... |
| grpinvex 18502 | Every member of a group ha... |
| grpideu 18503 | The two-sided identity ele... |
| grpmndd 18504 | A group is a monoid. (Con... |
| grpcld 18505 | Closure of the operation o... |
| grpplusf 18506 | The group addition operati... |
| grpplusfo 18507 | The group addition operati... |
| resgrpplusfrn 18508 | The underlying set of a gr... |
| grppropd 18509 | If two structures have the... |
| grpprop 18510 | If two structures have the... |
| grppropstr 18511 | Generalize a specific 2-el... |
| grpss 18512 | Show that a structure exte... |
| isgrpd2e 18513 | Deduce a group from its pr... |
| isgrpd2 18514 | Deduce a group from its pr... |
| isgrpde 18515 | Deduce a group from its pr... |
| isgrpd 18516 | Deduce a group from its pr... |
| isgrpi 18517 | Properties that determine ... |
| grpsgrp 18518 | A group is a semigroup. (... |
| dfgrp2 18519 | Alternate definition of a ... |
| dfgrp2e 18520 | Alternate definition of a ... |
| isgrpix 18521 | Properties that determine ... |
| grpidcl 18522 | The identity element of a ... |
| grpbn0 18523 | The base set of a group is... |
| grplid 18524 | The identity element of a ... |
| grprid 18525 | The identity element of a ... |
| grpn0 18526 | A group is not empty. (Co... |
| hashfingrpnn 18527 | A finite group has positiv... |
| grprcan 18528 | Right cancellation law for... |
| grpinveu 18529 | The left inverse element o... |
| grpid 18530 | Two ways of saying that an... |
| isgrpid2 18531 | Properties showing that an... |
| grpidd2 18532 | Deduce the identity elemen... |
| grpinvfval 18533 | The inverse function of a ... |
| grpinvfvalALT 18534 | Shorter proof of ~ grpinvf... |
| grpinvval 18535 | The inverse of a group ele... |
| grpinvfn 18536 | Functionality of the group... |
| grpinvfvi 18537 | The group inverse function... |
| grpsubfval 18538 | Group subtraction (divisio... |
| grpsubfvalALT 18539 | Shorter proof of ~ grpsubf... |
| grpsubval 18540 | Group subtraction (divisio... |
| grpinvf 18541 | The group inversion operat... |
| grpinvcl 18542 | A group element's inverse ... |
| grplinv 18543 | The left inverse of a grou... |
| grprinv 18544 | The right inverse of a gro... |
| grpinvid1 18545 | The inverse of a group ele... |
| grpinvid2 18546 | The inverse of a group ele... |
| isgrpinv 18547 | Properties showing that a ... |
| grplrinv 18548 | In a group, every member h... |
| grpidinv2 18549 | A group's properties using... |
| grpidinv 18550 | A group has a left and rig... |
| grpinvid 18551 | The inverse of the identit... |
| grplcan 18552 | Left cancellation law for ... |
| grpasscan1 18553 | An associative cancellatio... |
| grpasscan2 18554 | An associative cancellatio... |
| grpidrcan 18555 | If right adding an element... |
| grpidlcan 18556 | If left adding an element ... |
| grpinvinv 18557 | Double inverse law for gro... |
| grpinvcnv 18558 | The group inverse is its o... |
| grpinv11 18559 | The group inverse is one-t... |
| grpinvf1o 18560 | The group inverse is a one... |
| grpinvnz 18561 | The inverse of a nonzero g... |
| grpinvnzcl 18562 | The inverse of a nonzero g... |
| grpsubinv 18563 | Subtraction of an inverse.... |
| grplmulf1o 18564 | Left multiplication by a g... |
| grpinvpropd 18565 | If two structures have the... |
| grpidssd 18566 | If the base set of a group... |
| grpinvssd 18567 | If the base set of a group... |
| grpinvadd 18568 | The inverse of the group o... |
| grpsubf 18569 | Functionality of group sub... |
| grpsubcl 18570 | Closure of group subtracti... |
| grpsubrcan 18571 | Right cancellation law for... |
| grpinvsub 18572 | Inverse of a group subtrac... |
| grpinvval2 18573 | A ~ df-neg -like equation ... |
| grpsubid 18574 | Subtraction of a group ele... |
| grpsubid1 18575 | Subtraction of the identit... |
| grpsubeq0 18576 | If the difference between ... |
| grpsubadd0sub 18577 | Subtraction expressed as a... |
| grpsubadd 18578 | Relationship between group... |
| grpsubsub 18579 | Double group subtraction. ... |
| grpaddsubass 18580 | Associative-type law for g... |
| grppncan 18581 | Cancellation law for subtr... |
| grpnpcan 18582 | Cancellation law for subtr... |
| grpsubsub4 18583 | Double group subtraction (... |
| grppnpcan2 18584 | Cancellation law for mixed... |
| grpnpncan 18585 | Cancellation law for group... |
| grpnpncan0 18586 | Cancellation law for group... |
| grpnnncan2 18587 | Cancellation law for group... |
| dfgrp3lem 18588 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 18589 | Alternate definition of a ... |
| dfgrp3e 18590 | Alternate definition of a ... |
| grplactfval 18591 | The left group action of e... |
| grplactval 18592 | The value of the left grou... |
| grplactcnv 18593 | The left group action of e... |
| grplactf1o 18594 | The left group action of e... |
| grpsubpropd 18595 | Weak property deduction fo... |
| grpsubpropd2 18596 | Strong property deduction ... |
| grp1 18597 | The (smallest) structure r... |
| grp1inv 18598 | The inverse function of th... |
| prdsinvlem 18599 | Characterization of invers... |
| prdsgrpd 18600 | The product of a family of... |
| prdsinvgd 18601 | Negation in a product of g... |
| pwsgrp 18602 | A structure power of a gro... |
| pwsinvg 18603 | Negation in a group power.... |
| pwssub 18604 | Subtraction in a group pow... |
| imasgrp2 18605 | The image structure of a g... |
| imasgrp 18606 | The image structure of a g... |
| imasgrpf1 18607 | The image of a group under... |
| qusgrp2 18608 | Prove that a quotient stru... |
| xpsgrp 18609 | The binary product of grou... |
| mhmlem 18610 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 18611 | A surjective monoid morphi... |
| mhmmnd 18612 | The image of a monoid ` G ... |
| mhmfmhm 18613 | The function fulfilling th... |
| ghmgrp 18614 | The image of a group ` G `... |
| mulgfval 18617 | Group multiple (exponentia... |
| mulgfvalALT 18618 | Shorter proof of ~ mulgfva... |
| mulgval 18619 | Value of the group multipl... |
| mulgfn 18620 | Functionality of the group... |
| mulgfvi 18621 | The group multiple operati... |
| mulg0 18622 | Group multiple (exponentia... |
| mulgnn 18623 | Group multiple (exponentia... |
| mulgnngsum 18624 | Group multiple (exponentia... |
| mulgnn0gsum 18625 | Group multiple (exponentia... |
| mulg1 18626 | Group multiple (exponentia... |
| mulgnnp1 18627 | Group multiple (exponentia... |
| mulg2 18628 | Group multiple (exponentia... |
| mulgnegnn 18629 | Group multiple (exponentia... |
| mulgnn0p1 18630 | Group multiple (exponentia... |
| mulgnnsubcl 18631 | Closure of the group multi... |
| mulgnn0subcl 18632 | Closure of the group multi... |
| mulgsubcl 18633 | Closure of the group multi... |
| mulgnncl 18634 | Closure of the group multi... |
| mulgnn0cl 18635 | Closure of the group multi... |
| mulgcl 18636 | Closure of the group multi... |
| mulgneg 18637 | Group multiple (exponentia... |
| mulgnegneg 18638 | The inverse of a negative ... |
| mulgm1 18639 | Group multiple (exponentia... |
| mulgcld 18640 | Deduction associated with ... |
| mulgaddcomlem 18641 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 18642 | The group multiple operato... |
| mulginvcom 18643 | The group multiple operato... |
| mulginvinv 18644 | The group multiple operato... |
| mulgnn0z 18645 | A group multiple of the id... |
| mulgz 18646 | A group multiple of the id... |
| mulgnndir 18647 | Sum of group multiples, fo... |
| mulgnn0dir 18648 | Sum of group multiples, ge... |
| mulgdirlem 18649 | Lemma for ~ mulgdir . (Co... |
| mulgdir 18650 | Sum of group multiples, ge... |
| mulgp1 18651 | Group multiple (exponentia... |
| mulgneg2 18652 | Group multiple (exponentia... |
| mulgnnass 18653 | Product of group multiples... |
| mulgnn0ass 18654 | Product of group multiples... |
| mulgass 18655 | Product of group multiples... |
| mulgassr 18656 | Reversed product of group ... |
| mulgmodid 18657 | Casting out multiples of t... |
| mulgsubdir 18658 | Subtraction of a group ele... |
| mhmmulg 18659 | A homomorphism of monoids ... |
| mulgpropd 18660 | Two structures with the sa... |
| submmulgcl 18661 | Closure of the group multi... |
| submmulg 18662 | A group multiple is the sa... |
| pwsmulg 18663 | Value of a group multiple ... |
| issubg 18670 | The subgroup predicate. (... |
| subgss 18671 | A subgroup is a subset. (... |
| subgid 18672 | A group is a subgroup of i... |
| subggrp 18673 | A subgroup is a group. (C... |
| subgbas 18674 | The base of the restricted... |
| subgrcl 18675 | Reverse closure for the su... |
| subg0 18676 | A subgroup of a group must... |
| subginv 18677 | The inverse of an element ... |
| subg0cl 18678 | The group identity is an e... |
| subginvcl 18679 | The inverse of an element ... |
| subgcl 18680 | A subgroup is closed under... |
| subgsubcl 18681 | A subgroup is closed under... |
| subgsub 18682 | The subtraction of element... |
| subgmulgcl 18683 | Closure of the group multi... |
| subgmulg 18684 | A group multiple is the sa... |
| issubg2 18685 | Characterize the subgroups... |
| issubgrpd2 18686 | Prove a subgroup by closur... |
| issubgrpd 18687 | Prove a subgroup by closur... |
| issubg3 18688 | A subgroup is a symmetric ... |
| issubg4 18689 | A subgroup is a nonempty s... |
| grpissubg 18690 | If the base set of a group... |
| resgrpisgrp 18691 | If the base set of a group... |
| subgsubm 18692 | A subgroup is a submonoid.... |
| subsubg 18693 | A subgroup of a subgroup i... |
| subgint 18694 | The intersection of a none... |
| 0subg 18695 | The zero subgroup of an ar... |
| trivsubgd 18696 | The only subgroup of a tri... |
| trivsubgsnd 18697 | The only subgroup of a tri... |
| isnsg 18698 | Property of being a normal... |
| isnsg2 18699 | Weaken the condition of ~ ... |
| nsgbi 18700 | Defining property of a nor... |
| nsgsubg 18701 | A normal subgroup is a sub... |
| nsgconj 18702 | The conjugation of an elem... |
| isnsg3 18703 | A subgroup is normal iff t... |
| subgacs 18704 | Subgroups are an algebraic... |
| nsgacs 18705 | Normal subgroups form an a... |
| elnmz 18706 | Elementhood in the normali... |
| nmzbi 18707 | Defining property of the n... |
| nmzsubg 18708 | The normalizer N_G(S) of a... |
| ssnmz 18709 | A subgroup is a subset of ... |
| isnsg4 18710 | A subgroup is normal iff i... |
| nmznsg 18711 | Any subgroup is a normal s... |
| 0nsg 18712 | The zero subgroup is norma... |
| nsgid 18713 | The whole group is a norma... |
| 0idnsgd 18714 | The whole group and the ze... |
| trivnsgd 18715 | The only normal subgroup o... |
| triv1nsgd 18716 | A trivial group has exactl... |
| 1nsgtrivd 18717 | A group with exactly one n... |
| releqg 18718 | The left coset equivalence... |
| eqgfval 18719 | Value of the subgroup left... |
| eqgval 18720 | Value of the subgroup left... |
| eqger 18721 | The subgroup coset equival... |
| eqglact 18722 | A left coset can be expres... |
| eqgid 18723 | The left coset containing ... |
| eqgen 18724 | Each coset is equipotent t... |
| eqgcpbl 18725 | The subgroup coset equival... |
| qusgrp 18726 | If ` Y ` is a normal subgr... |
| quseccl 18727 | Closure of the quotient ma... |
| qusadd 18728 | Value of the group operati... |
| qus0 18729 | Value of the group identit... |
| qusinv 18730 | Value of the group inverse... |
| qussub 18731 | Value of the group subtrac... |
| lagsubg2 18732 | Lagrange's theorem for fin... |
| lagsubg 18733 | Lagrange's theorem for Gro... |
| cycsubmel 18734 | Characterization of an ele... |
| cycsubmcl 18735 | The set of nonnegative int... |
| cycsubm 18736 | The set of nonnegative int... |
| cyccom 18737 | Condition for an operation... |
| cycsubmcom 18738 | The operation of a monoid ... |
| cycsubggend 18739 | The cyclic subgroup genera... |
| cycsubgcl 18740 | The set of integer powers ... |
| cycsubgss 18741 | The cyclic subgroup genera... |
| cycsubg 18742 | The cyclic group generated... |
| cycsubgcld 18743 | The cyclic subgroup genera... |
| cycsubg2 18744 | The subgroup generated by ... |
| cycsubg2cl 18745 | Any multiple of an element... |
| reldmghm 18748 | Lemma for group homomorphi... |
| isghm 18749 | Property of being a homomo... |
| isghm3 18750 | Property of a group homomo... |
| ghmgrp1 18751 | A group homomorphism is on... |
| ghmgrp2 18752 | A group homomorphism is on... |
| ghmf 18753 | A group homomorphism is a ... |
| ghmlin 18754 | A homomorphism of groups i... |
| ghmid 18755 | A homomorphism of groups p... |
| ghminv 18756 | A homomorphism of groups p... |
| ghmsub 18757 | Linearity of subtraction t... |
| isghmd 18758 | Deduction for a group homo... |
| ghmmhm 18759 | A group homomorphism is a ... |
| ghmmhmb 18760 | Group homomorphisms and mo... |
| ghmmulg 18761 | A homomorphism of monoids ... |
| ghmrn 18762 | The range of a homomorphis... |
| 0ghm 18763 | The constant zero linear f... |
| idghm 18764 | The identity homomorphism ... |
| resghm 18765 | Restriction of a homomorph... |
| resghm2 18766 | One direction of ~ resghm2... |
| resghm2b 18767 | Restriction of the codomai... |
| ghmghmrn 18768 | A group homomorphism from ... |
| ghmco 18769 | The composition of group h... |
| ghmima 18770 | The image of a subgroup un... |
| ghmpreima 18771 | The inverse image of a sub... |
| ghmeql 18772 | The equalizer of two group... |
| ghmnsgima 18773 | The image of a normal subg... |
| ghmnsgpreima 18774 | The inverse image of a nor... |
| ghmker 18775 | The kernel of a homomorphi... |
| ghmeqker 18776 | Two source points map to t... |
| pwsdiagghm 18777 | Diagonal homomorphism into... |
| ghmf1 18778 | Two ways of saying a group... |
| ghmf1o 18779 | A bijective group homomorp... |
| conjghm 18780 | Conjugation is an automorp... |
| conjsubg 18781 | A conjugated subgroup is a... |
| conjsubgen 18782 | A conjugated subgroup is e... |
| conjnmz 18783 | A subgroup is unchanged un... |
| conjnmzb 18784 | Alternative condition for ... |
| conjnsg 18785 | A normal subgroup is uncha... |
| qusghm 18786 | If ` Y ` is a normal subgr... |
| ghmpropd 18787 | Group homomorphism depends... |
| gimfn 18792 | The group isomorphism func... |
| isgim 18793 | An isomorphism of groups i... |
| gimf1o 18794 | An isomorphism of groups i... |
| gimghm 18795 | An isomorphism of groups i... |
| isgim2 18796 | A group isomorphism is a h... |
| subggim 18797 | Behavior of subgroups unde... |
| gimcnv 18798 | The converse of a bijectiv... |
| gimco 18799 | The composition of group i... |
| brgic 18800 | The relation "is isomorphi... |
| brgici 18801 | Prove isomorphic by an exp... |
| gicref 18802 | Isomorphism is reflexive. ... |
| giclcl 18803 | Isomorphism implies the le... |
| gicrcl 18804 | Isomorphism implies the ri... |
| gicsym 18805 | Isomorphism is symmetric. ... |
| gictr 18806 | Isomorphism is transitive.... |
| gicer 18807 | Isomorphism is an equivale... |
| gicen 18808 | Isomorphic groups have equ... |
| gicsubgen 18809 | A less trivial example of ... |
| isga 18812 | The predicate "is a (left)... |
| gagrp 18813 | The left argument of a gro... |
| gaset 18814 | The right argument of a gr... |
| gagrpid 18815 | The identity of the group ... |
| gaf 18816 | The mapping of the group a... |
| gafo 18817 | A group action is onto its... |
| gaass 18818 | An "associative" property ... |
| ga0 18819 | The action of a group on t... |
| gaid 18820 | The trivial action of a gr... |
| subgga 18821 | A subgroup acts on its par... |
| gass 18822 | A subset of a group action... |
| gasubg 18823 | The restriction of a group... |
| gaid2 18824 | A group operation is a lef... |
| galcan 18825 | The action of a particular... |
| gacan 18826 | Group inverses cancel in a... |
| gapm 18827 | The action of a particular... |
| gaorb 18828 | The orbit equivalence rela... |
| gaorber 18829 | The orbit equivalence rela... |
| gastacl 18830 | The stabilizer subgroup in... |
| gastacos 18831 | Write the coset relation f... |
| orbstafun 18832 | Existence and uniqueness f... |
| orbstaval 18833 | Value of the function at a... |
| orbsta 18834 | The Orbit-Stabilizer theor... |
| orbsta2 18835 | Relation between the size ... |
| cntrval 18840 | Substitute definition of t... |
| cntzfval 18841 | First level substitution f... |
| cntzval 18842 | Definition substitution fo... |
| elcntz 18843 | Elementhood in the central... |
| cntzel 18844 | Membership in a centralize... |
| cntzsnval 18845 | Special substitution for t... |
| elcntzsn 18846 | Value of the centralizer o... |
| sscntz 18847 | A centralizer expression f... |
| cntzrcl 18848 | Reverse closure for elemen... |
| cntzssv 18849 | The centralizer is uncondi... |
| cntzi 18850 | Membership in a centralize... |
| cntrss 18851 | The center is a subset of ... |
| cntri 18852 | Defining property of the c... |
| resscntz 18853 | Centralizer in a substruct... |
| cntz2ss 18854 | Centralizers reverse the s... |
| cntzrec 18855 | Reciprocity relationship f... |
| cntziinsn 18856 | Express any centralizer as... |
| cntzsubm 18857 | Centralizers in a monoid a... |
| cntzsubg 18858 | Centralizers in a group ar... |
| cntzidss 18859 | If the elements of ` S ` c... |
| cntzmhm 18860 | Centralizers in a monoid a... |
| cntzmhm2 18861 | Centralizers in a monoid a... |
| cntrsubgnsg 18862 | A central subgroup is norm... |
| cntrnsg 18863 | The center of a group is a... |
| oppgval 18866 | Value of the opposite grou... |
| oppgplusfval 18867 | Value of the addition oper... |
| oppgplus 18868 | Value of the addition oper... |
| setsplusg 18869 | The other components of an... |
| oppglemOLD 18870 | Obsolete version of ~ sets... |
| oppgbas 18871 | Base set of an opposite gr... |
| oppgbasOLD 18872 | Obsolete version of ~ oppg... |
| oppgtset 18873 | Topology of an opposite gr... |
| oppgtsetOLD 18874 | Obsolete version of ~ oppg... |
| oppgtopn 18875 | Topology of an opposite gr... |
| oppgmnd 18876 | The opposite of a monoid i... |
| oppgmndb 18877 | Bidirectional form of ~ op... |
| oppgid 18878 | Zero in a monoid is a symm... |
| oppggrp 18879 | The opposite of a group is... |
| oppggrpb 18880 | Bidirectional form of ~ op... |
| oppginv 18881 | Inverses in a group are a ... |
| invoppggim 18882 | The inverse is an antiauto... |
| oppggic 18883 | Every group is (naturally)... |
| oppgsubm 18884 | Being a submonoid is a sym... |
| oppgsubg 18885 | Being a subgroup is a symm... |
| oppgcntz 18886 | A centralizer in a group i... |
| oppgcntr 18887 | The center of a group is t... |
| gsumwrev 18888 | A sum in an opposite monoi... |
| symgval 18891 | The value of the symmetric... |
| permsetexOLD 18892 | Obsolete version of ~ f1os... |
| symgbas 18893 | The base set of the symmet... |
| symgbasexOLD 18894 | Obsolete as of 8-Aug-2024.... |
| elsymgbas2 18895 | Two ways of saying a funct... |
| elsymgbas 18896 | Two ways of saying a funct... |
| symgbasf1o 18897 | Elements in the symmetric ... |
| symgbasf 18898 | A permutation (element of ... |
| symgbasmap 18899 | A permutation (element of ... |
| symghash 18900 | The symmetric group on ` n... |
| symgbasfi 18901 | The symmetric group on a f... |
| symgfv 18902 | The function value of a pe... |
| symgfvne 18903 | The function values of a p... |
| symgressbas 18904 | The symmetric group on ` A... |
| symgplusg 18905 | The group operation of a s... |
| symgov 18906 | The value of the group ope... |
| symgcl 18907 | The group operation of the... |
| idresperm 18908 | The identity function rest... |
| symgmov1 18909 | For a permutation of a set... |
| symgmov2 18910 | For a permutation of a set... |
| symgbas0 18911 | The base set of the symmet... |
| symg1hash 18912 | The symmetric group on a s... |
| symg1bas 18913 | The symmetric group on a s... |
| symg2hash 18914 | The symmetric group on a (... |
| symg2bas 18915 | The symmetric group on a p... |
| 0symgefmndeq 18916 | The symmetric group on the... |
| snsymgefmndeq 18917 | The symmetric group on a s... |
| symgpssefmnd 18918 | For a set ` A ` with more ... |
| symgvalstruct 18919 | The value of the symmetric... |
| symgvalstructOLD 18920 | Obsolete proof of ~ symgva... |
| symgsubmefmnd 18921 | The symmetric group on a s... |
| symgtset 18922 | The topology of the symmet... |
| symggrp 18923 | The symmetric group on a s... |
| symgid 18924 | The group identity element... |
| symginv 18925 | The group inverse in the s... |
| symgsubmefmndALT 18926 | The symmetric group on a s... |
| galactghm 18927 | The currying of a group ac... |
| lactghmga 18928 | The converse of ~ galactgh... |
| symgtopn 18929 | The topology of the symmet... |
| symgga 18930 | The symmetric group induce... |
| pgrpsubgsymgbi 18931 | Every permutation group is... |
| pgrpsubgsymg 18932 | Every permutation group is... |
| idressubgsymg 18933 | The singleton containing o... |
| idrespermg 18934 | The structure with the sin... |
| cayleylem1 18935 | Lemma for ~ cayley . (Con... |
| cayleylem2 18936 | Lemma for ~ cayley . (Con... |
| cayley 18937 | Cayley's Theorem (construc... |
| cayleyth 18938 | Cayley's Theorem (existenc... |
| symgfix2 18939 | If a permutation does not ... |
| symgextf 18940 | The extension of a permuta... |
| symgextfv 18941 | The function value of the ... |
| symgextfve 18942 | The function value of the ... |
| symgextf1lem 18943 | Lemma for ~ symgextf1 . (... |
| symgextf1 18944 | The extension of a permuta... |
| symgextfo 18945 | The extension of a permuta... |
| symgextf1o 18946 | The extension of a permuta... |
| symgextsymg 18947 | The extension of a permuta... |
| symgextres 18948 | The restriction of the ext... |
| gsumccatsymgsn 18949 | Homomorphic property of co... |
| gsmsymgrfixlem1 18950 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 18951 | The composition of permuta... |
| fvcosymgeq 18952 | The values of two composit... |
| gsmsymgreqlem1 18953 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 18954 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 18955 | Two combination of permuta... |
| symgfixelq 18956 | A permutation of a set fix... |
| symgfixels 18957 | The restriction of a permu... |
| symgfixelsi 18958 | The restriction of a permu... |
| symgfixf 18959 | The mapping of a permutati... |
| symgfixf1 18960 | The mapping of a permutati... |
| symgfixfolem1 18961 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 18962 | The mapping of a permutati... |
| symgfixf1o 18963 | The mapping of a permutati... |
| f1omvdmvd 18966 | A permutation of any class... |
| f1omvdcnv 18967 | A permutation and its inve... |
| mvdco 18968 | Composing two permutations... |
| f1omvdconj 18969 | Conjugation of a permutati... |
| f1otrspeq 18970 | A transposition is charact... |
| f1omvdco2 18971 | If exactly one of two perm... |
| f1omvdco3 18972 | If a point is moved by exa... |
| pmtrfval 18973 | The function generating tr... |
| pmtrval 18974 | A generated transposition,... |
| pmtrfv 18975 | General value of mapping a... |
| pmtrprfv 18976 | In a transposition of two ... |
| pmtrprfv3 18977 | In a transposition of two ... |
| pmtrf 18978 | Functionality of a transpo... |
| pmtrmvd 18979 | A transposition moves prec... |
| pmtrrn 18980 | Transposing two points giv... |
| pmtrfrn 18981 | A transposition (as a kind... |
| pmtrffv 18982 | Mapping of a point under a... |
| pmtrrn2 18983 | For any transposition ther... |
| pmtrfinv 18984 | A transposition function i... |
| pmtrfmvdn0 18985 | A transposition moves at l... |
| pmtrff1o 18986 | A transposition function i... |
| pmtrfcnv 18987 | A transposition function i... |
| pmtrfb 18988 | An intrinsic characterizat... |
| pmtrfconj 18989 | Any conjugate of a transpo... |
| symgsssg 18990 | The symmetric group has su... |
| symgfisg 18991 | The symmetric group has a ... |
| symgtrf 18992 | Transpositions are element... |
| symggen 18993 | The span of the transposit... |
| symggen2 18994 | A finite permutation group... |
| symgtrinv 18995 | To invert a permutation re... |
| pmtr3ncomlem1 18996 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 18997 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 18998 | Transpositions over sets w... |
| pmtrdifellem1 18999 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 19000 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 19001 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 19002 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 19003 | A transposition of element... |
| pmtrdifwrdellem1 19004 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 19005 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 19006 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 19007 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 19008 | A sequence of transpositio... |
| pmtrdifwrdel2 19009 | A sequence of transpositio... |
| pmtrprfval 19010 | The transpositions on a pa... |
| pmtrprfvalrn 19011 | The range of the transposi... |
| psgnunilem1 19016 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 19017 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 19018 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 19019 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 19020 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 19021 | Addition and subtraction o... |
| psgnuni 19022 | If the same permutation ca... |
| psgnfval 19023 | Function definition of the... |
| psgnfn 19024 | Functionality and domain o... |
| psgndmsubg 19025 | The finitary permutations ... |
| psgneldm 19026 | Property of being a finita... |
| psgneldm2 19027 | The finitary permutations ... |
| psgneldm2i 19028 | A sequence of transpositio... |
| psgneu 19029 | A finitary permutation has... |
| psgnval 19030 | Value of the permutation s... |
| psgnvali 19031 | A finitary permutation has... |
| psgnvalii 19032 | Any representation of a pe... |
| psgnpmtr 19033 | All transpositions are odd... |
| psgn0fv0 19034 | The permutation sign funct... |
| sygbasnfpfi 19035 | The class of non-fixed poi... |
| psgnfvalfi 19036 | Function definition of the... |
| psgnvalfi 19037 | Value of the permutation s... |
| psgnran 19038 | The range of the permutati... |
| gsmtrcl 19039 | The group sum of transposi... |
| psgnfitr 19040 | A permutation of a finite ... |
| psgnfieu 19041 | A permutation of a finite ... |
| pmtrsn 19042 | The value of the transposi... |
| psgnsn 19043 | The permutation sign funct... |
| psgnprfval 19044 | The permutation sign funct... |
| psgnprfval1 19045 | The permutation sign of th... |
| psgnprfval2 19046 | The permutation sign of th... |
| odfval 19055 | Value of the order functio... |
| odfvalALT 19056 | Shorter proof of ~ odfval ... |
| odval 19057 | Second substitution for th... |
| odlem1 19058 | The group element order is... |
| odcl 19059 | The order of a group eleme... |
| odf 19060 | Functionality of the group... |
| odid 19061 | Any element to the power o... |
| odlem2 19062 | Any positive annihilator o... |
| odmodnn0 19063 | Reduce the argument of a g... |
| mndodconglem 19064 | Lemma for ~ mndodcong . (... |
| mndodcong 19065 | If two multipliers are con... |
| mndodcongi 19066 | If two multipliers are con... |
| oddvdsnn0 19067 | The only multiples of ` A ... |
| odnncl 19068 | If a nonzero multiple of a... |
| odmod 19069 | Reduce the argument of a g... |
| oddvds 19070 | The only multiples of ` A ... |
| oddvdsi 19071 | Any group element is annih... |
| odcong 19072 | If two multipliers are con... |
| odeq 19073 | The ~ oddvds property uniq... |
| odval2 19074 | A non-conditional definiti... |
| odcld 19075 | The order of a group eleme... |
| odmulgid 19076 | A relationship between the... |
| odmulg2 19077 | The order of a multiple di... |
| odmulg 19078 | Relationship between the o... |
| odmulgeq 19079 | A multiple of a point of f... |
| odbezout 19080 | If ` N ` is coprime to the... |
| od1 19081 | The order of the group ide... |
| odeq1 19082 | The group identity is the ... |
| odinv 19083 | The order of the inverse o... |
| odf1 19084 | The multiples of an elemen... |
| odinf 19085 | The multiples of an elemen... |
| dfod2 19086 | An alternative definition ... |
| odcl2 19087 | The order of an element of... |
| oddvds2 19088 | The order of an element of... |
| submod 19089 | The order of an element is... |
| subgod 19090 | The order of an element is... |
| odsubdvds 19091 | The order of an element of... |
| odf1o1 19092 | An element with zero order... |
| odf1o2 19093 | An element with nonzero or... |
| odhash 19094 | An element of zero order g... |
| odhash2 19095 | If an element has nonzero ... |
| odhash3 19096 | An element which generates... |
| odngen 19097 | A cyclic subgroup of size ... |
| gexval 19098 | Value of the exponent of a... |
| gexlem1 19099 | The group element order is... |
| gexcl 19100 | The exponent of a group is... |
| gexid 19101 | Any element to the power o... |
| gexlem2 19102 | Any positive annihilator o... |
| gexdvdsi 19103 | Any group element is annih... |
| gexdvds 19104 | The only ` N ` that annihi... |
| gexdvds2 19105 | An integer divides the gro... |
| gexod 19106 | Any group element is annih... |
| gexcl3 19107 | If the order of every grou... |
| gexnnod 19108 | Every group element has fi... |
| gexcl2 19109 | The exponent of a finite g... |
| gexdvds3 19110 | The exponent of a finite g... |
| gex1 19111 | A group or monoid has expo... |
| ispgp 19112 | A group is a ` P ` -group ... |
| pgpprm 19113 | Reverse closure for the fi... |
| pgpgrp 19114 | Reverse closure for the se... |
| pgpfi1 19115 | A finite group with order ... |
| pgp0 19116 | The identity subgroup is a... |
| subgpgp 19117 | A subgroup of a p-group is... |
| sylow1lem1 19118 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 19119 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 19120 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 19121 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 19122 | Lemma for ~ sylow1 . Usin... |
| sylow1 19123 | Sylow's first theorem. If... |
| odcau 19124 | Cauchy's theorem for the o... |
| pgpfi 19125 | The converse to ~ pgpfi1 .... |
| pgpfi2 19126 | Alternate version of ~ pgp... |
| pgphash 19127 | The order of a p-group. (... |
| isslw 19128 | The property of being a Sy... |
| slwprm 19129 | Reverse closure for the fi... |
| slwsubg 19130 | A Sylow ` P ` -subgroup is... |
| slwispgp 19131 | Defining property of a Syl... |
| slwpss 19132 | A proper superset of a Syl... |
| slwpgp 19133 | A Sylow ` P ` -subgroup is... |
| pgpssslw 19134 | Every ` P ` -subgroup is c... |
| slwn0 19135 | Every finite group contain... |
| subgslw 19136 | A Sylow subgroup that is c... |
| sylow2alem1 19137 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 19138 | Lemma for ~ sylow2a . All... |
| sylow2a 19139 | A named lemma of Sylow's s... |
| sylow2blem1 19140 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 19141 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 19142 | Sylow's second theorem. P... |
| sylow2b 19143 | Sylow's second theorem. A... |
| slwhash 19144 | A sylow subgroup has cardi... |
| fislw 19145 | The sylow subgroups of a f... |
| sylow2 19146 | Sylow's second theorem. S... |
| sylow3lem1 19147 | Lemma for ~ sylow3 , first... |
| sylow3lem2 19148 | Lemma for ~ sylow3 , first... |
| sylow3lem3 19149 | Lemma for ~ sylow3 , first... |
| sylow3lem4 19150 | Lemma for ~ sylow3 , first... |
| sylow3lem5 19151 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 19152 | Lemma for ~ sylow3 , secon... |
| sylow3 19153 | Sylow's third theorem. Th... |
| lsmfval 19158 | The subgroup sum function ... |
| lsmvalx 19159 | Subspace sum value (for a ... |
| lsmelvalx 19160 | Subspace sum membership (f... |
| lsmelvalix 19161 | Subspace sum membership (f... |
| oppglsm 19162 | The subspace sum operation... |
| lsmssv 19163 | Subgroup sum is a subset o... |
| lsmless1x 19164 | Subset implies subgroup su... |
| lsmless2x 19165 | Subset implies subgroup su... |
| lsmub1x 19166 | Subgroup sum is an upper b... |
| lsmub2x 19167 | Subgroup sum is an upper b... |
| lsmval 19168 | Subgroup sum value (for a ... |
| lsmelval 19169 | Subgroup sum membership (f... |
| lsmelvali 19170 | Subgroup sum membership (f... |
| lsmelvalm 19171 | Subgroup sum membership an... |
| lsmelvalmi 19172 | Membership of vector subtr... |
| lsmsubm 19173 | The sum of two commuting s... |
| lsmsubg 19174 | The sum of two commuting s... |
| lsmcom2 19175 | Subgroup sum commutes. (C... |
| smndlsmidm 19176 | The direct product is idem... |
| lsmub1 19177 | Subgroup sum is an upper b... |
| lsmub2 19178 | Subgroup sum is an upper b... |
| lsmunss 19179 | Union of subgroups is a su... |
| lsmless1 19180 | Subset implies subgroup su... |
| lsmless2 19181 | Subset implies subgroup su... |
| lsmless12 19182 | Subset implies subgroup su... |
| lsmidm 19183 | Subgroup sum is idempotent... |
| lsmidmOLD 19184 | Obsolete proof of ~ lsmidm... |
| lsmlub 19185 | The least upper bound prop... |
| lsmss1 19186 | Subgroup sum with a subset... |
| lsmss1b 19187 | Subgroup sum with a subset... |
| lsmss2 19188 | Subgroup sum with a subset... |
| lsmss2b 19189 | Subgroup sum with a subset... |
| lsmass 19190 | Subgroup sum is associativ... |
| mndlsmidm 19191 | Subgroup sum is idempotent... |
| lsm01 19192 | Subgroup sum with the zero... |
| lsm02 19193 | Subgroup sum with the zero... |
| subglsm 19194 | The subgroup sum evaluated... |
| lssnle 19195 | Equivalent expressions for... |
| lsmmod 19196 | The modular law holds for ... |
| lsmmod2 19197 | Modular law dual for subgr... |
| lsmpropd 19198 | If two structures have the... |
| cntzrecd 19199 | Commute the "subgroups com... |
| lsmcntz 19200 | The "subgroups commute" pr... |
| lsmcntzr 19201 | The "subgroups commute" pr... |
| lsmdisj 19202 | Disjointness from a subgro... |
| lsmdisj2 19203 | Association of the disjoin... |
| lsmdisj3 19204 | Association of the disjoin... |
| lsmdisjr 19205 | Disjointness from a subgro... |
| lsmdisj2r 19206 | Association of the disjoin... |
| lsmdisj3r 19207 | Association of the disjoin... |
| lsmdisj2a 19208 | Association of the disjoin... |
| lsmdisj2b 19209 | Association of the disjoin... |
| lsmdisj3a 19210 | Association of the disjoin... |
| lsmdisj3b 19211 | Association of the disjoin... |
| subgdisj1 19212 | Vectors belonging to disjo... |
| subgdisj2 19213 | Vectors belonging to disjo... |
| subgdisjb 19214 | Vectors belonging to disjo... |
| pj1fval 19215 | The left projection functi... |
| pj1val 19216 | The left projection functi... |
| pj1eu 19217 | Uniqueness of a left proje... |
| pj1f 19218 | The left projection functi... |
| pj2f 19219 | The right projection funct... |
| pj1id 19220 | Any element of a direct su... |
| pj1eq 19221 | Any element of a direct su... |
| pj1lid 19222 | The left projection functi... |
| pj1rid 19223 | The left projection functi... |
| pj1ghm 19224 | The left projection functi... |
| pj1ghm2 19225 | The left projection functi... |
| lsmhash 19226 | The order of the direct pr... |
| efgmval 19233 | Value of the formal invers... |
| efgmf 19234 | The formal inverse operati... |
| efgmnvl 19235 | The inversion function on ... |
| efgrcl 19236 | Lemma for ~ efgval . (Con... |
| efglem 19237 | Lemma for ~ efgval . (Con... |
| efgval 19238 | Value of the free group co... |
| efger 19239 | Value of the free group co... |
| efgi 19240 | Value of the free group co... |
| efgi0 19241 | Value of the free group co... |
| efgi1 19242 | Value of the free group co... |
| efgtf 19243 | Value of the free group co... |
| efgtval 19244 | Value of the extension fun... |
| efgval2 19245 | Value of the free group co... |
| efgi2 19246 | Value of the free group co... |
| efgtlen 19247 | Value of the free group co... |
| efginvrel2 19248 | The inverse of the reverse... |
| efginvrel1 19249 | The inverse of the reverse... |
| efgsf 19250 | Value of the auxiliary fun... |
| efgsdm 19251 | Elementhood in the domain ... |
| efgsval 19252 | Value of the auxiliary fun... |
| efgsdmi 19253 | Property of the last link ... |
| efgsval2 19254 | Value of the auxiliary fun... |
| efgsrel 19255 | The start and end of any e... |
| efgs1 19256 | A singleton of an irreduci... |
| efgs1b 19257 | Every extension sequence e... |
| efgsp1 19258 | If ` F ` is an extension s... |
| efgsres 19259 | An initial segment of an e... |
| efgsfo 19260 | For any word, there is a s... |
| efgredlema 19261 | The reduced word that form... |
| efgredlemf 19262 | Lemma for ~ efgredleme . ... |
| efgredlemg 19263 | Lemma for ~ efgred . (Con... |
| efgredleme 19264 | Lemma for ~ efgred . (Con... |
| efgredlemd 19265 | The reduced word that form... |
| efgredlemc 19266 | The reduced word that form... |
| efgredlemb 19267 | The reduced word that form... |
| efgredlem 19268 | The reduced word that form... |
| efgred 19269 | The reduced word that form... |
| efgrelexlema 19270 | If two words ` A , B ` are... |
| efgrelexlemb 19271 | If two words ` A , B ` are... |
| efgrelex 19272 | If two words ` A , B ` are... |
| efgredeu 19273 | There is a unique reduced ... |
| efgred2 19274 | Two extension sequences ha... |
| efgcpbllema 19275 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 19276 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 19277 | Two extension sequences ha... |
| efgcpbl2 19278 | Two extension sequences ha... |
| frgpval 19279 | Value of the free group co... |
| frgpcpbl 19280 | Compatibility of the group... |
| frgp0 19281 | The free group is a group.... |
| frgpeccl 19282 | Closure of the quotient ma... |
| frgpgrp 19283 | The free group is a group.... |
| frgpadd 19284 | Addition in the free group... |
| frgpinv 19285 | The inverse of an element ... |
| frgpmhm 19286 | The "natural map" from wor... |
| vrgpfval 19287 | The canonical injection fr... |
| vrgpval 19288 | The value of the generatin... |
| vrgpf 19289 | The mapping from the index... |
| vrgpinv 19290 | The inverse of a generatin... |
| frgpuptf 19291 | Any assignment of the gene... |
| frgpuptinv 19292 | Any assignment of the gene... |
| frgpuplem 19293 | Any assignment of the gene... |
| frgpupf 19294 | Any assignment of the gene... |
| frgpupval 19295 | Any assignment of the gene... |
| frgpup1 19296 | Any assignment of the gene... |
| frgpup2 19297 | The evaluation map has the... |
| frgpup3lem 19298 | The evaluation map has the... |
| frgpup3 19299 | Universal property of the ... |
| 0frgp 19300 | The free group on zero gen... |
| isabl 19305 | The predicate "is an Abeli... |
| ablgrp 19306 | An Abelian group is a grou... |
| ablgrpd 19307 | An Abelian group is a grou... |
| ablcmn 19308 | An Abelian group is a comm... |
| iscmn 19309 | The predicate "is a commut... |
| isabl2 19310 | The predicate "is an Abeli... |
| cmnpropd 19311 | If two structures have the... |
| ablpropd 19312 | If two structures have the... |
| ablprop 19313 | If two structures have the... |
| iscmnd 19314 | Properties that determine ... |
| isabld 19315 | Properties that determine ... |
| isabli 19316 | Properties that determine ... |
| cmnmnd 19317 | A commutative monoid is a ... |
| cmncom 19318 | A commutative monoid is co... |
| ablcom 19319 | An Abelian group operation... |
| cmn32 19320 | Commutative/associative la... |
| cmn4 19321 | Commutative/associative la... |
| cmn12 19322 | Commutative/associative la... |
| abl32 19323 | Commutative/associative la... |
| cmnmndd 19324 | A commutative monoid is a ... |
| rinvmod 19325 | Uniqueness of a right inve... |
| ablinvadd 19326 | The inverse of an Abelian ... |
| ablsub2inv 19327 | Abelian group subtraction ... |
| ablsubadd 19328 | Relationship between Abeli... |
| ablsub4 19329 | Commutative/associative su... |
| abladdsub4 19330 | Abelian group addition/sub... |
| abladdsub 19331 | Associative-type law for g... |
| ablpncan2 19332 | Cancellation law for subtr... |
| ablpncan3 19333 | A cancellation law for com... |
| ablsubsub 19334 | Law for double subtraction... |
| ablsubsub4 19335 | Law for double subtraction... |
| ablpnpcan 19336 | Cancellation law for mixed... |
| ablnncan 19337 | Cancellation law for group... |
| ablsub32 19338 | Swap the second and third ... |
| ablnnncan 19339 | Cancellation law for group... |
| ablnnncan1 19340 | Cancellation law for group... |
| ablsubsub23 19341 | Swap subtrahend and result... |
| mulgnn0di 19342 | Group multiple of a sum, f... |
| mulgdi 19343 | Group multiple of a sum. ... |
| mulgmhm 19344 | The map from ` x ` to ` n ... |
| mulgghm 19345 | The map from ` x ` to ` n ... |
| mulgsubdi 19346 | Group multiple of a differ... |
| ghmfghm 19347 | The function fulfilling th... |
| ghmcmn 19348 | The image of a commutative... |
| ghmabl 19349 | The image of an abelian gr... |
| invghm 19350 | The inversion map is a gro... |
| eqgabl 19351 | Value of the subgroup cose... |
| subgabl 19352 | A subgroup of an abelian g... |
| subcmn 19353 | A submonoid of a commutati... |
| submcmn 19354 | A submonoid of a commutati... |
| submcmn2 19355 | A submonoid is commutative... |
| cntzcmn 19356 | The centralizer of any sub... |
| cntzcmnss 19357 | Any subset in a commutativ... |
| cntrcmnd 19358 | The center of a monoid is ... |
| cntrabl 19359 | The center of a group is a... |
| cntzspan 19360 | If the generators commute,... |
| cntzcmnf 19361 | Discharge the centralizer ... |
| ghmplusg 19362 | The pointwise sum of two l... |
| ablnsg 19363 | Every subgroup of an abeli... |
| odadd1 19364 | The order of a product in ... |
| odadd2 19365 | The order of a product in ... |
| odadd 19366 | The order of a product is ... |
| gex2abl 19367 | A group with exponent 2 (o... |
| gexexlem 19368 | Lemma for ~ gexex . (Cont... |
| gexex 19369 | In an abelian group with f... |
| torsubg 19370 | The set of all elements of... |
| oddvdssubg 19371 | The set of all elements wh... |
| lsmcomx 19372 | Subgroup sum commutes (ext... |
| ablcntzd 19373 | All subgroups in an abelia... |
| lsmcom 19374 | Subgroup sum commutes. (C... |
| lsmsubg2 19375 | The sum of two subgroups i... |
| lsm4 19376 | Commutative/associative la... |
| prdscmnd 19377 | The product of a family of... |
| prdsabld 19378 | The product of a family of... |
| pwscmn 19379 | The structure power on a c... |
| pwsabl 19380 | The structure power on an ... |
| qusabl 19381 | If ` Y ` is a subgroup of ... |
| abl1 19382 | The (smallest) structure r... |
| abln0 19383 | Abelian groups (and theref... |
| cnaddablx 19384 | The complex numbers are an... |
| cnaddabl 19385 | The complex numbers are an... |
| cnaddid 19386 | The group identity element... |
| cnaddinv 19387 | Value of the group inverse... |
| zaddablx 19388 | The integers are an Abelia... |
| frgpnabllem1 19389 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 19390 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 19391 | The free group on two or m... |
| iscyg 19394 | Definition of a cyclic gro... |
| iscyggen 19395 | The property of being a cy... |
| iscyggen2 19396 | The property of being a cy... |
| iscyg2 19397 | A cyclic group is a group ... |
| cyggeninv 19398 | The inverse of a cyclic ge... |
| cyggenod 19399 | An element is the generato... |
| cyggenod2 19400 | In an infinite cyclic grou... |
| iscyg3 19401 | Definition of a cyclic gro... |
| iscygd 19402 | Definition of a cyclic gro... |
| iscygodd 19403 | Show that a group with an ... |
| cycsubmcmn 19404 | The set of nonnegative int... |
| cyggrp 19405 | A cyclic group is a group.... |
| cygabl 19406 | A cyclic group is abelian.... |
| cygablOLD 19407 | Obsolete proof of ~ cygabl... |
| cygctb 19408 | A cyclic group is countabl... |
| 0cyg 19409 | The trivial group is cycli... |
| prmcyg 19410 | A group with prime order i... |
| lt6abl 19411 | A group with fewer than ` ... |
| ghmcyg 19412 | The image of a cyclic grou... |
| cyggex2 19413 | The exponent of a cyclic g... |
| cyggex 19414 | The exponent of a finite c... |
| cyggexb 19415 | A finite abelian group is ... |
| giccyg 19416 | Cyclicity is a group prope... |
| cycsubgcyg 19417 | The cyclic subgroup genera... |
| cycsubgcyg2 19418 | The cyclic subgroup genera... |
| gsumval3a 19419 | Value of the group sum ope... |
| gsumval3eu 19420 | The group sum as defined i... |
| gsumval3lem1 19421 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19422 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19423 | Value of the group sum ope... |
| gsumcllem 19424 | Lemma for ~ gsumcl and rel... |
| gsumzres 19425 | Extend a finite group sum ... |
| gsumzcl2 19426 | Closure of a finite group ... |
| gsumzcl 19427 | Closure of a finite group ... |
| gsumzf1o 19428 | Re-index a finite group su... |
| gsumres 19429 | Extend a finite group sum ... |
| gsumcl2 19430 | Closure of a finite group ... |
| gsumcl 19431 | Closure of a finite group ... |
| gsumf1o 19432 | Re-index a finite group su... |
| gsumreidx 19433 | Re-index a finite group su... |
| gsumzsubmcl 19434 | Closure of a group sum in ... |
| gsumsubmcl 19435 | Closure of a group sum in ... |
| gsumsubgcl 19436 | Closure of a group sum in ... |
| gsumzaddlem 19437 | The sum of two group sums.... |
| gsumzadd 19438 | The sum of two group sums.... |
| gsumadd 19439 | The sum of two group sums.... |
| gsummptfsadd 19440 | The sum of two group sums ... |
| gsummptfidmadd 19441 | The sum of two group sums ... |
| gsummptfidmadd2 19442 | The sum of two group sums ... |
| gsumzsplit 19443 | Split a group sum into two... |
| gsumsplit 19444 | Split a group sum into two... |
| gsumsplit2 19445 | Split a group sum into two... |
| gsummptfidmsplit 19446 | Split a group sum expresse... |
| gsummptfidmsplitres 19447 | Split a group sum expresse... |
| gsummptfzsplit 19448 | Split a group sum expresse... |
| gsummptfzsplitl 19449 | Split a group sum expresse... |
| gsumconst 19450 | Sum of a constant series. ... |
| gsumconstf 19451 | Sum of a constant series. ... |
| gsummptshft 19452 | Index shift of a finite gr... |
| gsumzmhm 19453 | Apply a group homomorphism... |
| gsummhm 19454 | Apply a group homomorphism... |
| gsummhm2 19455 | Apply a group homomorphism... |
| gsummptmhm 19456 | Apply a group homomorphism... |
| gsummulglem 19457 | Lemma for ~ gsummulg and ~... |
| gsummulg 19458 | Nonnegative multiple of a ... |
| gsummulgz 19459 | Integer multiple of a grou... |
| gsumzoppg 19460 | The opposite of a group su... |
| gsumzinv 19461 | Inverse of a group sum. (... |
| gsuminv 19462 | Inverse of a group sum. (... |
| gsummptfidminv 19463 | Inverse of a group sum exp... |
| gsumsub 19464 | The difference of two grou... |
| gsummptfssub 19465 | The difference of two grou... |
| gsummptfidmsub 19466 | The difference of two grou... |
| gsumsnfd 19467 | Group sum of a singleton, ... |
| gsumsnd 19468 | Group sum of a singleton, ... |
| gsumsnf 19469 | Group sum of a singleton, ... |
| gsumsn 19470 | Group sum of a singleton. ... |
| gsumpr 19471 | Group sum of a pair. (Con... |
| gsumzunsnd 19472 | Append an element to a fin... |
| gsumunsnfd 19473 | Append an element to a fin... |
| gsumunsnd 19474 | Append an element to a fin... |
| gsumunsnf 19475 | Append an element to a fin... |
| gsumunsn 19476 | Append an element to a fin... |
| gsumdifsnd 19477 | Extract a summand from a f... |
| gsumpt 19478 | Sum of a family that is no... |
| gsummptf1o 19479 | Re-index a finite group su... |
| gsummptun 19480 | Group sum of a disjoint un... |
| gsummpt1n0 19481 | If only one summand in a f... |
| gsummptif1n0 19482 | If only one summand in a f... |
| gsummptcl 19483 | Closure of a finite group ... |
| gsummptfif1o 19484 | Re-index a finite group su... |
| gsummptfzcl 19485 | Closure of a finite group ... |
| gsum2dlem1 19486 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 19487 | Lemma for ~ gsum2d . (Con... |
| gsum2d 19488 | Write a sum over a two-dim... |
| gsum2d2lem 19489 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 19490 | Write a group sum over a t... |
| gsumcom2 19491 | Two-dimensional commutatio... |
| gsumxp 19492 | Write a group sum over a c... |
| gsumcom 19493 | Commute the arguments of a... |
| gsumcom3 19494 | A commutative law for fini... |
| gsumcom3fi 19495 | A commutative law for fini... |
| gsumxp2 19496 | Write a group sum over a c... |
| prdsgsum 19497 | Finite commutative sums in... |
| pwsgsum 19498 | Finite commutative sums in... |
| fsfnn0gsumfsffz 19499 | Replacing a finitely suppo... |
| nn0gsumfz 19500 | Replacing a finitely suppo... |
| nn0gsumfz0 19501 | Replacing a finitely suppo... |
| gsummptnn0fz 19502 | A final group sum over a f... |
| gsummptnn0fzfv 19503 | A final group sum over a f... |
| telgsumfzslem 19504 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 19505 | Telescoping group sum rang... |
| telgsumfz 19506 | Telescoping group sum rang... |
| telgsumfz0s 19507 | Telescoping finite group s... |
| telgsumfz0 19508 | Telescoping finite group s... |
| telgsums 19509 | Telescoping finitely suppo... |
| telgsum 19510 | Telescoping finitely suppo... |
| reldmdprd 19515 | The domain of the internal... |
| dmdprd 19516 | The domain of definition o... |
| dmdprdd 19517 | Show that a given family i... |
| dprddomprc 19518 | A family of subgroups inde... |
| dprddomcld 19519 | If a family of subgroups i... |
| dprdval0prc 19520 | The internal direct produc... |
| dprdval 19521 | The value of the internal ... |
| eldprd 19522 | A class ` A ` is an intern... |
| dprdgrp 19523 | Reverse closure for the in... |
| dprdf 19524 | The function ` S ` is a fa... |
| dprdf2 19525 | The function ` S ` is a fa... |
| dprdcntz 19526 | The function ` S ` is a fa... |
| dprddisj 19527 | The function ` S ` is a fa... |
| dprdw 19528 | The property of being a fi... |
| dprdwd 19529 | A mapping being a finitely... |
| dprdff 19530 | A finitely supported funct... |
| dprdfcl 19531 | A finitely supported funct... |
| dprdffsupp 19532 | A finitely supported funct... |
| dprdfcntz 19533 | A function on the elements... |
| dprdssv 19534 | The internal direct produc... |
| dprdfid 19535 | A function mapping all but... |
| eldprdi 19536 | The domain of definition o... |
| dprdfinv 19537 | Take the inverse of a grou... |
| dprdfadd 19538 | Take the sum of group sums... |
| dprdfsub 19539 | Take the difference of gro... |
| dprdfeq0 19540 | The zero function is the o... |
| dprdf11 19541 | Two group sums over a dire... |
| dprdsubg 19542 | The internal direct produc... |
| dprdub 19543 | Each factor is a subset of... |
| dprdlub 19544 | The direct product is smal... |
| dprdspan 19545 | The direct product is the ... |
| dprdres 19546 | Restriction of a direct pr... |
| dprdss 19547 | Create a direct product by... |
| dprdz 19548 | A family consisting entire... |
| dprd0 19549 | The empty family is an int... |
| dprdf1o 19550 | Rearrange the index set of... |
| dprdf1 19551 | Rearrange the index set of... |
| subgdmdprd 19552 | A direct product in a subg... |
| subgdprd 19553 | A direct product in a subg... |
| dprdsn 19554 | A singleton family is an i... |
| dmdprdsplitlem 19555 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 19556 | The function ` S ` is a fa... |
| dprddisj2 19557 | The function ` S ` is a fa... |
| dprd2dlem2 19558 | The direct product of a co... |
| dprd2dlem1 19559 | The direct product of a co... |
| dprd2da 19560 | The direct product of a co... |
| dprd2db 19561 | The direct product of a co... |
| dprd2d2 19562 | The direct product of a co... |
| dmdprdsplit2lem 19563 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 19564 | The direct product splits ... |
| dmdprdsplit 19565 | The direct product splits ... |
| dprdsplit 19566 | The direct product is the ... |
| dmdprdpr 19567 | A singleton family is an i... |
| dprdpr 19568 | A singleton family is an i... |
| dpjlem 19569 | Lemma for theorems about d... |
| dpjcntz 19570 | The two subgroups that app... |
| dpjdisj 19571 | The two subgroups that app... |
| dpjlsm 19572 | The two subgroups that app... |
| dpjfval 19573 | Value of the direct produc... |
| dpjval 19574 | Value of the direct produc... |
| dpjf 19575 | The ` X ` -th index projec... |
| dpjidcl 19576 | The key property of projec... |
| dpjeq 19577 | Decompose a group sum into... |
| dpjid 19578 | The key property of projec... |
| dpjlid 19579 | The ` X ` -th index projec... |
| dpjrid 19580 | The ` Y ` -th index projec... |
| dpjghm 19581 | The direct product is the ... |
| dpjghm2 19582 | The direct product is the ... |
| ablfacrplem 19583 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 19584 | A finite abelian group who... |
| ablfacrp2 19585 | The factors ` K , L ` of ~... |
| ablfac1lem 19586 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 19587 | The factors of ~ ablfac1b ... |
| ablfac1b 19588 | Any abelian group is the d... |
| ablfac1c 19589 | The factors of ~ ablfac1b ... |
| ablfac1eulem 19590 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 19591 | The factorization of ~ abl... |
| pgpfac1lem1 19592 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 19593 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 19594 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 19595 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 19596 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 19597 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 19598 | Factorization of a finite ... |
| pgpfaclem1 19599 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 19600 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 19601 | Lemma for ~ pgpfac . (Con... |
| pgpfac 19602 | Full factorization of a fi... |
| ablfaclem1 19603 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 19604 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 19605 | Lemma for ~ ablfac . (Con... |
| ablfac 19606 | The Fundamental Theorem of... |
| ablfac2 19607 | Choose generators for each... |
| issimpg 19610 | The predicate "is a simple... |
| issimpgd 19611 | Deduce a simple group from... |
| simpggrp 19612 | A simple group is a group.... |
| simpggrpd 19613 | A simple group is a group.... |
| simpg2nsg 19614 | A simple group has two nor... |
| trivnsimpgd 19615 | Trivial groups are not sim... |
| simpgntrivd 19616 | Simple groups are nontrivi... |
| simpgnideld 19617 | A simple group contains a ... |
| simpgnsgd 19618 | The only normal subgroups ... |
| simpgnsgeqd 19619 | A normal subgroup of a sim... |
| 2nsgsimpgd 19620 | If any normal subgroup of ... |
| simpgnsgbid 19621 | A nontrivial group is simp... |
| ablsimpnosubgd 19622 | A subgroup of an abelian s... |
| ablsimpg1gend 19623 | An abelian simple group is... |
| ablsimpgcygd 19624 | An abelian simple group is... |
| ablsimpgfindlem1 19625 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 19626 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 19627 | Relationship between the o... |
| ablsimpgfind 19628 | An abelian simple group is... |
| fincygsubgd 19629 | The subgroup referenced in... |
| fincygsubgodd 19630 | Calculate the order of a s... |
| fincygsubgodexd 19631 | A finite cyclic group has ... |
| prmgrpsimpgd 19632 | A group of prime order is ... |
| ablsimpgprmd 19633 | An abelian simple group ha... |
| ablsimpgd 19634 | An abelian group is simple... |
| fnmgp 19637 | The multiplicative group o... |
| mgpval 19638 | Value of the multiplicatio... |
| mgpplusg 19639 | Value of the group operati... |
| mgplemOLD 19640 | Obsolete version of ~ sets... |
| mgpbas 19641 | Base set of the multiplica... |
| mgpbasOLD 19642 | Obsolete version of ~ mgpb... |
| mgpsca 19643 | The multiplication monoid ... |
| mgpscaOLD 19644 | Obsolete version of ~ mgps... |
| mgptset 19645 | Topology component of the ... |
| mgptsetOLD 19646 | Obsolete version of ~ mgpt... |
| mgptopn 19647 | Topology of the multiplica... |
| mgpds 19648 | Distance function of the m... |
| mgpdsOLD 19649 | Obsolete version of ~ mgpd... |
| mgpress 19650 | Subgroup commutes with the... |
| mgpressOLD 19651 | Obsolete version of ~ mgpr... |
| ringidval 19654 | The value of the unity ele... |
| dfur2 19655 | The multiplicative identit... |
| issrg 19658 | The predicate "is a semiri... |
| srgcmn 19659 | A semiring is a commutativ... |
| srgmnd 19660 | A semiring is a monoid. (... |
| srgmgp 19661 | A semiring is a monoid und... |
| srgi 19662 | Properties of a semiring. ... |
| srgcl 19663 | Closure of the multiplicat... |
| srgass 19664 | Associative law for the mu... |
| srgideu 19665 | The unit element of a semi... |
| srgfcl 19666 | Functionality of the multi... |
| srgdi 19667 | Distributive law for the m... |
| srgdir 19668 | Distributive law for the m... |
| srgidcl 19669 | The unit element of a semi... |
| srg0cl 19670 | The zero element of a semi... |
| srgidmlem 19671 | Lemma for ~ srglidm and ~ ... |
| srglidm 19672 | The unit element of a semi... |
| srgridm 19673 | The unit element of a semi... |
| issrgid 19674 | Properties showing that an... |
| srgacl 19675 | Closure of the addition op... |
| srgcom 19676 | Commutativity of the addit... |
| srgrz 19677 | The zero of a semiring is ... |
| srglz 19678 | The zero of a semiring is ... |
| srgisid 19679 | In a semiring, the only le... |
| srg1zr 19680 | The only semiring with a b... |
| srgen1zr 19681 | The only semiring with one... |
| srgmulgass 19682 | An associative property be... |
| srgpcomp 19683 | If two elements of a semir... |
| srgpcompp 19684 | If two elements of a semir... |
| srgpcomppsc 19685 | If two elements of a semir... |
| srglmhm 19686 | Left-multiplication in a s... |
| srgrmhm 19687 | Right-multiplication in a ... |
| srgsummulcr 19688 | A finite semiring sum mult... |
| sgsummulcl 19689 | A finite semiring sum mult... |
| srg1expzeq1 19690 | The exponentiation (by a n... |
| srgbinomlem1 19691 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 19692 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 19693 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 19694 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 19695 | Lemma for ~ srgbinom . In... |
| srgbinom 19696 | The binomial theorem for c... |
| csrgbinom 19697 | The binomial theorem for c... |
| isring 19702 | The predicate "is a (unita... |
| ringgrp 19703 | A ring is a group. (Contr... |
| ringmgp 19704 | A ring is a monoid under m... |
| iscrng 19705 | A commutative ring is a ri... |
| crngmgp 19706 | A commutative ring's multi... |
| ringgrpd 19707 | A ring is a group. (Contr... |
| ringmnd 19708 | A ring is a monoid under a... |
| ringmgm 19709 | A ring is a magma. (Contr... |
| crngring 19710 | A commutative ring is a ri... |
| crngringd 19711 | A commutative ring is a ri... |
| crnggrpd 19712 | A commutative ring is a gr... |
| mgpf 19713 | Restricted functionality o... |
| ringi 19714 | Properties of a unital rin... |
| ringcl 19715 | Closure of the multiplicat... |
| crngcom 19716 | A commutative ring's multi... |
| iscrng2 19717 | A commutative ring is a ri... |
| ringass 19718 | Associative law for multip... |
| ringideu 19719 | The unit element of a ring... |
| ringdi 19720 | Distributive law for the m... |
| ringdir 19721 | Distributive law for the m... |
| ringidcl 19722 | The unit element of a ring... |
| ring0cl 19723 | The zero element of a ring... |
| ringidmlem 19724 | Lemma for ~ ringlidm and ~... |
| ringlidm 19725 | The unit element of a ring... |
| ringridm 19726 | The unit element of a ring... |
| isringid 19727 | Properties showing that an... |
| ringid 19728 | The multiplication operati... |
| ringadd2 19729 | A ring element plus itself... |
| rngo2times 19730 | A ring element plus itself... |
| ringidss 19731 | A subset of the multiplica... |
| ringacl 19732 | Closure of the addition op... |
| ringcom 19733 | Commutativity of the addit... |
| ringabl 19734 | A ring is an Abelian group... |
| ringcmn 19735 | A ring is a commutative mo... |
| ringpropd 19736 | If two structures have the... |
| crngpropd 19737 | If two structures have the... |
| ringprop 19738 | If two structures have the... |
| isringd 19739 | Properties that determine ... |
| iscrngd 19740 | Properties that determine ... |
| ringlz 19741 | The zero of a unital ring ... |
| ringrz 19742 | The zero of a unital ring ... |
| ringsrg 19743 | Any ring is also a semirin... |
| ring1eq0 19744 | If one and zero are equal,... |
| ring1ne0 19745 | If a ring has at least two... |
| ringinvnz1ne0 19746 | In a unitary ring, a left ... |
| ringinvnzdiv 19747 | In a unitary ring, a left ... |
| ringnegl 19748 | Negation in a ring is the ... |
| rngnegr 19749 | Negation in a ring is the ... |
| ringmneg1 19750 | Negation of a product in a... |
| ringmneg2 19751 | Negation of a product in a... |
| ringm2neg 19752 | Double negation of a produ... |
| ringsubdi 19753 | Ring multiplication distri... |
| rngsubdir 19754 | Ring multiplication distri... |
| mulgass2 19755 | An associative property be... |
| ring1 19756 | The (smallest) structure r... |
| ringn0 19757 | Rings exist. (Contributed... |
| ringlghm 19758 | Left-multiplication in a r... |
| ringrghm 19759 | Right-multiplication in a ... |
| gsummulc1 19760 | A finite ring sum multipli... |
| gsummulc2 19761 | A finite ring sum multipli... |
| gsummgp0 19762 | If one factor in a finite ... |
| gsumdixp 19763 | Distribute a binary produc... |
| prdsmgp 19764 | The multiplicative monoid ... |
| prdsmulrcl 19765 | A structure product of rin... |
| prdsringd 19766 | A product of rings is a ri... |
| prdscrngd 19767 | A product of commutative r... |
| prds1 19768 | Value of the ring unit in ... |
| pwsring 19769 | A structure power of a rin... |
| pws1 19770 | Value of the ring unit in ... |
| pwscrng 19771 | A structure power of a com... |
| pwsmgp 19772 | The multiplicative group o... |
| imasring 19773 | The image structure of a r... |
| qusring2 19774 | The quotient structure of ... |
| crngbinom 19775 | The binomial theorem for c... |
| opprval 19778 | Value of the opposite ring... |
| opprmulfval 19779 | Value of the multiplicatio... |
| opprmul 19780 | Value of the multiplicatio... |
| crngoppr 19781 | In a commutative ring, the... |
| opprlem 19782 | Lemma for ~ opprbas and ~ ... |
| opprlemOLD 19783 | Obsolete version of ~ oppr... |
| opprbas 19784 | Base set of an opposite ri... |
| opprbasOLD 19785 | Obsolete proof of ~ opprba... |
| oppradd 19786 | Addition operation of an o... |
| oppraddOLD 19787 | Obsolete proof of ~ opprba... |
| opprring 19788 | An opposite ring is a ring... |
| opprringb 19789 | Bidirectional form of ~ op... |
| oppr0 19790 | Additive identity of an op... |
| oppr1 19791 | Multiplicative identity of... |
| opprneg 19792 | The negative function in a... |
| opprsubg 19793 | Being a subgroup is a symm... |
| mulgass3 19794 | An associative property be... |
| reldvdsr 19801 | The divides relation is a ... |
| dvdsrval 19802 | Value of the divides relat... |
| dvdsr 19803 | Value of the divides relat... |
| dvdsr2 19804 | Value of the divides relat... |
| dvdsrmul 19805 | A left-multiple of ` X ` i... |
| dvdsrcl 19806 | Closure of a dividing elem... |
| dvdsrcl2 19807 | Closure of a dividing elem... |
| dvdsrid 19808 | An element in a (unital) r... |
| dvdsrtr 19809 | Divisibility is transitive... |
| dvdsrmul1 19810 | The divisibility relation ... |
| dvdsrneg 19811 | An element divides its neg... |
| dvdsr01 19812 | In a ring, zero is divisib... |
| dvdsr02 19813 | Only zero is divisible by ... |
| isunit 19814 | Property of being a unit o... |
| 1unit 19815 | The multiplicative identit... |
| unitcl 19816 | A unit is an element of th... |
| unitss 19817 | The set of units is contai... |
| opprunit 19818 | Being a unit is a symmetri... |
| crngunit 19819 | Property of being a unit i... |
| dvdsunit 19820 | A divisor of a unit is a u... |
| unitmulcl 19821 | The product of units is a ... |
| unitmulclb 19822 | Reversal of ~ unitmulcl in... |
| unitgrpbas 19823 | The base set of the group ... |
| unitgrp 19824 | The group of units is a gr... |
| unitabl 19825 | The group of units of a co... |
| unitgrpid 19826 | The identity of the multip... |
| unitsubm 19827 | The group of units is a su... |
| invrfval 19830 | Multiplicative inverse fun... |
| unitinvcl 19831 | The inverse of a unit exis... |
| unitinvinv 19832 | The inverse of the inverse... |
| ringinvcl 19833 | The inverse of a unit is a... |
| unitlinv 19834 | A unit times its inverse i... |
| unitrinv 19835 | A unit times its inverse i... |
| 1rinv 19836 | The inverse of the identit... |
| 0unit 19837 | The additive identity is a... |
| unitnegcl 19838 | The negative of a unit is ... |
| dvrfval 19841 | Division operation in a ri... |
| dvrval 19842 | Division operation in a ri... |
| dvrcl 19843 | Closure of division operat... |
| unitdvcl 19844 | The units are closed under... |
| dvrid 19845 | A cancellation law for div... |
| dvr1 19846 | A cancellation law for div... |
| dvrass 19847 | An associative law for div... |
| dvrcan1 19848 | A cancellation law for div... |
| dvrcan3 19849 | A cancellation law for div... |
| dvreq1 19850 | A cancellation law for div... |
| ringinvdv 19851 | Write the inverse function... |
| rngidpropd 19852 | The ring identity depends ... |
| dvdsrpropd 19853 | The divisibility relation ... |
| unitpropd 19854 | The set of units depends o... |
| invrpropd 19855 | The ring inverse function ... |
| isirred 19856 | An irreducible element of ... |
| isnirred 19857 | The property of being a no... |
| isirred2 19858 | Expand out the class diffe... |
| opprirred 19859 | Irreducibility is symmetri... |
| irredn0 19860 | The additive identity is n... |
| irredcl 19861 | An irreducible element is ... |
| irrednu 19862 | An irreducible element is ... |
| irredn1 19863 | The multiplicative identit... |
| irredrmul 19864 | The product of an irreduci... |
| irredlmul 19865 | The product of a unit and ... |
| irredmul 19866 | If product of two elements... |
| irredneg 19867 | The negative of an irreduc... |
| irrednegb 19868 | An element is irreducible ... |
| dfrhm2 19876 | The property of a ring hom... |
| rhmrcl1 19878 | Reverse closure of a ring ... |
| rhmrcl2 19879 | Reverse closure of a ring ... |
| isrhm 19880 | A function is a ring homom... |
| rhmmhm 19881 | A ring homomorphism is a h... |
| isrim0 19882 | An isomorphism of rings is... |
| rimrcl 19883 | Reverse closure for an iso... |
| rhmghm 19884 | A ring homomorphism is an ... |
| rhmf 19885 | A ring homomorphism is a f... |
| rhmmul 19886 | A homomorphism of rings pr... |
| isrhm2d 19887 | Demonstration of ring homo... |
| isrhmd 19888 | Demonstration of ring homo... |
| rhm1 19889 | Ring homomorphisms are req... |
| idrhm 19890 | The identity homomorphism ... |
| rhmf1o 19891 | A ring homomorphism is bij... |
| isrim 19892 | An isomorphism of rings is... |
| rimf1o 19893 | An isomorphism of rings is... |
| rimrhm 19894 | An isomorphism of rings is... |
| rimgim 19895 | An isomorphism of rings is... |
| rhmco 19896 | The composition of ring ho... |
| pwsco1rhm 19897 | Right composition with a f... |
| pwsco2rhm 19898 | Left composition with a ri... |
| f1ghm0to0 19899 | If a group homomorphism ` ... |
| f1rhm0to0ALT 19900 | Alternate proof for ~ f1gh... |
| gim0to0 19901 | A group isomorphism maps t... |
| kerf1ghm 19902 | A group homomorphism ` F `... |
| brric 19903 | The relation "is isomorphi... |
| brric2 19904 | The relation "is isomorphi... |
| ricgic 19905 | If two rings are (ring) is... |
| isdrng 19910 | The predicate "is a divisi... |
| drngunit 19911 | Elementhood in the set of ... |
| drngui 19912 | The set of units of a divi... |
| drngring 19913 | A division ring is a ring.... |
| drnggrp 19914 | A division ring is a group... |
| isfld 19915 | A field is a commutative d... |
| isdrng2 19916 | A division ring can equiva... |
| drngprop 19917 | If two structures have the... |
| drngmgp 19918 | A division ring contains a... |
| drngmcl 19919 | The product of two nonzero... |
| drngid 19920 | A division ring's unit is ... |
| drngunz 19921 | A division ring's unit is ... |
| drngid2 19922 | Properties showing that an... |
| drnginvrcl 19923 | Closure of the multiplicat... |
| drnginvrn0 19924 | The multiplicative inverse... |
| drnginvrl 19925 | Property of the multiplica... |
| drnginvrr 19926 | Property of the multiplica... |
| drngmul0or 19927 | A product is zero iff one ... |
| drngmulne0 19928 | A product is nonzero iff b... |
| drngmuleq0 19929 | An element is zero iff its... |
| opprdrng 19930 | The opposite of a division... |
| isdrngd 19931 | Properties that characteri... |
| isdrngrd 19932 | Properties that characteri... |
| drngpropd 19933 | If two structures have the... |
| fldpropd 19934 | If two structures have the... |
| issubrg 19939 | The subring predicate. (C... |
| subrgss 19940 | A subring is a subset. (C... |
| subrgid 19941 | Every ring is a subring of... |
| subrgring 19942 | A subring is a ring. (Con... |
| subrgcrng 19943 | A subring of a commutative... |
| subrgrcl 19944 | Reverse closure for a subr... |
| subrgsubg 19945 | A subring is a subgroup. ... |
| subrg0 19946 | A subring always has the s... |
| subrg1cl 19947 | A subring contains the mul... |
| subrgbas 19948 | Base set of a subring stru... |
| subrg1 19949 | A subring always has the s... |
| subrgacl 19950 | A subring is closed under ... |
| subrgmcl 19951 | A subgroup is closed under... |
| subrgsubm 19952 | A subring is a submonoid o... |
| subrgdvds 19953 | If an element divides anot... |
| subrguss 19954 | A unit of a subring is a u... |
| subrginv 19955 | A subring always has the s... |
| subrgdv 19956 | A subring always has the s... |
| subrgunit 19957 | An element of a ring is a ... |
| subrgugrp 19958 | The units of a subring for... |
| issubrg2 19959 | Characterize the subrings ... |
| opprsubrg 19960 | Being a subring is a symme... |
| subrgint 19961 | The intersection of a none... |
| subrgin 19962 | The intersection of two su... |
| subrgmre 19963 | The subrings of a ring are... |
| issubdrg 19964 | Characterize the subfields... |
| subsubrg 19965 | A subring of a subring is ... |
| subsubrg2 19966 | The set of subrings of a s... |
| issubrg3 19967 | A subring is an additive s... |
| resrhm 19968 | Restriction of a ring homo... |
| rhmeql 19969 | The equalizer of two ring ... |
| rhmima 19970 | The homomorphic image of a... |
| rnrhmsubrg 19971 | The range of a ring homomo... |
| cntzsubr 19972 | Centralizers in a ring are... |
| pwsdiagrhm 19973 | Diagonal homomorphism into... |
| subrgpropd 19974 | If two structures have the... |
| rhmpropd 19975 | Ring homomorphism depends ... |
| issdrg 19978 | Property of a division sub... |
| sdrgid 19979 | Every division ring is a d... |
| sdrgss 19980 | A division subring is a su... |
| issdrg2 19981 | Property of a division sub... |
| acsfn1p 19982 | Construction of a closure ... |
| subrgacs 19983 | Closure property of subrin... |
| sdrgacs 19984 | Closure property of divisi... |
| cntzsdrg 19985 | Centralizers in division r... |
| subdrgint 19986 | The intersection of a none... |
| sdrgint 19987 | The intersection of a none... |
| primefld 19988 | The smallest sub division ... |
| primefld0cl 19989 | The prime field contains t... |
| primefld1cl 19990 | The prime field contains t... |
| abvfval 19993 | Value of the set of absolu... |
| isabv 19994 | Elementhood in the set of ... |
| isabvd 19995 | Properties that determine ... |
| abvrcl 19996 | Reverse closure for the ab... |
| abvfge0 19997 | An absolute value is a fun... |
| abvf 19998 | An absolute value is a fun... |
| abvcl 19999 | An absolute value is a fun... |
| abvge0 20000 | The absolute value of a nu... |
| abveq0 20001 | The value of an absolute v... |
| abvne0 20002 | The absolute value of a no... |
| abvgt0 20003 | The absolute value of a no... |
| abvmul 20004 | An absolute value distribu... |
| abvtri 20005 | An absolute value satisfie... |
| abv0 20006 | The absolute value of zero... |
| abv1z 20007 | The absolute value of one ... |
| abv1 20008 | The absolute value of one ... |
| abvneg 20009 | The absolute value of a ne... |
| abvsubtri 20010 | An absolute value satisfie... |
| abvrec 20011 | The absolute value distrib... |
| abvdiv 20012 | The absolute value distrib... |
| abvdom 20013 | Any ring with an absolute ... |
| abvres 20014 | The restriction of an abso... |
| abvtrivd 20015 | The trivial absolute value... |
| abvtriv 20016 | The trivial absolute value... |
| abvpropd 20017 | If two structures have the... |
| staffval 20022 | The functionalization of t... |
| stafval 20023 | The functionalization of t... |
| staffn 20024 | The functionalization is e... |
| issrng 20025 | The predicate "is a star r... |
| srngrhm 20026 | The involution function in... |
| srngring 20027 | A star ring is a ring. (C... |
| srngcnv 20028 | The involution function in... |
| srngf1o 20029 | The involution function in... |
| srngcl 20030 | The involution function in... |
| srngnvl 20031 | The involution function in... |
| srngadd 20032 | The involution function in... |
| srngmul 20033 | The involution function in... |
| srng1 20034 | The conjugate of the ring ... |
| srng0 20035 | The conjugate of the ring ... |
| issrngd 20036 | Properties that determine ... |
| idsrngd 20037 | A commutative ring is a st... |
| islmod 20042 | The predicate "is a left m... |
| lmodlema 20043 | Lemma for properties of a ... |
| islmodd 20044 | Properties that determine ... |
| lmodgrp 20045 | A left module is a group. ... |
| lmodring 20046 | The scalar component of a ... |
| lmodfgrp 20047 | The scalar component of a ... |
| lmodbn0 20048 | The base set of a left mod... |
| lmodacl 20049 | Closure of ring addition f... |
| lmodmcl 20050 | Closure of ring multiplica... |
| lmodsn0 20051 | The set of scalars in a le... |
| lmodvacl 20052 | Closure of vector addition... |
| lmodass 20053 | Left module vector sum is ... |
| lmodlcan 20054 | Left cancellation law for ... |
| lmodvscl 20055 | Closure of scalar product ... |
| scaffval 20056 | The scalar multiplication ... |
| scafval 20057 | The scalar multiplication ... |
| scafeq 20058 | If the scalar multiplicati... |
| scaffn 20059 | The scalar multiplication ... |
| lmodscaf 20060 | The scalar multiplication ... |
| lmodvsdi 20061 | Distributive law for scala... |
| lmodvsdir 20062 | Distributive law for scala... |
| lmodvsass 20063 | Associative law for scalar... |
| lmod0cl 20064 | The ring zero in a left mo... |
| lmod1cl 20065 | The ring unit in a left mo... |
| lmodvs1 20066 | Scalar product with ring u... |
| lmod0vcl 20067 | The zero vector is a vecto... |
| lmod0vlid 20068 | Left identity law for the ... |
| lmod0vrid 20069 | Right identity law for the... |
| lmod0vid 20070 | Identity equivalent to the... |
| lmod0vs 20071 | Zero times a vector is the... |
| lmodvs0 20072 | Anything times the zero ve... |
| lmodvsmmulgdi 20073 | Distributive law for a gro... |
| lmodfopnelem1 20074 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 20075 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 20076 | The (functionalized) opera... |
| lcomf 20077 | A linear-combination sum i... |
| lcomfsupp 20078 | A linear-combination sum i... |
| lmodvnegcl 20079 | Closure of vector negative... |
| lmodvnegid 20080 | Addition of a vector with ... |
| lmodvneg1 20081 | Minus 1 times a vector is ... |
| lmodvsneg 20082 | Multiplication of a vector... |
| lmodvsubcl 20083 | Closure of vector subtract... |
| lmodcom 20084 | Left module vector sum is ... |
| lmodabl 20085 | A left module is an abelia... |
| lmodcmn 20086 | A left module is a commuta... |
| lmodnegadd 20087 | Distribute negation throug... |
| lmod4 20088 | Commutative/associative la... |
| lmodvsubadd 20089 | Relationship between vecto... |
| lmodvaddsub4 20090 | Vector addition/subtractio... |
| lmodvpncan 20091 | Addition/subtraction cance... |
| lmodvnpcan 20092 | Cancellation law for vecto... |
| lmodvsubval2 20093 | Value of vector subtractio... |
| lmodsubvs 20094 | Subtraction of a scalar pr... |
| lmodsubdi 20095 | Scalar multiplication dist... |
| lmodsubdir 20096 | Scalar multiplication dist... |
| lmodsubeq0 20097 | If the difference between ... |
| lmodsubid 20098 | Subtraction of a vector fr... |
| lmodvsghm 20099 | Scalar multiplication of t... |
| lmodprop2d 20100 | If two structures have the... |
| lmodpropd 20101 | If two structures have the... |
| gsumvsmul 20102 | Pull a scalar multiplicati... |
| mptscmfsupp0 20103 | A mapping to a scalar prod... |
| mptscmfsuppd 20104 | A function mapping to a sc... |
| rmodislmodlem 20105 | Lemma for ~ rmodislmod . ... |
| rmodislmod 20106 | The right module ` R ` ind... |
| rmodislmodOLD 20107 | Obsolete version of ~ rmod... |
| lssset 20110 | The set of all (not necess... |
| islss 20111 | The predicate "is a subspa... |
| islssd 20112 | Properties that determine ... |
| lssss 20113 | A subspace is a set of vec... |
| lssel 20114 | A subspace member is a vec... |
| lss1 20115 | The set of vectors in a le... |
| lssuni 20116 | The union of all subspaces... |
| lssn0 20117 | A subspace is not empty. ... |
| 00lss 20118 | The empty structure has no... |
| lsscl 20119 | Closure property of a subs... |
| lssvsubcl 20120 | Closure of vector subtract... |
| lssvancl1 20121 | Non-closure: if one vector... |
| lssvancl2 20122 | Non-closure: if one vector... |
| lss0cl 20123 | The zero vector belongs to... |
| lsssn0 20124 | The singleton of the zero ... |
| lss0ss 20125 | The zero subspace is inclu... |
| lssle0 20126 | No subspace is smaller tha... |
| lssne0 20127 | A nonzero subspace has a n... |
| lssvneln0 20128 | A vector ` X ` which doesn... |
| lssneln0 20129 | A vector ` X ` which doesn... |
| lssssr 20130 | Conclude subspace ordering... |
| lssvacl 20131 | Closure of vector addition... |
| lssvscl 20132 | Closure of scalar product ... |
| lssvnegcl 20133 | Closure of negative vector... |
| lsssubg 20134 | All subspaces are subgroup... |
| lsssssubg 20135 | All subspaces are subgroup... |
| islss3 20136 | A linear subspace of a mod... |
| lsslmod 20137 | A submodule is a module. ... |
| lsslss 20138 | The subspaces of a subspac... |
| islss4 20139 | A linear subspace is a sub... |
| lss1d 20140 | One-dimensional subspace (... |
| lssintcl 20141 | The intersection of a none... |
| lssincl 20142 | The intersection of two su... |
| lssmre 20143 | The subspaces of a module ... |
| lssacs 20144 | Submodules are an algebrai... |
| prdsvscacl 20145 | Pointwise scalar multiplic... |
| prdslmodd 20146 | The product of a family of... |
| pwslmod 20147 | A structure power of a lef... |
| lspfval 20150 | The span function for a le... |
| lspf 20151 | The span operator on a lef... |
| lspval 20152 | The span of a set of vecto... |
| lspcl 20153 | The span of a set of vecto... |
| lspsncl 20154 | The span of a singleton is... |
| lspprcl 20155 | The span of a pair is a su... |
| lsptpcl 20156 | The span of an unordered t... |
| lspsnsubg 20157 | The span of a singleton is... |
| 00lsp 20158 | ~ fvco4i lemma for linear ... |
| lspid 20159 | The span of a subspace is ... |
| lspssv 20160 | A span is a set of vectors... |
| lspss 20161 | Span preserves subset orde... |
| lspssid 20162 | A set of vectors is a subs... |
| lspidm 20163 | The span of a set of vecto... |
| lspun 20164 | The span of union is the s... |
| lspssp 20165 | If a set of vectors is a s... |
| mrclsp 20166 | Moore closure generalizes ... |
| lspsnss 20167 | The span of the singleton ... |
| lspsnel3 20168 | A member of the span of th... |
| lspprss 20169 | The span of a pair of vect... |
| lspsnid 20170 | A vector belongs to the sp... |
| lspsnel6 20171 | Relationship between a vec... |
| lspsnel5 20172 | Relationship between a vec... |
| lspsnel5a 20173 | Relationship between a vec... |
| lspprid1 20174 | A member of a pair of vect... |
| lspprid2 20175 | A member of a pair of vect... |
| lspprvacl 20176 | The sum of two vectors bel... |
| lssats2 20177 | A way to express atomistic... |
| lspsneli 20178 | A scalar product with a ve... |
| lspsn 20179 | Span of the singleton of a... |
| lspsnel 20180 | Member of span of the sing... |
| lspsnvsi 20181 | Span of a scalar product o... |
| lspsnss2 20182 | Comparable spans of single... |
| lspsnneg 20183 | Negation does not change t... |
| lspsnsub 20184 | Swapping subtraction order... |
| lspsn0 20185 | Span of the singleton of t... |
| lsp0 20186 | Span of the empty set. (C... |
| lspuni0 20187 | Union of the span of the e... |
| lspun0 20188 | The span of a union with t... |
| lspsneq0 20189 | Span of the singleton is t... |
| lspsneq0b 20190 | Equal singleton spans impl... |
| lmodindp1 20191 | Two independent (non-colin... |
| lsslsp 20192 | Spans in submodules corres... |
| lss0v 20193 | The zero vector in a submo... |
| lsspropd 20194 | If two structures have the... |
| lsppropd 20195 | If two structures have the... |
| reldmlmhm 20202 | Lemma for module homomorph... |
| lmimfn 20203 | Lemma for module isomorphi... |
| islmhm 20204 | Property of being a homomo... |
| islmhm3 20205 | Property of a module homom... |
| lmhmlem 20206 | Non-quantified consequence... |
| lmhmsca 20207 | A homomorphism of left mod... |
| lmghm 20208 | A homomorphism of left mod... |
| lmhmlmod2 20209 | A homomorphism of left mod... |
| lmhmlmod1 20210 | A homomorphism of left mod... |
| lmhmf 20211 | A homomorphism of left mod... |
| lmhmlin 20212 | A homomorphism of left mod... |
| lmodvsinv 20213 | Multiplication of a vector... |
| lmodvsinv2 20214 | Multiplying a negated vect... |
| islmhm2 20215 | A one-equation proof of li... |
| islmhmd 20216 | Deduction for a module hom... |
| 0lmhm 20217 | The constant zero linear f... |
| idlmhm 20218 | The identity function on a... |
| invlmhm 20219 | The negative function on a... |
| lmhmco 20220 | The composition of two mod... |
| lmhmplusg 20221 | The pointwise sum of two l... |
| lmhmvsca 20222 | The pointwise scalar produ... |
| lmhmf1o 20223 | A bijective module homomor... |
| lmhmima 20224 | The image of a subspace un... |
| lmhmpreima 20225 | The inverse image of a sub... |
| lmhmlsp 20226 | Homomorphisms preserve spa... |
| lmhmrnlss 20227 | The range of a homomorphis... |
| lmhmkerlss 20228 | The kernel of a homomorphi... |
| reslmhm 20229 | Restriction of a homomorph... |
| reslmhm2 20230 | Expansion of the codomain ... |
| reslmhm2b 20231 | Expansion of the codomain ... |
| lmhmeql 20232 | The equalizer of two modul... |
| lspextmo 20233 | A linear function is compl... |
| pwsdiaglmhm 20234 | Diagonal homomorphism into... |
| pwssplit0 20235 | Splitting for structure po... |
| pwssplit1 20236 | Splitting for structure po... |
| pwssplit2 20237 | Splitting for structure po... |
| pwssplit3 20238 | Splitting for structure po... |
| islmim 20239 | An isomorphism of left mod... |
| lmimf1o 20240 | An isomorphism of left mod... |
| lmimlmhm 20241 | An isomorphism of modules ... |
| lmimgim 20242 | An isomorphism of modules ... |
| islmim2 20243 | An isomorphism of left mod... |
| lmimcnv 20244 | The converse of a bijectiv... |
| brlmic 20245 | The relation "is isomorphi... |
| brlmici 20246 | Prove isomorphic by an exp... |
| lmiclcl 20247 | Isomorphism implies the le... |
| lmicrcl 20248 | Isomorphism implies the ri... |
| lmicsym 20249 | Module isomorphism is symm... |
| lmhmpropd 20250 | Module homomorphism depend... |
| islbs 20253 | The predicate " ` B ` is a... |
| lbsss 20254 | A basis is a set of vector... |
| lbsel 20255 | An element of a basis is a... |
| lbssp 20256 | The span of a basis is the... |
| lbsind 20257 | A basis is linearly indepe... |
| lbsind2 20258 | A basis is linearly indepe... |
| lbspss 20259 | No proper subset of a basi... |
| lsmcl 20260 | The sum of two subspaces i... |
| lsmspsn 20261 | Member of subspace sum of ... |
| lsmelval2 20262 | Subspace sum membership in... |
| lsmsp 20263 | Subspace sum in terms of s... |
| lsmsp2 20264 | Subspace sum of spans of s... |
| lsmssspx 20265 | Subspace sum (in its exten... |
| lsmpr 20266 | The span of a pair of vect... |
| lsppreli 20267 | A vector expressed as a su... |
| lsmelpr 20268 | Two ways to say that a vec... |
| lsppr0 20269 | The span of a vector paire... |
| lsppr 20270 | Span of a pair of vectors.... |
| lspprel 20271 | Member of the span of a pa... |
| lspprabs 20272 | Absorption of vector sum i... |
| lspvadd 20273 | The span of a vector sum i... |
| lspsntri 20274 | Triangle-type inequality f... |
| lspsntrim 20275 | Triangle-type inequality f... |
| lbspropd 20276 | If two structures have the... |
| pj1lmhm 20277 | The left projection functi... |
| pj1lmhm2 20278 | The left projection functi... |
| islvec 20281 | The predicate "is a left v... |
| lvecdrng 20282 | The set of scalars of a le... |
| lveclmod 20283 | A left vector space is a l... |
| lsslvec 20284 | A vector subspace is a vec... |
| lvecvs0or 20285 | If a scalar product is zer... |
| lvecvsn0 20286 | A scalar product is nonzer... |
| lssvs0or 20287 | If a scalar product belong... |
| lvecvscan 20288 | Cancellation law for scala... |
| lvecvscan2 20289 | Cancellation law for scala... |
| lvecinv 20290 | Invert coefficient of scal... |
| lspsnvs 20291 | A nonzero scalar product d... |
| lspsneleq 20292 | Membership relation that i... |
| lspsncmp 20293 | Comparable spans of nonzer... |
| lspsnne1 20294 | Two ways to express that v... |
| lspsnne2 20295 | Two ways to express that v... |
| lspsnnecom 20296 | Swap two vectors with diff... |
| lspabs2 20297 | Absorption law for span of... |
| lspabs3 20298 | Absorption law for span of... |
| lspsneq 20299 | Equal spans of singletons ... |
| lspsneu 20300 | Nonzero vectors with equal... |
| lspsnel4 20301 | A member of the span of th... |
| lspdisj 20302 | The span of a vector not i... |
| lspdisjb 20303 | A nonzero vector is not in... |
| lspdisj2 20304 | Unequal spans are disjoint... |
| lspfixed 20305 | Show membership in the spa... |
| lspexch 20306 | Exchange property for span... |
| lspexchn1 20307 | Exchange property for span... |
| lspexchn2 20308 | Exchange property for span... |
| lspindpi 20309 | Partial independence prope... |
| lspindp1 20310 | Alternate way to say 3 vec... |
| lspindp2l 20311 | Alternate way to say 3 vec... |
| lspindp2 20312 | Alternate way to say 3 vec... |
| lspindp3 20313 | Independence of 2 vectors ... |
| lspindp4 20314 | (Partial) independence of ... |
| lvecindp 20315 | Compute the ` X ` coeffici... |
| lvecindp2 20316 | Sums of independent vector... |
| lspsnsubn0 20317 | Unequal singleton spans im... |
| lsmcv 20318 | Subspace sum has the cover... |
| lspsolvlem 20319 | Lemma for ~ lspsolv . (Co... |
| lspsolv 20320 | If ` X ` is in the span of... |
| lssacsex 20321 | In a vector space, subspac... |
| lspsnat 20322 | There is no subspace stric... |
| lspsncv0 20323 | The span of a singleton co... |
| lsppratlem1 20324 | Lemma for ~ lspprat . Let... |
| lsppratlem2 20325 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 20326 | Lemma for ~ lspprat . In ... |
| lsppratlem4 20327 | Lemma for ~ lspprat . In ... |
| lsppratlem5 20328 | Lemma for ~ lspprat . Com... |
| lsppratlem6 20329 | Lemma for ~ lspprat . Neg... |
| lspprat 20330 | A proper subspace of the s... |
| islbs2 20331 | An equivalent formulation ... |
| islbs3 20332 | An equivalent formulation ... |
| lbsacsbs 20333 | Being a basis in a vector ... |
| lvecdim 20334 | The dimension theorem for ... |
| lbsextlem1 20335 | Lemma for ~ lbsext . The ... |
| lbsextlem2 20336 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 20337 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 20338 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 20339 | For any linearly independe... |
| lbsext 20340 | For any linearly independe... |
| lbsexg 20341 | Every vector space has a b... |
| lbsex 20342 | Every vector space has a b... |
| lvecprop2d 20343 | If two structures have the... |
| lvecpropd 20344 | If two structures have the... |
| sraval 20353 | Lemma for ~ srabase throug... |
| sralem 20354 | Lemma for ~ srabase and si... |
| sralemOLD 20355 | Obsolete version of ~ sral... |
| srabase 20356 | Base set of a subring alge... |
| srabaseOLD 20357 | Obsolete proof of ~ srabas... |
| sraaddg 20358 | Additive operation of a su... |
| sraaddgOLD 20359 | Obsolete proof of ~ sraadd... |
| sramulr 20360 | Multiplicative operation o... |
| sramulrOLD 20361 | Obsolete proof of ~ sramul... |
| srasca 20362 | The set of scalars of a su... |
| sravsca 20363 | The scalar product operati... |
| sraip 20364 | The inner product operatio... |
| sratset 20365 | Topology component of a su... |
| sratsetOLD 20366 | Obsolete proof of ~ sratse... |
| sratopn 20367 | Topology component of a su... |
| srads 20368 | Distance function of a sub... |
| sradsOLD 20369 | Obsolete proof of ~ srads ... |
| sralmod 20370 | The subring algebra is a l... |
| sralmod0 20371 | The subring module inherit... |
| issubrngd2 20372 | Prove a subring by closure... |
| rlmfn 20373 | ` ringLMod ` is a function... |
| rlmval 20374 | Value of the ring module. ... |
| lidlval 20375 | Value of the set of ring i... |
| rspval 20376 | Value of the ring span fun... |
| rlmval2 20377 | Value of the ring module e... |
| rlmbas 20378 | Base set of the ring modul... |
| rlmplusg 20379 | Vector addition in the rin... |
| rlm0 20380 | Zero vector in the ring mo... |
| rlmsub 20381 | Subtraction in the ring mo... |
| rlmmulr 20382 | Ring multiplication in the... |
| rlmsca 20383 | Scalars in the ring module... |
| rlmsca2 20384 | Scalars in the ring module... |
| rlmvsca 20385 | Scalar multiplication in t... |
| rlmtopn 20386 | Topology component of the ... |
| rlmds 20387 | Metric component of the ri... |
| rlmlmod 20388 | The ring module is a modul... |
| rlmlvec 20389 | The ring module over a div... |
| rlmlsm 20390 | Subgroup sum of the ring m... |
| rlmvneg 20391 | Vector negation in the rin... |
| rlmscaf 20392 | Functionalized scalar mult... |
| ixpsnbasval 20393 | The value of an infinite C... |
| lidlss 20394 | An ideal is a subset of th... |
| islidl 20395 | Predicate of being a (left... |
| lidl0cl 20396 | An ideal contains 0. (Con... |
| lidlacl 20397 | An ideal is closed under a... |
| lidlnegcl 20398 | An ideal contains negative... |
| lidlsubg 20399 | An ideal is a subgroup of ... |
| lidlsubcl 20400 | An ideal is closed under s... |
| lidlmcl 20401 | An ideal is closed under l... |
| lidl1el 20402 | An ideal contains 1 iff it... |
| lidl0 20403 | Every ring contains a zero... |
| lidl1 20404 | Every ring contains a unit... |
| lidlacs 20405 | The ideal system is an alg... |
| rspcl 20406 | The span of a set of ring ... |
| rspssid 20407 | The span of a set of ring ... |
| rsp1 20408 | The span of the identity e... |
| rsp0 20409 | The span of the zero eleme... |
| rspssp 20410 | The ideal span of a set of... |
| mrcrsp 20411 | Moore closure generalizes ... |
| lidlnz 20412 | A nonzero ideal contains a... |
| drngnidl 20413 | A division ring has only t... |
| lidlrsppropd 20414 | The left ideals and ring s... |
| 2idlval 20417 | Definition of a two-sided ... |
| 2idlcpbl 20418 | The coset equivalence rela... |
| qus1 20419 | The multiplicative identit... |
| qusring 20420 | If ` S ` is a two-sided id... |
| qusrhm 20421 | If ` S ` is a two-sided id... |
| crngridl 20422 | In a commutative ring, the... |
| crng2idl 20423 | In a commutative ring, a t... |
| quscrng 20424 | The quotient of a commutat... |
| lpival 20429 | Value of the set of princi... |
| islpidl 20430 | Property of being a princi... |
| lpi0 20431 | The zero ideal is always p... |
| lpi1 20432 | The unit ideal is always p... |
| islpir 20433 | Principal ideal rings are ... |
| lpiss 20434 | Principal ideals are a sub... |
| islpir2 20435 | Principal ideal rings are ... |
| lpirring 20436 | Principal ideal rings are ... |
| drnglpir 20437 | Division rings are princip... |
| rspsn 20438 | Membership in principal id... |
| lidldvgen 20439 | An element generates an id... |
| lpigen 20440 | An ideal is principal iff ... |
| isnzr 20443 | Property of a nonzero ring... |
| nzrnz 20444 | One and zero are different... |
| nzrring 20445 | A nonzero ring is a ring. ... |
| drngnzr 20446 | All division rings are non... |
| isnzr2 20447 | Equivalent characterizatio... |
| isnzr2hash 20448 | Equivalent characterizatio... |
| opprnzr 20449 | The opposite of a nonzero ... |
| ringelnzr 20450 | A ring is nonzero if it ha... |
| nzrunit 20451 | A unit is nonzero in any n... |
| subrgnzr 20452 | A subring of a nonzero rin... |
| 0ringnnzr 20453 | A ring is a zero ring iff ... |
| 0ring 20454 | If a ring has only one ele... |
| 0ring01eq 20455 | In a ring with only one el... |
| 01eq0ring 20456 | If the zero and the identi... |
| 0ring01eqbi 20457 | In a unital ring the zero ... |
| rng1nnzr 20458 | The (smallest) structure r... |
| ring1zr 20459 | The only (unital) ring wit... |
| rngen1zr 20460 | The only (unital) ring wit... |
| ringen1zr 20461 | The only unital ring with ... |
| rng1nfld 20462 | The zero ring is not a fie... |
| rrgval 20471 | Value of the set or left-r... |
| isrrg 20472 | Membership in the set of l... |
| rrgeq0i 20473 | Property of a left-regular... |
| rrgeq0 20474 | Left-multiplication by a l... |
| rrgsupp 20475 | Left multiplication by a l... |
| rrgss 20476 | Left-regular elements are ... |
| unitrrg 20477 | Units are regular elements... |
| isdomn 20478 | Expand definition of a dom... |
| domnnzr 20479 | A domain is a nonzero ring... |
| domnring 20480 | A domain is a ring. (Cont... |
| domneq0 20481 | In a domain, a product is ... |
| domnmuln0 20482 | In a domain, a product of ... |
| isdomn2 20483 | A ring is a domain iff all... |
| domnrrg 20484 | In a domain, any nonzero e... |
| opprdomn 20485 | The opposite of a domain i... |
| abvn0b 20486 | Another characterization o... |
| drngdomn 20487 | A division ring is a domai... |
| isidom 20488 | An integral domain is a co... |
| fldidom 20489 | A field is an integral dom... |
| fldidomOLD 20490 | Obsolete version of ~ fldi... |
| fidomndrnglem 20491 | Lemma for ~ fidomndrng . ... |
| fidomndrng 20492 | A finite domain is a divis... |
| fiidomfld 20493 | A finite integral domain i... |
| cnfldstr 20512 | The field of complex numbe... |
| cnfldex 20513 | The field of complex numbe... |
| cnfldbas 20514 | The base set of the field ... |
| cnfldadd 20515 | The addition operation of ... |
| cnfldmul 20516 | The multiplication operati... |
| cnfldcj 20517 | The conjugation operation ... |
| cnfldtset 20518 | The topology component of ... |
| cnfldle 20519 | The ordering of the field ... |
| cnfldds 20520 | The metric of the field of... |
| cnfldunif 20521 | The uniform structure comp... |
| cnfldfun 20522 | The field of complex numbe... |
| cnfldfunALT 20523 | Alternate proof of ~ cnfld... |
| xrsstr 20524 | The extended real structur... |
| xrsex 20525 | The extended real structur... |
| xrsbas 20526 | The base set of the extend... |
| xrsadd 20527 | The addition operation of ... |
| xrsmul 20528 | The multiplication operati... |
| xrstset 20529 | The topology component of ... |
| xrsle 20530 | The ordering of the extend... |
| cncrng 20531 | The complex numbers form a... |
| cnring 20532 | The complex numbers form a... |
| xrsmcmn 20533 | The "multiplicative group"... |
| cnfld0 20534 | Zero is the zero element o... |
| cnfld1 20535 | One is the unit element of... |
| cnfldneg 20536 | The additive inverse in th... |
| cnfldplusf 20537 | The functionalized additio... |
| cnfldsub 20538 | The subtraction operator i... |
| cndrng 20539 | The complex numbers form a... |
| cnflddiv 20540 | The division operation in ... |
| cnfldinv 20541 | The multiplicative inverse... |
| cnfldmulg 20542 | The group multiple functio... |
| cnfldexp 20543 | The exponentiation operato... |
| cnsrng 20544 | The complex numbers form a... |
| xrsmgm 20545 | The "additive group" of th... |
| xrsnsgrp 20546 | The "additive group" of th... |
| xrsmgmdifsgrp 20547 | The "additive group" of th... |
| xrs1mnd 20548 | The extended real numbers,... |
| xrs10 20549 | The zero of the extended r... |
| xrs1cmn 20550 | The extended real numbers ... |
| xrge0subm 20551 | The nonnegative extended r... |
| xrge0cmn 20552 | The nonnegative extended r... |
| xrsds 20553 | The metric of the extended... |
| xrsdsval 20554 | The metric of the extended... |
| xrsdsreval 20555 | The metric of the extended... |
| xrsdsreclblem 20556 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 20557 | The metric of the extended... |
| cnsubmlem 20558 | Lemma for ~ nn0subm and fr... |
| cnsubglem 20559 | Lemma for ~ resubdrg and f... |
| cnsubrglem 20560 | Lemma for ~ resubdrg and f... |
| cnsubdrglem 20561 | Lemma for ~ resubdrg and f... |
| qsubdrg 20562 | The rational numbers form ... |
| zsubrg 20563 | The integers form a subrin... |
| gzsubrg 20564 | The gaussian integers form... |
| nn0subm 20565 | The nonnegative integers f... |
| rege0subm 20566 | The nonnegative reals form... |
| absabv 20567 | The regular absolute value... |
| zsssubrg 20568 | The integers are a subset ... |
| qsssubdrg 20569 | The rational numbers are a... |
| cnsubrg 20570 | There are no subrings of t... |
| cnmgpabl 20571 | The unit group of the comp... |
| cnmgpid 20572 | The group identity element... |
| cnmsubglem 20573 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 20574 | The positive reals form a ... |
| gzrngunitlem 20575 | Lemma for ~ gzrngunit . (... |
| gzrngunit 20576 | The units on ` ZZ [ _i ] `... |
| gsumfsum 20577 | Relate a group sum on ` CC... |
| regsumfsum 20578 | Relate a group sum on ` ( ... |
| expmhm 20579 | Exponentiation is a monoid... |
| nn0srg 20580 | The nonnegative integers f... |
| rge0srg 20581 | The nonnegative real numbe... |
| zringcrng 20584 | The ring of integers is a ... |
| zringring 20585 | The ring of integers is a ... |
| zringabl 20586 | The ring of integers is an... |
| zringgrp 20587 | The ring of integers is an... |
| zringbas 20588 | The integers are the base ... |
| zringplusg 20589 | The addition operation of ... |
| zringmulg 20590 | The multiplication (group ... |
| zringmulr 20591 | The multiplication operati... |
| zring0 20592 | The neutral element of the... |
| zring1 20593 | The multiplicative neutral... |
| zringnzr 20594 | The ring of integers is a ... |
| dvdsrzring 20595 | Ring divisibility in the r... |
| zringlpirlem1 20596 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 20597 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 20598 | Lemma for ~ zringlpir . A... |
| zringinvg 20599 | The additive inverse of an... |
| zringunit 20600 | The units of ` ZZ ` are th... |
| zringlpir 20601 | The integers are a princip... |
| zringndrg 20602 | The integers are not a div... |
| zringcyg 20603 | The integers are a cyclic ... |
| zringsubgval 20604 | Subtraction in the ring of... |
| zringmpg 20605 | The multiplication group o... |
| prmirredlem 20606 | A positive integer is irre... |
| dfprm2 20607 | The positive irreducible e... |
| prmirred 20608 | The irreducible elements o... |
| expghm 20609 | Exponentiation is a group ... |
| mulgghm2 20610 | The powers of a group elem... |
| mulgrhm 20611 | The powers of the element ... |
| mulgrhm2 20612 | The powers of the element ... |
| zrhval 20621 | Define the unique homomorp... |
| zrhval2 20622 | Alternate value of the ` Z... |
| zrhmulg 20623 | Value of the ` ZRHom ` hom... |
| zrhrhmb 20624 | The ` ZRHom ` homomorphism... |
| zrhrhm 20625 | The ` ZRHom ` homomorphism... |
| zrh1 20626 | Interpretation of 1 in a r... |
| zrh0 20627 | Interpretation of 0 in a r... |
| zrhpropd 20628 | The ` ZZ ` ring homomorphi... |
| zlmval 20629 | Augment an abelian group w... |
| zlmlem 20630 | Lemma for ~ zlmbas and ~ z... |
| zlmlemOLD 20631 | Obsolete version of ~ zlml... |
| zlmbas 20632 | Base set of a ` ZZ ` -modu... |
| zlmbasOLD 20633 | Obsolete version of ~ zlmb... |
| zlmplusg 20634 | Group operation of a ` ZZ ... |
| zlmplusgOLD 20635 | Obsolete version of ~ zlmb... |
| zlmmulr 20636 | Ring operation of a ` ZZ `... |
| zlmmulrOLD 20637 | Obsolete version of ~ zlmb... |
| zlmsca 20638 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 20639 | Scalar multiplication oper... |
| zlmlmod 20640 | The ` ZZ ` -module operati... |
| chrval 20641 | Definition substitution of... |
| chrcl 20642 | Closure of the characteris... |
| chrid 20643 | The canonical ` ZZ ` ring ... |
| chrdvds 20644 | The ` ZZ ` ring homomorphi... |
| chrcong 20645 | If two integers are congru... |
| chrnzr 20646 | Nonzero rings are precisel... |
| chrrhm 20647 | The characteristic restric... |
| domnchr 20648 | The characteristic of a do... |
| znlidl 20649 | The set ` n ZZ ` is an ide... |
| zncrng2 20650 | The value of the ` Z/nZ ` ... |
| znval 20651 | The value of the ` Z/nZ ` ... |
| znle 20652 | The value of the ` Z/nZ ` ... |
| znval2 20653 | Self-referential expressio... |
| znbaslem 20654 | Lemma for ~ znbas . (Cont... |
| znbaslemOLD 20655 | Obsolete version of ~ znba... |
| znbas2 20656 | The base set of ` Z/nZ ` i... |
| znbas2OLD 20657 | Obsolete version of ~ znba... |
| znadd 20658 | The additive structure of ... |
| znaddOLD 20659 | Obsolete version of ~ znad... |
| znmul 20660 | The multiplicative structu... |
| znmulOLD 20661 | Obsolete version of ~ znad... |
| znzrh 20662 | The ` ZZ ` ring homomorphi... |
| znbas 20663 | The base set of ` Z/nZ ` s... |
| zncrng 20664 | ` Z/nZ ` is a commutative ... |
| znzrh2 20665 | The ` ZZ ` ring homomorphi... |
| znzrhval 20666 | The ` ZZ ` ring homomorphi... |
| znzrhfo 20667 | The ` ZZ ` ring homomorphi... |
| zncyg 20668 | The group ` ZZ / n ZZ ` is... |
| zndvds 20669 | Express equality of equiva... |
| zndvds0 20670 | Special case of ~ zndvds w... |
| znf1o 20671 | The function ` F ` enumera... |
| zzngim 20672 | The ` ZZ ` ring homomorphi... |
| znle2 20673 | The ordering of the ` Z/nZ... |
| znleval 20674 | The ordering of the ` Z/nZ... |
| znleval2 20675 | The ordering of the ` Z/nZ... |
| zntoslem 20676 | Lemma for ~ zntos . (Cont... |
| zntos 20677 | The ` Z/nZ ` structure is ... |
| znhash 20678 | The ` Z/nZ ` structure has... |
| znfi 20679 | The ` Z/nZ ` structure is ... |
| znfld 20680 | The ` Z/nZ ` structure is ... |
| znidomb 20681 | The ` Z/nZ ` structure is ... |
| znchr 20682 | Cyclic rings are defined b... |
| znunit 20683 | The units of ` Z/nZ ` are ... |
| znunithash 20684 | The size of the unit group... |
| znrrg 20685 | The regular elements of ` ... |
| cygznlem1 20686 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 20687 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 20688 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 20689 | A cyclic group with ` n ` ... |
| cygzn 20690 | A cyclic group with ` n ` ... |
| cygth 20691 | The "fundamental theorem o... |
| cyggic 20692 | Cyclic groups are isomorph... |
| frgpcyg 20693 | A free group is cyclic iff... |
| cnmsgnsubg 20694 | The signs form a multiplic... |
| cnmsgnbas 20695 | The base set of the sign s... |
| cnmsgngrp 20696 | The group of signs under m... |
| psgnghm 20697 | The sign is a homomorphism... |
| psgnghm2 20698 | The sign is a homomorphism... |
| psgninv 20699 | The sign of a permutation ... |
| psgnco 20700 | Multiplicativity of the pe... |
| zrhpsgnmhm 20701 | Embedding of permutation s... |
| zrhpsgninv 20702 | The embedded sign of a per... |
| evpmss 20703 | Even permutations are perm... |
| psgnevpmb 20704 | A class is an even permuta... |
| psgnodpm 20705 | A permutation which is odd... |
| psgnevpm 20706 | A permutation which is eve... |
| psgnodpmr 20707 | If a permutation has sign ... |
| zrhpsgnevpm 20708 | The sign of an even permut... |
| zrhpsgnodpm 20709 | The sign of an odd permuta... |
| cofipsgn 20710 | Composition of any class `... |
| zrhpsgnelbas 20711 | Embedding of permutation s... |
| zrhcopsgnelbas 20712 | Embedding of permutation s... |
| evpmodpmf1o 20713 | The function for performin... |
| pmtrodpm 20714 | A transposition is an odd ... |
| psgnfix1 20715 | A permutation of a finite ... |
| psgnfix2 20716 | A permutation of a finite ... |
| psgndiflemB 20717 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 20718 | Lemma 2 for ~ psgndif . (... |
| psgndif 20719 | Embedding of permutation s... |
| copsgndif 20720 | Embedding of permutation s... |
| rebase 20723 | The base of the field of r... |
| remulg 20724 | The multiplication (group ... |
| resubdrg 20725 | The real numbers form a di... |
| resubgval 20726 | Subtraction in the field o... |
| replusg 20727 | The addition operation of ... |
| remulr 20728 | The multiplication operati... |
| re0g 20729 | The neutral element of the... |
| re1r 20730 | The multiplicative neutral... |
| rele2 20731 | The ordering relation of t... |
| relt 20732 | The ordering relation of t... |
| reds 20733 | The distance of the field ... |
| redvr 20734 | The division operation of ... |
| retos 20735 | The real numbers are a tot... |
| refld 20736 | The real numbers form a fi... |
| refldcj 20737 | The conjugation operation ... |
| recrng 20738 | The real numbers form a st... |
| regsumsupp 20739 | The group sum over the rea... |
| rzgrp 20740 | The quotient group ` RR / ... |
| isphl 20745 | The predicate "is a genera... |
| phllvec 20746 | A pre-Hilbert space is a l... |
| phllmod 20747 | A pre-Hilbert space is a l... |
| phlsrng 20748 | The scalar ring of a pre-H... |
| phllmhm 20749 | The inner product of a pre... |
| ipcl 20750 | Closure of the inner produ... |
| ipcj 20751 | Conjugate of an inner prod... |
| iporthcom 20752 | Orthogonality (meaning inn... |
| ip0l 20753 | Inner product with a zero ... |
| ip0r 20754 | Inner product with a zero ... |
| ipeq0 20755 | The inner product of a vec... |
| ipdir 20756 | Distributive law for inner... |
| ipdi 20757 | Distributive law for inner... |
| ip2di 20758 | Distributive law for inner... |
| ipsubdir 20759 | Distributive law for inner... |
| ipsubdi 20760 | Distributive law for inner... |
| ip2subdi 20761 | Distributive law for inner... |
| ipass 20762 | Associative law for inner ... |
| ipassr 20763 | "Associative" law for seco... |
| ipassr2 20764 | "Associative" law for inne... |
| ipffval 20765 | The inner product operatio... |
| ipfval 20766 | The inner product operatio... |
| ipfeq 20767 | If the inner product opera... |
| ipffn 20768 | The inner product operatio... |
| phlipf 20769 | The inner product operatio... |
| ip2eq 20770 | Two vectors are equal iff ... |
| isphld 20771 | Properties that determine ... |
| phlpropd 20772 | If two structures have the... |
| ssipeq 20773 | The inner product on a sub... |
| phssipval 20774 | The inner product on a sub... |
| phssip 20775 | The inner product (as a fu... |
| phlssphl 20776 | A subspace of an inner pro... |
| ocvfval 20783 | The orthocomplement operat... |
| ocvval 20784 | Value of the orthocompleme... |
| elocv 20785 | Elementhood in the orthoco... |
| ocvi 20786 | Property of a member of th... |
| ocvss 20787 | The orthocomplement of a s... |
| ocvocv 20788 | A set is contained in its ... |
| ocvlss 20789 | The orthocomplement of a s... |
| ocv2ss 20790 | Orthocomplements reverse s... |
| ocvin 20791 | An orthocomplement has tri... |
| ocvsscon 20792 | Two ways to say that ` S `... |
| ocvlsp 20793 | The orthocomplement of a l... |
| ocv0 20794 | The orthocomplement of the... |
| ocvz 20795 | The orthocomplement of the... |
| ocv1 20796 | The orthocomplement of the... |
| unocv 20797 | The orthocomplement of a u... |
| iunocv 20798 | The orthocomplement of an ... |
| cssval 20799 | The set of closed subspace... |
| iscss 20800 | The predicate "is a closed... |
| cssi 20801 | Property of a closed subsp... |
| cssss 20802 | A closed subspace is a sub... |
| iscss2 20803 | It is sufficient to prove ... |
| ocvcss 20804 | The orthocomplement of any... |
| cssincl 20805 | The zero subspace is a clo... |
| css0 20806 | The zero subspace is a clo... |
| css1 20807 | The whole space is a close... |
| csslss 20808 | A closed subspace of a pre... |
| lsmcss 20809 | A subset of a pre-Hilbert ... |
| cssmre 20810 | The closed subspaces of a ... |
| mrccss 20811 | The Moore closure correspo... |
| thlval 20812 | Value of the Hilbert latti... |
| thlbas 20813 | Base set of the Hilbert la... |
| thlle 20814 | Ordering on the Hilbert la... |
| thlleval 20815 | Ordering on the Hilbert la... |
| thloc 20816 | Orthocomplement on the Hil... |
| pjfval 20823 | The value of the projectio... |
| pjdm 20824 | A subspace is in the domai... |
| pjpm 20825 | The projection map is a pa... |
| pjfval2 20826 | Value of the projection ma... |
| pjval 20827 | Value of the projection ma... |
| pjdm2 20828 | A subspace is in the domai... |
| pjff 20829 | A projection is a linear o... |
| pjf 20830 | A projection is a function... |
| pjf2 20831 | A projection is a function... |
| pjfo 20832 | A projection is a surjecti... |
| pjcss 20833 | A projection subspace is a... |
| ocvpj 20834 | The orthocomplement of a p... |
| ishil 20835 | The predicate "is a Hilber... |
| ishil2 20836 | The predicate "is a Hilber... |
| isobs 20837 | The predicate "is an ortho... |
| obsip 20838 | The inner product of two e... |
| obsipid 20839 | A basis element has unit l... |
| obsrcl 20840 | Reverse closure for an ort... |
| obsss 20841 | An orthonormal basis is a ... |
| obsne0 20842 | A basis element is nonzero... |
| obsocv 20843 | An orthonormal basis has t... |
| obs2ocv 20844 | The double orthocomplement... |
| obselocv 20845 | A basis element is in the ... |
| obs2ss 20846 | A basis has no proper subs... |
| obslbs 20847 | An orthogonal basis is a l... |
| reldmdsmm 20850 | The direct sum is a well-b... |
| dsmmval 20851 | Value of the module direct... |
| dsmmbase 20852 | Base set of the module dir... |
| dsmmval2 20853 | Self-referential definitio... |
| dsmmbas2 20854 | Base set of the direct sum... |
| dsmmfi 20855 | For finite products, the d... |
| dsmmelbas 20856 | Membership in the finitely... |
| dsmm0cl 20857 | The all-zero vector is con... |
| dsmmacl 20858 | The finite hull is closed ... |
| prdsinvgd2 20859 | Negation of a single coord... |
| dsmmsubg 20860 | The finite hull of a produ... |
| dsmmlss 20861 | The finite hull of a produ... |
| dsmmlmod 20862 | The direct sum of a family... |
| frlmval 20865 | Value of the "free module"... |
| frlmlmod 20866 | The free module is a modul... |
| frlmpws 20867 | The free module as a restr... |
| frlmlss 20868 | The base set of the free m... |
| frlmpwsfi 20869 | The finite free module is ... |
| frlmsca 20870 | The ring of scalars of a f... |
| frlm0 20871 | Zero in a free module (rin... |
| frlmbas 20872 | Base set of the free modul... |
| frlmelbas 20873 | Membership in the base set... |
| frlmrcl 20874 | If a free module is inhabi... |
| frlmbasfsupp 20875 | Elements of the free modul... |
| frlmbasmap 20876 | Elements of the free modul... |
| frlmbasf 20877 | Elements of the free modul... |
| frlmlvec 20878 | The free module over a div... |
| frlmfibas 20879 | The base set of the finite... |
| elfrlmbasn0 20880 | If the dimension of a free... |
| frlmplusgval 20881 | Addition in a free module.... |
| frlmsubgval 20882 | Subtraction in a free modu... |
| frlmvscafval 20883 | Scalar multiplication in a... |
| frlmvplusgvalc 20884 | Coordinates of a sum with ... |
| frlmvscaval 20885 | Coordinates of a scalar mu... |
| frlmplusgvalb 20886 | Addition in a free module ... |
| frlmvscavalb 20887 | Scalar multiplication in a... |
| frlmvplusgscavalb 20888 | Addition combined with sca... |
| frlmgsum 20889 | Finite commutative sums in... |
| frlmsplit2 20890 | Restriction is homomorphic... |
| frlmsslss 20891 | A subset of a free module ... |
| frlmsslss2 20892 | A subset of a free module ... |
| frlmbas3 20893 | An element of the base set... |
| mpofrlmd 20894 | Elements of the free modul... |
| frlmip 20895 | The inner product of a fre... |
| frlmipval 20896 | The inner product of a fre... |
| frlmphllem 20897 | Lemma for ~ frlmphl . (Co... |
| frlmphl 20898 | Conditions for a free modu... |
| uvcfval 20901 | Value of the unit-vector g... |
| uvcval 20902 | Value of a single unit vec... |
| uvcvval 20903 | Value of a unit vector coo... |
| uvcvvcl 20904 | A coordinate of a unit vec... |
| uvcvvcl2 20905 | A unit vector coordinate i... |
| uvcvv1 20906 | The unit vector is one at ... |
| uvcvv0 20907 | The unit vector is zero at... |
| uvcff 20908 | Domain and range of the un... |
| uvcf1 20909 | In a nonzero ring, each un... |
| uvcresum 20910 | Any element of a free modu... |
| frlmssuvc1 20911 | A scalar multiple of a uni... |
| frlmssuvc2 20912 | A nonzero scalar multiple ... |
| frlmsslsp 20913 | A subset of a free module ... |
| frlmlbs 20914 | The unit vectors comprise ... |
| frlmup1 20915 | Any assignment of unit vec... |
| frlmup2 20916 | The evaluation map has the... |
| frlmup3 20917 | The range of such an evalu... |
| frlmup4 20918 | Universal property of the ... |
| ellspd 20919 | The elements of the span o... |
| elfilspd 20920 | Simplified version of ~ el... |
| rellindf 20925 | The independent-family pre... |
| islinds 20926 | Property of an independent... |
| linds1 20927 | An independent set of vect... |
| linds2 20928 | An independent set of vect... |
| islindf 20929 | Property of an independent... |
| islinds2 20930 | Expanded property of an in... |
| islindf2 20931 | Property of an independent... |
| lindff 20932 | Functional property of a l... |
| lindfind 20933 | A linearly independent fam... |
| lindsind 20934 | A linearly independent set... |
| lindfind2 20935 | In a linearly independent ... |
| lindsind2 20936 | In a linearly independent ... |
| lindff1 20937 | A linearly independent fam... |
| lindfrn 20938 | The range of an independen... |
| f1lindf 20939 | Rearranging and deleting e... |
| lindfres 20940 | Any restriction of an inde... |
| lindsss 20941 | Any subset of an independe... |
| f1linds 20942 | A family constructed from ... |
| islindf3 20943 | In a nonzero ring, indepen... |
| lindfmm 20944 | Linear independence of a f... |
| lindsmm 20945 | Linear independence of a s... |
| lindsmm2 20946 | The monomorphic image of a... |
| lsslindf 20947 | Linear independence is unc... |
| lsslinds 20948 | Linear independence is unc... |
| islbs4 20949 | A basis is an independent ... |
| lbslinds 20950 | A basis is independent. (... |
| islinds3 20951 | A subset is linearly indep... |
| islinds4 20952 | A set is independent in a ... |
| lmimlbs 20953 | The isomorphic image of a ... |
| lmiclbs 20954 | Having a basis is an isomo... |
| islindf4 20955 | A family is independent if... |
| islindf5 20956 | A family is independent if... |
| indlcim 20957 | An independent, spanning f... |
| lbslcic 20958 | A module with a basis is i... |
| lmisfree 20959 | A module has a basis iff i... |
| lvecisfrlm 20960 | Every vector space is isom... |
| lmimco 20961 | The composition of two iso... |
| lmictra 20962 | Module isomorphism is tran... |
| uvcf1o 20963 | In a nonzero ring, the map... |
| uvcendim 20964 | In a nonzero ring, the num... |
| frlmisfrlm 20965 | A free module is isomorphi... |
| frlmiscvec 20966 | Every free module is isomo... |
| isassa 20973 | The properties of an assoc... |
| assalem 20974 | The properties of an assoc... |
| assaass 20975 | Left-associative property ... |
| assaassr 20976 | Right-associative property... |
| assalmod 20977 | An associative algebra is ... |
| assaring 20978 | An associative algebra is ... |
| assasca 20979 | An associative algebra's s... |
| assa2ass 20980 | Left- and right-associativ... |
| isassad 20981 | Sufficient condition for b... |
| issubassa3 20982 | A subring that is also a s... |
| issubassa 20983 | The subalgebras of an asso... |
| sraassa 20984 | The subring algebra over a... |
| rlmassa 20985 | The ring module over a com... |
| assapropd 20986 | If two structures have the... |
| aspval 20987 | Value of the algebraic clo... |
| asplss 20988 | The algebraic span of a se... |
| aspid 20989 | The algebraic span of a su... |
| aspsubrg 20990 | The algebraic span of a se... |
| aspss 20991 | Span preserves subset orde... |
| aspssid 20992 | A set of vectors is a subs... |
| asclfval 20993 | Function value of the alge... |
| asclval 20994 | Value of a mapped algebra ... |
| asclfn 20995 | Unconditional functionalit... |
| asclf 20996 | The algebra scalars functi... |
| asclghm 20997 | The algebra scalars functi... |
| ascl0 20998 | The scalar 0 embedded into... |
| ascl1 20999 | The scalar 1 embedded into... |
| asclmul1 21000 | Left multiplication by a l... |
| asclmul2 21001 | Right multiplication by a ... |
| ascldimul 21002 | The algebra scalars functi... |
| asclinvg 21003 | The group inverse (negatio... |
| asclrhm 21004 | The scalar injection is a ... |
| rnascl 21005 | The set of injected scalar... |
| issubassa2 21006 | A subring of a unital alge... |
| rnasclsubrg 21007 | The scalar multiples of th... |
| rnasclmulcl 21008 | (Vector) multiplication is... |
| rnasclassa 21009 | The scalar multiples of th... |
| ressascl 21010 | The injection of scalars i... |
| asclpropd 21011 | If two structures have the... |
| aspval2 21012 | The algebraic closure is t... |
| assamulgscmlem1 21013 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 21014 | Lemma for ~ assamulgscm (i... |
| assamulgscm 21015 | Exponentiation of a scalar... |
| zlmassa 21016 | The ` ZZ ` -module operati... |
| reldmpsr 21027 | The multivariate power ser... |
| psrval 21028 | Value of the multivariate ... |
| psrvalstr 21029 | The multivariate power ser... |
| psrbag 21030 | Elementhood in the set of ... |
| psrbagf 21031 | A finite bag is a function... |
| psrbagfOLD 21032 | Obsolete version of ~ psrb... |
| psrbagfsupp 21033 | Finite bags have finite su... |
| psrbagfsuppOLD 21034 | Obsolete version of ~ psrb... |
| snifpsrbag 21035 | A bag containing one eleme... |
| fczpsrbag 21036 | The constant function equa... |
| psrbaglesupp 21037 | The support of a dominated... |
| psrbaglesuppOLD 21038 | Obsolete version of ~ psrb... |
| psrbaglecl 21039 | The set of finite bags is ... |
| psrbagleclOLD 21040 | Obsolete version of ~ psrb... |
| psrbagaddcl 21041 | The sum of two finite bags... |
| psrbagaddclOLD 21042 | Obsolete version of ~ psrb... |
| psrbagcon 21043 | The analogue of the statem... |
| psrbagconOLD 21044 | Obsolete version of ~ psrb... |
| psrbaglefi 21045 | There are finitely many ba... |
| psrbaglefiOLD 21046 | Obsolete version of ~ psrb... |
| psrbagconcl 21047 | The complement of a bag is... |
| psrbagconclOLD 21048 | Obsolete version of ~ psrb... |
| psrbagconf1o 21049 | Bag complementation is a b... |
| psrbagconf1oOLD 21050 | Obsolete version of ~ psrb... |
| gsumbagdiaglemOLD 21051 | Obsolete version of ~ gsum... |
| gsumbagdiagOLD 21052 | Obsolete version of ~ gsum... |
| psrass1lemOLD 21053 | Obsolete version of ~ psra... |
| gsumbagdiaglem 21054 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 21055 | Two-dimensional commutatio... |
| psrass1lem 21056 | A group sum commutation us... |
| psrbas 21057 | The base set of the multiv... |
| psrelbas 21058 | An element of the set of p... |
| psrelbasfun 21059 | An element of the set of p... |
| psrplusg 21060 | The addition operation of ... |
| psradd 21061 | The addition operation of ... |
| psraddcl 21062 | Closure of the power serie... |
| psrmulr 21063 | The multiplication operati... |
| psrmulfval 21064 | The multiplication operati... |
| psrmulval 21065 | The multiplication operati... |
| psrmulcllem 21066 | Closure of the power serie... |
| psrmulcl 21067 | Closure of the power serie... |
| psrsca 21068 | The scalar field of the mu... |
| psrvscafval 21069 | The scalar multiplication ... |
| psrvsca 21070 | The scalar multiplication ... |
| psrvscaval 21071 | The scalar multiplication ... |
| psrvscacl 21072 | Closure of the power serie... |
| psr0cl 21073 | The zero element of the ri... |
| psr0lid 21074 | The zero element of the ri... |
| psrnegcl 21075 | The negative function in t... |
| psrlinv 21076 | The negative function in t... |
| psrgrp 21077 | The ring of power series i... |
| psr0 21078 | The zero element of the ri... |
| psrneg 21079 | The negative function of t... |
| psrlmod 21080 | The ring of power series i... |
| psr1cl 21081 | The identity element of th... |
| psrlidm 21082 | The identity element of th... |
| psrridm 21083 | The identity element of th... |
| psrass1 21084 | Associative identity for t... |
| psrdi 21085 | Distributive law for the r... |
| psrdir 21086 | Distributive law for the r... |
| psrass23l 21087 | Associative identity for t... |
| psrcom 21088 | Commutative law for the ri... |
| psrass23 21089 | Associative identities for... |
| psrring 21090 | The ring of power series i... |
| psr1 21091 | The identity element of th... |
| psrcrng 21092 | The ring of power series i... |
| psrassa 21093 | The ring of power series i... |
| resspsrbas 21094 | A restricted power series ... |
| resspsradd 21095 | A restricted power series ... |
| resspsrmul 21096 | A restricted power series ... |
| resspsrvsca 21097 | A restricted power series ... |
| subrgpsr 21098 | A subring of the base ring... |
| mvrfval 21099 | Value of the generating el... |
| mvrval 21100 | Value of the generating el... |
| mvrval2 21101 | Value of the generating el... |
| mvrid 21102 | The ` X i ` -th coefficien... |
| mvrf 21103 | The power series variable ... |
| mvrf1 21104 | The power series variable ... |
| mvrcl2 21105 | A power series variable is... |
| reldmmpl 21106 | The multivariate polynomia... |
| mplval 21107 | Value of the set of multiv... |
| mplbas 21108 | Base set of the set of mul... |
| mplelbas 21109 | Property of being a polyno... |
| mplrcl 21110 | Reverse closure for the po... |
| mplelsfi 21111 | A polynomial treated as a ... |
| mplval2 21112 | Self-referential expressio... |
| mplbasss 21113 | The set of polynomials is ... |
| mplelf 21114 | A polynomial is defined as... |
| mplsubglem 21115 | If ` A ` is an ideal of se... |
| mpllsslem 21116 | If ` A ` is an ideal of su... |
| mplsubglem2 21117 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 21118 | The set of polynomials is ... |
| mpllss 21119 | The set of polynomials is ... |
| mplsubrglem 21120 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 21121 | The set of polynomials is ... |
| mpl0 21122 | The zero polynomial. (Con... |
| mpladd 21123 | The addition operation on ... |
| mplneg 21124 | The negative function on m... |
| mplmul 21125 | The multiplication operati... |
| mpl1 21126 | The identity element of th... |
| mplsca 21127 | The scalar field of a mult... |
| mplvsca2 21128 | The scalar multiplication ... |
| mplvsca 21129 | The scalar multiplication ... |
| mplvscaval 21130 | The scalar multiplication ... |
| mvrcl 21131 | A power series variable is... |
| mplgrp 21132 | The polynomial ring is a g... |
| mpllmod 21133 | The polynomial ring is a l... |
| mplring 21134 | The polynomial ring is a r... |
| mpllvec 21135 | The polynomial ring is a v... |
| mplcrng 21136 | The polynomial ring is a c... |
| mplassa 21137 | The polynomial ring is an ... |
| ressmplbas2 21138 | The base set of a restrict... |
| ressmplbas 21139 | A restricted polynomial al... |
| ressmpladd 21140 | A restricted polynomial al... |
| ressmplmul 21141 | A restricted polynomial al... |
| ressmplvsca 21142 | A restricted power series ... |
| subrgmpl 21143 | A subring of the base ring... |
| subrgmvr 21144 | The variables in a subring... |
| subrgmvrf 21145 | The variables in a polynom... |
| mplmon 21146 | A monomial is a polynomial... |
| mplmonmul 21147 | The product of two monomia... |
| mplcoe1 21148 | Decompose a polynomial int... |
| mplcoe3 21149 | Decompose a monomial in on... |
| mplcoe5lem 21150 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 21151 | Decompose a monomial into ... |
| mplcoe2 21152 | Decompose a monomial into ... |
| mplbas2 21153 | An alternative expression ... |
| ltbval 21154 | Value of the well-order on... |
| ltbwe 21155 | The finite bag order is a ... |
| reldmopsr 21156 | Lemma for ordered power se... |
| opsrval 21157 | The value of the "ordered ... |
| opsrle 21158 | An alternative expression ... |
| opsrval2 21159 | Self-referential expressio... |
| opsrbaslem 21160 | Get a component of the ord... |
| opsrbaslemOLD 21161 | Obsolete version of ~ opsr... |
| opsrbas 21162 | The base set of the ordere... |
| opsrbasOLD 21163 | Obsolete version of ~ opsr... |
| opsrplusg 21164 | The addition operation of ... |
| opsrplusgOLD 21165 | Obsolete version of ~ opsr... |
| opsrmulr 21166 | The multiplication operati... |
| opsrmulrOLD 21167 | Obsolete version of ~ opsr... |
| opsrvsca 21168 | The scalar product operati... |
| opsrvscaOLD 21169 | Obsolete version of ~ opsr... |
| opsrsca 21170 | The scalar ring of the ord... |
| opsrscaOLD 21171 | Obsolete version of ~ opsr... |
| opsrtoslem1 21172 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 21173 | Lemma for ~ opsrtos . (Co... |
| opsrtos 21174 | The ordered power series s... |
| opsrso 21175 | The ordered power series s... |
| opsrcrng 21176 | The ring of ordered power ... |
| opsrassa 21177 | The ring of ordered power ... |
| mvrf2 21178 | The power series/polynomia... |
| mplmon2 21179 | Express a scaled monomial.... |
| psrbag0 21180 | The empty bag is a bag. (... |
| psrbagsn 21181 | A singleton bag is a bag. ... |
| mplascl 21182 | Value of the scalar inject... |
| mplasclf 21183 | The scalar injection is a ... |
| subrgascl 21184 | The scalar injection funct... |
| subrgasclcl 21185 | The scalars in a polynomia... |
| mplmon2cl 21186 | A scaled monomial is a pol... |
| mplmon2mul 21187 | Product of scaled monomial... |
| mplind 21188 | Prove a property of polyno... |
| mplcoe4 21189 | Decompose a polynomial int... |
| evlslem4 21194 | The support of a tensor pr... |
| psrbagev1 21195 | A bag of multipliers provi... |
| psrbagev1OLD 21196 | Obsolete version of ~ psrb... |
| psrbagev2 21197 | Closure of a sum using a b... |
| psrbagev2OLD 21198 | Obsolete version of ~ psrb... |
| evlslem2 21199 | A linear function on the p... |
| evlslem3 21200 | Lemma for ~ evlseu . Poly... |
| evlslem6 21201 | Lemma for ~ evlseu . Fini... |
| evlslem1 21202 | Lemma for ~ evlseu , give ... |
| evlseu 21203 | For a given interpretation... |
| reldmevls 21204 | Well-behaved binary operat... |
| mpfrcl 21205 | Reverse closure for the se... |
| evlsval 21206 | Value of the polynomial ev... |
| evlsval2 21207 | Characterizing properties ... |
| evlsrhm 21208 | Polynomial evaluation is a... |
| evlssca 21209 | Polynomial evaluation maps... |
| evlsvar 21210 | Polynomial evaluation maps... |
| evlsgsumadd 21211 | Polynomial evaluation maps... |
| evlsgsummul 21212 | Polynomial evaluation maps... |
| evlspw 21213 | Polynomial evaluation for ... |
| evlsvarpw 21214 | Polynomial evaluation for ... |
| evlval 21215 | Value of the simple/same r... |
| evlrhm 21216 | The simple evaluation map ... |
| evlsscasrng 21217 | The evaluation of a scalar... |
| evlsca 21218 | Simple polynomial evaluati... |
| evlsvarsrng 21219 | The evaluation of the vari... |
| evlvar 21220 | Simple polynomial evaluati... |
| mpfconst 21221 | Constants are multivariate... |
| mpfproj 21222 | Projections are multivaria... |
| mpfsubrg 21223 | Polynomial functions are a... |
| mpff 21224 | Polynomial functions are f... |
| mpfaddcl 21225 | The sum of multivariate po... |
| mpfmulcl 21226 | The product of multivariat... |
| mpfind 21227 | Prove a property of polyno... |
| selvffval 21236 | Value of the "variable sel... |
| selvfval 21237 | Value of the "variable sel... |
| selvval 21238 | Value of the "variable sel... |
| mhpfval 21239 | Value of the "homogeneous ... |
| mhpval 21240 | Value of the "homogeneous ... |
| ismhp 21241 | Property of being a homoge... |
| ismhp2 21242 | Deduce a homogeneous polyn... |
| ismhp3 21243 | A polynomial is homogeneou... |
| mhpmpl 21244 | A homogeneous polynomial i... |
| mhpdeg 21245 | All nonzero terms of a hom... |
| mhp0cl 21246 | The zero polynomial is hom... |
| mhpsclcl 21247 | A scalar (or constant) pol... |
| mhpvarcl 21248 | A power series variable is... |
| mhpmulcl 21249 | A product of homogeneous p... |
| mhppwdeg 21250 | Degree of a homogeneous po... |
| mhpaddcl 21251 | Homogeneous polynomials ar... |
| mhpinvcl 21252 | Homogeneous polynomials ar... |
| mhpsubg 21253 | Homogeneous polynomials fo... |
| mhpvscacl 21254 | Homogeneous polynomials ar... |
| mhplss 21255 | Homogeneous polynomials fo... |
| psr1baslem 21266 | The set of finite bags on ... |
| psr1val 21267 | Value of the ring of univa... |
| psr1crng 21268 | The ring of univariate pow... |
| psr1assa 21269 | The ring of univariate pow... |
| psr1tos 21270 | The ordered power series s... |
| psr1bas2 21271 | The base set of the ring o... |
| psr1bas 21272 | The base set of the ring o... |
| vr1val 21273 | The value of the generator... |
| vr1cl2 21274 | The variable ` X ` is a me... |
| ply1val 21275 | The value of the set of un... |
| ply1bas 21276 | The value of the base set ... |
| ply1lss 21277 | Univariate polynomials for... |
| ply1subrg 21278 | Univariate polynomials for... |
| ply1crng 21279 | The ring of univariate pol... |
| ply1assa 21280 | The ring of univariate pol... |
| psr1bascl 21281 | A univariate power series ... |
| psr1basf 21282 | Univariate power series ba... |
| ply1basf 21283 | Univariate polynomial base... |
| ply1bascl 21284 | A univariate polynomial is... |
| ply1bascl2 21285 | A univariate polynomial is... |
| coe1fval 21286 | Value of the univariate po... |
| coe1fv 21287 | Value of an evaluated coef... |
| fvcoe1 21288 | Value of a multivariate co... |
| coe1fval3 21289 | Univariate power series co... |
| coe1f2 21290 | Functionality of univariat... |
| coe1fval2 21291 | Univariate polynomial coef... |
| coe1f 21292 | Functionality of univariat... |
| coe1fvalcl 21293 | A coefficient of a univari... |
| coe1sfi 21294 | Finite support of univaria... |
| coe1fsupp 21295 | The coefficient vector of ... |
| mptcoe1fsupp 21296 | A mapping involving coeffi... |
| coe1ae0 21297 | The coefficient vector of ... |
| vr1cl 21298 | The generator of a univari... |
| opsr0 21299 | Zero in the ordered power ... |
| opsr1 21300 | One in the ordered power s... |
| mplplusg 21301 | Value of addition in a pol... |
| mplmulr 21302 | Value of multiplication in... |
| psr1plusg 21303 | Value of addition in a uni... |
| psr1vsca 21304 | Value of scalar multiplica... |
| psr1mulr 21305 | Value of multiplication in... |
| ply1plusg 21306 | Value of addition in a uni... |
| ply1vsca 21307 | Value of scalar multiplica... |
| ply1mulr 21308 | Value of multiplication in... |
| ressply1bas2 21309 | The base set of a restrict... |
| ressply1bas 21310 | A restricted polynomial al... |
| ressply1add 21311 | A restricted polynomial al... |
| ressply1mul 21312 | A restricted polynomial al... |
| ressply1vsca 21313 | A restricted power series ... |
| subrgply1 21314 | A subring of the base ring... |
| gsumply1subr 21315 | Evaluate a group sum in a ... |
| psrbaspropd 21316 | Property deduction for pow... |
| psrplusgpropd 21317 | Property deduction for pow... |
| mplbaspropd 21318 | Property deduction for pol... |
| psropprmul 21319 | Reversing multiplication i... |
| ply1opprmul 21320 | Reversing multiplication i... |
| 00ply1bas 21321 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 21322 | Protection compatibility o... |
| ply1plusgfvi 21323 | Protection compatibility o... |
| ply1baspropd 21324 | Property deduction for uni... |
| ply1plusgpropd 21325 | Property deduction for uni... |
| opsrring 21326 | Ordered power series form ... |
| opsrlmod 21327 | Ordered power series form ... |
| psr1ring 21328 | Univariate power series fo... |
| ply1ring 21329 | Univariate polynomials for... |
| psr1lmod 21330 | Univariate power series fo... |
| psr1sca 21331 | Scalars of a univariate po... |
| psr1sca2 21332 | Scalars of a univariate po... |
| ply1lmod 21333 | Univariate polynomials for... |
| ply1sca 21334 | Scalars of a univariate po... |
| ply1sca2 21335 | Scalars of a univariate po... |
| ply1mpl0 21336 | The univariate polynomial ... |
| ply10s0 21337 | Zero times a univariate po... |
| ply1mpl1 21338 | The univariate polynomial ... |
| ply1ascl 21339 | The univariate polynomial ... |
| subrg1ascl 21340 | The scalar injection funct... |
| subrg1asclcl 21341 | The scalars in a polynomia... |
| subrgvr1 21342 | The variables in a subring... |
| subrgvr1cl 21343 | The variables in a polynom... |
| coe1z 21344 | The coefficient vector of ... |
| coe1add 21345 | The coefficient vector of ... |
| coe1addfv 21346 | A particular coefficient o... |
| coe1subfv 21347 | A particular coefficient o... |
| coe1mul2lem1 21348 | An equivalence for ~ coe1m... |
| coe1mul2lem2 21349 | An equivalence for ~ coe1m... |
| coe1mul2 21350 | The coefficient vector of ... |
| coe1mul 21351 | The coefficient vector of ... |
| ply1moncl 21352 | Closure of the expression ... |
| ply1tmcl 21353 | Closure of the expression ... |
| coe1tm 21354 | Coefficient vector of a po... |
| coe1tmfv1 21355 | Nonzero coefficient of a p... |
| coe1tmfv2 21356 | Zero coefficient of a poly... |
| coe1tmmul2 21357 | Coefficient vector of a po... |
| coe1tmmul 21358 | Coefficient vector of a po... |
| coe1tmmul2fv 21359 | Function value of a right-... |
| coe1pwmul 21360 | Coefficient vector of a po... |
| coe1pwmulfv 21361 | Function value of a right-... |
| ply1scltm 21362 | A scalar is a term with ze... |
| coe1sclmul 21363 | Coefficient vector of a po... |
| coe1sclmulfv 21364 | A single coefficient of a ... |
| coe1sclmul2 21365 | Coefficient vector of a po... |
| ply1sclf 21366 | A scalar polynomial is a p... |
| ply1sclcl 21367 | The value of the algebra s... |
| coe1scl 21368 | Coefficient vector of a sc... |
| ply1sclid 21369 | Recover the base scalar fr... |
| ply1sclf1 21370 | The polynomial scalar func... |
| ply1scl0 21371 | The zero scalar is zero. ... |
| ply1scln0 21372 | Nonzero scalars create non... |
| ply1scl1 21373 | The one scalar is the unit... |
| ply1idvr1 21374 | The identity of a polynomi... |
| cply1mul 21375 | The product of two constan... |
| ply1coefsupp 21376 | The decomposition of a uni... |
| ply1coe 21377 | Decompose a univariate pol... |
| eqcoe1ply1eq 21378 | Two polynomials over the s... |
| ply1coe1eq 21379 | Two polynomials over the s... |
| cply1coe0 21380 | All but the first coeffici... |
| cply1coe0bi 21381 | A polynomial is constant (... |
| coe1fzgsumdlem 21382 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 21383 | Value of an evaluated coef... |
| gsumsmonply1 21384 | A finite group sum of scal... |
| gsummoncoe1 21385 | A coefficient of the polyn... |
| gsumply1eq 21386 | Two univariate polynomials... |
| lply1binom 21387 | The binomial theorem for l... |
| lply1binomsc 21388 | The binomial theorem for l... |
| reldmevls1 21393 | Well-behaved binary operat... |
| ply1frcl 21394 | Reverse closure for the se... |
| evls1fval 21395 | Value of the univariate po... |
| evls1val 21396 | Value of the univariate po... |
| evls1rhmlem 21397 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 21398 | Polynomial evaluation is a... |
| evls1sca 21399 | Univariate polynomial eval... |
| evls1gsumadd 21400 | Univariate polynomial eval... |
| evls1gsummul 21401 | Univariate polynomial eval... |
| evls1pw 21402 | Univariate polynomial eval... |
| evls1varpw 21403 | Univariate polynomial eval... |
| evl1fval 21404 | Value of the simple/same r... |
| evl1val 21405 | Value of the simple/same r... |
| evl1fval1lem 21406 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 21407 | Value of the simple/same r... |
| evl1rhm 21408 | Polynomial evaluation is a... |
| fveval1fvcl 21409 | The function value of the ... |
| evl1sca 21410 | Polynomial evaluation maps... |
| evl1scad 21411 | Polynomial evaluation buil... |
| evl1var 21412 | Polynomial evaluation maps... |
| evl1vard 21413 | Polynomial evaluation buil... |
| evls1var 21414 | Univariate polynomial eval... |
| evls1scasrng 21415 | The evaluation of a scalar... |
| evls1varsrng 21416 | The evaluation of the vari... |
| evl1addd 21417 | Polynomial evaluation buil... |
| evl1subd 21418 | Polynomial evaluation buil... |
| evl1muld 21419 | Polynomial evaluation buil... |
| evl1vsd 21420 | Polynomial evaluation buil... |
| evl1expd 21421 | Polynomial evaluation buil... |
| pf1const 21422 | Constants are polynomial f... |
| pf1id 21423 | The identity is a polynomi... |
| pf1subrg 21424 | Polynomial functions are a... |
| pf1rcl 21425 | Reverse closure for the se... |
| pf1f 21426 | Polynomial functions are f... |
| mpfpf1 21427 | Convert a multivariate pol... |
| pf1mpf 21428 | Convert a univariate polyn... |
| pf1addcl 21429 | The sum of multivariate po... |
| pf1mulcl 21430 | The product of multivariat... |
| pf1ind 21431 | Prove a property of polyno... |
| evl1gsumdlem 21432 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 21433 | Polynomial evaluation buil... |
| evl1gsumadd 21434 | Univariate polynomial eval... |
| evl1gsumaddval 21435 | Value of a univariate poly... |
| evl1gsummul 21436 | Univariate polynomial eval... |
| evl1varpw 21437 | Univariate polynomial eval... |
| evl1varpwval 21438 | Value of a univariate poly... |
| evl1scvarpw 21439 | Univariate polynomial eval... |
| evl1scvarpwval 21440 | Value of a univariate poly... |
| evl1gsummon 21441 | Value of a univariate poly... |
| mamufval 21444 | Functional value of the ma... |
| mamuval 21445 | Multiplication of two matr... |
| mamufv 21446 | A cell in the multiplicati... |
| mamudm 21447 | The domain of the matrix m... |
| mamufacex 21448 | Every solution of the equa... |
| mamures 21449 | Rows in a matrix product a... |
| mndvcl 21450 | Tuple-wise additive closur... |
| mndvass 21451 | Tuple-wise associativity i... |
| mndvlid 21452 | Tuple-wise left identity i... |
| mndvrid 21453 | Tuple-wise right identity ... |
| grpvlinv 21454 | Tuple-wise left inverse in... |
| grpvrinv 21455 | Tuple-wise right inverse i... |
| mhmvlin 21456 | Tuple extension of monoid ... |
| ringvcl 21457 | Tuple-wise multiplication ... |
| mamucl 21458 | Operation closure of matri... |
| mamuass 21459 | Matrix multiplication is a... |
| mamudi 21460 | Matrix multiplication dist... |
| mamudir 21461 | Matrix multiplication dist... |
| mamuvs1 21462 | Matrix multiplication dist... |
| mamuvs2 21463 | Matrix multiplication dist... |
| matbas0pc 21466 | There is no matrix with a ... |
| matbas0 21467 | There is no matrix for a n... |
| matval 21468 | Value of the matrix algebr... |
| matrcl 21469 | Reverse closure for the ma... |
| matbas 21470 | The matrix ring has the sa... |
| matplusg 21471 | The matrix ring has the sa... |
| matsca 21472 | The matrix ring has the sa... |
| matvsca 21473 | The matrix ring has the sa... |
| mat0 21474 | The matrix ring has the sa... |
| matinvg 21475 | The matrix ring has the sa... |
| mat0op 21476 | Value of a zero matrix as ... |
| matsca2 21477 | The scalars of the matrix ... |
| matbas2 21478 | The base set of the matrix... |
| matbas2i 21479 | A matrix is a function. (... |
| matbas2d 21480 | The base set of the matrix... |
| eqmat 21481 | Two square matrices of the... |
| matecl 21482 | Each entry (according to W... |
| matecld 21483 | Each entry (according to W... |
| matplusg2 21484 | Addition in the matrix rin... |
| matvsca2 21485 | Scalar multiplication in t... |
| matlmod 21486 | The matrix ring is a linea... |
| matgrp 21487 | The matrix ring is a group... |
| matvscl 21488 | Closure of the scalar mult... |
| matsubg 21489 | The matrix ring has the sa... |
| matplusgcell 21490 | Addition in the matrix rin... |
| matsubgcell 21491 | Subtraction in the matrix ... |
| matinvgcell 21492 | Additive inversion in the ... |
| matvscacell 21493 | Scalar multiplication in t... |
| matgsum 21494 | Finite commutative sums in... |
| matmulr 21495 | Multiplication in the matr... |
| mamumat1cl 21496 | The identity matrix (as op... |
| mat1comp 21497 | The components of the iden... |
| mamulid 21498 | The identity matrix (as op... |
| mamurid 21499 | The identity matrix (as op... |
| matring 21500 | Existence of the matrix ri... |
| matassa 21501 | Existence of the matrix al... |
| matmulcell 21502 | Multiplication in the matr... |
| mpomatmul 21503 | Multiplication of two N x ... |
| mat1 21504 | Value of an identity matri... |
| mat1ov 21505 | Entries of an identity mat... |
| mat1bas 21506 | The identity matrix is a m... |
| matsc 21507 | The identity matrix multip... |
| ofco2 21508 | Distribution law for the f... |
| oftpos 21509 | The transposition of the v... |
| mattposcl 21510 | The transpose of a square ... |
| mattpostpos 21511 | The transpose of the trans... |
| mattposvs 21512 | The transposition of a mat... |
| mattpos1 21513 | The transposition of the i... |
| tposmap 21514 | The transposition of an I ... |
| mamutpos 21515 | Behavior of transposes in ... |
| mattposm 21516 | Multiplying two transposed... |
| matgsumcl 21517 | Closure of a group sum ove... |
| madetsumid 21518 | The identity summand in th... |
| matepmcl 21519 | Each entry of a matrix wit... |
| matepm2cl 21520 | Each entry of a matrix wit... |
| madetsmelbas 21521 | A summand of the determina... |
| madetsmelbas2 21522 | A summand of the determina... |
| mat0dimbas0 21523 | The empty set is the one a... |
| mat0dim0 21524 | The zero of the algebra of... |
| mat0dimid 21525 | The identity of the algebr... |
| mat0dimscm 21526 | The scalar multiplication ... |
| mat0dimcrng 21527 | The algebra of matrices wi... |
| mat1dimelbas 21528 | A matrix with dimension 1 ... |
| mat1dimbas 21529 | A matrix with dimension 1 ... |
| mat1dim0 21530 | The zero of the algebra of... |
| mat1dimid 21531 | The identity of the algebr... |
| mat1dimscm 21532 | The scalar multiplication ... |
| mat1dimmul 21533 | The ring multiplication in... |
| mat1dimcrng 21534 | The algebra of matrices wi... |
| mat1f1o 21535 | There is a 1-1 function fr... |
| mat1rhmval 21536 | The value of the ring homo... |
| mat1rhmelval 21537 | The value of the ring homo... |
| mat1rhmcl 21538 | The value of the ring homo... |
| mat1f 21539 | There is a function from a... |
| mat1ghm 21540 | There is a group homomorph... |
| mat1mhm 21541 | There is a monoid homomorp... |
| mat1rhm 21542 | There is a ring homomorphi... |
| mat1rngiso 21543 | There is a ring isomorphis... |
| mat1ric 21544 | A ring is isomorphic to th... |
| dmatval 21549 | The set of ` N ` x ` N ` d... |
| dmatel 21550 | A ` N ` x ` N ` diagonal m... |
| dmatmat 21551 | An ` N ` x ` N ` diagonal ... |
| dmatid 21552 | The identity matrix is a d... |
| dmatelnd 21553 | An extradiagonal entry of ... |
| dmatmul 21554 | The product of two diagona... |
| dmatsubcl 21555 | The difference of two diag... |
| dmatsgrp 21556 | The set of diagonal matric... |
| dmatmulcl 21557 | The product of two diagona... |
| dmatsrng 21558 | The set of diagonal matric... |
| dmatcrng 21559 | The subring of diagonal ma... |
| dmatscmcl 21560 | The multiplication of a di... |
| scmatval 21561 | The set of ` N ` x ` N ` s... |
| scmatel 21562 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 21563 | A scalar matrix can be exp... |
| scmatscmide 21564 | An entry of a scalar matri... |
| scmatscmiddistr 21565 | Distributive law for scala... |
| scmatmat 21566 | An ` N ` x ` N ` scalar ma... |
| scmate 21567 | An entry of an ` N ` x ` N... |
| scmatmats 21568 | The set of an ` N ` x ` N ... |
| scmateALT 21569 | Alternate proof of ~ scmat... |
| scmatscm 21570 | The multiplication of a ma... |
| scmatid 21571 | The identity matrix is a s... |
| scmatdmat 21572 | A scalar matrix is a diago... |
| scmataddcl 21573 | The sum of two scalar matr... |
| scmatsubcl 21574 | The difference of two scal... |
| scmatmulcl 21575 | The product of two scalar ... |
| scmatsgrp 21576 | The set of scalar matrices... |
| scmatsrng 21577 | The set of scalar matrices... |
| scmatcrng 21578 | The subring of scalar matr... |
| scmatsgrp1 21579 | The set of scalar matrices... |
| scmatsrng1 21580 | The set of scalar matrices... |
| smatvscl 21581 | Closure of the scalar mult... |
| scmatlss 21582 | The set of scalar matrices... |
| scmatstrbas 21583 | The set of scalar matrices... |
| scmatrhmval 21584 | The value of the ring homo... |
| scmatrhmcl 21585 | The value of the ring homo... |
| scmatf 21586 | There is a function from a... |
| scmatfo 21587 | There is a function from a... |
| scmatf1 21588 | There is a 1-1 function fr... |
| scmatf1o 21589 | There is a bijection betwe... |
| scmatghm 21590 | There is a group homomorph... |
| scmatmhm 21591 | There is a monoid homomorp... |
| scmatrhm 21592 | There is a ring homomorphi... |
| scmatrngiso 21593 | There is a ring isomorphis... |
| scmatric 21594 | A ring is isomorphic to ev... |
| mat0scmat 21595 | The empty matrix over a ri... |
| mat1scmat 21596 | A 1-dimensional matrix ove... |
| mvmulfval 21599 | Functional value of the ma... |
| mvmulval 21600 | Multiplication of a vector... |
| mvmulfv 21601 | A cell/element in the vect... |
| mavmulval 21602 | Multiplication of a vector... |
| mavmulfv 21603 | A cell/element in the vect... |
| mavmulcl 21604 | Multiplication of an NxN m... |
| 1mavmul 21605 | Multiplication of the iden... |
| mavmulass 21606 | Associativity of the multi... |
| mavmuldm 21607 | The domain of the matrix v... |
| mavmulsolcl 21608 | Every solution of the equa... |
| mavmul0 21609 | Multiplication of a 0-dime... |
| mavmul0g 21610 | The result of the 0-dimens... |
| mvmumamul1 21611 | The multiplication of an M... |
| mavmumamul1 21612 | The multiplication of an N... |
| marrepfval 21617 | First substitution for the... |
| marrepval0 21618 | Second substitution for th... |
| marrepval 21619 | Third substitution for the... |
| marrepeval 21620 | An entry of a matrix with ... |
| marrepcl 21621 | Closure of the row replace... |
| marepvfval 21622 | First substitution for the... |
| marepvval0 21623 | Second substitution for th... |
| marepvval 21624 | Third substitution for the... |
| marepveval 21625 | An entry of a matrix with ... |
| marepvcl 21626 | Closure of the column repl... |
| ma1repvcl 21627 | Closure of the column repl... |
| ma1repveval 21628 | An entry of an identity ma... |
| mulmarep1el 21629 | Element by element multipl... |
| mulmarep1gsum1 21630 | The sum of element by elem... |
| mulmarep1gsum2 21631 | The sum of element by elem... |
| 1marepvmarrepid 21632 | Replacing the ith row by 0... |
| submabas 21635 | Any subset of the index se... |
| submafval 21636 | First substitution for a s... |
| submaval0 21637 | Second substitution for a ... |
| submaval 21638 | Third substitution for a s... |
| submaeval 21639 | An entry of a submatrix of... |
| 1marepvsma1 21640 | The submatrix of the ident... |
| mdetfval 21643 | First substitution for the... |
| mdetleib 21644 | Full substitution of our d... |
| mdetleib2 21645 | Leibniz' formula can also ... |
| nfimdetndef 21646 | The determinant is not def... |
| mdetfval1 21647 | First substitution of an a... |
| mdetleib1 21648 | Full substitution of an al... |
| mdet0pr 21649 | The determinant function f... |
| mdet0f1o 21650 | The determinant function f... |
| mdet0fv0 21651 | The determinant of the emp... |
| mdetf 21652 | Functionality of the deter... |
| mdetcl 21653 | The determinant evaluates ... |
| m1detdiag 21654 | The determinant of a 1-dim... |
| mdetdiaglem 21655 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 21656 | The determinant of a diago... |
| mdetdiagid 21657 | The determinant of a diago... |
| mdet1 21658 | The determinant of the ide... |
| mdetrlin 21659 | The determinant function i... |
| mdetrsca 21660 | The determinant function i... |
| mdetrsca2 21661 | The determinant function i... |
| mdetr0 21662 | The determinant of a matri... |
| mdet0 21663 | The determinant of the zer... |
| mdetrlin2 21664 | The determinant function i... |
| mdetralt 21665 | The determinant function i... |
| mdetralt2 21666 | The determinant function i... |
| mdetero 21667 | The determinant function i... |
| mdettpos 21668 | Determinant is invariant u... |
| mdetunilem1 21669 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 21670 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 21671 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 21672 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 21673 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 21674 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 21675 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 21676 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 21677 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 21678 | Lemma for ~ mdetuni . (Co... |
| mdetuni 21679 | According to the definitio... |
| mdetmul 21680 | Multiplicativity of the de... |
| m2detleiblem1 21681 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 21682 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 21683 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 21684 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 21685 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 21686 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 21687 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 21688 | Leibniz' Formula for 2x2-m... |
| mndifsplit 21693 | Lemma for ~ maducoeval2 . ... |
| madufval 21694 | First substitution for the... |
| maduval 21695 | Second substitution for th... |
| maducoeval 21696 | An entry of the adjunct (c... |
| maducoeval2 21697 | An entry of the adjunct (c... |
| maduf 21698 | Creating the adjunct of ma... |
| madutpos 21699 | The adjuct of a transposed... |
| madugsum 21700 | The determinant of a matri... |
| madurid 21701 | Multiplying a matrix with ... |
| madulid 21702 | Multiplying the adjunct of... |
| minmar1fval 21703 | First substitution for the... |
| minmar1val0 21704 | Second substitution for th... |
| minmar1val 21705 | Third substitution for the... |
| minmar1eval 21706 | An entry of a matrix for a... |
| minmar1marrep 21707 | The minor matrix is a spec... |
| minmar1cl 21708 | Closure of the row replace... |
| maducoevalmin1 21709 | The coefficients of an adj... |
| symgmatr01lem 21710 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 21711 | Applying a permutation tha... |
| gsummatr01lem1 21712 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 21713 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 21714 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 21715 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 21716 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 21717 | Replacing a row of a squar... |
| smadiadetlem0 21718 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 21719 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 21720 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 21721 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 21722 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 21723 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 21724 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 21725 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 21726 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 21727 | The determinant of a subma... |
| smadiadetglem1 21728 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 21729 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 21730 | The determinant of a squar... |
| smadiadetg0 21731 | Lemma for ~ smadiadetr : v... |
| smadiadetr 21732 | The determinant of a squar... |
| invrvald 21733 | If a matrix multiplied wit... |
| matinv 21734 | The inverse of a matrix is... |
| matunit 21735 | A matrix is a unit in the ... |
| slesolvec 21736 | Every solution of a system... |
| slesolinv 21737 | The solution of a system o... |
| slesolinvbi 21738 | The solution of a system o... |
| slesolex 21739 | Every system of linear equ... |
| cramerimplem1 21740 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 21741 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 21742 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 21743 | One direction of Cramer's ... |
| cramerlem1 21744 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 21745 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 21746 | Lemma 3 for ~ cramer . (C... |
| cramer0 21747 | Special case of Cramer's r... |
| cramer 21748 | Cramer's rule. According ... |
| pmatring 21749 | The set of polynomial matr... |
| pmatlmod 21750 | The set of polynomial matr... |
| pmatassa 21751 | The set of polynomial matr... |
| pmat0op 21752 | The zero polynomial matrix... |
| pmat1op 21753 | The identity polynomial ma... |
| pmat1ovd 21754 | Entries of the identity po... |
| pmat0opsc 21755 | The zero polynomial matrix... |
| pmat1opsc 21756 | The identity polynomial ma... |
| pmat1ovscd 21757 | Entries of the identity po... |
| pmatcoe1fsupp 21758 | For a polynomial matrix th... |
| 1pmatscmul 21759 | The scalar product of the ... |
| cpmat 21766 | Value of the constructor o... |
| cpmatpmat 21767 | A constant polynomial matr... |
| cpmatel 21768 | Property of a constant pol... |
| cpmatelimp 21769 | Implication of a set being... |
| cpmatel2 21770 | Another property of a cons... |
| cpmatelimp2 21771 | Another implication of a s... |
| 1elcpmat 21772 | The identity of the ring o... |
| cpmatacl 21773 | The set of all constant po... |
| cpmatinvcl 21774 | The set of all constant po... |
| cpmatmcllem 21775 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 21776 | The set of all constant po... |
| cpmatsubgpmat 21777 | The set of all constant po... |
| cpmatsrgpmat 21778 | The set of all constant po... |
| 0elcpmat 21779 | The zero of the ring of al... |
| mat2pmatfval 21780 | Value of the matrix transf... |
| mat2pmatval 21781 | The result of a matrix tra... |
| mat2pmatvalel 21782 | A (matrix) element of the ... |
| mat2pmatbas 21783 | The result of a matrix tra... |
| mat2pmatbas0 21784 | The result of a matrix tra... |
| mat2pmatf 21785 | The matrix transformation ... |
| mat2pmatf1 21786 | The matrix transformation ... |
| mat2pmatghm 21787 | The transformation of matr... |
| mat2pmatmul 21788 | The transformation of matr... |
| mat2pmat1 21789 | The transformation of the ... |
| mat2pmatmhm 21790 | The transformation of matr... |
| mat2pmatrhm 21791 | The transformation of matr... |
| mat2pmatlin 21792 | The transformation of matr... |
| 0mat2pmat 21793 | The transformed zero matri... |
| idmatidpmat 21794 | The transformed identity m... |
| d0mat2pmat 21795 | The transformed empty set ... |
| d1mat2pmat 21796 | The transformation of a ma... |
| mat2pmatscmxcl 21797 | A transformed matrix multi... |
| m2cpm 21798 | The result of a matrix tra... |
| m2cpmf 21799 | The matrix transformation ... |
| m2cpmf1 21800 | The matrix transformation ... |
| m2cpmghm 21801 | The transformation of matr... |
| m2cpmmhm 21802 | The transformation of matr... |
| m2cpmrhm 21803 | The transformation of matr... |
| m2pmfzmap 21804 | The transformed values of ... |
| m2pmfzgsumcl 21805 | Closure of the sum of scal... |
| cpm2mfval 21806 | Value of the inverse matri... |
| cpm2mval 21807 | The result of an inverse m... |
| cpm2mvalel 21808 | A (matrix) element of the ... |
| cpm2mf 21809 | The inverse matrix transfo... |
| m2cpminvid 21810 | The inverse transformation... |
| m2cpminvid2lem 21811 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 21812 | The transformation applied... |
| m2cpmfo 21813 | The matrix transformation ... |
| m2cpmf1o 21814 | The matrix transformation ... |
| m2cpmrngiso 21815 | The transformation of matr... |
| matcpmric 21816 | The ring of matrices over ... |
| m2cpminv 21817 | The inverse matrix transfo... |
| m2cpminv0 21818 | The inverse matrix transfo... |
| decpmatval0 21821 | The matrix consisting of t... |
| decpmatval 21822 | The matrix consisting of t... |
| decpmate 21823 | An entry of the matrix con... |
| decpmatcl 21824 | Closure of the decompositi... |
| decpmataa0 21825 | The matrix consisting of t... |
| decpmatfsupp 21826 | The mapping to the matrice... |
| decpmatid 21827 | The matrix consisting of t... |
| decpmatmullem 21828 | Lemma for ~ decpmatmul . ... |
| decpmatmul 21829 | The matrix consisting of t... |
| decpmatmulsumfsupp 21830 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 21831 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 21832 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 21833 | Write a polynomial matrix ... |
| pmatcollpw2lem 21834 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 21835 | Write a polynomial matrix ... |
| monmatcollpw 21836 | The matrix consisting of t... |
| pmatcollpwlem 21837 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 21838 | Write a polynomial matrix ... |
| pmatcollpwfi 21839 | Write a polynomial matrix ... |
| pmatcollpw3lem 21840 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 21841 | Write a polynomial matrix ... |
| pmatcollpw3fi 21842 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 21843 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 21844 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 21845 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 21846 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 21847 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 21848 | Write a scalar matrix over... |
| pm2mpf1lem 21851 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 21852 | Value of the transformatio... |
| pm2mpfval 21853 | A polynomial matrix transf... |
| pm2mpcl 21854 | The transformation of poly... |
| pm2mpf 21855 | The transformation of poly... |
| pm2mpf1 21856 | The transformation of poly... |
| pm2mpcoe1 21857 | A coefficient of the polyn... |
| idpm2idmp 21858 | The transformation of the ... |
| mptcoe1matfsupp 21859 | The mapping extracting the... |
| mply1topmatcllem 21860 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 21861 | A polynomial over matrices... |
| mply1topmatcl 21862 | A polynomial over matrices... |
| mp2pm2mplem1 21863 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 21864 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 21865 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 21866 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 21867 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 21868 | A polynomial over matrices... |
| pm2mpghmlem2 21869 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 21870 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 21871 | The transformation of poly... |
| pm2mpf1o 21872 | The transformation of poly... |
| pm2mpghm 21873 | The transformation of poly... |
| pm2mpgrpiso 21874 | The transformation of poly... |
| pm2mpmhmlem1 21875 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 21876 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 21877 | The transformation of poly... |
| pm2mprhm 21878 | The transformation of poly... |
| pm2mprngiso 21879 | The transformation of poly... |
| pmmpric 21880 | The ring of polynomial mat... |
| monmat2matmon 21881 | The transformation of a po... |
| pm2mp 21882 | The transformation of a su... |
| chmatcl 21885 | Closure of the characteris... |
| chmatval 21886 | The entries of the charact... |
| chpmatfval 21887 | Value of the characteristi... |
| chpmatval 21888 | The characteristic polynom... |
| chpmatply1 21889 | The characteristic polynom... |
| chpmatval2 21890 | The characteristic polynom... |
| chpmat0d 21891 | The characteristic polynom... |
| chpmat1dlem 21892 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 21893 | The characteristic polynom... |
| chpdmatlem0 21894 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 21895 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 21896 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 21897 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 21898 | The characteristic polynom... |
| chpscmat 21899 | The characteristic polynom... |
| chpscmat0 21900 | The characteristic polynom... |
| chpscmatgsumbin 21901 | The characteristic polynom... |
| chpscmatgsummon 21902 | The characteristic polynom... |
| chp0mat 21903 | The characteristic polynom... |
| chpidmat 21904 | The characteristic polynom... |
| chmaidscmat 21905 | The characteristic polynom... |
| fvmptnn04if 21906 | The function values of a m... |
| fvmptnn04ifa 21907 | The function value of a ma... |
| fvmptnn04ifb 21908 | The function value of a ma... |
| fvmptnn04ifc 21909 | The function value of a ma... |
| fvmptnn04ifd 21910 | The function value of a ma... |
| chfacfisf 21911 | The "characteristic factor... |
| chfacfisfcpmat 21912 | The "characteristic factor... |
| chfacffsupp 21913 | The "characteristic factor... |
| chfacfscmulcl 21914 | Closure of a scaled value ... |
| chfacfscmul0 21915 | A scaled value of the "cha... |
| chfacfscmulfsupp 21916 | A mapping of scaled values... |
| chfacfscmulgsum 21917 | Breaking up a sum of value... |
| chfacfpmmulcl 21918 | Closure of the value of th... |
| chfacfpmmul0 21919 | The value of the "characte... |
| chfacfpmmulfsupp 21920 | A mapping of values of the... |
| chfacfpmmulgsum 21921 | Breaking up a sum of value... |
| chfacfpmmulgsum2 21922 | Breaking up a sum of value... |
| cayhamlem1 21923 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 21924 | The right-hand fundamental... |
| cpmidgsum 21925 | Representation of the iden... |
| cpmidgsumm2pm 21926 | Representation of the iden... |
| cpmidpmatlem1 21927 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 21928 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 21929 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 21930 | Representation of the iden... |
| cpmadugsumlemB 21931 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 21932 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 21933 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 21934 | The product of the charact... |
| cpmadugsum 21935 | The product of the charact... |
| cpmidgsum2 21936 | Representation of the iden... |
| cpmidg2sum 21937 | Equality of two sums repre... |
| cpmadumatpolylem1 21938 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 21939 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 21940 | The product of the charact... |
| cayhamlem2 21941 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 21942 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 21943 | The coefficients of the ch... |
| cayhamlem3 21944 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 21945 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 21946 | The Cayley-Hamilton theore... |
| cayleyhamilton 21947 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 21948 | Alternate proof of ~ cayle... |
| cayleyhamilton1 21949 | The Cayley-Hamilton theore... |
| istopg 21952 | Express the predicate " ` ... |
| istop2g 21953 | Express the predicate " ` ... |
| uniopn 21954 | The union of a subset of a... |
| iunopn 21955 | The indexed union of a sub... |
| inopn 21956 | The intersection of two op... |
| fitop 21957 | A topology is closed under... |
| fiinopn 21958 | The intersection of a none... |
| iinopn 21959 | The intersection of a none... |
| unopn 21960 | The union of two open sets... |
| 0opn 21961 | The empty set is an open s... |
| 0ntop 21962 | The empty set is not a top... |
| topopn 21963 | The underlying set of a to... |
| eltopss 21964 | A member of a topology is ... |
| riinopn 21965 | A finite indexed relative ... |
| rintopn 21966 | A finite relative intersec... |
| istopon 21969 | Property of being a topolo... |
| topontop 21970 | A topology on a given base... |
| toponuni 21971 | The base set of a topology... |
| topontopi 21972 | A topology on a given base... |
| toponunii 21973 | The base set of a topology... |
| toptopon 21974 | Alternative definition of ... |
| toptopon2 21975 | A topology is the same thi... |
| topontopon 21976 | A topology on a set is a t... |
| funtopon 21977 | The class ` TopOn ` is a f... |
| toponrestid 21978 | Given a topology on a set,... |
| toponsspwpw 21979 | The set of topologies on a... |
| dmtopon 21980 | The domain of ` TopOn ` is... |
| fntopon 21981 | The class ` TopOn ` is a f... |
| toprntopon 21982 | A topology is the same thi... |
| toponmax 21983 | The base set of a topology... |
| toponss 21984 | A member of a topology is ... |
| toponcom 21985 | If ` K ` is a topology on ... |
| toponcomb 21986 | Biconditional form of ~ to... |
| topgele 21987 | The topologies over the sa... |
| topsn 21988 | The only topology on a sin... |
| istps 21991 | Express the predicate "is ... |
| istps2 21992 | Express the predicate "is ... |
| tpsuni 21993 | The base set of a topologi... |
| tpstop 21994 | The topology extractor on ... |
| tpspropd 21995 | A topological space depend... |
| tpsprop2d 21996 | A topological space depend... |
| topontopn 21997 | Express the predicate "is ... |
| tsettps 21998 | If the topology component ... |
| istpsi 21999 | Properties that determine ... |
| eltpsg 22000 | Properties that determine ... |
| eltpsgOLD 22001 | Obsolete version of ~ eltp... |
| eltpsi 22002 | Properties that determine ... |
| isbasisg 22005 | Express the predicate "the... |
| isbasis2g 22006 | Express the predicate "the... |
| isbasis3g 22007 | Express the predicate "the... |
| basis1 22008 | Property of a basis. (Con... |
| basis2 22009 | Property of a basis. (Con... |
| fiinbas 22010 | If a set is closed under f... |
| basdif0 22011 | A basis is not affected by... |
| baspartn 22012 | A disjoint system of sets ... |
| tgval 22013 | The topology generated by ... |
| tgval2 22014 | Definition of a topology g... |
| eltg 22015 | Membership in a topology g... |
| eltg2 22016 | Membership in a topology g... |
| eltg2b 22017 | Membership in a topology g... |
| eltg4i 22018 | An open set in a topology ... |
| eltg3i 22019 | The union of a set of basi... |
| eltg3 22020 | Membership in a topology g... |
| tgval3 22021 | Alternate expression for t... |
| tg1 22022 | Property of a member of a ... |
| tg2 22023 | Property of a member of a ... |
| bastg 22024 | A member of a basis is a s... |
| unitg 22025 | The topology generated by ... |
| tgss 22026 | Subset relation for genera... |
| tgcl 22027 | Show that a basis generate... |
| tgclb 22028 | The property ~ tgcl can be... |
| tgtopon 22029 | A basis generates a topolo... |
| topbas 22030 | A topology is its own basi... |
| tgtop 22031 | A topology is its own basi... |
| eltop 22032 | Membership in a topology, ... |
| eltop2 22033 | Membership in a topology. ... |
| eltop3 22034 | Membership in a topology. ... |
| fibas 22035 | A collection of finite int... |
| tgdom 22036 | A space has no more open s... |
| tgiun 22037 | The indexed union of a set... |
| tgidm 22038 | The topology generator fun... |
| bastop 22039 | Two ways to express that a... |
| tgtop11 22040 | The topology generation fu... |
| 0top 22041 | The singleton of the empty... |
| en1top 22042 | ` { (/) } ` is the only to... |
| en2top 22043 | If a topology has two elem... |
| tgss3 22044 | A criterion for determinin... |
| tgss2 22045 | A criterion for determinin... |
| basgen 22046 | Given a topology ` J ` , s... |
| basgen2 22047 | Given a topology ` J ` , s... |
| 2basgen 22048 | Conditions that determine ... |
| tgfiss 22049 | If a subbase is included i... |
| tgdif0 22050 | A generated topology is no... |
| bastop1 22051 | A subset of a topology is ... |
| bastop2 22052 | A version of ~ bastop1 tha... |
| distop 22053 | The discrete topology on a... |
| topnex 22054 | The class of all topologie... |
| distopon 22055 | The discrete topology on a... |
| sn0topon 22056 | The singleton of the empty... |
| sn0top 22057 | The singleton of the empty... |
| indislem 22058 | A lemma to eliminate some ... |
| indistopon 22059 | The indiscrete topology on... |
| indistop 22060 | The indiscrete topology on... |
| indisuni 22061 | The base set of the indisc... |
| fctop 22062 | The finite complement topo... |
| fctop2 22063 | The finite complement topo... |
| cctop 22064 | The countable complement t... |
| ppttop 22065 | The particular point topol... |
| pptbas 22066 | The particular point topol... |
| epttop 22067 | The excluded point topolog... |
| indistpsx 22068 | The indiscrete topology on... |
| indistps 22069 | The indiscrete topology on... |
| indistps2 22070 | The indiscrete topology on... |
| indistpsALT 22071 | The indiscrete topology on... |
| indistpsALTOLD 22072 | Obsolete proof of ~ indist... |
| indistps2ALT 22073 | The indiscrete topology on... |
| distps 22074 | The discrete topology on a... |
| fncld 22081 | The closed-set generator i... |
| cldval 22082 | The set of closed sets of ... |
| ntrfval 22083 | The interior function on t... |
| clsfval 22084 | The closure function on th... |
| cldrcl 22085 | Reverse closure of the clo... |
| iscld 22086 | The predicate "the class `... |
| iscld2 22087 | A subset of the underlying... |
| cldss 22088 | A closed set is a subset o... |
| cldss2 22089 | The set of closed sets is ... |
| cldopn 22090 | The complement of a closed... |
| isopn2 22091 | A subset of the underlying... |
| opncld 22092 | The complement of an open ... |
| difopn 22093 | The difference of a closed... |
| topcld 22094 | The underlying set of a to... |
| ntrval 22095 | The interior of a subset o... |
| clsval 22096 | The closure of a subset of... |
| 0cld 22097 | The empty set is closed. ... |
| iincld 22098 | The indexed intersection o... |
| intcld 22099 | The intersection of a set ... |
| uncld 22100 | The union of two closed se... |
| cldcls 22101 | A closed subset equals its... |
| incld 22102 | The intersection of two cl... |
| riincld 22103 | An indexed relative inters... |
| iuncld 22104 | A finite indexed union of ... |
| unicld 22105 | A finite union of closed s... |
| clscld 22106 | The closure of a subset of... |
| clsf 22107 | The closure function is a ... |
| ntropn 22108 | The interior of a subset o... |
| clsval2 22109 | Express closure in terms o... |
| ntrval2 22110 | Interior expressed in term... |
| ntrdif 22111 | An interior of a complemen... |
| clsdif 22112 | A closure of a complement ... |
| clsss 22113 | Subset relationship for cl... |
| ntrss 22114 | Subset relationship for in... |
| sscls 22115 | A subset of a topology's u... |
| ntrss2 22116 | A subset includes its inte... |
| ssntr 22117 | An open subset of a set is... |
| clsss3 22118 | The closure of a subset of... |
| ntrss3 22119 | The interior of a subset o... |
| ntrin 22120 | A pairwise intersection of... |
| cmclsopn 22121 | The complement of a closur... |
| cmntrcld 22122 | The complement of an inter... |
| iscld3 22123 | A subset is closed iff it ... |
| iscld4 22124 | A subset is closed iff it ... |
| isopn3 22125 | A subset is open iff it eq... |
| clsidm 22126 | The closure operation is i... |
| ntridm 22127 | The interior operation is ... |
| clstop 22128 | The closure of a topology'... |
| ntrtop 22129 | The interior of a topology... |
| 0ntr 22130 | A subset with an empty int... |
| clsss2 22131 | If a subset is included in... |
| elcls 22132 | Membership in a closure. ... |
| elcls2 22133 | Membership in a closure. ... |
| clsndisj 22134 | Any open set containing a ... |
| ntrcls0 22135 | A subset whose closure has... |
| ntreq0 22136 | Two ways to say that a sub... |
| cldmre 22137 | The closed sets of a topol... |
| mrccls 22138 | Moore closure generalizes ... |
| cls0 22139 | The closure of the empty s... |
| ntr0 22140 | The interior of the empty ... |
| isopn3i 22141 | An open subset equals its ... |
| elcls3 22142 | Membership in a closure in... |
| opncldf1 22143 | A bijection useful for con... |
| opncldf2 22144 | The values of the open-clo... |
| opncldf3 22145 | The values of the converse... |
| isclo 22146 | A set ` A ` is clopen iff ... |
| isclo2 22147 | A set ` A ` is clopen iff ... |
| discld 22148 | The open sets of a discret... |
| sn0cld 22149 | The closed sets of the top... |
| indiscld 22150 | The closed sets of an indi... |
| mretopd 22151 | A Moore collection which i... |
| toponmre 22152 | The topologies over a give... |
| cldmreon 22153 | The closed sets of a topol... |
| iscldtop 22154 | A family is the closed set... |
| mreclatdemoBAD 22155 | The closed subspaces of a ... |
| neifval 22158 | Value of the neighborhood ... |
| neif 22159 | The neighborhood function ... |
| neiss2 22160 | A set with a neighborhood ... |
| neival 22161 | Value of the set of neighb... |
| isnei 22162 | The predicate "the class `... |
| neiint 22163 | An intuitive definition of... |
| isneip 22164 | The predicate "the class `... |
| neii1 22165 | A neighborhood is included... |
| neisspw 22166 | The neighborhoods of any s... |
| neii2 22167 | Property of a neighborhood... |
| neiss 22168 | Any neighborhood of a set ... |
| ssnei 22169 | A set is included in any o... |
| elnei 22170 | A point belongs to any of ... |
| 0nnei 22171 | The empty set is not a nei... |
| neips 22172 | A neighborhood of a set is... |
| opnneissb 22173 | An open set is a neighborh... |
| opnssneib 22174 | Any superset of an open se... |
| ssnei2 22175 | Any subset ` M ` of ` X ` ... |
| neindisj 22176 | Any neighborhood of an ele... |
| opnneiss 22177 | An open set is a neighborh... |
| opnneip 22178 | An open set is a neighborh... |
| opnnei 22179 | A set is open iff it is a ... |
| tpnei 22180 | The underlying set of a to... |
| neiuni 22181 | The union of the neighborh... |
| neindisj2 22182 | A point ` P ` belongs to t... |
| topssnei 22183 | A finer topology has more ... |
| innei 22184 | The intersection of two ne... |
| opnneiid 22185 | Only an open set is a neig... |
| neissex 22186 | For any neighborhood ` N `... |
| 0nei 22187 | The empty set is a neighbo... |
| neipeltop 22188 | Lemma for ~ neiptopreu . ... |
| neiptopuni 22189 | Lemma for ~ neiptopreu . ... |
| neiptoptop 22190 | Lemma for ~ neiptopreu . ... |
| neiptopnei 22191 | Lemma for ~ neiptopreu . ... |
| neiptopreu 22192 | If, to each element ` P ` ... |
| lpfval 22197 | The limit point function o... |
| lpval 22198 | The set of limit points of... |
| islp 22199 | The predicate "the class `... |
| lpsscls 22200 | The limit points of a subs... |
| lpss 22201 | The limit points of a subs... |
| lpdifsn 22202 | ` P ` is a limit point of ... |
| lpss3 22203 | Subset relationship for li... |
| islp2 22204 | The predicate " ` P ` is a... |
| islp3 22205 | The predicate " ` P ` is a... |
| maxlp 22206 | A point is a limit point o... |
| clslp 22207 | The closure of a subset of... |
| islpi 22208 | A point belonging to a set... |
| cldlp 22209 | A subset of a topological ... |
| isperf 22210 | Definition of a perfect sp... |
| isperf2 22211 | Definition of a perfect sp... |
| isperf3 22212 | A perfect space is a topol... |
| perflp 22213 | The limit points of a perf... |
| perfi 22214 | Property of a perfect spac... |
| perftop 22215 | A perfect space is a topol... |
| restrcl 22216 | Reverse closure for the su... |
| restbas 22217 | A subspace topology basis ... |
| tgrest 22218 | A subspace can be generate... |
| resttop 22219 | A subspace topology is a t... |
| resttopon 22220 | A subspace topology is a t... |
| restuni 22221 | The underlying set of a su... |
| stoig 22222 | The topological space buil... |
| restco 22223 | Composition of subspaces. ... |
| restabs 22224 | Equivalence of being a sub... |
| restin 22225 | When the subspace region i... |
| restuni2 22226 | The underlying set of a su... |
| resttopon2 22227 | The underlying set of a su... |
| rest0 22228 | The subspace topology indu... |
| restsn 22229 | The only subspace topology... |
| restsn2 22230 | The subspace topology indu... |
| restcld 22231 | A closed set of a subspace... |
| restcldi 22232 | A closed set is closed in ... |
| restcldr 22233 | A set which is closed in t... |
| restopnb 22234 | If ` B ` is an open subset... |
| ssrest 22235 | If ` K ` is a finer topolo... |
| restopn2 22236 | If ` A ` is open, then ` B... |
| restdis 22237 | A subspace of a discrete t... |
| restfpw 22238 | The restriction of the set... |
| neitr 22239 | The neighborhood of a trac... |
| restcls 22240 | A closure in a subspace to... |
| restntr 22241 | An interior in a subspace ... |
| restlp 22242 | The limit points of a subs... |
| restperf 22243 | Perfection of a subspace. ... |
| perfopn 22244 | An open subset of a perfec... |
| resstopn 22245 | The topology of a restrict... |
| resstps 22246 | A restricted topological s... |
| ordtbaslem 22247 | Lemma for ~ ordtbas . In ... |
| ordtval 22248 | Value of the order topolog... |
| ordtuni 22249 | Value of the order topolog... |
| ordtbas2 22250 | Lemma for ~ ordtbas . (Co... |
| ordtbas 22251 | In a total order, the fini... |
| ordttopon 22252 | Value of the order topolog... |
| ordtopn1 22253 | An upward ray ` ( P , +oo ... |
| ordtopn2 22254 | A downward ray ` ( -oo , P... |
| ordtopn3 22255 | An open interval ` ( A , B... |
| ordtcld1 22256 | A downward ray ` ( -oo , P... |
| ordtcld2 22257 | An upward ray ` [ P , +oo ... |
| ordtcld3 22258 | A closed interval ` [ A , ... |
| ordttop 22259 | The order topology is a to... |
| ordtcnv 22260 | The order dual generates t... |
| ordtrest 22261 | The subspace topology of a... |
| ordtrest2lem 22262 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 22263 | An interval-closed set ` A... |
| letopon 22264 | The topology of the extend... |
| letop 22265 | The topology of the extend... |
| letopuni 22266 | The topology of the extend... |
| xrstopn 22267 | The topology component of ... |
| xrstps 22268 | The extended real number s... |
| leordtvallem1 22269 | Lemma for ~ leordtval . (... |
| leordtvallem2 22270 | Lemma for ~ leordtval . (... |
| leordtval2 22271 | The topology of the extend... |
| leordtval 22272 | The topology of the extend... |
| iccordt 22273 | A closed interval is close... |
| iocpnfordt 22274 | An unbounded above open in... |
| icomnfordt 22275 | An unbounded above open in... |
| iooordt 22276 | An open interval is open i... |
| reordt 22277 | The real numbers are an op... |
| lecldbas 22278 | The set of closed interval... |
| pnfnei 22279 | A neighborhood of ` +oo ` ... |
| mnfnei 22280 | A neighborhood of ` -oo ` ... |
| ordtrestixx 22281 | The restriction of the les... |
| ordtresticc 22282 | The restriction of the les... |
| lmrel 22289 | The topological space conv... |
| lmrcl 22290 | Reverse closure for the co... |
| lmfval 22291 | The relation "sequence ` f... |
| cnfval 22292 | The set of all continuous ... |
| cnpfval 22293 | The function mapping the p... |
| iscn 22294 | The predicate "the class `... |
| cnpval 22295 | The set of all functions f... |
| iscnp 22296 | The predicate "the class `... |
| iscn2 22297 | The predicate "the class `... |
| iscnp2 22298 | The predicate "the class `... |
| cntop1 22299 | Reverse closure for a cont... |
| cntop2 22300 | Reverse closure for a cont... |
| cnptop1 22301 | Reverse closure for a func... |
| cnptop2 22302 | Reverse closure for a func... |
| iscnp3 22303 | The predicate "the class `... |
| cnprcl 22304 | Reverse closure for a func... |
| cnf 22305 | A continuous function is a... |
| cnpf 22306 | A continuous function at p... |
| cnpcl 22307 | The value of a continuous ... |
| cnf2 22308 | A continuous function is a... |
| cnpf2 22309 | A continuous function at p... |
| cnprcl2 22310 | Reverse closure for a func... |
| tgcn 22311 | The continuity predicate w... |
| tgcnp 22312 | The "continuous at a point... |
| subbascn 22313 | The continuity predicate w... |
| ssidcn 22314 | The identity function is a... |
| cnpimaex 22315 | Property of a function con... |
| idcn 22316 | A restricted identity func... |
| lmbr 22317 | Express the binary relatio... |
| lmbr2 22318 | Express the binary relatio... |
| lmbrf 22319 | Express the binary relatio... |
| lmconst 22320 | A constant sequence conver... |
| lmcvg 22321 | Convergence property of a ... |
| iscnp4 22322 | The predicate "the class `... |
| cnpnei 22323 | A condition for continuity... |
| cnima 22324 | An open subset of the codo... |
| cnco 22325 | The composition of two con... |
| cnpco 22326 | The composition of a funct... |
| cnclima 22327 | A closed subset of the cod... |
| iscncl 22328 | A characterization of a co... |
| cncls2i 22329 | Property of the preimage o... |
| cnntri 22330 | Property of the preimage o... |
| cnclsi 22331 | Property of the image of a... |
| cncls2 22332 | Continuity in terms of clo... |
| cncls 22333 | Continuity in terms of clo... |
| cnntr 22334 | Continuity in terms of int... |
| cnss1 22335 | If the topology ` K ` is f... |
| cnss2 22336 | If the topology ` K ` is f... |
| cncnpi 22337 | A continuous function is c... |
| cnsscnp 22338 | The set of continuous func... |
| cncnp 22339 | A continuous function is c... |
| cncnp2 22340 | A continuous function is c... |
| cnnei 22341 | Continuity in terms of nei... |
| cnconst2 22342 | A constant function is con... |
| cnconst 22343 | A constant function is con... |
| cnrest 22344 | Continuity of a restrictio... |
| cnrest2 22345 | Equivalence of continuity ... |
| cnrest2r 22346 | Equivalence of continuity ... |
| cnpresti 22347 | One direction of ~ cnprest... |
| cnprest 22348 | Equivalence of continuity ... |
| cnprest2 22349 | Equivalence of point-conti... |
| cndis 22350 | Every function is continuo... |
| cnindis 22351 | Every function is continuo... |
| cnpdis 22352 | If ` A ` is an isolated po... |
| paste 22353 | Pasting lemma. If ` A ` a... |
| lmfpm 22354 | If ` F ` converges, then `... |
| lmfss 22355 | Inclusion of a function ha... |
| lmcl 22356 | Closure of a limit. (Cont... |
| lmss 22357 | Limit on a subspace. (Con... |
| sslm 22358 | A finer topology has fewer... |
| lmres 22359 | A function converges iff i... |
| lmff 22360 | If ` F ` converges, there ... |
| lmcls 22361 | Any convergent sequence of... |
| lmcld 22362 | Any convergent sequence of... |
| lmcnp 22363 | The image of a convergent ... |
| lmcn 22364 | The image of a convergent ... |
| ist0 22379 | The predicate "is a T_0 sp... |
| ist1 22380 | The predicate "is a T_1 sp... |
| ishaus 22381 | The predicate "is a Hausdo... |
| iscnrm 22382 | The property of being comp... |
| t0sep 22383 | Any two topologically indi... |
| t0dist 22384 | Any two distinct points in... |
| t1sncld 22385 | In a T_1 space, singletons... |
| t1ficld 22386 | In a T_1 space, finite set... |
| hausnei 22387 | Neighborhood property of a... |
| t0top 22388 | A T_0 space is a topologic... |
| t1top 22389 | A T_1 space is a topologic... |
| haustop 22390 | A Hausdorff space is a top... |
| isreg 22391 | The predicate "is a regula... |
| regtop 22392 | A regular space is a topol... |
| regsep 22393 | In a regular space, every ... |
| isnrm 22394 | The predicate "is a normal... |
| nrmtop 22395 | A normal space is a topolo... |
| cnrmtop 22396 | A completely normal space ... |
| iscnrm2 22397 | The property of being comp... |
| ispnrm 22398 | The property of being perf... |
| pnrmnrm 22399 | A perfectly normal space i... |
| pnrmtop 22400 | A perfectly normal space i... |
| pnrmcld 22401 | A closed set in a perfectl... |
| pnrmopn 22402 | An open set in a perfectly... |
| ist0-2 22403 | The predicate "is a T_0 sp... |
| ist0-3 22404 | The predicate "is a T_0 sp... |
| cnt0 22405 | The preimage of a T_0 topo... |
| ist1-2 22406 | An alternate characterizat... |
| t1t0 22407 | A T_1 space is a T_0 space... |
| ist1-3 22408 | A space is T_1 iff every p... |
| cnt1 22409 | The preimage of a T_1 topo... |
| ishaus2 22410 | Express the predicate " ` ... |
| haust1 22411 | A Hausdorff space is a T_1... |
| hausnei2 22412 | The Hausdorff condition st... |
| cnhaus 22413 | The preimage of a Hausdorf... |
| nrmsep3 22414 | In a normal space, given a... |
| nrmsep2 22415 | In a normal space, any two... |
| nrmsep 22416 | In a normal space, disjoin... |
| isnrm2 22417 | An alternate characterizat... |
| isnrm3 22418 | A topological space is nor... |
| cnrmi 22419 | A subspace of a completely... |
| cnrmnrm 22420 | A completely normal space ... |
| restcnrm 22421 | A subspace of a completely... |
| resthauslem 22422 | Lemma for ~ resthaus and s... |
| lpcls 22423 | The limit points of the cl... |
| perfcls 22424 | A subset of a perfect spac... |
| restt0 22425 | A subspace of a T_0 topolo... |
| restt1 22426 | A subspace of a T_1 topolo... |
| resthaus 22427 | A subspace of a Hausdorff ... |
| t1sep2 22428 | Any two points in a T_1 sp... |
| t1sep 22429 | Any two distinct points in... |
| sncld 22430 | A singleton is closed in a... |
| sshauslem 22431 | Lemma for ~ sshaus and sim... |
| sst0 22432 | A topology finer than a T_... |
| sst1 22433 | A topology finer than a T_... |
| sshaus 22434 | A topology finer than a Ha... |
| regsep2 22435 | In a regular space, a clos... |
| isreg2 22436 | A topological space is reg... |
| dnsconst 22437 | If a continuous mapping to... |
| ordtt1 22438 | The order topology is T_1 ... |
| lmmo 22439 | A sequence in a Hausdorff ... |
| lmfun 22440 | The convergence relation i... |
| dishaus 22441 | A discrete topology is Hau... |
| ordthauslem 22442 | Lemma for ~ ordthaus . (C... |
| ordthaus 22443 | The order topology of a to... |
| xrhaus 22444 | The topology of the extend... |
| iscmp 22447 | The predicate "is a compac... |
| cmpcov 22448 | An open cover of a compact... |
| cmpcov2 22449 | Rewrite ~ cmpcov for the c... |
| cmpcovf 22450 | Combine ~ cmpcov with ~ ac... |
| cncmp 22451 | Compactness is respected b... |
| fincmp 22452 | A finite topology is compa... |
| 0cmp 22453 | The singleton of the empty... |
| cmptop 22454 | A compact topology is a to... |
| rncmp 22455 | The image of a compact set... |
| imacmp 22456 | The image of a compact set... |
| discmp 22457 | A discrete topology is com... |
| cmpsublem 22458 | Lemma for ~ cmpsub . (Con... |
| cmpsub 22459 | Two equivalent ways of des... |
| tgcmp 22460 | A topology generated by a ... |
| cmpcld 22461 | A closed subset of a compa... |
| uncmp 22462 | The union of two compact s... |
| fiuncmp 22463 | A finite union of compact ... |
| sscmp 22464 | A subset of a compact topo... |
| hauscmplem 22465 | Lemma for ~ hauscmp . (Co... |
| hauscmp 22466 | A compact subspace of a T2... |
| cmpfi 22467 | If a topology is compact a... |
| cmpfii 22468 | In a compact topology, a s... |
| bwth 22469 | The glorious Bolzano-Weier... |
| isconn 22472 | The predicate ` J ` is a c... |
| isconn2 22473 | The predicate ` J ` is a c... |
| connclo 22474 | The only nonempty clopen s... |
| conndisj 22475 | If a topology is connected... |
| conntop 22476 | A connected topology is a ... |
| indisconn 22477 | The indiscrete topology (o... |
| dfconn2 22478 | An alternate definition of... |
| connsuba 22479 | Connectedness for a subspa... |
| connsub 22480 | Two equivalent ways of say... |
| cnconn 22481 | Connectedness is respected... |
| nconnsubb 22482 | Disconnectedness for a sub... |
| connsubclo 22483 | If a clopen set meets a co... |
| connima 22484 | The image of a connected s... |
| conncn 22485 | A continuous function from... |
| iunconnlem 22486 | Lemma for ~ iunconn . (Co... |
| iunconn 22487 | The indexed union of conne... |
| unconn 22488 | The union of two connected... |
| clsconn 22489 | The closure of a connected... |
| conncompid 22490 | The connected component co... |
| conncompconn 22491 | The connected component co... |
| conncompss 22492 | The connected component co... |
| conncompcld 22493 | The connected component co... |
| conncompclo 22494 | The connected component co... |
| t1connperf 22495 | A connected T_1 space is p... |
| is1stc 22500 | The predicate "is a first-... |
| is1stc2 22501 | An equivalent way of sayin... |
| 1stctop 22502 | A first-countable topology... |
| 1stcclb 22503 | A property of points in a ... |
| 1stcfb 22504 | For any point ` A ` in a f... |
| is2ndc 22505 | The property of being seco... |
| 2ndctop 22506 | A second-countable topolog... |
| 2ndci 22507 | A countable basis generate... |
| 2ndcsb 22508 | Having a countable subbase... |
| 2ndcredom 22509 | A second-countable space h... |
| 2ndc1stc 22510 | A second-countable space i... |
| 1stcrestlem 22511 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 22512 | A subspace of a first-coun... |
| 2ndcrest 22513 | A subspace of a second-cou... |
| 2ndcctbss 22514 | If a topology is second-co... |
| 2ndcdisj 22515 | Any disjoint family of ope... |
| 2ndcdisj2 22516 | Any disjoint collection of... |
| 2ndcomap 22517 | A surjective continuous op... |
| 2ndcsep 22518 | A second-countable topolog... |
| dis2ndc 22519 | A discrete space is second... |
| 1stcelcls 22520 | A point belongs to the clo... |
| 1stccnp 22521 | A mapping is continuous at... |
| 1stccn 22522 | A mapping ` X --> Y ` , wh... |
| islly 22527 | The property of being a lo... |
| isnlly 22528 | The property of being an n... |
| llyeq 22529 | Equality theorem for the `... |
| nllyeq 22530 | Equality theorem for the `... |
| llytop 22531 | A locally ` A ` space is a... |
| nllytop 22532 | A locally ` A ` space is a... |
| llyi 22533 | The property of a locally ... |
| nllyi 22534 | The property of an n-local... |
| nlly2i 22535 | Eliminate the neighborhood... |
| llynlly 22536 | A locally ` A ` space is n... |
| llyssnlly 22537 | A locally ` A ` space is n... |
| llyss 22538 | The "locally" predicate re... |
| nllyss 22539 | The "n-locally" predicate ... |
| subislly 22540 | The property of a subspace... |
| restnlly 22541 | If the property ` A ` pass... |
| restlly 22542 | If the property ` A ` pass... |
| islly2 22543 | An alternative expression ... |
| llyrest 22544 | An open subspace of a loca... |
| nllyrest 22545 | An open subspace of an n-l... |
| loclly 22546 | If ` A ` is a local proper... |
| llyidm 22547 | Idempotence of the "locall... |
| nllyidm 22548 | Idempotence of the "n-loca... |
| toplly 22549 | A topology is locally a to... |
| topnlly 22550 | A topology is n-locally a ... |
| hauslly 22551 | A Hausdorff space is local... |
| hausnlly 22552 | A Hausdorff space is n-loc... |
| hausllycmp 22553 | A compact Hausdorff space ... |
| cldllycmp 22554 | A closed subspace of a loc... |
| lly1stc 22555 | First-countability is a lo... |
| dislly 22556 | The discrete space ` ~P X ... |
| disllycmp 22557 | A discrete space is locall... |
| dis1stc 22558 | A discrete space is first-... |
| hausmapdom 22559 | If ` X ` is a first-counta... |
| hauspwdom 22560 | Simplify the cardinal ` A ... |
| refrel 22567 | Refinement is a relation. ... |
| isref 22568 | The property of being a re... |
| refbas 22569 | A refinement covers the sa... |
| refssex 22570 | Every set in a refinement ... |
| ssref 22571 | A subcover is a refinement... |
| refref 22572 | Reflexivity of refinement.... |
| reftr 22573 | Refinement is transitive. ... |
| refun0 22574 | Adding the empty set prese... |
| isptfin 22575 | The statement "is a point-... |
| islocfin 22576 | The statement "is a locall... |
| finptfin 22577 | A finite cover is a point-... |
| ptfinfin 22578 | A point covered by a point... |
| finlocfin 22579 | A finite cover of a topolo... |
| locfintop 22580 | A locally finite cover cov... |
| locfinbas 22581 | A locally finite cover mus... |
| locfinnei 22582 | A point covered by a local... |
| lfinpfin 22583 | A locally finite cover is ... |
| lfinun 22584 | Adding a finite set preser... |
| locfincmp 22585 | For a compact space, the l... |
| unisngl 22586 | Taking the union of the se... |
| dissnref 22587 | The set of singletons is a... |
| dissnlocfin 22588 | The set of singletons is l... |
| locfindis 22589 | The locally finite covers ... |
| locfincf 22590 | A locally finite cover in ... |
| comppfsc 22591 | A space where every open c... |
| kgenval 22594 | Value of the compact gener... |
| elkgen 22595 | Value of the compact gener... |
| kgeni 22596 | Property of the open sets ... |
| kgentopon 22597 | The compact generator gene... |
| kgenuni 22598 | The base set of the compac... |
| kgenftop 22599 | The compact generator gene... |
| kgenf 22600 | The compact generator is a... |
| kgentop 22601 | A compactly generated spac... |
| kgenss 22602 | The compact generator gene... |
| kgenhaus 22603 | The compact generator gene... |
| kgencmp 22604 | The compact generator topo... |
| kgencmp2 22605 | The compact generator topo... |
| kgenidm 22606 | The compact generator is i... |
| iskgen2 22607 | A space is compactly gener... |
| iskgen3 22608 | Derive the usual definitio... |
| llycmpkgen2 22609 | A locally compact space is... |
| cmpkgen 22610 | A compact space is compact... |
| llycmpkgen 22611 | A locally compact space is... |
| 1stckgenlem 22612 | The one-point compactifica... |
| 1stckgen 22613 | A first-countable space is... |
| kgen2ss 22614 | The compact generator pres... |
| kgencn 22615 | A function from a compactl... |
| kgencn2 22616 | A function ` F : J --> K `... |
| kgencn3 22617 | The set of continuous func... |
| kgen2cn 22618 | A continuous function is a... |
| txval 22623 | Value of the binary topolo... |
| txuni2 22624 | The underlying set of the ... |
| txbasex 22625 | The basis for the product ... |
| txbas 22626 | The set of Cartesian produ... |
| eltx 22627 | A set in a product is open... |
| txtop 22628 | The product of two topolog... |
| ptval 22629 | The value of the product t... |
| ptpjpre1 22630 | The preimage of a projecti... |
| elpt 22631 | Elementhood in the bases o... |
| elptr 22632 | A basic open set in the pr... |
| elptr2 22633 | A basic open set in the pr... |
| ptbasid 22634 | The base set of the produc... |
| ptuni2 22635 | The base set for the produ... |
| ptbasin 22636 | The basis for a product to... |
| ptbasin2 22637 | The basis for a product to... |
| ptbas 22638 | The basis for a product to... |
| ptpjpre2 22639 | The basis for a product to... |
| ptbasfi 22640 | The basis for the product ... |
| pttop 22641 | The product topology is a ... |
| ptopn 22642 | A basic open set in the pr... |
| ptopn2 22643 | A sub-basic open set in th... |
| xkotf 22644 | Functionality of function ... |
| xkobval 22645 | Alternative expression for... |
| xkoval 22646 | Value of the compact-open ... |
| xkotop 22647 | The compact-open topology ... |
| xkoopn 22648 | A basic open set of the co... |
| txtopi 22649 | The product of two topolog... |
| txtopon 22650 | The underlying set of the ... |
| txuni 22651 | The underlying set of the ... |
| txunii 22652 | The underlying set of the ... |
| ptuni 22653 | The base set for the produ... |
| ptunimpt 22654 | Base set of a product topo... |
| pttopon 22655 | The base set for the produ... |
| pttoponconst 22656 | The base set for a product... |
| ptuniconst 22657 | The base set for a product... |
| xkouni 22658 | The base set of the compac... |
| xkotopon 22659 | The base set of the compac... |
| ptval2 22660 | The value of the product t... |
| txopn 22661 | The product of two open se... |
| txcld 22662 | The product of two closed ... |
| txcls 22663 | Closure of a rectangle in ... |
| txss12 22664 | Subset property of the top... |
| txbasval 22665 | It is sufficient to consid... |
| neitx 22666 | The Cartesian product of t... |
| txcnpi 22667 | Continuity of a two-argume... |
| tx1cn 22668 | Continuity of the first pr... |
| tx2cn 22669 | Continuity of the second p... |
| ptpjcn 22670 | Continuity of a projection... |
| ptpjopn 22671 | The projection map is an o... |
| ptcld 22672 | A closed box in the produc... |
| ptcldmpt 22673 | A closed box in the produc... |
| ptclsg 22674 | The closure of a box in th... |
| ptcls 22675 | The closure of a box in th... |
| dfac14lem 22676 | Lemma for ~ dfac14 . By e... |
| dfac14 22677 | Theorem ~ ptcls is an equi... |
| xkoccn 22678 | The "constant function" fu... |
| txcnp 22679 | If two functions are conti... |
| ptcnplem 22680 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 22681 | If every projection of a f... |
| upxp 22682 | Universal property of the ... |
| txcnmpt 22683 | A map into the product of ... |
| uptx 22684 | Universal property of the ... |
| txcn 22685 | A map into the product of ... |
| ptcn 22686 | If every projection of a f... |
| prdstopn 22687 | Topology of a structure pr... |
| prdstps 22688 | A structure product of top... |
| pwstps 22689 | A structure power of a top... |
| txrest 22690 | The subspace of a topologi... |
| txdis 22691 | The topological product of... |
| txindislem 22692 | Lemma for ~ txindis . (Co... |
| txindis 22693 | The topological product of... |
| txdis1cn 22694 | A function is jointly cont... |
| txlly 22695 | If the property ` A ` is p... |
| txnlly 22696 | If the property ` A ` is p... |
| pthaus 22697 | The product of a collectio... |
| ptrescn 22698 | Restriction is a continuou... |
| txtube 22699 | The "tube lemma". If ` X ... |
| txcmplem1 22700 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 22701 | Lemma for ~ txcmp . (Cont... |
| txcmp 22702 | The topological product of... |
| txcmpb 22703 | The topological product of... |
| hausdiag 22704 | A topology is Hausdorff if... |
| hauseqlcld 22705 | In a Hausdorff topology, t... |
| txhaus 22706 | The topological product of... |
| txlm 22707 | Two sequences converge iff... |
| lmcn2 22708 | The image of a convergent ... |
| tx1stc 22709 | The topological product of... |
| tx2ndc 22710 | The topological product of... |
| txkgen 22711 | The topological product of... |
| xkohaus 22712 | If the codomain space is H... |
| xkoptsub 22713 | The compact-open topology ... |
| xkopt 22714 | The compact-open topology ... |
| xkopjcn 22715 | Continuity of a projection... |
| xkoco1cn 22716 | If ` F ` is a continuous f... |
| xkoco2cn 22717 | If ` F ` is a continuous f... |
| xkococnlem 22718 | Continuity of the composit... |
| xkococn 22719 | Continuity of the composit... |
| cnmptid 22720 | The identity function is c... |
| cnmptc 22721 | A constant function is con... |
| cnmpt11 22722 | The composition of continu... |
| cnmpt11f 22723 | The composition of continu... |
| cnmpt1t 22724 | The composition of continu... |
| cnmpt12f 22725 | The composition of continu... |
| cnmpt12 22726 | The composition of continu... |
| cnmpt1st 22727 | The projection onto the fi... |
| cnmpt2nd 22728 | The projection onto the se... |
| cnmpt2c 22729 | A constant function is con... |
| cnmpt21 22730 | The composition of continu... |
| cnmpt21f 22731 | The composition of continu... |
| cnmpt2t 22732 | The composition of continu... |
| cnmpt22 22733 | The composition of continu... |
| cnmpt22f 22734 | The composition of continu... |
| cnmpt1res 22735 | The restriction of a conti... |
| cnmpt2res 22736 | The restriction of a conti... |
| cnmptcom 22737 | The argument converse of a... |
| cnmptkc 22738 | The curried first projecti... |
| cnmptkp 22739 | The evaluation of the inne... |
| cnmptk1 22740 | The composition of a curri... |
| cnmpt1k 22741 | The composition of a one-a... |
| cnmptkk 22742 | The composition of two cur... |
| xkofvcn 22743 | Joint continuity of the fu... |
| cnmptk1p 22744 | The evaluation of a currie... |
| cnmptk2 22745 | The uncurrying of a currie... |
| xkoinjcn 22746 | Continuity of "injection",... |
| cnmpt2k 22747 | The currying of a two-argu... |
| txconn 22748 | The topological product of... |
| imasnopn 22749 | If a relation graph is ope... |
| imasncld 22750 | If a relation graph is clo... |
| imasncls 22751 | If a relation graph is clo... |
| qtopval 22754 | Value of the quotient topo... |
| qtopval2 22755 | Value of the quotient topo... |
| elqtop 22756 | Value of the quotient topo... |
| qtopres 22757 | The quotient topology is u... |
| qtoptop2 22758 | The quotient topology is a... |
| qtoptop 22759 | The quotient topology is a... |
| elqtop2 22760 | Value of the quotient topo... |
| qtopuni 22761 | The base set of the quotie... |
| elqtop3 22762 | Value of the quotient topo... |
| qtoptopon 22763 | The base set of the quotie... |
| qtopid 22764 | A quotient map is a contin... |
| idqtop 22765 | The quotient topology indu... |
| qtopcmplem 22766 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 22767 | A quotient of a compact sp... |
| qtopconn 22768 | A quotient of a connected ... |
| qtopkgen 22769 | A quotient of a compactly ... |
| basqtop 22770 | An injection maps bases to... |
| tgqtop 22771 | An injection maps generate... |
| qtopcld 22772 | The property of being a cl... |
| qtopcn 22773 | Universal property of a qu... |
| qtopss 22774 | A surjective continuous fu... |
| qtopeu 22775 | Universal property of the ... |
| qtoprest 22776 | If ` A ` is a saturated op... |
| qtopomap 22777 | If ` F ` is a surjective c... |
| qtopcmap 22778 | If ` F ` is a surjective c... |
| imastopn 22779 | The topology of an image s... |
| imastps 22780 | The image of a topological... |
| qustps 22781 | A quotient structure is a ... |
| kqfval 22782 | Value of the function appe... |
| kqfeq 22783 | Two points in the Kolmogor... |
| kqffn 22784 | The topological indistingu... |
| kqval 22785 | Value of the quotient topo... |
| kqtopon 22786 | The Kolmogorov quotient is... |
| kqid 22787 | The topological indistingu... |
| ist0-4 22788 | The topological indistingu... |
| kqfvima 22789 | When the image set is open... |
| kqsat 22790 | Any open set is saturated ... |
| kqdisj 22791 | A version of ~ imain for t... |
| kqcldsat 22792 | Any closed set is saturate... |
| kqopn 22793 | The topological indistingu... |
| kqcld 22794 | The topological indistingu... |
| kqt0lem 22795 | Lemma for ~ kqt0 . (Contr... |
| isr0 22796 | The property " ` J ` is an... |
| r0cld 22797 | The analogue of the T_1 ax... |
| regr1lem 22798 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 22799 | A Kolmogorov quotient of a... |
| kqreglem1 22800 | A Kolmogorov quotient of a... |
| kqreglem2 22801 | If the Kolmogorov quotient... |
| kqnrmlem1 22802 | A Kolmogorov quotient of a... |
| kqnrmlem2 22803 | If the Kolmogorov quotient... |
| kqtop 22804 | The Kolmogorov quotient is... |
| kqt0 22805 | The Kolmogorov quotient is... |
| kqf 22806 | The Kolmogorov quotient is... |
| r0sep 22807 | The separation property of... |
| nrmr0reg 22808 | A normal R_0 space is also... |
| regr1 22809 | A regular space is R_1, wh... |
| kqreg 22810 | The Kolmogorov quotient of... |
| kqnrm 22811 | The Kolmogorov quotient of... |
| hmeofn 22816 | The set of homeomorphisms ... |
| hmeofval 22817 | The set of all the homeomo... |
| ishmeo 22818 | The predicate F is a homeo... |
| hmeocn 22819 | A homeomorphism is continu... |
| hmeocnvcn 22820 | The converse of a homeomor... |
| hmeocnv 22821 | The converse of a homeomor... |
| hmeof1o2 22822 | A homeomorphism is a 1-1-o... |
| hmeof1o 22823 | A homeomorphism is a 1-1-o... |
| hmeoima 22824 | The image of an open set b... |
| hmeoopn 22825 | Homeomorphisms preserve op... |
| hmeocld 22826 | Homeomorphisms preserve cl... |
| hmeocls 22827 | Homeomorphisms preserve cl... |
| hmeontr 22828 | Homeomorphisms preserve in... |
| hmeoimaf1o 22829 | The function mapping open ... |
| hmeores 22830 | The restriction of a homeo... |
| hmeoco 22831 | The composite of two homeo... |
| idhmeo 22832 | The identity function is a... |
| hmeocnvb 22833 | The converse of a homeomor... |
| hmeoqtop 22834 | A homeomorphism is a quoti... |
| hmph 22835 | Express the predicate ` J ... |
| hmphi 22836 | If there is a homeomorphis... |
| hmphtop 22837 | Reverse closure for the ho... |
| hmphtop1 22838 | The relation "being homeom... |
| hmphtop2 22839 | The relation "being homeom... |
| hmphref 22840 | "Is homeomorphic to" is re... |
| hmphsym 22841 | "Is homeomorphic to" is sy... |
| hmphtr 22842 | "Is homeomorphic to" is tr... |
| hmpher 22843 | "Is homeomorphic to" is an... |
| hmphen 22844 | Homeomorphisms preserve th... |
| hmphsymb 22845 | "Is homeomorphic to" is sy... |
| haushmphlem 22846 | Lemma for ~ haushmph and s... |
| cmphmph 22847 | Compactness is a topologic... |
| connhmph 22848 | Connectedness is a topolog... |
| t0hmph 22849 | T_0 is a topological prope... |
| t1hmph 22850 | T_1 is a topological prope... |
| haushmph 22851 | Hausdorff-ness is a topolo... |
| reghmph 22852 | Regularity is a topologica... |
| nrmhmph 22853 | Normality is a topological... |
| hmph0 22854 | A topology homeomorphic to... |
| hmphdis 22855 | Homeomorphisms preserve to... |
| hmphindis 22856 | Homeomorphisms preserve to... |
| indishmph 22857 | Equinumerous sets equipped... |
| hmphen2 22858 | Homeomorphisms preserve th... |
| cmphaushmeo 22859 | A continuous bijection fro... |
| ordthmeolem 22860 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 22861 | An order isomorphism is a ... |
| txhmeo 22862 | Lift a pair of homeomorphi... |
| txswaphmeolem 22863 | Show inverse for the "swap... |
| txswaphmeo 22864 | There is a homeomorphism f... |
| pt1hmeo 22865 | The canonical homeomorphis... |
| ptuncnv 22866 | Exhibit the converse funct... |
| ptunhmeo 22867 | Define a homeomorphism fro... |
| xpstopnlem1 22868 | The function ` F ` used in... |
| xpstps 22869 | A binary product of topolo... |
| xpstopnlem2 22870 | Lemma for ~ xpstopn . (Co... |
| xpstopn 22871 | The topology on a binary p... |
| ptcmpfi 22872 | A topological product of f... |
| xkocnv 22873 | The inverse of the "curryi... |
| xkohmeo 22874 | The Exponential Law for to... |
| qtopf1 22875 | If a quotient map is injec... |
| qtophmeo 22876 | If two functions on a base... |
| t0kq 22877 | A topological space is T_0... |
| kqhmph 22878 | A topological space is T_0... |
| ist1-5lem 22879 | Lemma for ~ ist1-5 and sim... |
| t1r0 22880 | A T_1 space is R_0. That ... |
| ist1-5 22881 | A topological space is T_1... |
| ishaus3 22882 | A topological space is Hau... |
| nrmreg 22883 | A normal T_1 space is regu... |
| reghaus 22884 | A regular T_0 space is Hau... |
| nrmhaus 22885 | A T_1 normal space is Haus... |
| elmptrab 22886 | Membership in a one-parame... |
| elmptrab2 22887 | Membership in a one-parame... |
| isfbas 22888 | The predicate " ` F ` is a... |
| fbasne0 22889 | There are no empty filter ... |
| 0nelfb 22890 | No filter base contains th... |
| fbsspw 22891 | A filter base on a set is ... |
| fbelss 22892 | An element of the filter b... |
| fbdmn0 22893 | The domain of a filter bas... |
| isfbas2 22894 | The predicate " ` F ` is a... |
| fbasssin 22895 | A filter base contains sub... |
| fbssfi 22896 | A filter base contains sub... |
| fbssint 22897 | A filter base contains sub... |
| fbncp 22898 | A filter base does not con... |
| fbun 22899 | A necessary and sufficient... |
| fbfinnfr 22900 | No filter base containing ... |
| opnfbas 22901 | The collection of open sup... |
| trfbas2 22902 | Conditions for the trace o... |
| trfbas 22903 | Conditions for the trace o... |
| isfil 22906 | The predicate "is a filter... |
| filfbas 22907 | A filter is a filter base.... |
| 0nelfil 22908 | The empty set doesn't belo... |
| fileln0 22909 | An element of a filter is ... |
| filsspw 22910 | A filter is a subset of th... |
| filelss 22911 | An element of a filter is ... |
| filss 22912 | A filter is closed under t... |
| filin 22913 | A filter is closed under t... |
| filtop 22914 | The underlying set belongs... |
| isfil2 22915 | Derive the standard axioms... |
| isfildlem 22916 | Lemma for ~ isfild . (Con... |
| isfild 22917 | Sufficient condition for a... |
| filfi 22918 | A filter is closed under t... |
| filinn0 22919 | The intersection of two el... |
| filintn0 22920 | A filter has the finite in... |
| filn0 22921 | The empty set is not a fil... |
| infil 22922 | The intersection of two fi... |
| snfil 22923 | A singleton is a filter. ... |
| fbasweak 22924 | A filter base on any set i... |
| snfbas 22925 | Condition for a singleton ... |
| fsubbas 22926 | A condition for a set to g... |
| fbasfip 22927 | A filter base has the fini... |
| fbunfip 22928 | A helpful lemma for showin... |
| fgval 22929 | The filter generating clas... |
| elfg 22930 | A condition for elements o... |
| ssfg 22931 | A filter base is a subset ... |
| fgss 22932 | A bigger base generates a ... |
| fgss2 22933 | A condition for a filter t... |
| fgfil 22934 | A filter generates itself.... |
| elfilss 22935 | An element belongs to a fi... |
| filfinnfr 22936 | No filter containing a fin... |
| fgcl 22937 | A generated filter is a fi... |
| fgabs 22938 | Absorption law for filter ... |
| neifil 22939 | The neighborhoods of a non... |
| filunibas 22940 | Recover the base set from ... |
| filunirn 22941 | Two ways to express a filt... |
| filconn 22942 | A filter gives rise to a c... |
| fbasrn 22943 | Given a filter on a domain... |
| filuni 22944 | The union of a nonempty se... |
| trfil1 22945 | Conditions for the trace o... |
| trfil2 22946 | Conditions for the trace o... |
| trfil3 22947 | Conditions for the trace o... |
| trfilss 22948 | If ` A ` is a member of th... |
| fgtr 22949 | If ` A ` is a member of th... |
| trfg 22950 | The trace operation and th... |
| trnei 22951 | The trace, over a set ` A ... |
| cfinfil 22952 | Relative complements of th... |
| csdfil 22953 | The set of all elements wh... |
| supfil 22954 | The supersets of a nonempt... |
| zfbas 22955 | The set of upper sets of i... |
| uzrest 22956 | The restriction of the set... |
| uzfbas 22957 | The set of upper sets of i... |
| isufil 22962 | The property of being an u... |
| ufilfil 22963 | An ultrafilter is a filter... |
| ufilss 22964 | For any subset of the base... |
| ufilb 22965 | The complement is in an ul... |
| ufilmax 22966 | Any filter finer than an u... |
| isufil2 22967 | The maximal property of an... |
| ufprim 22968 | An ultrafilter is a prime ... |
| trufil 22969 | Conditions for the trace o... |
| filssufilg 22970 | A filter is contained in s... |
| filssufil 22971 | A filter is contained in s... |
| isufl 22972 | Define the (strong) ultraf... |
| ufli 22973 | Property of a set that sat... |
| numufl 22974 | Consequence of ~ filssufil... |
| fiufl 22975 | A finite set satisfies the... |
| acufl 22976 | The axiom of choice implie... |
| ssufl 22977 | If ` Y ` is a subset of ` ... |
| ufileu 22978 | If the ultrafilter contain... |
| filufint 22979 | A filter is equal to the i... |
| uffix 22980 | Lemma for ~ fixufil and ~ ... |
| fixufil 22981 | The condition describing a... |
| uffixfr 22982 | An ultrafilter is either f... |
| uffix2 22983 | A classification of fixed ... |
| uffixsn 22984 | The singleton of the gener... |
| ufildom1 22985 | An ultrafilter is generate... |
| uffinfix 22986 | An ultrafilter containing ... |
| cfinufil 22987 | An ultrafilter is free iff... |
| ufinffr 22988 | An infinite subset is cont... |
| ufilen 22989 | Any infinite set has an ul... |
| ufildr 22990 | An ultrafilter gives rise ... |
| fin1aufil 22991 | There are no definable fre... |
| fmval 23002 | Introduce a function that ... |
| fmfil 23003 | A mapping filter is a filt... |
| fmf 23004 | Pushing-forward via a func... |
| fmss 23005 | A finer filter produces a ... |
| elfm 23006 | An element of a mapping fi... |
| elfm2 23007 | An element of a mapping fi... |
| fmfg 23008 | The image filter of a filt... |
| elfm3 23009 | An alternate formulation o... |
| imaelfm 23010 | An image of a filter eleme... |
| rnelfmlem 23011 | Lemma for ~ rnelfm . (Con... |
| rnelfm 23012 | A condition for a filter t... |
| fmfnfmlem1 23013 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 23014 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 23015 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 23016 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 23017 | A filter finer than an ima... |
| fmufil 23018 | An image filter of an ultr... |
| fmid 23019 | The filter map applied to ... |
| fmco 23020 | Composition of image filte... |
| ufldom 23021 | The ultrafilter lemma prop... |
| flimval 23022 | The set of limit points of... |
| elflim2 23023 | The predicate "is a limit ... |
| flimtop 23024 | Reverse closure for the li... |
| flimneiss 23025 | A filter contains the neig... |
| flimnei 23026 | A filter contains all of t... |
| flimelbas 23027 | A limit point of a filter ... |
| flimfil 23028 | Reverse closure for the li... |
| flimtopon 23029 | Reverse closure for the li... |
| elflim 23030 | The predicate "is a limit ... |
| flimss2 23031 | A limit point of a filter ... |
| flimss1 23032 | A limit point of a filter ... |
| neiflim 23033 | A point is a limit point o... |
| flimopn 23034 | The condition for being a ... |
| fbflim 23035 | A condition for a filter t... |
| fbflim2 23036 | A condition for a filter b... |
| flimclsi 23037 | The convergent points of a... |
| hausflimlem 23038 | If ` A ` and ` B ` are bot... |
| hausflimi 23039 | One direction of ~ hausfli... |
| hausflim 23040 | A condition for a topology... |
| flimcf 23041 | Fineness is properly chara... |
| flimrest 23042 | The set of limit points in... |
| flimclslem 23043 | Lemma for ~ flimcls . (Co... |
| flimcls 23044 | Closure in terms of filter... |
| flimsncls 23045 | If ` A ` is a limit point ... |
| hauspwpwf1 23046 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 23047 | If ` X ` is a Hausdorff sp... |
| flffval 23048 | Given a topology and a fil... |
| flfval 23049 | Given a function from a fi... |
| flfnei 23050 | The property of being a li... |
| flfneii 23051 | A neighborhood of a limit ... |
| isflf 23052 | The property of being a li... |
| flfelbas 23053 | A limit point of a functio... |
| flffbas 23054 | Limit points of a function... |
| flftg 23055 | Limit points of a function... |
| hausflf 23056 | If a function has its valu... |
| hausflf2 23057 | If a convergent function h... |
| cnpflfi 23058 | Forward direction of ~ cnp... |
| cnpflf2 23059 | ` F ` is continuous at poi... |
| cnpflf 23060 | Continuity of a function a... |
| cnflf 23061 | A function is continuous i... |
| cnflf2 23062 | A function is continuous i... |
| flfcnp 23063 | A continuous function pres... |
| lmflf 23064 | The topological limit rela... |
| txflf 23065 | Two sequences converge in ... |
| flfcnp2 23066 | The image of a convergent ... |
| fclsval 23067 | The set of all cluster poi... |
| isfcls 23068 | A cluster point of a filte... |
| fclsfil 23069 | Reverse closure for the cl... |
| fclstop 23070 | Reverse closure for the cl... |
| fclstopon 23071 | Reverse closure for the cl... |
| isfcls2 23072 | A cluster point of a filte... |
| fclsopn 23073 | Write the cluster point co... |
| fclsopni 23074 | An open neighborhood of a ... |
| fclselbas 23075 | A cluster point is in the ... |
| fclsneii 23076 | A neighborhood of a cluste... |
| fclssscls 23077 | The set of cluster points ... |
| fclsnei 23078 | Cluster points in terms of... |
| supnfcls 23079 | The filter of supersets of... |
| fclsbas 23080 | Cluster points in terms of... |
| fclsss1 23081 | A finer topology has fewer... |
| fclsss2 23082 | A finer filter has fewer c... |
| fclsrest 23083 | The set of cluster points ... |
| fclscf 23084 | Characterization of finene... |
| flimfcls 23085 | A limit point is a cluster... |
| fclsfnflim 23086 | A filter clusters at a poi... |
| flimfnfcls 23087 | A filter converges to a po... |
| fclscmpi 23088 | Forward direction of ~ fcl... |
| fclscmp 23089 | A space is compact iff eve... |
| uffclsflim 23090 | The cluster points of an u... |
| ufilcmp 23091 | A space is compact iff eve... |
| fcfval 23092 | The set of cluster points ... |
| isfcf 23093 | The property of being a cl... |
| fcfnei 23094 | The property of being a cl... |
| fcfelbas 23095 | A cluster point of a funct... |
| fcfneii 23096 | A neighborhood of a cluste... |
| flfssfcf 23097 | A limit point of a functio... |
| uffcfflf 23098 | If the domain filter is an... |
| cnpfcfi 23099 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 23100 | A function ` F ` is contin... |
| cnfcf 23101 | Continuity of a function i... |
| flfcntr 23102 | A continuous function's va... |
| alexsublem 23103 | Lemma for ~ alexsub . (Co... |
| alexsub 23104 | The Alexander Subbase Theo... |
| alexsubb 23105 | Biconditional form of the ... |
| alexsubALTlem1 23106 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 23107 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 23108 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 23109 | Lemma for ~ alexsubALT . ... |
| alexsubALT 23110 | The Alexander Subbase Theo... |
| ptcmplem1 23111 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 23112 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 23113 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 23114 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 23115 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 23116 | Tychonoff's theorem: The ... |
| ptcmp 23117 | Tychonoff's theorem: The ... |
| cnextval 23120 | The function applying cont... |
| cnextfval 23121 | The continuous extension o... |
| cnextrel 23122 | In the general case, a con... |
| cnextfun 23123 | If the target space is Hau... |
| cnextfvval 23124 | The value of the continuou... |
| cnextf 23125 | Extension by continuity. ... |
| cnextcn 23126 | Extension by continuity. ... |
| cnextfres1 23127 | ` F ` and its extension by... |
| cnextfres 23128 | ` F ` and its extension by... |
| istmd 23133 | The predicate "is a topolo... |
| tmdmnd 23134 | A topological monoid is a ... |
| tmdtps 23135 | A topological monoid is a ... |
| istgp 23136 | The predicate "is a topolo... |
| tgpgrp 23137 | A topological group is a g... |
| tgptmd 23138 | A topological group is a t... |
| tgptps 23139 | A topological group is a t... |
| tmdtopon 23140 | The topology of a topologi... |
| tgptopon 23141 | The topology of a topologi... |
| tmdcn 23142 | In a topological monoid, t... |
| tgpcn 23143 | In a topological group, th... |
| tgpinv 23144 | In a topological group, th... |
| grpinvhmeo 23145 | The inverse function in a ... |
| cnmpt1plusg 23146 | Continuity of the group su... |
| cnmpt2plusg 23147 | Continuity of the group su... |
| tmdcn2 23148 | Write out the definition o... |
| tgpsubcn 23149 | In a topological group, th... |
| istgp2 23150 | A group with a topology is... |
| tmdmulg 23151 | In a topological monoid, t... |
| tgpmulg 23152 | In a topological group, th... |
| tgpmulg2 23153 | In a topological monoid, t... |
| tmdgsum 23154 | In a topological monoid, t... |
| tmdgsum2 23155 | For any neighborhood ` U `... |
| oppgtmd 23156 | The opposite of a topologi... |
| oppgtgp 23157 | The opposite of a topologi... |
| distgp 23158 | Any group equipped with th... |
| indistgp 23159 | Any group equipped with th... |
| efmndtmd 23160 | The monoid of endofunction... |
| tmdlactcn 23161 | The left group action of e... |
| tgplacthmeo 23162 | The left group action of e... |
| submtmd 23163 | A submonoid of a topologic... |
| subgtgp 23164 | A subgroup of a topologica... |
| symgtgp 23165 | The symmetric group is a t... |
| subgntr 23166 | A subgroup of a topologica... |
| opnsubg 23167 | An open subgroup of a topo... |
| clssubg 23168 | The closure of a subgroup ... |
| clsnsg 23169 | The closure of a normal su... |
| cldsubg 23170 | A subgroup of finite index... |
| tgpconncompeqg 23171 | The connected component co... |
| tgpconncomp 23172 | The identity component, th... |
| tgpconncompss 23173 | The identity component is ... |
| ghmcnp 23174 | A group homomorphism on to... |
| snclseqg 23175 | The coset of the closure o... |
| tgphaus 23176 | A topological group is Hau... |
| tgpt1 23177 | Hausdorff and T1 are equiv... |
| tgpt0 23178 | Hausdorff and T0 are equiv... |
| qustgpopn 23179 | A quotient map in a topolo... |
| qustgplem 23180 | Lemma for ~ qustgp . (Con... |
| qustgp 23181 | The quotient of a topologi... |
| qustgphaus 23182 | The quotient of a topologi... |
| prdstmdd 23183 | The product of a family of... |
| prdstgpd 23184 | The product of a family of... |
| tsmsfbas 23187 | The collection of all sets... |
| tsmslem1 23188 | The finite partial sums of... |
| tsmsval2 23189 | Definition of the topologi... |
| tsmsval 23190 | Definition of the topologi... |
| tsmspropd 23191 | The group sum depends only... |
| eltsms 23192 | The property of being a su... |
| tsmsi 23193 | The property of being a su... |
| tsmscl 23194 | A sum in a topological gro... |
| haustsms 23195 | In a Hausdorff topological... |
| haustsms2 23196 | In a Hausdorff topological... |
| tsmscls 23197 | One half of ~ tgptsmscls ,... |
| tsmsgsum 23198 | The convergent points of a... |
| tsmsid 23199 | If a sum is finite, the us... |
| haustsmsid 23200 | In a Hausdorff topological... |
| tsms0 23201 | The sum of zero is zero. ... |
| tsmssubm 23202 | Evaluate an infinite group... |
| tsmsres 23203 | Extend an infinite group s... |
| tsmsf1o 23204 | Re-index an infinite group... |
| tsmsmhm 23205 | Apply a continuous group h... |
| tsmsadd 23206 | The sum of two infinite gr... |
| tsmsinv 23207 | Inverse of an infinite gro... |
| tsmssub 23208 | The difference of two infi... |
| tgptsmscls 23209 | A sum in a topological gro... |
| tgptsmscld 23210 | The set of limit points to... |
| tsmssplit 23211 | Split a topological group ... |
| tsmsxplem1 23212 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 23213 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 23214 | Write a sum over a two-dim... |
| istrg 23223 | Express the predicate " ` ... |
| trgtmd 23224 | The multiplicative monoid ... |
| istdrg 23225 | Express the predicate " ` ... |
| tdrgunit 23226 | The unit group of a topolo... |
| trgtgp 23227 | A topological ring is a to... |
| trgtmd2 23228 | A topological ring is a to... |
| trgtps 23229 | A topological ring is a to... |
| trgring 23230 | A topological ring is a ri... |
| trggrp 23231 | A topological ring is a gr... |
| tdrgtrg 23232 | A topological division rin... |
| tdrgdrng 23233 | A topological division rin... |
| tdrgring 23234 | A topological division rin... |
| tdrgtmd 23235 | A topological division rin... |
| tdrgtps 23236 | A topological division rin... |
| istdrg2 23237 | A topological-ring divisio... |
| mulrcn 23238 | The functionalization of t... |
| invrcn2 23239 | The multiplicative inverse... |
| invrcn 23240 | The multiplicative inverse... |
| cnmpt1mulr 23241 | Continuity of ring multipl... |
| cnmpt2mulr 23242 | Continuity of ring multipl... |
| dvrcn 23243 | The division function is c... |
| istlm 23244 | The predicate " ` W ` is a... |
| vscacn 23245 | The scalar multiplication ... |
| tlmtmd 23246 | A topological module is a ... |
| tlmtps 23247 | A topological module is a ... |
| tlmlmod 23248 | A topological module is a ... |
| tlmtrg 23249 | The scalar ring of a topol... |
| tlmscatps 23250 | The scalar ring of a topol... |
| istvc 23251 | A topological vector space... |
| tvctdrg 23252 | The scalar field of a topo... |
| cnmpt1vsca 23253 | Continuity of scalar multi... |
| cnmpt2vsca 23254 | Continuity of scalar multi... |
| tlmtgp 23255 | A topological vector space... |
| tvctlm 23256 | A topological vector space... |
| tvclmod 23257 | A topological vector space... |
| tvclvec 23258 | A topological vector space... |
| ustfn 23261 | The defined uniform struct... |
| ustval 23262 | The class of all uniform s... |
| isust 23263 | The predicate " ` U ` is a... |
| ustssxp 23264 | Entourages are subsets of ... |
| ustssel 23265 | A uniform structure is upw... |
| ustbasel 23266 | The full set is always an ... |
| ustincl 23267 | A uniform structure is clo... |
| ustdiag 23268 | The diagonal set is includ... |
| ustinvel 23269 | If ` V ` is an entourage, ... |
| ustexhalf 23270 | For each entourage ` V ` t... |
| ustrel 23271 | The elements of uniform st... |
| ustfilxp 23272 | A uniform structure on a n... |
| ustne0 23273 | A uniform structure cannot... |
| ustssco 23274 | In an uniform structure, a... |
| ustexsym 23275 | In an uniform structure, f... |
| ustex2sym 23276 | In an uniform structure, f... |
| ustex3sym 23277 | In an uniform structure, f... |
| ustref 23278 | Any element of the base se... |
| ust0 23279 | The unique uniform structu... |
| ustn0 23280 | The empty set is not an un... |
| ustund 23281 | If two intersecting sets `... |
| ustelimasn 23282 | Any point ` A ` is near en... |
| ustneism 23283 | For a point ` A ` in ` X `... |
| elrnust 23284 | First direction for ~ ustb... |
| ustbas2 23285 | Second direction for ~ ust... |
| ustuni 23286 | The set union of a uniform... |
| ustbas 23287 | Recover the base of an uni... |
| ustimasn 23288 | Lemma for ~ ustuqtop . (C... |
| trust 23289 | The trace of a uniform str... |
| utopval 23292 | The topology induced by a ... |
| elutop 23293 | Open sets in the topology ... |
| utoptop 23294 | The topology induced by a ... |
| utopbas 23295 | The base of the topology i... |
| utoptopon 23296 | Topology induced by a unif... |
| restutop 23297 | Restriction of a topology ... |
| restutopopn 23298 | The restriction of the top... |
| ustuqtoplem 23299 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 23300 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 23301 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 23302 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 23303 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 23304 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 23305 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 23306 | For a given uniform struct... |
| utopsnneiplem 23307 | The neighborhoods of a poi... |
| utopsnneip 23308 | The neighborhoods of a poi... |
| utopsnnei 23309 | Images of singletons by en... |
| utop2nei 23310 | For any symmetrical entour... |
| utop3cls 23311 | Relation between a topolog... |
| utopreg 23312 | All Hausdorff uniform spac... |
| ussval 23319 | The uniform structure on u... |
| ussid 23320 | In case the base of the ` ... |
| isusp 23321 | The predicate ` W ` is a u... |
| ressuss 23322 | Value of the uniform struc... |
| ressust 23323 | The uniform structure of a... |
| ressusp 23324 | The restriction of a unifo... |
| tusval 23325 | The value of the uniform s... |
| tuslem 23326 | Lemma for ~ tusbas , ~ tus... |
| tuslemOLD 23327 | Obsolete proof of ~ tuslem... |
| tusbas 23328 | The base set of a construc... |
| tusunif 23329 | The uniform structure of a... |
| tususs 23330 | The uniform structure of a... |
| tustopn 23331 | The topology induced by a ... |
| tususp 23332 | A constructed uniform spac... |
| tustps 23333 | A constructed uniform spac... |
| uspreg 23334 | If a uniform space is Haus... |
| ucnval 23337 | The set of all uniformly c... |
| isucn 23338 | The predicate " ` F ` is a... |
| isucn2 23339 | The predicate " ` F ` is a... |
| ucnimalem 23340 | Reformulate the ` G ` func... |
| ucnima 23341 | An equivalent statement of... |
| ucnprima 23342 | The preimage by a uniforml... |
| iducn 23343 | The identity is uniformly ... |
| cstucnd 23344 | A constant function is uni... |
| ucncn 23345 | Uniform continuity implies... |
| iscfilu 23348 | The predicate " ` F ` is a... |
| cfilufbas 23349 | A Cauchy filter base is a ... |
| cfiluexsm 23350 | For a Cauchy filter base a... |
| fmucndlem 23351 | Lemma for ~ fmucnd . (Con... |
| fmucnd 23352 | The image of a Cauchy filt... |
| cfilufg 23353 | The filter generated by a ... |
| trcfilu 23354 | Condition for the trace of... |
| cfiluweak 23355 | A Cauchy filter base is al... |
| neipcfilu 23356 | In an uniform space, a nei... |
| iscusp 23359 | The predicate " ` W ` is a... |
| cuspusp 23360 | A complete uniform space i... |
| cuspcvg 23361 | In a complete uniform spac... |
| iscusp2 23362 | The predicate " ` W ` is a... |
| cnextucn 23363 | Extension by continuity. ... |
| ucnextcn 23364 | Extension by continuity. ... |
| ispsmet 23365 | Express the predicate " ` ... |
| psmetdmdm 23366 | Recover the base set from ... |
| psmetf 23367 | The distance function of a... |
| psmetcl 23368 | Closure of the distance fu... |
| psmet0 23369 | The distance function of a... |
| psmettri2 23370 | Triangle inequality for th... |
| psmetsym 23371 | The distance function of a... |
| psmettri 23372 | Triangle inequality for th... |
| psmetge0 23373 | The distance function of a... |
| psmetxrge0 23374 | The distance function of a... |
| psmetres2 23375 | Restriction of a pseudomet... |
| psmetlecl 23376 | Real closure of an extende... |
| distspace 23377 | A set ` X ` together with ... |
| ismet 23384 | Express the predicate " ` ... |
| isxmet 23385 | Express the predicate " ` ... |
| ismeti 23386 | Properties that determine ... |
| isxmetd 23387 | Properties that determine ... |
| isxmet2d 23388 | It is safe to only require... |
| metflem 23389 | Lemma for ~ metf and other... |
| xmetf 23390 | Mapping of the distance fu... |
| metf 23391 | Mapping of the distance fu... |
| xmetcl 23392 | Closure of the distance fu... |
| metcl 23393 | Closure of the distance fu... |
| ismet2 23394 | An extended metric is a me... |
| metxmet 23395 | A metric is an extended me... |
| xmetdmdm 23396 | Recover the base set from ... |
| metdmdm 23397 | Recover the base set from ... |
| xmetunirn 23398 | Two ways to express an ext... |
| xmeteq0 23399 | The value of an extended m... |
| meteq0 23400 | The value of a metric is z... |
| xmettri2 23401 | Triangle inequality for th... |
| mettri2 23402 | Triangle inequality for th... |
| xmet0 23403 | The distance function of a... |
| met0 23404 | The distance function of a... |
| xmetge0 23405 | The distance function of a... |
| metge0 23406 | The distance function of a... |
| xmetlecl 23407 | Real closure of an extende... |
| xmetsym 23408 | The distance function of a... |
| xmetpsmet 23409 | An extended metric is a ps... |
| xmettpos 23410 | The distance function of a... |
| metsym 23411 | The distance function of a... |
| xmettri 23412 | Triangle inequality for th... |
| mettri 23413 | Triangle inequality for th... |
| xmettri3 23414 | Triangle inequality for th... |
| mettri3 23415 | Triangle inequality for th... |
| xmetrtri 23416 | One half of the reverse tr... |
| xmetrtri2 23417 | The reverse triangle inequ... |
| metrtri 23418 | Reverse triangle inequalit... |
| xmetgt0 23419 | The distance function of a... |
| metgt0 23420 | The distance function of a... |
| metn0 23421 | A metric space is nonempty... |
| xmetres2 23422 | Restriction of an extended... |
| metreslem 23423 | Lemma for ~ metres . (Con... |
| metres2 23424 | Lemma for ~ metres . (Con... |
| xmetres 23425 | A restriction of an extend... |
| metres 23426 | A restriction of a metric ... |
| 0met 23427 | The empty metric. (Contri... |
| prdsdsf 23428 | The product metric is a fu... |
| prdsxmetlem 23429 | The product metric is an e... |
| prdsxmet 23430 | The product metric is an e... |
| prdsmet 23431 | The product metric is a me... |
| ressprdsds 23432 | Restriction of a product m... |
| resspwsds 23433 | Restriction of a power met... |
| imasdsf1olem 23434 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 23435 | The distance function is t... |
| imasf1oxmet 23436 | The image of an extended m... |
| imasf1omet 23437 | The image of a metric is a... |
| xpsdsfn 23438 | Closure of the metric in a... |
| xpsdsfn2 23439 | Closure of the metric in a... |
| xpsxmetlem 23440 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 23441 | A product metric of extend... |
| xpsdsval 23442 | Value of the metric in a b... |
| xpsmet 23443 | The direct product of two ... |
| blfvalps 23444 | The value of the ball func... |
| blfval 23445 | The value of the ball func... |
| blvalps 23446 | The ball around a point ` ... |
| blval 23447 | The ball around a point ` ... |
| elblps 23448 | Membership in a ball. (Co... |
| elbl 23449 | Membership in a ball. (Co... |
| elbl2ps 23450 | Membership in a ball. (Co... |
| elbl2 23451 | Membership in a ball. (Co... |
| elbl3ps 23452 | Membership in a ball, with... |
| elbl3 23453 | Membership in a ball, with... |
| blcomps 23454 | Commute the arguments to t... |
| blcom 23455 | Commute the arguments to t... |
| xblpnfps 23456 | The infinity ball in an ex... |
| xblpnf 23457 | The infinity ball in an ex... |
| blpnf 23458 | The infinity ball in a sta... |
| bldisj 23459 | Two balls are disjoint if ... |
| blgt0 23460 | A nonempty ball implies th... |
| bl2in 23461 | Two balls are disjoint if ... |
| xblss2ps 23462 | One ball is contained in a... |
| xblss2 23463 | One ball is contained in a... |
| blss2ps 23464 | One ball is contained in a... |
| blss2 23465 | One ball is contained in a... |
| blhalf 23466 | A ball of radius ` R / 2 `... |
| blfps 23467 | Mapping of a ball. (Contr... |
| blf 23468 | Mapping of a ball. (Contr... |
| blrnps 23469 | Membership in the range of... |
| blrn 23470 | Membership in the range of... |
| xblcntrps 23471 | A ball contains its center... |
| xblcntr 23472 | A ball contains its center... |
| blcntrps 23473 | A ball contains its center... |
| blcntr 23474 | A ball contains its center... |
| xbln0 23475 | A ball is nonempty iff the... |
| bln0 23476 | A ball is not empty. (Con... |
| blelrnps 23477 | A ball belongs to the set ... |
| blelrn 23478 | A ball belongs to the set ... |
| blssm 23479 | A ball is a subset of the ... |
| unirnblps 23480 | The union of the set of ba... |
| unirnbl 23481 | The union of the set of ba... |
| blin 23482 | The intersection of two ba... |
| ssblps 23483 | The size of a ball increas... |
| ssbl 23484 | The size of a ball increas... |
| blssps 23485 | Any point ` P ` in a ball ... |
| blss 23486 | Any point ` P ` in a ball ... |
| blssexps 23487 | Two ways to express the ex... |
| blssex 23488 | Two ways to express the ex... |
| ssblex 23489 | A nested ball exists whose... |
| blin2 23490 | Given any two balls and a ... |
| blbas 23491 | The balls of a metric spac... |
| blres 23492 | A ball in a restricted met... |
| xmeterval 23493 | Value of the "finitely sep... |
| xmeter 23494 | The "finitely separated" r... |
| xmetec 23495 | The equivalence classes un... |
| blssec 23496 | A ball centered at ` P ` i... |
| blpnfctr 23497 | The infinity ball in an ex... |
| xmetresbl 23498 | An extended metric restric... |
| mopnval 23499 | An open set is a subset of... |
| mopntopon 23500 | The set of open sets of a ... |
| mopntop 23501 | The set of open sets of a ... |
| mopnuni 23502 | The union of all open sets... |
| elmopn 23503 | The defining property of a... |
| mopnfss 23504 | The family of open sets of... |
| mopnm 23505 | The base set of a metric s... |
| elmopn2 23506 | A defining property of an ... |
| mopnss 23507 | An open set of a metric sp... |
| isxms 23508 | Express the predicate " ` ... |
| isxms2 23509 | Express the predicate " ` ... |
| isms 23510 | Express the predicate " ` ... |
| isms2 23511 | Express the predicate " ` ... |
| xmstopn 23512 | The topology component of ... |
| mstopn 23513 | The topology component of ... |
| xmstps 23514 | An extended metric space i... |
| msxms 23515 | A metric space is an exten... |
| mstps 23516 | A metric space is a topolo... |
| xmsxmet 23517 | The distance function, sui... |
| msmet 23518 | The distance function, sui... |
| msf 23519 | The distance function of a... |
| xmsxmet2 23520 | The distance function, sui... |
| msmet2 23521 | The distance function, sui... |
| mscl 23522 | Closure of the distance fu... |
| xmscl 23523 | Closure of the distance fu... |
| xmsge0 23524 | The distance function in a... |
| xmseq0 23525 | The distance between two p... |
| xmssym 23526 | The distance function in a... |
| xmstri2 23527 | Triangle inequality for th... |
| mstri2 23528 | Triangle inequality for th... |
| xmstri 23529 | Triangle inequality for th... |
| mstri 23530 | Triangle inequality for th... |
| xmstri3 23531 | Triangle inequality for th... |
| mstri3 23532 | Triangle inequality for th... |
| msrtri 23533 | Reverse triangle inequalit... |
| xmspropd 23534 | Property deduction for an ... |
| mspropd 23535 | Property deduction for a m... |
| setsmsbas 23536 | The base set of a construc... |
| setsmsds 23537 | The distance function of a... |
| setsmstset 23538 | The topology of a construc... |
| setsmstopn 23539 | The topology of a construc... |
| setsxms 23540 | The constructed metric spa... |
| setsms 23541 | The constructed metric spa... |
| tmsval 23542 | For any metric there is an... |
| tmslem 23543 | Lemma for ~ tmsbas , ~ tms... |
| tmslemOLD 23544 | Obsolete version of ~ tmsl... |
| tmsbas 23545 | The base set of a construc... |
| tmsds 23546 | The metric of a constructe... |
| tmstopn 23547 | The topology of a construc... |
| tmsxms 23548 | The constructed metric spa... |
| tmsms 23549 | The constructed metric spa... |
| imasf1obl 23550 | The image of a metric spac... |
| imasf1oxms 23551 | The image of a metric spac... |
| imasf1oms 23552 | The image of a metric spac... |
| prdsbl 23553 | A ball in the product metr... |
| mopni 23554 | An open set of a metric sp... |
| mopni2 23555 | An open set of a metric sp... |
| mopni3 23556 | An open set of a metric sp... |
| blssopn 23557 | The balls of a metric spac... |
| unimopn 23558 | The union of a collection ... |
| mopnin 23559 | The intersection of two op... |
| mopn0 23560 | The empty set is an open s... |
| rnblopn 23561 | A ball of a metric space i... |
| blopn 23562 | A ball of a metric space i... |
| neibl 23563 | The neighborhoods around a... |
| blnei 23564 | A ball around a point is a... |
| lpbl 23565 | Every ball around a limit ... |
| blsscls2 23566 | A smaller closed ball is c... |
| blcld 23567 | A "closed ball" in a metri... |
| blcls 23568 | The closure of an open bal... |
| blsscls 23569 | If two concentric balls ha... |
| metss 23570 | Two ways of saying that me... |
| metequiv 23571 | Two ways of saying that tw... |
| metequiv2 23572 | If there is a sequence of ... |
| metss2lem 23573 | Lemma for ~ metss2 . (Con... |
| metss2 23574 | If the metric ` D ` is "st... |
| comet 23575 | The composition of an exte... |
| stdbdmetval 23576 | Value of the standard boun... |
| stdbdxmet 23577 | The standard bounded metri... |
| stdbdmet 23578 | The standard bounded metri... |
| stdbdbl 23579 | The standard bounded metri... |
| stdbdmopn 23580 | The standard bounded metri... |
| mopnex 23581 | The topology generated by ... |
| methaus 23582 | The topology generated by ... |
| met1stc 23583 | The topology generated by ... |
| met2ndci 23584 | A separable metric space (... |
| met2ndc 23585 | A metric space is second-c... |
| metrest 23586 | Two alternate formulations... |
| ressxms 23587 | The restriction of a metri... |
| ressms 23588 | The restriction of a metri... |
| prdsmslem1 23589 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 23590 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 23591 | Lemma for ~ prdsxms . The... |
| prdsxms 23592 | The indexed product struct... |
| prdsms 23593 | The indexed product struct... |
| pwsxms 23594 | A power of an extended met... |
| pwsms 23595 | A power of a metric space ... |
| xpsxms 23596 | A binary product of metric... |
| xpsms 23597 | A binary product of metric... |
| tmsxps 23598 | Express the product of two... |
| tmsxpsmopn 23599 | Express the product of two... |
| tmsxpsval 23600 | Value of the product of tw... |
| tmsxpsval2 23601 | Value of the product of tw... |
| metcnp3 23602 | Two ways to express that `... |
| metcnp 23603 | Two ways to say a mapping ... |
| metcnp2 23604 | Two ways to say a mapping ... |
| metcn 23605 | Two ways to say a mapping ... |
| metcnpi 23606 | Epsilon-delta property of ... |
| metcnpi2 23607 | Epsilon-delta property of ... |
| metcnpi3 23608 | Epsilon-delta property of ... |
| txmetcnp 23609 | Continuity of a binary ope... |
| txmetcn 23610 | Continuity of a binary ope... |
| metuval 23611 | Value of the uniform struc... |
| metustel 23612 | Define a filter base ` F `... |
| metustss 23613 | Range of the elements of t... |
| metustrel 23614 | Elements of the filter bas... |
| metustto 23615 | Any two elements of the fi... |
| metustid 23616 | The identity diagonal is i... |
| metustsym 23617 | Elements of the filter bas... |
| metustexhalf 23618 | For any element ` A ` of t... |
| metustfbas 23619 | The filter base generated ... |
| metust 23620 | The uniform structure gene... |
| cfilucfil 23621 | Given a metric ` D ` and a... |
| metuust 23622 | The uniform structure gene... |
| cfilucfil2 23623 | Given a metric ` D ` and a... |
| blval2 23624 | The ball around a point ` ... |
| elbl4 23625 | Membership in a ball, alte... |
| metuel 23626 | Elementhood in the uniform... |
| metuel2 23627 | Elementhood in the uniform... |
| metustbl 23628 | The "section" image of an ... |
| psmetutop 23629 | The topology induced by a ... |
| xmetutop 23630 | The topology induced by a ... |
| xmsusp 23631 | If the uniform set of a me... |
| restmetu 23632 | The uniform structure gene... |
| metucn 23633 | Uniform continuity in metr... |
| dscmet 23634 | The discrete metric on any... |
| dscopn 23635 | The discrete metric genera... |
| nrmmetd 23636 | Show that a group norm gen... |
| abvmet 23637 | An absolute value ` F ` ge... |
| nmfval 23650 | The value of the norm func... |
| nmval 23651 | The value of the norm as t... |
| nmfval0 23652 | The value of the norm func... |
| nmfval2 23653 | The value of the norm func... |
| nmval2 23654 | The value of the norm on a... |
| nmf2 23655 | The norm on a metric group... |
| nmpropd 23656 | Weak property deduction fo... |
| nmpropd2 23657 | Strong property deduction ... |
| isngp 23658 | The property of being a no... |
| isngp2 23659 | The property of being a no... |
| isngp3 23660 | The property of being a no... |
| ngpgrp 23661 | A normed group is a group.... |
| ngpms 23662 | A normed group is a metric... |
| ngpxms 23663 | A normed group is an exten... |
| ngptps 23664 | A normed group is a topolo... |
| ngpmet 23665 | The (induced) metric of a ... |
| ngpds 23666 | Value of the distance func... |
| ngpdsr 23667 | Value of the distance func... |
| ngpds2 23668 | Write the distance between... |
| ngpds2r 23669 | Write the distance between... |
| ngpds3 23670 | Write the distance between... |
| ngpds3r 23671 | Write the distance between... |
| ngprcan 23672 | Cancel right addition insi... |
| ngplcan 23673 | Cancel left addition insid... |
| isngp4 23674 | Express the property of be... |
| ngpinvds 23675 | Two elements are the same ... |
| ngpsubcan 23676 | Cancel right subtraction i... |
| nmf 23677 | The norm on a normed group... |
| nmcl 23678 | The norm of a normed group... |
| nmge0 23679 | The norm of a normed group... |
| nmeq0 23680 | The identity is the only e... |
| nmne0 23681 | The norm of a nonzero elem... |
| nmrpcl 23682 | The norm of a nonzero elem... |
| nminv 23683 | The norm of a negated elem... |
| nmmtri 23684 | The triangle inequality fo... |
| nmsub 23685 | The norm of the difference... |
| nmrtri 23686 | Reverse triangle inequalit... |
| nm2dif 23687 | Inequality for the differe... |
| nmtri 23688 | The triangle inequality fo... |
| nmtri2 23689 | Triangle inequality for th... |
| ngpi 23690 | The properties of a normed... |
| nm0 23691 | Norm of the identity eleme... |
| nmgt0 23692 | The norm of a nonzero elem... |
| sgrim 23693 | The induced metric on a su... |
| sgrimval 23694 | The induced metric on a su... |
| subgnm 23695 | The norm in a subgroup. (... |
| subgnm2 23696 | A substructure assigns the... |
| subgngp 23697 | A normed group restricted ... |
| ngptgp 23698 | A normed abelian group is ... |
| ngppropd 23699 | Property deduction for a n... |
| reldmtng 23700 | The function ` toNrmGrp ` ... |
| tngval 23701 | Value of the function whic... |
| tnglem 23702 | Lemma for ~ tngbas and sim... |
| tnglemOLD 23703 | Obsolete version of ~ tngl... |
| tngbas 23704 | The base set of a structur... |
| tngbasOLD 23705 | Obsolete proof of ~ tngbas... |
| tngplusg 23706 | The group addition of a st... |
| tngplusgOLD 23707 | Obsolete proof of ~ tngplu... |
| tng0 23708 | The group identity of a st... |
| tngmulr 23709 | The ring multiplication of... |
| tngmulrOLD 23710 | Obsolete proof of ~ tngmul... |
| tngsca 23711 | The scalar ring of a struc... |
| tngscaOLD 23712 | Obsolete proof of ~ tngsca... |
| tngvsca 23713 | The scalar multiplication ... |
| tngvscaOLD 23714 | Obsolete proof of ~ tngvsc... |
| tngip 23715 | The inner product operatio... |
| tngipOLD 23716 | Obsolete proof of ~ tngip ... |
| tngds 23717 | The metric function of a s... |
| tngdsOLD 23718 | Obsolete proof of ~ tngds ... |
| tngtset 23719 | The topology generated by ... |
| tngtopn 23720 | The topology generated by ... |
| tngnm 23721 | The topology generated by ... |
| tngngp2 23722 | A norm turns a group into ... |
| tngngpd 23723 | Derive the axioms for a no... |
| tngngp 23724 | Derive the axioms for a no... |
| tnggrpr 23725 | If a structure equipped wi... |
| tngngp3 23726 | Alternate definition of a ... |
| nrmtngdist 23727 | The augmentation of a norm... |
| nrmtngnrm 23728 | The augmentation of a norm... |
| tngngpim 23729 | The induced metric of a no... |
| isnrg 23730 | A normed ring is a ring wi... |
| nrgabv 23731 | The norm of a normed ring ... |
| nrgngp 23732 | A normed ring is a normed ... |
| nrgring 23733 | A normed ring is a ring. ... |
| nmmul 23734 | The norm of a product in a... |
| nrgdsdi 23735 | Distribute a distance calc... |
| nrgdsdir 23736 | Distribute a distance calc... |
| nm1 23737 | The norm of one in a nonze... |
| unitnmn0 23738 | The norm of a unit is nonz... |
| nminvr 23739 | The norm of an inverse in ... |
| nmdvr 23740 | The norm of a division in ... |
| nrgdomn 23741 | A nonzero normed ring is a... |
| nrgtgp 23742 | A normed ring is a topolog... |
| subrgnrg 23743 | A normed ring restricted t... |
| tngnrg 23744 | Given any absolute value o... |
| isnlm 23745 | A normed (left) module is ... |
| nmvs 23746 | Defining property of a nor... |
| nlmngp 23747 | A normed module is a norme... |
| nlmlmod 23748 | A normed module is a left ... |
| nlmnrg 23749 | The scalar component of a ... |
| nlmngp2 23750 | The scalar component of a ... |
| nlmdsdi 23751 | Distribute a distance calc... |
| nlmdsdir 23752 | Distribute a distance calc... |
| nlmmul0or 23753 | If a scalar product is zer... |
| sranlm 23754 | The subring algebra over a... |
| nlmvscnlem2 23755 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 23756 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 23757 | The scalar multiplication ... |
| rlmnlm 23758 | The ring module over a nor... |
| rlmnm 23759 | The norm function in the r... |
| nrgtrg 23760 | A normed ring is a topolog... |
| nrginvrcnlem 23761 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 23762 | The ring inverse function ... |
| nrgtdrg 23763 | A normed division ring is ... |
| nlmtlm 23764 | A normed module is a topol... |
| isnvc 23765 | A normed vector space is j... |
| nvcnlm 23766 | A normed vector space is a... |
| nvclvec 23767 | A normed vector space is a... |
| nvclmod 23768 | A normed vector space is a... |
| isnvc2 23769 | A normed vector space is j... |
| nvctvc 23770 | A normed vector space is a... |
| lssnlm 23771 | A subspace of a normed mod... |
| lssnvc 23772 | A subspace of a normed vec... |
| rlmnvc 23773 | The ring module over a nor... |
| ngpocelbl 23774 | Membership of an off-cente... |
| nmoffn 23781 | The function producing ope... |
| reldmnghm 23782 | Lemma for normed group hom... |
| reldmnmhm 23783 | Lemma for module homomorph... |
| nmofval 23784 | Value of the operator norm... |
| nmoval 23785 | Value of the operator norm... |
| nmogelb 23786 | Property of the operator n... |
| nmolb 23787 | Any upper bound on the val... |
| nmolb2d 23788 | Any upper bound on the val... |
| nmof 23789 | The operator norm is a fun... |
| nmocl 23790 | The operator norm of an op... |
| nmoge0 23791 | The operator norm of an op... |
| nghmfval 23792 | A normed group homomorphis... |
| isnghm 23793 | A normed group homomorphis... |
| isnghm2 23794 | A normed group homomorphis... |
| isnghm3 23795 | A normed group homomorphis... |
| bddnghm 23796 | A bounded group homomorphi... |
| nghmcl 23797 | A normed group homomorphis... |
| nmoi 23798 | The operator norm achieves... |
| nmoix 23799 | The operator norm is a bou... |
| nmoi2 23800 | The operator norm is a bou... |
| nmoleub 23801 | The operator norm, defined... |
| nghmrcl1 23802 | Reverse closure for a norm... |
| nghmrcl2 23803 | Reverse closure for a norm... |
| nghmghm 23804 | A normed group homomorphis... |
| nmo0 23805 | The operator norm of the z... |
| nmoeq0 23806 | The operator norm is zero ... |
| nmoco 23807 | An upper bound on the oper... |
| nghmco 23808 | The composition of normed ... |
| nmotri 23809 | Triangle inequality for th... |
| nghmplusg 23810 | The sum of two bounded lin... |
| 0nghm 23811 | The zero operator is a nor... |
| nmoid 23812 | The operator norm of the i... |
| idnghm 23813 | The identity operator is a... |
| nmods 23814 | Upper bound for the distan... |
| nghmcn 23815 | A normed group homomorphis... |
| isnmhm 23816 | A normed module homomorphi... |
| nmhmrcl1 23817 | Reverse closure for a norm... |
| nmhmrcl2 23818 | Reverse closure for a norm... |
| nmhmlmhm 23819 | A normed module homomorphi... |
| nmhmnghm 23820 | A normed module homomorphi... |
| nmhmghm 23821 | A normed module homomorphi... |
| isnmhm2 23822 | A normed module homomorphi... |
| nmhmcl 23823 | A normed module homomorphi... |
| idnmhm 23824 | The identity operator is a... |
| 0nmhm 23825 | The zero operator is a bou... |
| nmhmco 23826 | The composition of bounded... |
| nmhmplusg 23827 | The sum of two bounded lin... |
| qtopbaslem 23828 | The set of open intervals ... |
| qtopbas 23829 | The set of open intervals ... |
| retopbas 23830 | A basis for the standard t... |
| retop 23831 | The standard topology on t... |
| uniretop 23832 | The underlying set of the ... |
| retopon 23833 | The standard topology on t... |
| retps 23834 | The standard topological s... |
| iooretop 23835 | Open intervals are open se... |
| icccld 23836 | Closed intervals are close... |
| icopnfcld 23837 | Right-unbounded closed int... |
| iocmnfcld 23838 | Left-unbounded closed inte... |
| qdensere 23839 | ` QQ ` is dense in the sta... |
| cnmetdval 23840 | Value of the distance func... |
| cnmet 23841 | The absolute value metric ... |
| cnxmet 23842 | The absolute value metric ... |
| cnbl0 23843 | Two ways to write the open... |
| cnblcld 23844 | Two ways to write the clos... |
| cnfldms 23845 | The complex number field i... |
| cnfldxms 23846 | The complex number field i... |
| cnfldtps 23847 | The complex number field i... |
| cnfldnm 23848 | The norm of the field of c... |
| cnngp 23849 | The complex numbers form a... |
| cnnrg 23850 | The complex numbers form a... |
| cnfldtopn 23851 | The topology of the comple... |
| cnfldtopon 23852 | The topology of the comple... |
| cnfldtop 23853 | The topology of the comple... |
| cnfldhaus 23854 | The topology of the comple... |
| unicntop 23855 | The underlying set of the ... |
| cnopn 23856 | The set of complex numbers... |
| zringnrg 23857 | The ring of integers is a ... |
| remetdval 23858 | Value of the distance func... |
| remet 23859 | The absolute value metric ... |
| rexmet 23860 | The absolute value metric ... |
| bl2ioo 23861 | A ball in terms of an open... |
| ioo2bl 23862 | An open interval of reals ... |
| ioo2blex 23863 | An open interval of reals ... |
| blssioo 23864 | The balls of the standard ... |
| tgioo 23865 | The topology generated by ... |
| qdensere2 23866 | ` QQ ` is dense in ` RR ` ... |
| blcvx 23867 | An open ball in the comple... |
| rehaus 23868 | The standard topology on t... |
| tgqioo 23869 | The topology generated by ... |
| re2ndc 23870 | The standard topology on t... |
| resubmet 23871 | The subspace topology indu... |
| tgioo2 23872 | The standard topology on t... |
| rerest 23873 | The subspace topology indu... |
| tgioo3 23874 | The standard topology on t... |
| xrtgioo 23875 | The topology on the extend... |
| xrrest 23876 | The subspace topology indu... |
| xrrest2 23877 | The subspace topology indu... |
| xrsxmet 23878 | The metric on the extended... |
| xrsdsre 23879 | The metric on the extended... |
| xrsblre 23880 | Any ball of the metric of ... |
| xrsmopn 23881 | The metric on the extended... |
| zcld 23882 | The integers are a closed ... |
| recld2 23883 | The real numbers are a clo... |
| zcld2 23884 | The integers are a closed ... |
| zdis 23885 | The integers are a discret... |
| sszcld 23886 | Every subset of the intege... |
| reperflem 23887 | A subset of the real numbe... |
| reperf 23888 | The real numbers are a per... |
| cnperf 23889 | The complex numbers are a ... |
| iccntr 23890 | The interior of a closed i... |
| icccmplem1 23891 | Lemma for ~ icccmp . (Con... |
| icccmplem2 23892 | Lemma for ~ icccmp . (Con... |
| icccmplem3 23893 | Lemma for ~ icccmp . (Con... |
| icccmp 23894 | A closed interval in ` RR ... |
| reconnlem1 23895 | Lemma for ~ reconn . Conn... |
| reconnlem2 23896 | Lemma for ~ reconn . (Con... |
| reconn 23897 | A subset of the reals is c... |
| retopconn 23898 | Corollary of ~ reconn . T... |
| iccconn 23899 | A closed interval is conne... |
| opnreen 23900 | Every nonempty open set is... |
| rectbntr0 23901 | A countable subset of the ... |
| xrge0gsumle 23902 | A finite sum in the nonneg... |
| xrge0tsms 23903 | Any finite or infinite sum... |
| xrge0tsms2 23904 | Any finite or infinite sum... |
| metdcnlem 23905 | The metric function of a m... |
| xmetdcn2 23906 | The metric function of an ... |
| xmetdcn 23907 | The metric function of an ... |
| metdcn2 23908 | The metric function of a m... |
| metdcn 23909 | The metric function of a m... |
| msdcn 23910 | The metric function of a m... |
| cnmpt1ds 23911 | Continuity of the metric f... |
| cnmpt2ds 23912 | Continuity of the metric f... |
| nmcn 23913 | The norm of a normed group... |
| ngnmcncn 23914 | The norm of a normed group... |
| abscn 23915 | The absolute value functio... |
| metdsval 23916 | Value of the "distance to ... |
| metdsf 23917 | The distance from a point ... |
| metdsge 23918 | The distance from the poin... |
| metds0 23919 | If a point is in a set, it... |
| metdstri 23920 | A generalization of the tr... |
| metdsle 23921 | The distance from a point ... |
| metdsre 23922 | The distance from a point ... |
| metdseq0 23923 | The distance from a point ... |
| metdscnlem 23924 | Lemma for ~ metdscn . (Co... |
| metdscn 23925 | The function ` F ` which g... |
| metdscn2 23926 | The function ` F ` which g... |
| metnrmlem1a 23927 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 23928 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 23929 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 23930 | Lemma for ~ metnrm . (Con... |
| metnrm 23931 | A metric space is normal. ... |
| metreg 23932 | A metric space is regular.... |
| addcnlem 23933 | Lemma for ~ addcn , ~ subc... |
| addcn 23934 | Complex number addition is... |
| subcn 23935 | Complex number subtraction... |
| mulcn 23936 | Complex number multiplicat... |
| divcn 23937 | Complex number division is... |
| cnfldtgp 23938 | The complex numbers form a... |
| fsumcn 23939 | A finite sum of functions ... |
| fsum2cn 23940 | Version of ~ fsumcn for tw... |
| expcn 23941 | The power function on comp... |
| divccn 23942 | Division by a nonzero cons... |
| sqcn 23943 | The square function on com... |
| iitopon 23948 | The unit interval is a top... |
| iitop 23949 | The unit interval is a top... |
| iiuni 23950 | The base set of the unit i... |
| dfii2 23951 | Alternate definition of th... |
| dfii3 23952 | Alternate definition of th... |
| dfii4 23953 | Alternate definition of th... |
| dfii5 23954 | The unit interval expresse... |
| iicmp 23955 | The unit interval is compa... |
| iiconn 23956 | The unit interval is conne... |
| cncfval 23957 | The value of the continuou... |
| elcncf 23958 | Membership in the set of c... |
| elcncf2 23959 | Version of ~ elcncf with a... |
| cncfrss 23960 | Reverse closure of the con... |
| cncfrss2 23961 | Reverse closure of the con... |
| cncff 23962 | A continuous complex funct... |
| cncfi 23963 | Defining property of a con... |
| elcncf1di 23964 | Membership in the set of c... |
| elcncf1ii 23965 | Membership in the set of c... |
| rescncf 23966 | A continuous complex funct... |
| cncffvrn 23967 | Change the codomain of a c... |
| cncfss 23968 | The set of continuous func... |
| climcncf 23969 | Image of a limit under a c... |
| abscncf 23970 | Absolute value is continuo... |
| recncf 23971 | Real part is continuous. ... |
| imcncf 23972 | Imaginary part is continuo... |
| cjcncf 23973 | Complex conjugate is conti... |
| mulc1cncf 23974 | Multiplication by a consta... |
| divccncf 23975 | Division by a constant is ... |
| cncfco 23976 | The composition of two con... |
| cncfcompt2 23977 | Composition of continuous ... |
| cncfmet 23978 | Relate complex function co... |
| cncfcn 23979 | Relate complex function co... |
| cncfcn1 23980 | Relate complex function co... |
| cncfmptc 23981 | A constant function is a c... |
| cncfmptid 23982 | The identity function is a... |
| cncfmpt1f 23983 | Composition of continuous ... |
| cncfmpt2f 23984 | Composition of continuous ... |
| cncfmpt2ss 23985 | Composition of continuous ... |
| addccncf 23986 | Adding a constant is a con... |
| idcncf 23987 | The identity function is a... |
| sub1cncf 23988 | Subtracting a constant is ... |
| sub2cncf 23989 | Subtraction from a constan... |
| cdivcncf 23990 | Division with a constant n... |
| negcncf 23991 | The negative function is c... |
| negfcncf 23992 | The negative of a continuo... |
| abscncfALT 23993 | Absolute value is continuo... |
| cncfcnvcn 23994 | Rewrite ~ cmphaushmeo for ... |
| expcncf 23995 | The power function on comp... |
| cnmptre 23996 | Lemma for ~ iirevcn and re... |
| cnmpopc 23997 | Piecewise definition of a ... |
| iirev 23998 | Reverse the unit interval.... |
| iirevcn 23999 | The reversion function is ... |
| iihalf1 24000 | Map the first half of ` II... |
| iihalf1cn 24001 | The first half function is... |
| iihalf2 24002 | Map the second half of ` I... |
| iihalf2cn 24003 | The second half function i... |
| elii1 24004 | Divide the unit interval i... |
| elii2 24005 | Divide the unit interval i... |
| iimulcl 24006 | The unit interval is close... |
| iimulcn 24007 | Multiplication is a contin... |
| icoopnst 24008 | A half-open interval start... |
| iocopnst 24009 | A half-open interval endin... |
| icchmeo 24010 | The natural bijection from... |
| icopnfcnv 24011 | Define a bijection from ` ... |
| icopnfhmeo 24012 | The defined bijection from... |
| iccpnfcnv 24013 | Define a bijection from ` ... |
| iccpnfhmeo 24014 | The defined bijection from... |
| xrhmeo 24015 | The bijection from ` [ -u ... |
| xrhmph 24016 | The extended reals are hom... |
| xrcmp 24017 | The topology of the extend... |
| xrconn 24018 | The topology of the extend... |
| icccvx 24019 | A linear combination of tw... |
| oprpiece1res1 24020 | Restriction to the first p... |
| oprpiece1res2 24021 | Restriction to the second ... |
| cnrehmeo 24022 | The canonical bijection fr... |
| cnheiborlem 24023 | Lemma for ~ cnheibor . (C... |
| cnheibor 24024 | Heine-Borel theorem for co... |
| cnllycmp 24025 | The topology on the comple... |
| rellycmp 24026 | The topology on the reals ... |
| bndth 24027 | The Boundedness Theorem. ... |
| evth 24028 | The Extreme Value Theorem.... |
| evth2 24029 | The Extreme Value Theorem,... |
| lebnumlem1 24030 | Lemma for ~ lebnum . The ... |
| lebnumlem2 24031 | Lemma for ~ lebnum . As a... |
| lebnumlem3 24032 | Lemma for ~ lebnum . By t... |
| lebnum 24033 | The Lebesgue number lemma,... |
| xlebnum 24034 | Generalize ~ lebnum to ext... |
| lebnumii 24035 | Specialize the Lebesgue nu... |
| ishtpy 24041 | Membership in the class of... |
| htpycn 24042 | A homotopy is a continuous... |
| htpyi 24043 | A homotopy evaluated at it... |
| ishtpyd 24044 | Deduction for membership i... |
| htpycom 24045 | Given a homotopy from ` F ... |
| htpyid 24046 | A homotopy from a function... |
| htpyco1 24047 | Compose a homotopy with a ... |
| htpyco2 24048 | Compose a homotopy with a ... |
| htpycc 24049 | Concatenate two homotopies... |
| isphtpy 24050 | Membership in the class of... |
| phtpyhtpy 24051 | A path homotopy is a homot... |
| phtpycn 24052 | A path homotopy is a conti... |
| phtpyi 24053 | Membership in the class of... |
| phtpy01 24054 | Two path-homotopic paths h... |
| isphtpyd 24055 | Deduction for membership i... |
| isphtpy2d 24056 | Deduction for membership i... |
| phtpycom 24057 | Given a homotopy from ` F ... |
| phtpyid 24058 | A homotopy from a path to ... |
| phtpyco2 24059 | Compose a path homotopy wi... |
| phtpycc 24060 | Concatenate two path homot... |
| phtpcrel 24062 | The path homotopy relation... |
| isphtpc 24063 | The relation "is path homo... |
| phtpcer 24064 | Path homotopy is an equiva... |
| phtpc01 24065 | Path homotopic paths have ... |
| reparphti 24066 | Lemma for ~ reparpht . (C... |
| reparpht 24067 | Reparametrization lemma. ... |
| phtpcco2 24068 | Compose a path homotopy wi... |
| pcofval 24079 | The value of the path conc... |
| pcoval 24080 | The concatenation of two p... |
| pcovalg 24081 | Evaluate the concatenation... |
| pcoval1 24082 | Evaluate the concatenation... |
| pco0 24083 | The starting point of a pa... |
| pco1 24084 | The ending point of a path... |
| pcoval2 24085 | Evaluate the concatenation... |
| pcocn 24086 | The concatenation of two p... |
| copco 24087 | The composition of a conca... |
| pcohtpylem 24088 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 24089 | Homotopy invariance of pat... |
| pcoptcl 24090 | A constant function is a p... |
| pcopt 24091 | Concatenation with a point... |
| pcopt2 24092 | Concatenation with a point... |
| pcoass 24093 | Order of concatenation doe... |
| pcorevcl 24094 | Closure for a reversed pat... |
| pcorevlem 24095 | Lemma for ~ pcorev . Prov... |
| pcorev 24096 | Concatenation with the rev... |
| pcorev2 24097 | Concatenation with the rev... |
| pcophtb 24098 | The path homotopy equivale... |
| om1val 24099 | The definition of the loop... |
| om1bas 24100 | The base set of the loop s... |
| om1elbas 24101 | Elementhood in the base se... |
| om1addcl 24102 | Closure of the group opera... |
| om1plusg 24103 | The group operation (which... |
| om1tset 24104 | The topology of the loop s... |
| om1opn 24105 | The topology of the loop s... |
| pi1val 24106 | The definition of the fund... |
| pi1bas 24107 | The base set of the fundam... |
| pi1blem 24108 | Lemma for ~ pi1buni . (Co... |
| pi1buni 24109 | Another way to write the l... |
| pi1bas2 24110 | The base set of the fundam... |
| pi1eluni 24111 | Elementhood in the base se... |
| pi1bas3 24112 | The base set of the fundam... |
| pi1cpbl 24113 | The group operation, loop ... |
| elpi1 24114 | The elements of the fundam... |
| elpi1i 24115 | The elements of the fundam... |
| pi1addf 24116 | The group operation of ` p... |
| pi1addval 24117 | The concatenation of two p... |
| pi1grplem 24118 | Lemma for ~ pi1grp . (Con... |
| pi1grp 24119 | The fundamental group is a... |
| pi1id 24120 | The identity element of th... |
| pi1inv 24121 | An inverse in the fundamen... |
| pi1xfrf 24122 | Functionality of the loop ... |
| pi1xfrval 24123 | The value of the loop tran... |
| pi1xfr 24124 | Given a path ` F ` and its... |
| pi1xfrcnvlem 24125 | Given a path ` F ` between... |
| pi1xfrcnv 24126 | Given a path ` F ` between... |
| pi1xfrgim 24127 | The mapping ` G ` between ... |
| pi1cof 24128 | Functionality of the loop ... |
| pi1coval 24129 | The value of the loop tran... |
| pi1coghm 24130 | The mapping ` G ` between ... |
| isclm 24133 | A subcomplex module is a l... |
| clmsca 24134 | The ring of scalars ` F ` ... |
| clmsubrg 24135 | The base set of the ring o... |
| clmlmod 24136 | A subcomplex module is a l... |
| clmgrp 24137 | A subcomplex module is an ... |
| clmabl 24138 | A subcomplex module is an ... |
| clmring 24139 | The scalar ring of a subco... |
| clmfgrp 24140 | The scalar ring of a subco... |
| clm0 24141 | The zero of the scalar rin... |
| clm1 24142 | The identity of the scalar... |
| clmadd 24143 | The addition of the scalar... |
| clmmul 24144 | The multiplication of the ... |
| clmcj 24145 | The conjugation of the sca... |
| isclmi 24146 | Reverse direction of ~ isc... |
| clmzss 24147 | The scalar ring of a subco... |
| clmsscn 24148 | The scalar ring of a subco... |
| clmsub 24149 | Subtraction in the scalar ... |
| clmneg 24150 | Negation in the scalar rin... |
| clmneg1 24151 | Minus one is in the scalar... |
| clmabs 24152 | Norm in the scalar ring of... |
| clmacl 24153 | Closure of ring addition f... |
| clmmcl 24154 | Closure of ring multiplica... |
| clmsubcl 24155 | Closure of ring subtractio... |
| lmhmclm 24156 | The domain of a linear ope... |
| clmvscl 24157 | Closure of scalar product ... |
| clmvsass 24158 | Associative law for scalar... |
| clmvscom 24159 | Commutative law for the sc... |
| clmvsdir 24160 | Distributive law for scala... |
| clmvsdi 24161 | Distributive law for scala... |
| clmvs1 24162 | Scalar product with ring u... |
| clmvs2 24163 | A vector plus itself is tw... |
| clm0vs 24164 | Zero times a vector is the... |
| clmopfne 24165 | The (functionalized) opera... |
| isclmp 24166 | The predicate "is a subcom... |
| isclmi0 24167 | Properties that determine ... |
| clmvneg1 24168 | Minus 1 times a vector is ... |
| clmvsneg 24169 | Multiplication of a vector... |
| clmmulg 24170 | The group multiple functio... |
| clmsubdir 24171 | Scalar multiplication dist... |
| clmpm1dir 24172 | Subtractive distributive l... |
| clmnegneg 24173 | Double negative of a vecto... |
| clmnegsubdi2 24174 | Distribution of negative o... |
| clmsub4 24175 | Rearrangement of 4 terms i... |
| clmvsrinv 24176 | A vector minus itself. (C... |
| clmvslinv 24177 | Minus a vector plus itself... |
| clmvsubval 24178 | Value of vector subtractio... |
| clmvsubval2 24179 | Value of vector subtractio... |
| clmvz 24180 | Two ways to express the ne... |
| zlmclm 24181 | The ` ZZ ` -module operati... |
| clmzlmvsca 24182 | The scalar product of a su... |
| nmoleub2lem 24183 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 24184 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 24185 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 24186 | The operator norm is the s... |
| nmoleub2b 24187 | The operator norm is the s... |
| nmoleub3 24188 | The operator norm is the s... |
| nmhmcn 24189 | A linear operator over a n... |
| cmodscexp 24190 | The powers of ` _i ` belon... |
| cmodscmulexp 24191 | The scalar product of a ve... |
| cvslvec 24194 | A subcomplex vector space ... |
| cvsclm 24195 | A subcomplex vector space ... |
| iscvs 24196 | A subcomplex vector space ... |
| iscvsp 24197 | The predicate "is a subcom... |
| iscvsi 24198 | Properties that determine ... |
| cvsi 24199 | The properties of a subcom... |
| cvsunit 24200 | Unit group of the scalar r... |
| cvsdiv 24201 | Division of the scalar rin... |
| cvsdivcl 24202 | The scalar field of a subc... |
| cvsmuleqdivd 24203 | An equality involving rati... |
| cvsdiveqd 24204 | An equality involving rati... |
| cnlmodlem1 24205 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 24206 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 24207 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 24208 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 24209 | The set of complex numbers... |
| cnstrcvs 24210 | The set of complex numbers... |
| cnrbas 24211 | The set of complex numbers... |
| cnrlmod 24212 | The complex left module of... |
| cnrlvec 24213 | The complex left module of... |
| cncvs 24214 | The complex left module of... |
| recvs 24215 | The field of the real numb... |
| qcvs 24216 | The field of rational numb... |
| zclmncvs 24217 | The ring of integers as le... |
| isncvsngp 24218 | A normed subcomplex vector... |
| isncvsngpd 24219 | Properties that determine ... |
| ncvsi 24220 | The properties of a normed... |
| ncvsprp 24221 | Proportionality property o... |
| ncvsge0 24222 | The norm of a scalar produ... |
| ncvsm1 24223 | The norm of the opposite o... |
| ncvsdif 24224 | The norm of the difference... |
| ncvspi 24225 | The norm of a vector plus ... |
| ncvs1 24226 | From any nonzero vector of... |
| cnrnvc 24227 | The module of complex numb... |
| cnncvs 24228 | The module of complex numb... |
| cnnm 24229 | The norm of the normed sub... |
| ncvspds 24230 | Value of the distance func... |
| cnindmet 24231 | The metric induced on the ... |
| cnncvsaddassdemo 24232 | Derive the associative law... |
| cnncvsmulassdemo 24233 | Derive the associative law... |
| cnncvsabsnegdemo 24234 | Derive the absolute value ... |
| iscph 24239 | A subcomplex pre-Hilbert s... |
| cphphl 24240 | A subcomplex pre-Hilbert s... |
| cphnlm 24241 | A subcomplex pre-Hilbert s... |
| cphngp 24242 | A subcomplex pre-Hilbert s... |
| cphlmod 24243 | A subcomplex pre-Hilbert s... |
| cphlvec 24244 | A subcomplex pre-Hilbert s... |
| cphnvc 24245 | A subcomplex pre-Hilbert s... |
| cphsubrglem 24246 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 24247 | Lemma for ~ cphreccl . (C... |
| cphsca 24248 | A subcomplex pre-Hilbert s... |
| cphsubrg 24249 | The scalar field of a subc... |
| cphreccl 24250 | The scalar field of a subc... |
| cphdivcl 24251 | The scalar field of a subc... |
| cphcjcl 24252 | The scalar field of a subc... |
| cphsqrtcl 24253 | The scalar field of a subc... |
| cphabscl 24254 | The scalar field of a subc... |
| cphsqrtcl2 24255 | The scalar field of a subc... |
| cphsqrtcl3 24256 | If the scalar field of a s... |
| cphqss 24257 | The scalar field of a subc... |
| cphclm 24258 | A subcomplex pre-Hilbert s... |
| cphnmvs 24259 | Norm of a scalar product. ... |
| cphipcl 24260 | An inner product is a memb... |
| cphnmfval 24261 | The value of the norm in a... |
| cphnm 24262 | The square of the norm is ... |
| nmsq 24263 | The square of the norm is ... |
| cphnmf 24264 | The norm of a vector is a ... |
| cphnmcl 24265 | The norm of a vector is a ... |
| reipcl 24266 | An inner product of an ele... |
| ipge0 24267 | The inner product in a sub... |
| cphipcj 24268 | Conjugate of an inner prod... |
| cphipipcj 24269 | An inner product times its... |
| cphorthcom 24270 | Orthogonality (meaning inn... |
| cphip0l 24271 | Inner product with a zero ... |
| cphip0r 24272 | Inner product with a zero ... |
| cphipeq0 24273 | The inner product of a vec... |
| cphdir 24274 | Distributive law for inner... |
| cphdi 24275 | Distributive law for inner... |
| cph2di 24276 | Distributive law for inner... |
| cphsubdir 24277 | Distributive law for inner... |
| cphsubdi 24278 | Distributive law for inner... |
| cph2subdi 24279 | Distributive law for inner... |
| cphass 24280 | Associative law for inner ... |
| cphassr 24281 | "Associative" law for seco... |
| cph2ass 24282 | Move scalar multiplication... |
| cphassi 24283 | Associative law for the fi... |
| cphassir 24284 | "Associative" law for the ... |
| cphpyth 24285 | The pythagorean theorem fo... |
| tcphex 24286 | Lemma for ~ tcphbas and si... |
| tcphval 24287 | Define a function to augme... |
| tcphbas 24288 | The base set of a subcompl... |
| tchplusg 24289 | The addition operation of ... |
| tcphsub 24290 | The subtraction operation ... |
| tcphmulr 24291 | The ring operation of a su... |
| tcphsca 24292 | The scalar field of a subc... |
| tcphvsca 24293 | The scalar multiplication ... |
| tcphip 24294 | The inner product of a sub... |
| tcphtopn 24295 | The topology of a subcompl... |
| tcphphl 24296 | Augmentation of a subcompl... |
| tchnmfval 24297 | The norm of a subcomplex p... |
| tcphnmval 24298 | The norm of a subcomplex p... |
| cphtcphnm 24299 | The norm of a norm-augment... |
| tcphds 24300 | The distance of a pre-Hilb... |
| phclm 24301 | A pre-Hilbert space whose ... |
| tcphcphlem3 24302 | Lemma for ~ tcphcph : real... |
| ipcau2 24303 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 24304 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 24305 | Lemma for ~ tcphcph : homo... |
| tcphcph 24306 | The standard definition of... |
| ipcau 24307 | The Cauchy-Schwarz inequal... |
| nmparlem 24308 | Lemma for ~ nmpar . (Cont... |
| nmpar 24309 | A subcomplex pre-Hilbert s... |
| cphipval2 24310 | Value of the inner product... |
| 4cphipval2 24311 | Four times the inner produ... |
| cphipval 24312 | Value of the inner product... |
| ipcnlem2 24313 | The inner product operatio... |
| ipcnlem1 24314 | The inner product operatio... |
| ipcn 24315 | The inner product operatio... |
| cnmpt1ip 24316 | Continuity of inner produc... |
| cnmpt2ip 24317 | Continuity of inner produc... |
| csscld 24318 | A "closed subspace" in a s... |
| clsocv 24319 | The orthogonal complement ... |
| cphsscph 24320 | A subspace of a subcomplex... |
| lmmbr 24327 | Express the binary relatio... |
| lmmbr2 24328 | Express the binary relatio... |
| lmmbr3 24329 | Express the binary relatio... |
| lmmcvg 24330 | Convergence property of a ... |
| lmmbrf 24331 | Express the binary relatio... |
| lmnn 24332 | A condition that implies c... |
| cfilfval 24333 | The set of Cauchy filters ... |
| iscfil 24334 | The property of being a Ca... |
| iscfil2 24335 | The property of being a Ca... |
| cfilfil 24336 | A Cauchy filter is a filte... |
| cfili 24337 | Property of a Cauchy filte... |
| cfil3i 24338 | A Cauchy filter contains b... |
| cfilss 24339 | A filter finer than a Cauc... |
| fgcfil 24340 | The Cauchy filter conditio... |
| fmcfil 24341 | The Cauchy filter conditio... |
| iscfil3 24342 | A filter is Cauchy iff it ... |
| cfilfcls 24343 | Similar to ultrafilters ( ... |
| caufval 24344 | The set of Cauchy sequence... |
| iscau 24345 | Express the property " ` F... |
| iscau2 24346 | Express the property " ` F... |
| iscau3 24347 | Express the Cauchy sequenc... |
| iscau4 24348 | Express the property " ` F... |
| iscauf 24349 | Express the property " ` F... |
| caun0 24350 | A metric with a Cauchy seq... |
| caufpm 24351 | Inclusion of a Cauchy sequ... |
| caucfil 24352 | A Cauchy sequence predicat... |
| iscmet 24353 | The property " ` D ` is a ... |
| cmetcvg 24354 | The convergence of a Cauch... |
| cmetmet 24355 | A complete metric space is... |
| cmetmeti 24356 | A complete metric space is... |
| cmetcaulem 24357 | Lemma for ~ cmetcau . (Co... |
| cmetcau 24358 | The convergence of a Cauch... |
| iscmet3lem3 24359 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 24360 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 24361 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 24362 | The property " ` D ` is a ... |
| iscmet2 24363 | A metric ` D ` is complete... |
| cfilresi 24364 | A Cauchy filter on a metri... |
| cfilres 24365 | Cauchy filter on a metric ... |
| caussi 24366 | Cauchy sequence on a metri... |
| causs 24367 | Cauchy sequence on a metri... |
| equivcfil 24368 | If the metric ` D ` is "st... |
| equivcau 24369 | If the metric ` D ` is "st... |
| lmle 24370 | If the distance from each ... |
| nglmle 24371 | If the norm of each member... |
| lmclim 24372 | Relate a limit on the metr... |
| lmclimf 24373 | Relate a limit on the metr... |
| metelcls 24374 | A point belongs to the clo... |
| metcld 24375 | A subset of a metric space... |
| metcld2 24376 | A subset of a metric space... |
| caubl 24377 | Sufficient condition to en... |
| caublcls 24378 | The convergent point of a ... |
| metcnp4 24379 | Two ways to say a mapping ... |
| metcn4 24380 | Two ways to say a mapping ... |
| iscmet3i 24381 | Properties that determine ... |
| lmcau 24382 | Every convergent sequence ... |
| flimcfil 24383 | Every convergent filter in... |
| metsscmetcld 24384 | A complete subspace of a m... |
| cmetss 24385 | A subspace of a complete m... |
| equivcmet 24386 | If two metrics are strongl... |
| relcmpcmet 24387 | If ` D ` is a metric space... |
| cmpcmet 24388 | A compact metric space is ... |
| cfilucfil3 24389 | Given a metric ` D ` and a... |
| cfilucfil4 24390 | Given a metric ` D ` and a... |
| cncmet 24391 | The set of complex numbers... |
| recmet 24392 | The real numbers are a com... |
| bcthlem1 24393 | Lemma for ~ bcth . Substi... |
| bcthlem2 24394 | Lemma for ~ bcth . The ba... |
| bcthlem3 24395 | Lemma for ~ bcth . The li... |
| bcthlem4 24396 | Lemma for ~ bcth . Given ... |
| bcthlem5 24397 | Lemma for ~ bcth . The pr... |
| bcth 24398 | Baire's Category Theorem. ... |
| bcth2 24399 | Baire's Category Theorem, ... |
| bcth3 24400 | Baire's Category Theorem, ... |
| isbn 24407 | A Banach space is a normed... |
| bnsca 24408 | The scalar field of a Bana... |
| bnnvc 24409 | A Banach space is a normed... |
| bnnlm 24410 | A Banach space is a normed... |
| bnngp 24411 | A Banach space is a normed... |
| bnlmod 24412 | A Banach space is a left m... |
| bncms 24413 | A Banach space is a comple... |
| iscms 24414 | A complete metric space is... |
| cmscmet 24415 | The induced metric on a co... |
| bncmet 24416 | The induced metric on Bana... |
| cmsms 24417 | A complete metric space is... |
| cmspropd 24418 | Property deduction for a c... |
| cmssmscld 24419 | The restriction of a metri... |
| cmsss 24420 | The restriction of a compl... |
| lssbn 24421 | A subspace of a Banach spa... |
| cmetcusp1 24422 | If the uniform set of a co... |
| cmetcusp 24423 | The uniform space generate... |
| cncms 24424 | The field of complex numbe... |
| cnflduss 24425 | The uniform structure of t... |
| cnfldcusp 24426 | The field of complex numbe... |
| resscdrg 24427 | The real numbers are a sub... |
| cncdrg 24428 | The only complete subfield... |
| srabn 24429 | The subring algebra over a... |
| rlmbn 24430 | The ring module over a com... |
| ishl 24431 | The predicate "is a subcom... |
| hlbn 24432 | Every subcomplex Hilbert s... |
| hlcph 24433 | Every subcomplex Hilbert s... |
| hlphl 24434 | Every subcomplex Hilbert s... |
| hlcms 24435 | Every subcomplex Hilbert s... |
| hlprlem 24436 | Lemma for ~ hlpr . (Contr... |
| hlress 24437 | The scalar field of a subc... |
| hlpr 24438 | The scalar field of a subc... |
| ishl2 24439 | A Hilbert space is a compl... |
| cphssphl 24440 | A Banach subspace of a sub... |
| cmslssbn 24441 | A complete linear subspace... |
| cmscsscms 24442 | A closed subspace of a com... |
| bncssbn 24443 | A closed subspace of a Ban... |
| cssbn 24444 | A complete subspace of a n... |
| csschl 24445 | A complete subspace of a c... |
| cmslsschl 24446 | A complete linear subspace... |
| chlcsschl 24447 | A closed subspace of a sub... |
| retopn 24448 | The topology of the real n... |
| recms 24449 | The real numbers form a co... |
| reust 24450 | The Uniform structure of t... |
| recusp 24451 | The real numbers form a co... |
| rrxval 24456 | Value of the generalized E... |
| rrxbase 24457 | The base of the generalize... |
| rrxprds 24458 | Expand the definition of t... |
| rrxip 24459 | The inner product of the g... |
| rrxnm 24460 | The norm of the generalize... |
| rrxcph 24461 | Generalized Euclidean real... |
| rrxds 24462 | The distance over generali... |
| rrxvsca 24463 | The scalar product over ge... |
| rrxplusgvscavalb 24464 | The result of the addition... |
| rrxsca 24465 | The field of real numbers ... |
| rrx0 24466 | The zero ("origin") in a g... |
| rrx0el 24467 | The zero ("origin") in a g... |
| csbren 24468 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 24469 | Triangle inequality in R^n... |
| rrxf 24470 | Euclidean vectors as funct... |
| rrxfsupp 24471 | Euclidean vectors are of f... |
| rrxsuppss 24472 | Support of Euclidean vecto... |
| rrxmvallem 24473 | Support of the function us... |
| rrxmval 24474 | The value of the Euclidean... |
| rrxmfval 24475 | The value of the Euclidean... |
| rrxmetlem 24476 | Lemma for ~ rrxmet . (Con... |
| rrxmet 24477 | Euclidean space is a metri... |
| rrxdstprj1 24478 | The distance between two p... |
| rrxbasefi 24479 | The base of the generalize... |
| rrxdsfi 24480 | The distance over generali... |
| rrxmetfi 24481 | Euclidean space is a metri... |
| rrxdsfival 24482 | The value of the Euclidean... |
| ehlval 24483 | Value of the Euclidean spa... |
| ehlbase 24484 | The base of the Euclidean ... |
| ehl0base 24485 | The base of the Euclidean ... |
| ehl0 24486 | The Euclidean space of dim... |
| ehleudis 24487 | The Euclidean distance fun... |
| ehleudisval 24488 | The value of the Euclidean... |
| ehl1eudis 24489 | The Euclidean distance fun... |
| ehl1eudisval 24490 | The value of the Euclidean... |
| ehl2eudis 24491 | The Euclidean distance fun... |
| ehl2eudisval 24492 | The value of the Euclidean... |
| minveclem1 24493 | Lemma for ~ minvec . The ... |
| minveclem4c 24494 | Lemma for ~ minvec . The ... |
| minveclem2 24495 | Lemma for ~ minvec . Any ... |
| minveclem3a 24496 | Lemma for ~ minvec . ` D `... |
| minveclem3b 24497 | Lemma for ~ minvec . The ... |
| minveclem3 24498 | Lemma for ~ minvec . The ... |
| minveclem4a 24499 | Lemma for ~ minvec . ` F `... |
| minveclem4b 24500 | Lemma for ~ minvec . The ... |
| minveclem4 24501 | Lemma for ~ minvec . The ... |
| minveclem5 24502 | Lemma for ~ minvec . Disc... |
| minveclem6 24503 | Lemma for ~ minvec . Any ... |
| minveclem7 24504 | Lemma for ~ minvec . Sinc... |
| minvec 24505 | Minimizing vector theorem,... |
| pjthlem1 24506 | Lemma for ~ pjth . (Contr... |
| pjthlem2 24507 | Lemma for ~ pjth . (Contr... |
| pjth 24508 | Projection Theorem: Any H... |
| pjth2 24509 | Projection Theorem with ab... |
| cldcss 24510 | Corollary of the Projectio... |
| cldcss2 24511 | Corollary of the Projectio... |
| hlhil 24512 | Corollary of the Projectio... |
| addcncf 24513 | The addition of two contin... |
| subcncf 24514 | The addition of two contin... |
| mulcncf 24515 | The multiplication of two ... |
| divcncf 24516 | The quotient of two contin... |
| pmltpclem1 24517 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 24518 | Lemma for ~ pmltpc . (Con... |
| pmltpc 24519 | Any function on the reals ... |
| ivthlem1 24520 | Lemma for ~ ivth . The se... |
| ivthlem2 24521 | Lemma for ~ ivth . Show t... |
| ivthlem3 24522 | Lemma for ~ ivth , the int... |
| ivth 24523 | The intermediate value the... |
| ivth2 24524 | The intermediate value the... |
| ivthle 24525 | The intermediate value the... |
| ivthle2 24526 | The intermediate value the... |
| ivthicc 24527 | The interval between any t... |
| evthicc 24528 | Specialization of the Extr... |
| evthicc2 24529 | Combine ~ ivthicc with ~ e... |
| cniccbdd 24530 | A continuous function on a... |
| ovolfcl 24535 | Closure for the interval e... |
| ovolfioo 24536 | Unpack the interval coveri... |
| ovolficc 24537 | Unpack the interval coveri... |
| ovolficcss 24538 | Any (closed) interval cove... |
| ovolfsval 24539 | The value of the interval ... |
| ovolfsf 24540 | Closure for the interval l... |
| ovolsf 24541 | Closure for the partial su... |
| ovolval 24542 | The value of the outer mea... |
| elovolmlem 24543 | Lemma for ~ elovolm and re... |
| elovolm 24544 | Elementhood in the set ` M... |
| elovolmr 24545 | Sufficient condition for e... |
| ovolmge0 24546 | The set ` M ` is composed ... |
| ovolcl 24547 | The volume of a set is an ... |
| ovollb 24548 | The outer volume is a lowe... |
| ovolgelb 24549 | The outer volume is the gr... |
| ovolge0 24550 | The volume of a set is alw... |
| ovolf 24551 | The domain and range of th... |
| ovollecl 24552 | If an outer volume is boun... |
| ovolsslem 24553 | Lemma for ~ ovolss . (Con... |
| ovolss 24554 | The volume of a set is mon... |
| ovolsscl 24555 | If a set is contained in a... |
| ovolssnul 24556 | A subset of a nullset is n... |
| ovollb2lem 24557 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 24558 | It is often more convenien... |
| ovolctb 24559 | The volume of a denumerabl... |
| ovolq 24560 | The rational numbers have ... |
| ovolctb2 24561 | The volume of a countable ... |
| ovol0 24562 | The empty set has 0 outer ... |
| ovolfi 24563 | A finite set has 0 outer L... |
| ovolsn 24564 | A singleton has 0 outer Le... |
| ovolunlem1a 24565 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 24566 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 24567 | Lemma for ~ ovolun . (Con... |
| ovolun 24568 | The Lebesgue outer measure... |
| ovolunnul 24569 | Adding a nullset does not ... |
| ovolfiniun 24570 | The Lebesgue outer measure... |
| ovoliunlem1 24571 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 24572 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 24573 | Lemma for ~ ovoliun . (Co... |
| ovoliun 24574 | The Lebesgue outer measure... |
| ovoliun2 24575 | The Lebesgue outer measure... |
| ovoliunnul 24576 | A countable union of nulls... |
| shft2rab 24577 | If ` B ` is a shift of ` A... |
| ovolshftlem1 24578 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 24579 | Lemma for ~ ovolshft . (C... |
| ovolshft 24580 | The Lebesgue outer measure... |
| sca2rab 24581 | If ` B ` is a scale of ` A... |
| ovolscalem1 24582 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 24583 | Lemma for ~ ovolshft . (C... |
| ovolsca 24584 | The Lebesgue outer measure... |
| ovolicc1 24585 | The measure of a closed in... |
| ovolicc2lem1 24586 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 24587 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 24588 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 24589 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 24590 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 24591 | The measure of a closed in... |
| ovolicc 24592 | The measure of a closed in... |
| ovolicopnf 24593 | The measure of a right-unb... |
| ovolre 24594 | The measure of the real nu... |
| ismbl 24595 | The predicate " ` A ` is L... |
| ismbl2 24596 | From ~ ovolun , it suffice... |
| volres 24597 | A self-referencing abbrevi... |
| volf 24598 | The domain and range of th... |
| mblvol 24599 | The volume of a measurable... |
| mblss 24600 | A measurable set is a subs... |
| mblsplit 24601 | The defining property of m... |
| volss 24602 | The Lebesgue measure is mo... |
| cmmbl 24603 | The complement of a measur... |
| nulmbl 24604 | A nullset is measurable. ... |
| nulmbl2 24605 | A set of outer measure zer... |
| unmbl 24606 | A union of measurable sets... |
| shftmbl 24607 | A shift of a measurable se... |
| 0mbl 24608 | The empty set is measurabl... |
| rembl 24609 | The set of all real number... |
| unidmvol 24610 | The union of the Lebesgue ... |
| inmbl 24611 | An intersection of measura... |
| difmbl 24612 | A difference of measurable... |
| finiunmbl 24613 | A finite union of measurab... |
| volun 24614 | The Lebesgue measure funct... |
| volinun 24615 | Addition of non-disjoint s... |
| volfiniun 24616 | The volume of a disjoint f... |
| iundisj 24617 | Rewrite a countable union ... |
| iundisj2 24618 | A disjoint union is disjoi... |
| voliunlem1 24619 | Lemma for ~ voliun . (Con... |
| voliunlem2 24620 | Lemma for ~ voliun . (Con... |
| voliunlem3 24621 | Lemma for ~ voliun . (Con... |
| iunmbl 24622 | The measurable sets are cl... |
| voliun 24623 | The Lebesgue measure funct... |
| volsuplem 24624 | Lemma for ~ volsup . (Con... |
| volsup 24625 | The volume of the limit of... |
| iunmbl2 24626 | The measurable sets are cl... |
| ioombl1lem1 24627 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 24628 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 24629 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 24630 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 24631 | An open right-unbounded in... |
| icombl1 24632 | A closed unbounded-above i... |
| icombl 24633 | A closed-below, open-above... |
| ioombl 24634 | An open real interval is m... |
| iccmbl 24635 | A closed real interval is ... |
| iccvolcl 24636 | A closed real interval has... |
| ovolioo 24637 | The measure of an open int... |
| volioo 24638 | The measure of an open int... |
| ioovolcl 24639 | An open real interval has ... |
| ovolfs2 24640 | Alternative expression for... |
| ioorcl2 24641 | An open interval with fini... |
| ioorf 24642 | Define a function from ope... |
| ioorval 24643 | Define a function from ope... |
| ioorinv2 24644 | The function ` F ` is an "... |
| ioorinv 24645 | The function ` F ` is an "... |
| ioorcl 24646 | The function ` F ` does no... |
| uniiccdif 24647 | A union of closed interval... |
| uniioovol 24648 | A disjoint union of open i... |
| uniiccvol 24649 | An almost-disjoint union o... |
| uniioombllem1 24650 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 24651 | Lemma for ~ uniioombl . (... |
| uniioombllem2 24652 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 24653 | Lemma for ~ uniioombl . (... |
| uniioombllem3 24654 | Lemma for ~ uniioombl . (... |
| uniioombllem4 24655 | Lemma for ~ uniioombl . (... |
| uniioombllem5 24656 | Lemma for ~ uniioombl . (... |
| uniioombllem6 24657 | Lemma for ~ uniioombl . (... |
| uniioombl 24658 | A disjoint union of open i... |
| uniiccmbl 24659 | An almost-disjoint union o... |
| dyadf 24660 | The function ` F ` returns... |
| dyadval 24661 | Value of the dyadic ration... |
| dyadovol 24662 | Volume of a dyadic rationa... |
| dyadss 24663 | Two closed dyadic rational... |
| dyaddisjlem 24664 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 24665 | Two closed dyadic rational... |
| dyadmaxlem 24666 | Lemma for ~ dyadmax . (Co... |
| dyadmax 24667 | Any nonempty set of dyadic... |
| dyadmbllem 24668 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 24669 | Any union of dyadic ration... |
| opnmbllem 24670 | Lemma for ~ opnmbl . (Con... |
| opnmbl 24671 | All open sets are measurab... |
| opnmblALT 24672 | All open sets are measurab... |
| subopnmbl 24673 | Sets which are open in a m... |
| volsup2 24674 | The volume of ` A ` is the... |
| volcn 24675 | The function formed by res... |
| volivth 24676 | The Intermediate Value The... |
| vitalilem1 24677 | Lemma for ~ vitali . (Con... |
| vitalilem2 24678 | Lemma for ~ vitali . (Con... |
| vitalilem3 24679 | Lemma for ~ vitali . (Con... |
| vitalilem4 24680 | Lemma for ~ vitali . (Con... |
| vitalilem5 24681 | Lemma for ~ vitali . (Con... |
| vitali 24682 | If the reals can be well-o... |
| ismbf1 24693 | The predicate " ` F ` is a... |
| mbff 24694 | A measurable function is a... |
| mbfdm 24695 | The domain of a measurable... |
| mbfconstlem 24696 | Lemma for ~ mbfconst and r... |
| ismbf 24697 | The predicate " ` F ` is a... |
| ismbfcn 24698 | A complex function is meas... |
| mbfima 24699 | Definitional property of a... |
| mbfimaicc 24700 | The preimage of any closed... |
| mbfimasn 24701 | The preimage of a point un... |
| mbfconst 24702 | A constant function is mea... |
| mbf0 24703 | The empty function is meas... |
| mbfid 24704 | The identity function is m... |
| mbfmptcl 24705 | Lemma for the ` MblFn ` pr... |
| mbfdm2 24706 | The domain of a measurable... |
| ismbfcn2 24707 | A complex function is meas... |
| ismbfd 24708 | Deduction to prove measura... |
| ismbf2d 24709 | Deduction to prove measura... |
| mbfeqalem1 24710 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 24711 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 24712 | If two functions are equal... |
| mbfres 24713 | The restriction of a measu... |
| mbfres2 24714 | Measurability of a piecewi... |
| mbfss 24715 | Change the domain of a mea... |
| mbfmulc2lem 24716 | Multiplication by a consta... |
| mbfmulc2re 24717 | Multiplication by a consta... |
| mbfmax 24718 | The maximum of two functio... |
| mbfneg 24719 | The negative of a measurab... |
| mbfpos 24720 | The positive part of a mea... |
| mbfposr 24721 | Converse to ~ mbfpos . (C... |
| mbfposb 24722 | A function is measurable i... |
| ismbf3d 24723 | Simplified form of ~ ismbf... |
| mbfimaopnlem 24724 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 24725 | The preimage of any open s... |
| mbfimaopn2 24726 | The preimage of any set op... |
| cncombf 24727 | The composition of a conti... |
| cnmbf 24728 | A continuous function is m... |
| mbfaddlem 24729 | The sum of two measurable ... |
| mbfadd 24730 | The sum of two measurable ... |
| mbfsub 24731 | The difference of two meas... |
| mbfmulc2 24732 | A complex constant times a... |
| mbfsup 24733 | The supremum of a sequence... |
| mbfinf 24734 | The infimum of a sequence ... |
| mbflimsup 24735 | The limit supremum of a se... |
| mbflimlem 24736 | The pointwise limit of a s... |
| mbflim 24737 | The pointwise limit of a s... |
| 0pval 24740 | The zero function evaluate... |
| 0plef 24741 | Two ways to say that the f... |
| 0pledm 24742 | Adjust the domain of the l... |
| isi1f 24743 | The predicate " ` F ` is a... |
| i1fmbf 24744 | Simple functions are measu... |
| i1ff 24745 | A simple function is a fun... |
| i1frn 24746 | A simple function has fini... |
| i1fima 24747 | Any preimage of a simple f... |
| i1fima2 24748 | Any preimage of a simple f... |
| i1fima2sn 24749 | Preimage of a singleton. ... |
| i1fd 24750 | A simplified set of assump... |
| i1f0rn 24751 | Any simple function takes ... |
| itg1val 24752 | The value of the integral ... |
| itg1val2 24753 | The value of the integral ... |
| itg1cl 24754 | Closure of the integral on... |
| itg1ge0 24755 | Closure of the integral on... |
| i1f0 24756 | The zero function is simpl... |
| itg10 24757 | The zero function has zero... |
| i1f1lem 24758 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 24759 | Base case simple functions... |
| itg11 24760 | The integral of an indicat... |
| itg1addlem1 24761 | Decompose a preimage, whic... |
| i1faddlem 24762 | Decompose the preimage of ... |
| i1fmullem 24763 | Decompose the preimage of ... |
| i1fadd 24764 | The sum of two simple func... |
| i1fmul 24765 | The pointwise product of t... |
| itg1addlem2 24766 | Lemma for ~ itg1add . The... |
| itg1addlem3 24767 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 24768 | Lemma for ~ itg1add . (Co... |
| itg1addlem4OLD 24769 | Obsolete version of ~ itg1... |
| itg1addlem5 24770 | Lemma for ~ itg1add . (Co... |
| itg1add 24771 | The integral of a sum of s... |
| i1fmulclem 24772 | Decompose the preimage of ... |
| i1fmulc 24773 | A nonnegative constant tim... |
| itg1mulc 24774 | The integral of a constant... |
| i1fres 24775 | The "restriction" of a sim... |
| i1fpos 24776 | The positive part of a sim... |
| i1fposd 24777 | Deduction form of ~ i1fpos... |
| i1fsub 24778 | The difference of two simp... |
| itg1sub 24779 | The integral of a differen... |
| itg10a 24780 | The integral of a simple f... |
| itg1ge0a 24781 | The integral of an almost ... |
| itg1lea 24782 | Approximate version of ~ i... |
| itg1le 24783 | If one simple function dom... |
| itg1climres 24784 | Restricting the simple fun... |
| mbfi1fseqlem1 24785 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 24786 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 24787 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 24788 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 24789 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 24790 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 24791 | A characterization of meas... |
| mbfi1flimlem 24792 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 24793 | Any real measurable functi... |
| mbfmullem2 24794 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 24795 | Lemma for ~ mbfmul . (Con... |
| mbfmul 24796 | The product of two measura... |
| itg2lcl 24797 | The set of lower sums is a... |
| itg2val 24798 | Value of the integral on n... |
| itg2l 24799 | Elementhood in the set ` L... |
| itg2lr 24800 | Sufficient condition for e... |
| xrge0f 24801 | A real function is a nonne... |
| itg2cl 24802 | The integral of a nonnegat... |
| itg2ub 24803 | The integral of a nonnegat... |
| itg2leub 24804 | Any upper bound on the int... |
| itg2ge0 24805 | The integral of a nonnegat... |
| itg2itg1 24806 | The integral of a nonnegat... |
| itg20 24807 | The integral of the zero f... |
| itg2lecl 24808 | If an ` S.2 ` integral is ... |
| itg2le 24809 | If one function dominates ... |
| itg2const 24810 | Integral of a constant fun... |
| itg2const2 24811 | When the base set of a con... |
| itg2seq 24812 | Definitional property of t... |
| itg2uba 24813 | Approximate version of ~ i... |
| itg2lea 24814 | Approximate version of ~ i... |
| itg2eqa 24815 | Approximate equality of in... |
| itg2mulclem 24816 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 24817 | The integral of a nonnegat... |
| itg2splitlem 24818 | Lemma for ~ itg2split . (... |
| itg2split 24819 | The ` S.2 ` integral split... |
| itg2monolem1 24820 | Lemma for ~ itg2mono . We... |
| itg2monolem2 24821 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 24822 | Lemma for ~ itg2mono . (C... |
| itg2mono 24823 | The Monotone Convergence T... |
| itg2i1fseqle 24824 | Subject to the conditions ... |
| itg2i1fseq 24825 | Subject to the conditions ... |
| itg2i1fseq2 24826 | In an extension to the res... |
| itg2i1fseq3 24827 | Special case of ~ itg2i1fs... |
| itg2addlem 24828 | Lemma for ~ itg2add . (Co... |
| itg2add 24829 | The ` S.2 ` integral is li... |
| itg2gt0 24830 | If the function ` F ` is s... |
| itg2cnlem1 24831 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 24832 | Lemma for ~ itgcn . (Cont... |
| itg2cn 24833 | A sort of absolute continu... |
| ibllem 24834 | Conditioned equality theor... |
| isibl 24835 | The predicate " ` F ` is i... |
| isibl2 24836 | The predicate " ` F ` is i... |
| iblmbf 24837 | An integrable function is ... |
| iblitg 24838 | If a function is integrabl... |
| dfitg 24839 | Evaluate the class substit... |
| itgex 24840 | An integral is a set. (Co... |
| itgeq1f 24841 | Equality theorem for an in... |
| itgeq1 24842 | Equality theorem for an in... |
| nfitg1 24843 | Bound-variable hypothesis ... |
| nfitg 24844 | Bound-variable hypothesis ... |
| cbvitg 24845 | Change bound variable in a... |
| cbvitgv 24846 | Change bound variable in a... |
| itgeq2 24847 | Equality theorem for an in... |
| itgresr 24848 | The domain of an integral ... |
| itg0 24849 | The integral of anything o... |
| itgz 24850 | The integral of zero on an... |
| itgeq2dv 24851 | Equality theorem for an in... |
| itgmpt 24852 | Change bound variable in a... |
| itgcl 24853 | The integral of an integra... |
| itgvallem 24854 | Substitution lemma. (Cont... |
| itgvallem3 24855 | Lemma for ~ itgposval and ... |
| ibl0 24856 | The zero function is integ... |
| iblcnlem1 24857 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 24858 | Expand out the universal q... |
| itgcnlem 24859 | Expand out the sum in ~ df... |
| iblrelem 24860 | Integrability of a real fu... |
| iblposlem 24861 | Lemma for ~ iblpos . (Con... |
| iblpos 24862 | Integrability of a nonnega... |
| iblre 24863 | Integrability of a real fu... |
| itgrevallem1 24864 | Lemma for ~ itgposval and ... |
| itgposval 24865 | The integral of a nonnegat... |
| itgreval 24866 | Decompose the integral of ... |
| itgrecl 24867 | Real closure of an integra... |
| iblcn 24868 | Integrability of a complex... |
| itgcnval 24869 | Decompose the integral of ... |
| itgre 24870 | Real part of an integral. ... |
| itgim 24871 | Imaginary part of an integ... |
| iblneg 24872 | The negative of an integra... |
| itgneg 24873 | Negation of an integral. ... |
| iblss 24874 | A subset of an integrable ... |
| iblss2 24875 | Change the domain of an in... |
| itgitg2 24876 | Transfer an integral using... |
| i1fibl 24877 | A simple function is integ... |
| itgitg1 24878 | Transfer an integral using... |
| itgle 24879 | Monotonicity of an integra... |
| itgge0 24880 | The integral of a positive... |
| itgss 24881 | Expand the set of an integ... |
| itgss2 24882 | Expand the set of an integ... |
| itgeqa 24883 | Approximate equality of in... |
| itgss3 24884 | Expand the set of an integ... |
| itgioo 24885 | Equality of integrals on o... |
| itgless 24886 | Expand the integral of a n... |
| iblconst 24887 | A constant function is int... |
| itgconst 24888 | Integral of a constant fun... |
| ibladdlem 24889 | Lemma for ~ ibladd . (Con... |
| ibladd 24890 | Add two integrals over the... |
| iblsub 24891 | Subtract two integrals ove... |
| itgaddlem1 24892 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 24893 | Lemma for ~ itgadd . (Con... |
| itgadd 24894 | Add two integrals over the... |
| itgsub 24895 | Subtract two integrals ove... |
| itgfsum 24896 | Take a finite sum of integ... |
| iblabslem 24897 | Lemma for ~ iblabs . (Con... |
| iblabs 24898 | The absolute value of an i... |
| iblabsr 24899 | A measurable function is i... |
| iblmulc2 24900 | Multiply an integral by a ... |
| itgmulc2lem1 24901 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 24902 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 24903 | Multiply an integral by a ... |
| itgabs 24904 | The triangle inequality fo... |
| itgsplit 24905 | The ` S. ` integral splits... |
| itgspliticc 24906 | The ` S. ` integral splits... |
| itgsplitioo 24907 | The ` S. ` integral splits... |
| bddmulibl 24908 | A bounded function times a... |
| bddibl 24909 | A bounded function is inte... |
| cniccibl 24910 | A continuous function on a... |
| bddiblnc 24911 | Choice-free proof of ~ bdd... |
| cnicciblnc 24912 | Choice-free proof of ~ cni... |
| itggt0 24913 | The integral of a strictly... |
| itgcn 24914 | Transfer ~ itg2cn to the f... |
| ditgeq1 24917 | Equality theorem for the d... |
| ditgeq2 24918 | Equality theorem for the d... |
| ditgeq3 24919 | Equality theorem for the d... |
| ditgeq3dv 24920 | Equality theorem for the d... |
| ditgex 24921 | A directed integral is a s... |
| ditg0 24922 | Value of the directed inte... |
| cbvditg 24923 | Change bound variable in a... |
| cbvditgv 24924 | Change bound variable in a... |
| ditgpos 24925 | Value of the directed inte... |
| ditgneg 24926 | Value of the directed inte... |
| ditgcl 24927 | Closure of a directed inte... |
| ditgswap 24928 | Reverse a directed integra... |
| ditgsplitlem 24929 | Lemma for ~ ditgsplit . (... |
| ditgsplit 24930 | This theorem is the raison... |
| reldv 24939 | The derivative function is... |
| limcvallem 24940 | Lemma for ~ ellimc . (Con... |
| limcfval 24941 | Value and set bounds on th... |
| ellimc 24942 | Value of the limit predica... |
| limcrcl 24943 | Reverse closure for the li... |
| limccl 24944 | Closure of the limit opera... |
| limcdif 24945 | It suffices to consider fu... |
| ellimc2 24946 | Write the definition of a ... |
| limcnlp 24947 | If ` B ` is not a limit po... |
| ellimc3 24948 | Write the epsilon-delta de... |
| limcflflem 24949 | Lemma for ~ limcflf . (Co... |
| limcflf 24950 | The limit operator can be ... |
| limcmo 24951 | If ` B ` is a limit point ... |
| limcmpt 24952 | Express the limit operator... |
| limcmpt2 24953 | Express the limit operator... |
| limcresi 24954 | Any limit of ` F ` is also... |
| limcres 24955 | If ` B ` is an interior po... |
| cnplimc 24956 | A function is continuous a... |
| cnlimc 24957 | ` F ` is a continuous func... |
| cnlimci 24958 | If ` F ` is a continuous f... |
| cnmptlimc 24959 | If ` F ` is a continuous f... |
| limccnp 24960 | If the limit of ` F ` at `... |
| limccnp2 24961 | The image of a convergent ... |
| limcco 24962 | Composition of two limits.... |
| limciun 24963 | A point is a limit of ` F ... |
| limcun 24964 | A point is a limit of ` F ... |
| dvlem 24965 | Closure for a difference q... |
| dvfval 24966 | Value and set bounds on th... |
| eldv 24967 | The differentiable predica... |
| dvcl 24968 | The derivative function ta... |
| dvbssntr 24969 | The set of differentiable ... |
| dvbss 24970 | The set of differentiable ... |
| dvbsss 24971 | The set of differentiable ... |
| perfdvf 24972 | The derivative is a functi... |
| recnprss 24973 | Both ` RR ` and ` CC ` are... |
| recnperf 24974 | Both ` RR ` and ` CC ` are... |
| dvfg 24975 | Explicitly write out the f... |
| dvf 24976 | The derivative is a functi... |
| dvfcn 24977 | The derivative is a functi... |
| dvreslem 24978 | Lemma for ~ dvres . (Cont... |
| dvres2lem 24979 | Lemma for ~ dvres2 . (Con... |
| dvres 24980 | Restriction of a derivativ... |
| dvres2 24981 | Restriction of the base se... |
| dvres3 24982 | Restriction of a complex d... |
| dvres3a 24983 | Restriction of a complex d... |
| dvidlem 24984 | Lemma for ~ dvid and ~ dvc... |
| dvmptresicc 24985 | Derivative of a function r... |
| dvconst 24986 | Derivative of a constant f... |
| dvid 24987 | Derivative of the identity... |
| dvcnp 24988 | The difference quotient is... |
| dvcnp2 24989 | A function is continuous a... |
| dvcn 24990 | A differentiable function ... |
| dvnfval 24991 | Value of the iterated deri... |
| dvnff 24992 | The iterated derivative is... |
| dvn0 24993 | Zero times iterated deriva... |
| dvnp1 24994 | Successor iterated derivat... |
| dvn1 24995 | One times iterated derivat... |
| dvnf 24996 | The N-times derivative is ... |
| dvnbss 24997 | The set of N-times differe... |
| dvnadd 24998 | The ` N ` -th derivative o... |
| dvn2bss 24999 | An N-times differentiable ... |
| dvnres 25000 | Multiple derivative versio... |
| cpnfval 25001 | Condition for n-times cont... |
| fncpn 25002 | The ` C^n ` object is a fu... |
| elcpn 25003 | Condition for n-times cont... |
| cpnord 25004 | ` C^n ` conditions are ord... |
| cpncn 25005 | A ` C^n ` function is cont... |
| cpnres 25006 | The restriction of a ` C^n... |
| dvaddbr 25007 | The sum rule for derivativ... |
| dvmulbr 25008 | The product rule for deriv... |
| dvadd 25009 | The sum rule for derivativ... |
| dvmul 25010 | The product rule for deriv... |
| dvaddf 25011 | The sum rule for everywher... |
| dvmulf 25012 | The product rule for every... |
| dvcmul 25013 | The product rule when one ... |
| dvcmulf 25014 | The product rule when one ... |
| dvcobr 25015 | The chain rule for derivat... |
| dvco 25016 | The chain rule for derivat... |
| dvcof 25017 | The chain rule for everywh... |
| dvcjbr 25018 | The derivative of the conj... |
| dvcj 25019 | The derivative of the conj... |
| dvfre 25020 | The derivative of a real f... |
| dvnfre 25021 | The ` N ` -th derivative o... |
| dvexp 25022 | Derivative of a power func... |
| dvexp2 25023 | Derivative of an exponenti... |
| dvrec 25024 | Derivative of the reciproc... |
| dvmptres3 25025 | Function-builder for deriv... |
| dvmptid 25026 | Function-builder for deriv... |
| dvmptc 25027 | Function-builder for deriv... |
| dvmptcl 25028 | Closure lemma for ~ dvmptc... |
| dvmptadd 25029 | Function-builder for deriv... |
| dvmptmul 25030 | Function-builder for deriv... |
| dvmptres2 25031 | Function-builder for deriv... |
| dvmptres 25032 | Function-builder for deriv... |
| dvmptcmul 25033 | Function-builder for deriv... |
| dvmptdivc 25034 | Function-builder for deriv... |
| dvmptneg 25035 | Function-builder for deriv... |
| dvmptsub 25036 | Function-builder for deriv... |
| dvmptcj 25037 | Function-builder for deriv... |
| dvmptre 25038 | Function-builder for deriv... |
| dvmptim 25039 | Function-builder for deriv... |
| dvmptntr 25040 | Function-builder for deriv... |
| dvmptco 25041 | Function-builder for deriv... |
| dvrecg 25042 | Derivative of the reciproc... |
| dvmptdiv 25043 | Function-builder for deriv... |
| dvmptfsum 25044 | Function-builder for deriv... |
| dvcnvlem 25045 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 25046 | A weak version of ~ dvcnvr... |
| dvexp3 25047 | Derivative of an exponenti... |
| dveflem 25048 | Derivative of the exponent... |
| dvef 25049 | Derivative of the exponent... |
| dvsincos 25050 | Derivative of the sine and... |
| dvsin 25051 | Derivative of the sine fun... |
| dvcos 25052 | Derivative of the cosine f... |
| dvferm1lem 25053 | Lemma for ~ dvferm . (Con... |
| dvferm1 25054 | One-sided version of ~ dvf... |
| dvferm2lem 25055 | Lemma for ~ dvferm . (Con... |
| dvferm2 25056 | One-sided version of ~ dvf... |
| dvferm 25057 | Fermat's theorem on statio... |
| rollelem 25058 | Lemma for ~ rolle . (Cont... |
| rolle 25059 | Rolle's theorem. If ` F `... |
| cmvth 25060 | Cauchy's Mean Value Theore... |
| mvth 25061 | The Mean Value Theorem. I... |
| dvlip 25062 | A function with derivative... |
| dvlipcn 25063 | A complex function with de... |
| dvlip2 25064 | Combine the results of ~ d... |
| c1liplem1 25065 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 25066 | C^1 functions are Lipschit... |
| c1lip2 25067 | C^1 functions are Lipschit... |
| c1lip3 25068 | C^1 functions are Lipschit... |
| dveq0 25069 | If a continuous function h... |
| dv11cn 25070 | Two functions defined on a... |
| dvgt0lem1 25071 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 25072 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 25073 | A function on a closed int... |
| dvlt0 25074 | A function on a closed int... |
| dvge0 25075 | A function on a closed int... |
| dvle 25076 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 25077 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 25078 | Lemma for ~ dvivth . (Con... |
| dvivth 25079 | Darboux' theorem, or the i... |
| dvne0 25080 | A function on a closed int... |
| dvne0f1 25081 | A function on a closed int... |
| lhop1lem 25082 | Lemma for ~ lhop1 . (Cont... |
| lhop1 25083 | L'Hôpital's Rule for... |
| lhop2 25084 | L'Hôpital's Rule for... |
| lhop 25085 | L'Hôpital's Rule. I... |
| dvcnvrelem1 25086 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 25087 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 25088 | The derivative rule for in... |
| dvcvx 25089 | A real function with stric... |
| dvfsumle 25090 | Compare a finite sum to an... |
| dvfsumge 25091 | Compare a finite sum to an... |
| dvfsumabs 25092 | Compare a finite sum to an... |
| dvmptrecl 25093 | Real closure of a derivati... |
| dvfsumrlimf 25094 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 25095 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 25096 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem3 25097 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 25098 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 25099 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 25100 | Compare a finite sum to an... |
| dvfsumrlim2 25101 | Compare a finite sum to an... |
| dvfsumrlim3 25102 | Conjoin the statements of ... |
| dvfsum2 25103 | The reverse of ~ dvfsumrli... |
| ftc1lem1 25104 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 25105 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 25106 | The Fundamental Theorem of... |
| ftc1lem3 25107 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 25108 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 25109 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 25110 | Lemma for ~ ftc1 . (Contr... |
| ftc1 25111 | The Fundamental Theorem of... |
| ftc1cn 25112 | Strengthen the assumptions... |
| ftc2 25113 | The Fundamental Theorem of... |
| ftc2ditglem 25114 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 25115 | Directed integral analogue... |
| itgparts 25116 | Integration by parts. If ... |
| itgsubstlem 25117 | Lemma for ~ itgsubst . (C... |
| itgsubst 25118 | Integration by ` u ` -subs... |
| itgpowd 25119 | The integral of a monomial... |
| reldmmdeg 25124 | Multivariate degree is a b... |
| tdeglem1 25125 | Functionality of the total... |
| tdeglem1OLD 25126 | Obsolete version of ~ tdeg... |
| tdeglem3 25127 | Additivity of the total de... |
| tdeglem3OLD 25128 | Obsolete version of ~ tdeg... |
| tdeglem4 25129 | There is only one multi-in... |
| tdeglem4OLD 25130 | Obsolete version of ~ tdeg... |
| tdeglem2 25131 | Simplification of total de... |
| mdegfval 25132 | Value of the multivariate ... |
| mdegval 25133 | Value of the multivariate ... |
| mdegleb 25134 | Property of being of limit... |
| mdeglt 25135 | If there is an upper limit... |
| mdegldg 25136 | A nonzero polynomial has s... |
| mdegxrcl 25137 | Closure of polynomial degr... |
| mdegxrf 25138 | Functionality of polynomia... |
| mdegcl 25139 | Sharp closure for multivar... |
| mdeg0 25140 | Degree of the zero polynom... |
| mdegnn0cl 25141 | Degree of a nonzero polyno... |
| degltlem1 25142 | Theorem on arithmetic of e... |
| degltp1le 25143 | Theorem on arithmetic of e... |
| mdegaddle 25144 | The degree of a sum is at ... |
| mdegvscale 25145 | The degree of a scalar mul... |
| mdegvsca 25146 | The degree of a scalar mul... |
| mdegle0 25147 | A polynomial has nonpositi... |
| mdegmullem 25148 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 25149 | The multivariate degree of... |
| deg1fval 25150 | Relate univariate polynomi... |
| deg1xrf 25151 | Functionality of univariat... |
| deg1xrcl 25152 | Closure of univariate poly... |
| deg1cl 25153 | Sharp closure of univariat... |
| mdegpropd 25154 | Property deduction for pol... |
| deg1fvi 25155 | Univariate polynomial degr... |
| deg1propd 25156 | Property deduction for pol... |
| deg1z 25157 | Degree of the zero univari... |
| deg1nn0cl 25158 | Degree of a nonzero univar... |
| deg1n0ima 25159 | Degree image of a set of p... |
| deg1nn0clb 25160 | A polynomial is nonzero if... |
| deg1lt0 25161 | A polynomial is zero iff i... |
| deg1ldg 25162 | A nonzero univariate polyn... |
| deg1ldgn 25163 | An index at which a polyno... |
| deg1ldgdomn 25164 | A nonzero univariate polyn... |
| deg1leb 25165 | Property of being of limit... |
| deg1val 25166 | Value of the univariate de... |
| deg1lt 25167 | If the degree of a univari... |
| deg1ge 25168 | Conversely, a nonzero coef... |
| coe1mul3 25169 | The coefficient vector of ... |
| coe1mul4 25170 | Value of the "leading" coe... |
| deg1addle 25171 | The degree of a sum is at ... |
| deg1addle2 25172 | If both factors have degre... |
| deg1add 25173 | Exact degree of a sum of t... |
| deg1vscale 25174 | The degree of a scalar tim... |
| deg1vsca 25175 | The degree of a scalar tim... |
| deg1invg 25176 | The degree of the negated ... |
| deg1suble 25177 | The degree of a difference... |
| deg1sub 25178 | Exact degree of a differen... |
| deg1mulle2 25179 | Produce a bound on the pro... |
| deg1sublt 25180 | Subtraction of two polynom... |
| deg1le0 25181 | A polynomial has nonpositi... |
| deg1sclle 25182 | A scalar polynomial has no... |
| deg1scl 25183 | A nonzero scalar polynomia... |
| deg1mul2 25184 | Degree of multiplication o... |
| deg1mul3 25185 | Degree of multiplication o... |
| deg1mul3le 25186 | Degree of multiplication o... |
| deg1tmle 25187 | Limiting degree of a polyn... |
| deg1tm 25188 | Exact degree of a polynomi... |
| deg1pwle 25189 | Limiting degree of a varia... |
| deg1pw 25190 | Exact degree of a variable... |
| ply1nz 25191 | Univariate polynomials ove... |
| ply1nzb 25192 | Univariate polynomials are... |
| ply1domn 25193 | Corollary of ~ deg1mul2 : ... |
| ply1idom 25194 | The ring of univariate pol... |
| ply1divmo 25205 | Uniqueness of a quotient i... |
| ply1divex 25206 | Lemma for ~ ply1divalg : e... |
| ply1divalg 25207 | The division algorithm for... |
| ply1divalg2 25208 | Reverse the order of multi... |
| uc1pval 25209 | Value of the set of unitic... |
| isuc1p 25210 | Being a unitic polynomial.... |
| mon1pval 25211 | Value of the set of monic ... |
| ismon1p 25212 | Being a monic polynomial. ... |
| uc1pcl 25213 | Unitic polynomials are pol... |
| mon1pcl 25214 | Monic polynomials are poly... |
| uc1pn0 25215 | Unitic polynomials are not... |
| mon1pn0 25216 | Monic polynomials are not ... |
| uc1pdeg 25217 | Unitic polynomials have no... |
| uc1pldg 25218 | Unitic polynomials have un... |
| mon1pldg 25219 | Unitic polynomials have on... |
| mon1puc1p 25220 | Monic polynomials are unit... |
| uc1pmon1p 25221 | Make a unitic polynomial m... |
| deg1submon1p 25222 | The difference of two moni... |
| q1pval 25223 | Value of the univariate po... |
| q1peqb 25224 | Characterizing property of... |
| q1pcl 25225 | Closure of the quotient by... |
| r1pval 25226 | Value of the polynomial re... |
| r1pcl 25227 | Closure of remainder follo... |
| r1pdeglt 25228 | The remainder has a degree... |
| r1pid 25229 | Express the original polyn... |
| dvdsq1p 25230 | Divisibility in a polynomi... |
| dvdsr1p 25231 | Divisibility in a polynomi... |
| ply1remlem 25232 | A term of the form ` x - N... |
| ply1rem 25233 | The polynomial remainder t... |
| facth1 25234 | The factor theorem and its... |
| fta1glem1 25235 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 25236 | Lemma for ~ fta1g . (Cont... |
| fta1g 25237 | The one-sided fundamental ... |
| fta1blem 25238 | Lemma for ~ fta1b . (Cont... |
| fta1b 25239 | The assumption that ` R ` ... |
| drnguc1p 25240 | Over a division ring, all ... |
| ig1peu 25241 | There is a unique monic po... |
| ig1pval 25242 | Substitutions for the poly... |
| ig1pval2 25243 | Generator of the zero idea... |
| ig1pval3 25244 | Characterizing properties ... |
| ig1pcl 25245 | The monic generator of an ... |
| ig1pdvds 25246 | The monic generator of an ... |
| ig1prsp 25247 | Any ideal of polynomials o... |
| ply1lpir 25248 | The ring of polynomials ov... |
| ply1pid 25249 | The polynomials over a fie... |
| plyco0 25258 | Two ways to say that a fun... |
| plyval 25259 | Value of the polynomial se... |
| plybss 25260 | Reverse closure of the par... |
| elply 25261 | Definition of a polynomial... |
| elply2 25262 | The coefficient function c... |
| plyun0 25263 | The set of polynomials is ... |
| plyf 25264 | The polynomial is a functi... |
| plyss 25265 | The polynomial set functio... |
| plyssc 25266 | Every polynomial ring is c... |
| elplyr 25267 | Sufficient condition for e... |
| elplyd 25268 | Sufficient condition for e... |
| ply1termlem 25269 | Lemma for ~ ply1term . (C... |
| ply1term 25270 | A one-term polynomial. (C... |
| plypow 25271 | A power is a polynomial. ... |
| plyconst 25272 | A constant function is a p... |
| ne0p 25273 | A test to show that a poly... |
| ply0 25274 | The zero function is a pol... |
| plyid 25275 | The identity function is a... |
| plyeq0lem 25276 | Lemma for ~ plyeq0 . If `... |
| plyeq0 25277 | If a polynomial is zero at... |
| plypf1 25278 | Write the set of complex p... |
| plyaddlem1 25279 | Derive the coefficient fun... |
| plymullem1 25280 | Derive the coefficient fun... |
| plyaddlem 25281 | Lemma for ~ plyadd . (Con... |
| plymullem 25282 | Lemma for ~ plymul . (Con... |
| plyadd 25283 | The sum of two polynomials... |
| plymul 25284 | The product of two polynom... |
| plysub 25285 | The difference of two poly... |
| plyaddcl 25286 | The sum of two polynomials... |
| plymulcl 25287 | The product of two polynom... |
| plysubcl 25288 | The difference of two poly... |
| coeval 25289 | Value of the coefficient f... |
| coeeulem 25290 | Lemma for ~ coeeu . (Cont... |
| coeeu 25291 | Uniqueness of the coeffici... |
| coelem 25292 | Lemma for properties of th... |
| coeeq 25293 | If ` A ` satisfies the pro... |
| dgrval 25294 | Value of the degree functi... |
| dgrlem 25295 | Lemma for ~ dgrcl and simi... |
| coef 25296 | The domain and range of th... |
| coef2 25297 | The domain and range of th... |
| coef3 25298 | The domain and range of th... |
| dgrcl 25299 | The degree of any polynomi... |
| dgrub 25300 | If the ` M ` -th coefficie... |
| dgrub2 25301 | All the coefficients above... |
| dgrlb 25302 | If all the coefficients ab... |
| coeidlem 25303 | Lemma for ~ coeid . (Cont... |
| coeid 25304 | Reconstruct a polynomial a... |
| coeid2 25305 | Reconstruct a polynomial a... |
| coeid3 25306 | Reconstruct a polynomial a... |
| plyco 25307 | The composition of two pol... |
| coeeq2 25308 | Compute the coefficient fu... |
| dgrle 25309 | Given an explicit expressi... |
| dgreq 25310 | If the highest term in a p... |
| 0dgr 25311 | A constant function has de... |
| 0dgrb 25312 | A function has degree zero... |
| dgrnznn 25313 | A nonzero polynomial with ... |
| coefv0 25314 | The result of evaluating a... |
| coeaddlem 25315 | Lemma for ~ coeadd and ~ d... |
| coemullem 25316 | Lemma for ~ coemul and ~ d... |
| coeadd 25317 | The coefficient function o... |
| coemul 25318 | A coefficient of a product... |
| coe11 25319 | The coefficient function i... |
| coemulhi 25320 | The leading coefficient of... |
| coemulc 25321 | The coefficient function i... |
| coe0 25322 | The coefficients of the ze... |
| coesub 25323 | The coefficient function o... |
| coe1termlem 25324 | The coefficient function o... |
| coe1term 25325 | The coefficient function o... |
| dgr1term 25326 | The degree of a monomial. ... |
| plycn 25327 | A polynomial is a continuo... |
| dgr0 25328 | The degree of the zero pol... |
| coeidp 25329 | The coefficients of the id... |
| dgrid 25330 | The degree of the identity... |
| dgreq0 25331 | The leading coefficient of... |
| dgrlt 25332 | Two ways to say that the d... |
| dgradd 25333 | The degree of a sum of pol... |
| dgradd2 25334 | The degree of a sum of pol... |
| dgrmul2 25335 | The degree of a product of... |
| dgrmul 25336 | The degree of a product of... |
| dgrmulc 25337 | Scalar multiplication by a... |
| dgrsub 25338 | The degree of a difference... |
| dgrcolem1 25339 | The degree of a compositio... |
| dgrcolem2 25340 | Lemma for ~ dgrco . (Cont... |
| dgrco 25341 | The degree of a compositio... |
| plycjlem 25342 | Lemma for ~ plycj and ~ co... |
| plycj 25343 | The double conjugation of ... |
| coecj 25344 | Double conjugation of a po... |
| plyrecj 25345 | A polynomial with real coe... |
| plymul0or 25346 | Polynomial multiplication ... |
| ofmulrt 25347 | The set of roots of a prod... |
| plyreres 25348 | Real-coefficient polynomia... |
| dvply1 25349 | Derivative of a polynomial... |
| dvply2g 25350 | The derivative of a polyno... |
| dvply2 25351 | The derivative of a polyno... |
| dvnply2 25352 | Polynomials have polynomia... |
| dvnply 25353 | Polynomials have polynomia... |
| plycpn 25354 | Polynomials are smooth. (... |
| quotval 25357 | Value of the quotient func... |
| plydivlem1 25358 | Lemma for ~ plydivalg . (... |
| plydivlem2 25359 | Lemma for ~ plydivalg . (... |
| plydivlem3 25360 | Lemma for ~ plydivex . Ba... |
| plydivlem4 25361 | Lemma for ~ plydivex . In... |
| plydivex 25362 | Lemma for ~ plydivalg . (... |
| plydiveu 25363 | Lemma for ~ plydivalg . (... |
| plydivalg 25364 | The division algorithm on ... |
| quotlem 25365 | Lemma for properties of th... |
| quotcl 25366 | The quotient of two polyno... |
| quotcl2 25367 | Closure of the quotient fu... |
| quotdgr 25368 | Remainder property of the ... |
| plyremlem 25369 | Closure of a linear factor... |
| plyrem 25370 | The polynomial remainder t... |
| facth 25371 | The factor theorem. If a ... |
| fta1lem 25372 | Lemma for ~ fta1 . (Contr... |
| fta1 25373 | The easy direction of the ... |
| quotcan 25374 | Exact division with a mult... |
| vieta1lem1 25375 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 25376 | Lemma for ~ vieta1 : induc... |
| vieta1 25377 | The first-order Vieta's fo... |
| plyexmo 25378 | An infinite set of values ... |
| elaa 25381 | Elementhood in the set of ... |
| aacn 25382 | An algebraic number is a c... |
| aasscn 25383 | The algebraic numbers are ... |
| elqaalem1 25384 | Lemma for ~ elqaa . The f... |
| elqaalem2 25385 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 25386 | Lemma for ~ elqaa . (Cont... |
| elqaa 25387 | The set of numbers generat... |
| qaa 25388 | Every rational number is a... |
| qssaa 25389 | The rational numbers are c... |
| iaa 25390 | The imaginary unit is alge... |
| aareccl 25391 | The reciprocal of an algeb... |
| aacjcl 25392 | The conjugate of an algebr... |
| aannenlem1 25393 | Lemma for ~ aannen . (Con... |
| aannenlem2 25394 | Lemma for ~ aannen . (Con... |
| aannenlem3 25395 | The algebraic numbers are ... |
| aannen 25396 | The algebraic numbers are ... |
| aalioulem1 25397 | Lemma for ~ aaliou . An i... |
| aalioulem2 25398 | Lemma for ~ aaliou . (Con... |
| aalioulem3 25399 | Lemma for ~ aaliou . (Con... |
| aalioulem4 25400 | Lemma for ~ aaliou . (Con... |
| aalioulem5 25401 | Lemma for ~ aaliou . (Con... |
| aalioulem6 25402 | Lemma for ~ aaliou . (Con... |
| aaliou 25403 | Liouville's theorem on dio... |
| geolim3 25404 | Geometric series convergen... |
| aaliou2 25405 | Liouville's approximation ... |
| aaliou2b 25406 | Liouville's approximation ... |
| aaliou3lem1 25407 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 25408 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 25409 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 25410 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 25411 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 25412 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 25413 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 25414 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 25415 | Example of a "Liouville nu... |
| aaliou3 25416 | Example of a "Liouville nu... |
| taylfvallem1 25421 | Lemma for ~ taylfval . (C... |
| taylfvallem 25422 | Lemma for ~ taylfval . (C... |
| taylfval 25423 | Define the Taylor polynomi... |
| eltayl 25424 | Value of the Taylor series... |
| taylf 25425 | The Taylor series defines ... |
| tayl0 25426 | The Taylor series is alway... |
| taylplem1 25427 | Lemma for ~ taylpfval and ... |
| taylplem2 25428 | Lemma for ~ taylpfval and ... |
| taylpfval 25429 | Define the Taylor polynomi... |
| taylpf 25430 | The Taylor polynomial is a... |
| taylpval 25431 | Value of the Taylor polyno... |
| taylply2 25432 | The Taylor polynomial is a... |
| taylply 25433 | The Taylor polynomial is a... |
| dvtaylp 25434 | The derivative of the Tayl... |
| dvntaylp 25435 | The ` M ` -th derivative o... |
| dvntaylp0 25436 | The first ` N ` derivative... |
| taylthlem1 25437 | Lemma for ~ taylth . This... |
| taylthlem2 25438 | Lemma for ~ taylth . (Con... |
| taylth 25439 | Taylor's theorem. The Tay... |
| ulmrel 25442 | The uniform limit relation... |
| ulmscl 25443 | Closure of the base set in... |
| ulmval 25444 | Express the predicate: Th... |
| ulmcl 25445 | Closure of a uniform limit... |
| ulmf 25446 | Closure of a uniform limit... |
| ulmpm 25447 | Closure of a uniform limit... |
| ulmf2 25448 | Closure of a uniform limit... |
| ulm2 25449 | Simplify ~ ulmval when ` F... |
| ulmi 25450 | The uniform limit property... |
| ulmclm 25451 | A uniform limit of functio... |
| ulmres 25452 | A sequence of functions co... |
| ulmshftlem 25453 | Lemma for ~ ulmshft . (Co... |
| ulmshft 25454 | A sequence of functions co... |
| ulm0 25455 | Every function converges u... |
| ulmuni 25456 | A sequence of functions un... |
| ulmdm 25457 | Two ways to express that a... |
| ulmcaulem 25458 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 25459 | A sequence of functions co... |
| ulmcau2 25460 | A sequence of functions co... |
| ulmss 25461 | A uniform limit of functio... |
| ulmbdd 25462 | A uniform limit of bounded... |
| ulmcn 25463 | A uniform limit of continu... |
| ulmdvlem1 25464 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 25465 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 25466 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 25467 | If ` F ` is a sequence of ... |
| mtest 25468 | The Weierstrass M-test. I... |
| mtestbdd 25469 | Given the hypotheses of th... |
| mbfulm 25470 | A uniform limit of measura... |
| iblulm 25471 | A uniform limit of integra... |
| itgulm 25472 | A uniform limit of integra... |
| itgulm2 25473 | A uniform limit of integra... |
| pserval 25474 | Value of the function ` G ... |
| pserval2 25475 | Value of the function ` G ... |
| psergf 25476 | The sequence of terms in t... |
| radcnvlem1 25477 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 25478 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 25479 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 25480 | Zero is always a convergen... |
| radcnvcl 25481 | The radius of convergence ... |
| radcnvlt1 25482 | If ` X ` is within the ope... |
| radcnvlt2 25483 | If ` X ` is within the ope... |
| radcnvle 25484 | If ` X ` is a convergent p... |
| dvradcnv 25485 | The radius of convergence ... |
| pserulm 25486 | If ` S ` is a region conta... |
| psercn2 25487 | Since by ~ pserulm the ser... |
| psercnlem2 25488 | Lemma for ~ psercn . (Con... |
| psercnlem1 25489 | Lemma for ~ psercn . (Con... |
| psercn 25490 | An infinite series converg... |
| pserdvlem1 25491 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 25492 | Lemma for ~ pserdv . (Con... |
| pserdv 25493 | The derivative of a power ... |
| pserdv2 25494 | The derivative of a power ... |
| abelthlem1 25495 | Lemma for ~ abelth . (Con... |
| abelthlem2 25496 | Lemma for ~ abelth . The ... |
| abelthlem3 25497 | Lemma for ~ abelth . (Con... |
| abelthlem4 25498 | Lemma for ~ abelth . (Con... |
| abelthlem5 25499 | Lemma for ~ abelth . (Con... |
| abelthlem6 25500 | Lemma for ~ abelth . (Con... |
| abelthlem7a 25501 | Lemma for ~ abelth . (Con... |
| abelthlem7 25502 | Lemma for ~ abelth . (Con... |
| abelthlem8 25503 | Lemma for ~ abelth . (Con... |
| abelthlem9 25504 | Lemma for ~ abelth . By a... |
| abelth 25505 | Abel's theorem. If the po... |
| abelth2 25506 | Abel's theorem, restricted... |
| efcn 25507 | The exponential function i... |
| sincn 25508 | Sine is continuous. (Cont... |
| coscn 25509 | Cosine is continuous. (Co... |
| reeff1olem 25510 | Lemma for ~ reeff1o . (Co... |
| reeff1o 25511 | The real exponential funct... |
| reefiso 25512 | The exponential function o... |
| efcvx 25513 | The exponential function o... |
| reefgim 25514 | The exponential function i... |
| pilem1 25515 | Lemma for ~ pire , ~ pigt2... |
| pilem2 25516 | Lemma for ~ pire , ~ pigt2... |
| pilem3 25517 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 25518 | ` _pi ` is between 2 and 4... |
| sinpi 25519 | The sine of ` _pi ` is 0. ... |
| pire 25520 | ` _pi ` is a real number. ... |
| picn 25521 | ` _pi ` is a complex numbe... |
| pipos 25522 | ` _pi ` is positive. (Con... |
| pirp 25523 | ` _pi ` is a positive real... |
| negpicn 25524 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 25525 | Lemma for ~ sinhalfpi and ... |
| halfpire 25526 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 25527 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 25528 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 25529 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 25530 | The sine of ` _pi / 2 ` is... |
| coshalfpi 25531 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 25532 | The cosine of ` -u _pi / 2... |
| efhalfpi 25533 | The exponential of ` _i _p... |
| cospi 25534 | The cosine of ` _pi ` is `... |
| efipi 25535 | The exponential of ` _i x.... |
| eulerid 25536 | Euler's identity. (Contri... |
| sin2pi 25537 | The sine of ` 2 _pi ` is 0... |
| cos2pi 25538 | The cosine of ` 2 _pi ` is... |
| ef2pi 25539 | The exponential of ` 2 _pi... |
| ef2kpi 25540 | If ` K ` is an integer, th... |
| efper 25541 | The exponential function i... |
| sinperlem 25542 | Lemma for ~ sinper and ~ c... |
| sinper 25543 | The sine function is perio... |
| cosper 25544 | The cosine function is per... |
| sin2kpi 25545 | If ` K ` is an integer, th... |
| cos2kpi 25546 | If ` K ` is an integer, th... |
| sin2pim 25547 | Sine of a number subtracte... |
| cos2pim 25548 | Cosine of a number subtrac... |
| sinmpi 25549 | Sine of a number less ` _p... |
| cosmpi 25550 | Cosine of a number less ` ... |
| sinppi 25551 | Sine of a number plus ` _p... |
| cosppi 25552 | Cosine of a number plus ` ... |
| efimpi 25553 | The exponential function a... |
| sinhalfpip 25554 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 25555 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 25556 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 25557 | The cosine of ` _pi / 2 ` ... |
| ptolemy 25558 | Ptolemy's Theorem. This t... |
| sincosq1lem 25559 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 25560 | The signs of the sine and ... |
| sincosq2sgn 25561 | The signs of the sine and ... |
| sincosq3sgn 25562 | The signs of the sine and ... |
| sincosq4sgn 25563 | The signs of the sine and ... |
| coseq00topi 25564 | Location of the zeroes of ... |
| coseq0negpitopi 25565 | Location of the zeroes of ... |
| tanrpcl 25566 | Positive real closure of t... |
| tangtx 25567 | The tangent function is gr... |
| tanabsge 25568 | The tangent function is gr... |
| sinq12gt0 25569 | The sine of a number stric... |
| sinq12ge0 25570 | The sine of a number betwe... |
| sinq34lt0t 25571 | The sine of a number stric... |
| cosq14gt0 25572 | The cosine of a number str... |
| cosq14ge0 25573 | The cosine of a number bet... |
| sincosq1eq 25574 | Complementarity of the sin... |
| sincos4thpi 25575 | The sine and cosine of ` _... |
| tan4thpi 25576 | The tangent of ` _pi / 4 `... |
| sincos6thpi 25577 | The sine and cosine of ` _... |
| sincos3rdpi 25578 | The sine and cosine of ` _... |
| pigt3 25579 | ` _pi ` is greater than 3.... |
| pige3 25580 | ` _pi ` is greater than or... |
| pige3ALT 25581 | Alternate proof of ~ pige3... |
| abssinper 25582 | The absolute value of sine... |
| sinkpi 25583 | The sine of an integer mul... |
| coskpi 25584 | The absolute value of the ... |
| sineq0 25585 | A complex number whose sin... |
| coseq1 25586 | A complex number whose cos... |
| cos02pilt1 25587 | Cosine is less than one be... |
| cosq34lt1 25588 | Cosine is less than one in... |
| efeq1 25589 | A complex number whose exp... |
| cosne0 25590 | The cosine function has no... |
| cosordlem 25591 | Lemma for ~ cosord . (Con... |
| cosord 25592 | Cosine is decreasing over ... |
| cos0pilt1 25593 | Cosine is between minus on... |
| cos11 25594 | Cosine is one-to-one over ... |
| sinord 25595 | Sine is increasing over th... |
| recosf1o 25596 | The cosine function is a b... |
| resinf1o 25597 | The sine function is a bij... |
| tanord1 25598 | The tangent function is st... |
| tanord 25599 | The tangent function is st... |
| tanregt0 25600 | The real part of the tange... |
| negpitopissre 25601 | The interval ` ( -u _pi (,... |
| efgh 25602 | The exponential function o... |
| efif1olem1 25603 | Lemma for ~ efif1o . (Con... |
| efif1olem2 25604 | Lemma for ~ efif1o . (Con... |
| efif1olem3 25605 | Lemma for ~ efif1o . (Con... |
| efif1olem4 25606 | The exponential function o... |
| efif1o 25607 | The exponential function o... |
| efifo 25608 | The exponential function o... |
| eff1olem 25609 | The exponential function m... |
| eff1o 25610 | The exponential function m... |
| efabl 25611 | The image of a subgroup of... |
| efsubm 25612 | The image of a subgroup of... |
| circgrp 25613 | The circle group ` T ` is ... |
| circsubm 25614 | The circle group ` T ` is ... |
| logrn 25619 | The range of the natural l... |
| ellogrn 25620 | Write out the property ` A... |
| dflog2 25621 | The natural logarithm func... |
| relogrn 25622 | The range of the natural l... |
| logrncn 25623 | The range of the natural l... |
| eff1o2 25624 | The exponential function r... |
| logf1o 25625 | The natural logarithm func... |
| dfrelog 25626 | The natural logarithm func... |
| relogf1o 25627 | The natural logarithm func... |
| logrncl 25628 | Closure of the natural log... |
| logcl 25629 | Closure of the natural log... |
| logimcl 25630 | Closure of the imaginary p... |
| logcld 25631 | The logarithm of a nonzero... |
| logimcld 25632 | The imaginary part of the ... |
| logimclad 25633 | The imaginary part of the ... |
| abslogimle 25634 | The imaginary part of the ... |
| logrnaddcl 25635 | The range of the natural l... |
| relogcl 25636 | Closure of the natural log... |
| eflog 25637 | Relationship between the n... |
| logeq0im1 25638 | If the logarithm of a numb... |
| logccne0 25639 | The logarithm isn't 0 if i... |
| logne0 25640 | Logarithm of a non-1 posit... |
| reeflog 25641 | Relationship between the n... |
| logef 25642 | Relationship between the n... |
| relogef 25643 | Relationship between the n... |
| logeftb 25644 | Relationship between the n... |
| relogeftb 25645 | Relationship between the n... |
| log1 25646 | The natural logarithm of `... |
| loge 25647 | The natural logarithm of `... |
| logneg 25648 | The natural logarithm of a... |
| logm1 25649 | The natural logarithm of n... |
| lognegb 25650 | If a number has imaginary ... |
| relogoprlem 25651 | Lemma for ~ relogmul and ~... |
| relogmul 25652 | The natural logarithm of t... |
| relogdiv 25653 | The natural logarithm of t... |
| explog 25654 | Exponentiation of a nonzer... |
| reexplog 25655 | Exponentiation of a positi... |
| relogexp 25656 | The natural logarithm of p... |
| relog 25657 | Real part of a logarithm. ... |
| relogiso 25658 | The natural logarithm func... |
| reloggim 25659 | The natural logarithm is a... |
| logltb 25660 | The natural logarithm func... |
| logfac 25661 | The logarithm of a factori... |
| eflogeq 25662 | Solve an equation involvin... |
| logleb 25663 | Natural logarithm preserve... |
| rplogcl 25664 | Closure of the logarithm f... |
| logge0 25665 | The logarithm of a number ... |
| logcj 25666 | The natural logarithm dist... |
| efiarg 25667 | The exponential of the "ar... |
| cosargd 25668 | The cosine of the argument... |
| cosarg0d 25669 | The cosine of the argument... |
| argregt0 25670 | Closure of the argument of... |
| argrege0 25671 | Closure of the argument of... |
| argimgt0 25672 | Closure of the argument of... |
| argimlt0 25673 | Closure of the argument of... |
| logimul 25674 | Multiplying a number by ` ... |
| logneg2 25675 | The logarithm of the negat... |
| logmul2 25676 | Generalization of ~ relogm... |
| logdiv2 25677 | Generalization of ~ relogd... |
| abslogle 25678 | Bound on the magnitude of ... |
| tanarg 25679 | The basic relation between... |
| logdivlti 25680 | The ` log x / x ` function... |
| logdivlt 25681 | The ` log x / x ` function... |
| logdivle 25682 | The ` log x / x ` function... |
| relogcld 25683 | Closure of the natural log... |
| reeflogd 25684 | Relationship between the n... |
| relogmuld 25685 | The natural logarithm of t... |
| relogdivd 25686 | The natural logarithm of t... |
| logled 25687 | Natural logarithm preserve... |
| relogefd 25688 | Relationship between the n... |
| rplogcld 25689 | Closure of the logarithm f... |
| logge0d 25690 | The logarithm of a number ... |
| logge0b 25691 | The logarithm of a number ... |
| loggt0b 25692 | The logarithm of a number ... |
| logle1b 25693 | The logarithm of a number ... |
| loglt1b 25694 | The logarithm of a number ... |
| divlogrlim 25695 | The inverse logarithm func... |
| logno1 25696 | The logarithm function is ... |
| dvrelog 25697 | The derivative of the real... |
| relogcn 25698 | The real logarithm functio... |
| ellogdm 25699 | Elementhood in the "contin... |
| logdmn0 25700 | A number in the continuous... |
| logdmnrp 25701 | A number in the continuous... |
| logdmss 25702 | The continuity domain of `... |
| logcnlem2 25703 | Lemma for ~ logcn . (Cont... |
| logcnlem3 25704 | Lemma for ~ logcn . (Cont... |
| logcnlem4 25705 | Lemma for ~ logcn . (Cont... |
| logcnlem5 25706 | Lemma for ~ logcn . (Cont... |
| logcn 25707 | The logarithm function is ... |
| dvloglem 25708 | Lemma for ~ dvlog . (Cont... |
| logdmopn 25709 | The "continuous domain" of... |
| logf1o2 25710 | The logarithm maps its con... |
| dvlog 25711 | The derivative of the comp... |
| dvlog2lem 25712 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 25713 | The derivative of the comp... |
| advlog 25714 | The antiderivative of the ... |
| advlogexp 25715 | The antiderivative of a po... |
| efopnlem1 25716 | Lemma for ~ efopn . (Cont... |
| efopnlem2 25717 | Lemma for ~ efopn . (Cont... |
| efopn 25718 | The exponential map is an ... |
| logtayllem 25719 | Lemma for ~ logtayl . (Co... |
| logtayl 25720 | The Taylor series for ` -u... |
| logtaylsum 25721 | The Taylor series for ` -u... |
| logtayl2 25722 | Power series expression fo... |
| logccv 25723 | The natural logarithm func... |
| cxpval 25724 | Value of the complex power... |
| cxpef 25725 | Value of the complex power... |
| 0cxp 25726 | Value of the complex power... |
| cxpexpz 25727 | Relate the complex power f... |
| cxpexp 25728 | Relate the complex power f... |
| logcxp 25729 | Logarithm of a complex pow... |
| cxp0 25730 | Value of the complex power... |
| cxp1 25731 | Value of the complex power... |
| 1cxp 25732 | Value of the complex power... |
| ecxp 25733 | Write the exponential func... |
| cxpcl 25734 | Closure of the complex pow... |
| recxpcl 25735 | Real closure of the comple... |
| rpcxpcl 25736 | Positive real closure of t... |
| cxpne0 25737 | Complex exponentiation is ... |
| cxpeq0 25738 | Complex exponentiation is ... |
| cxpadd 25739 | Sum of exponents law for c... |
| cxpp1 25740 | Value of a nonzero complex... |
| cxpneg 25741 | Value of a complex number ... |
| cxpsub 25742 | Exponent subtraction law f... |
| cxpge0 25743 | Nonnegative exponentiation... |
| mulcxplem 25744 | Lemma for ~ mulcxp . (Con... |
| mulcxp 25745 | Complex exponentiation of ... |
| cxprec 25746 | Complex exponentiation of ... |
| divcxp 25747 | Complex exponentiation of ... |
| cxpmul 25748 | Product of exponents law f... |
| cxpmul2 25749 | Product of exponents law f... |
| cxproot 25750 | The complex power function... |
| cxpmul2z 25751 | Generalize ~ cxpmul2 to ne... |
| abscxp 25752 | Absolute value of a power,... |
| abscxp2 25753 | Absolute value of a power,... |
| cxplt 25754 | Ordering property for comp... |
| cxple 25755 | Ordering property for comp... |
| cxplea 25756 | Ordering property for comp... |
| cxple2 25757 | Ordering property for comp... |
| cxplt2 25758 | Ordering property for comp... |
| cxple2a 25759 | Ordering property for comp... |
| cxplt3 25760 | Ordering property for comp... |
| cxple3 25761 | Ordering property for comp... |
| cxpsqrtlem 25762 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 25763 | The complex exponential fu... |
| logsqrt 25764 | Logarithm of a square root... |
| cxp0d 25765 | Value of the complex power... |
| cxp1d 25766 | Value of the complex power... |
| 1cxpd 25767 | Value of the complex power... |
| cxpcld 25768 | Closure of the complex pow... |
| cxpmul2d 25769 | Product of exponents law f... |
| 0cxpd 25770 | Value of the complex power... |
| cxpexpzd 25771 | Relate the complex power f... |
| cxpefd 25772 | Value of the complex power... |
| cxpne0d 25773 | Complex exponentiation is ... |
| cxpp1d 25774 | Value of a nonzero complex... |
| cxpnegd 25775 | Value of a complex number ... |
| cxpmul2zd 25776 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 25777 | Sum of exponents law for c... |
| cxpsubd 25778 | Exponent subtraction law f... |
| cxpltd 25779 | Ordering property for comp... |
| cxpled 25780 | Ordering property for comp... |
| cxplead 25781 | Ordering property for comp... |
| divcxpd 25782 | Complex exponentiation of ... |
| recxpcld 25783 | Positive real closure of t... |
| cxpge0d 25784 | Nonnegative exponentiation... |
| cxple2ad 25785 | Ordering property for comp... |
| cxplt2d 25786 | Ordering property for comp... |
| cxple2d 25787 | Ordering property for comp... |
| mulcxpd 25788 | Complex exponentiation of ... |
| cxpsqrtth 25789 | Square root theorem over t... |
| 2irrexpq 25790 | There exist irrational num... |
| cxprecd 25791 | Complex exponentiation of ... |
| rpcxpcld 25792 | Positive real closure of t... |
| logcxpd 25793 | Logarithm of a complex pow... |
| cxplt3d 25794 | Ordering property for comp... |
| cxple3d 25795 | Ordering property for comp... |
| cxpmuld 25796 | Product of exponents law f... |
| cxpcom 25797 | Commutative law for real e... |
| dvcxp1 25798 | The derivative of a comple... |
| dvcxp2 25799 | The derivative of a comple... |
| dvsqrt 25800 | The derivative of the real... |
| dvcncxp1 25801 | Derivative of complex powe... |
| dvcnsqrt 25802 | Derivative of square root ... |
| cxpcn 25803 | Domain of continuity of th... |
| cxpcn2 25804 | Continuity of the complex ... |
| cxpcn3lem 25805 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 25806 | Extend continuity of the c... |
| resqrtcn 25807 | Continuity of the real squ... |
| sqrtcn 25808 | Continuity of the square r... |
| cxpaddlelem 25809 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 25810 | Ordering property for comp... |
| abscxpbnd 25811 | Bound on the absolute valu... |
| root1id 25812 | Property of an ` N ` -th r... |
| root1eq1 25813 | The only powers of an ` N ... |
| root1cj 25814 | Within the ` N ` -th roots... |
| cxpeq 25815 | Solve an equation involvin... |
| loglesqrt 25816 | An upper bound on the loga... |
| logreclem 25817 | Symmetry of the natural lo... |
| logrec 25818 | Logarithm of a reciprocal ... |
| logbval 25821 | Define the value of the ` ... |
| logbcl 25822 | General logarithm closure.... |
| logbid1 25823 | General logarithm is 1 whe... |
| logb1 25824 | The logarithm of ` 1 ` to ... |
| elogb 25825 | The general logarithm of a... |
| logbchbase 25826 | Change of base for logarit... |
| relogbval 25827 | Value of the general logar... |
| relogbcl 25828 | Closure of the general log... |
| relogbzcl 25829 | Closure of the general log... |
| relogbreexp 25830 | Power law for the general ... |
| relogbzexp 25831 | Power law for the general ... |
| relogbmul 25832 | The logarithm of the produ... |
| relogbmulexp 25833 | The logarithm of the produ... |
| relogbdiv 25834 | The logarithm of the quoti... |
| relogbexp 25835 | Identity law for general l... |
| nnlogbexp 25836 | Identity law for general l... |
| logbrec 25837 | Logarithm of a reciprocal ... |
| logbleb 25838 | The general logarithm func... |
| logblt 25839 | The general logarithm func... |
| relogbcxp 25840 | Identity law for the gener... |
| cxplogb 25841 | Identity law for the gener... |
| relogbcxpb 25842 | The logarithm is the inver... |
| logbmpt 25843 | The general logarithm to a... |
| logbf 25844 | The general logarithm to a... |
| logbfval 25845 | The general logarithm of a... |
| relogbf 25846 | The general logarithm to a... |
| logblog 25847 | The general logarithm to t... |
| logbgt0b 25848 | The logarithm of a positiv... |
| logbgcd1irr 25849 | The logarithm of an intege... |
| 2logb9irr 25850 | Example for ~ logbgcd1irr ... |
| logbprmirr 25851 | The logarithm of a prime t... |
| 2logb3irr 25852 | Example for ~ logbprmirr .... |
| 2logb9irrALT 25853 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 25854 | The square root of two to ... |
| 2irrexpqALT 25855 | Alternate proof of ~ 2irre... |
| angval 25856 | Define the angle function,... |
| angcan 25857 | Cancel a constant multipli... |
| angneg 25858 | Cancel a negative sign in ... |
| angvald 25859 | The (signed) angle between... |
| angcld 25860 | The (signed) angle between... |
| angrteqvd 25861 | Two vectors are at a right... |
| cosangneg2d 25862 | The cosine of the angle be... |
| angrtmuld 25863 | Perpendicularity of two ve... |
| ang180lem1 25864 | Lemma for ~ ang180 . Show... |
| ang180lem2 25865 | Lemma for ~ ang180 . Show... |
| ang180lem3 25866 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 25867 | Lemma for ~ ang180 . Redu... |
| ang180lem5 25868 | Lemma for ~ ang180 : Redu... |
| ang180 25869 | The sum of angles ` m A B ... |
| lawcoslem1 25870 | Lemma for ~ lawcos . Here... |
| lawcos 25871 | Law of cosines (also known... |
| pythag 25872 | Pythagorean theorem. Give... |
| isosctrlem1 25873 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 25874 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 25875 | Lemma for ~ isosctr . Cor... |
| isosctr 25876 | Isosceles triangle theorem... |
| ssscongptld 25877 | If two triangles have equa... |
| affineequiv 25878 | Equivalence between two wa... |
| affineequiv2 25879 | Equivalence between two wa... |
| affineequiv3 25880 | Equivalence between two wa... |
| affineequiv4 25881 | Equivalence between two wa... |
| affineequivne 25882 | Equivalence between two wa... |
| angpieqvdlem 25883 | Equivalence used in the pr... |
| angpieqvdlem2 25884 | Equivalence used in ~ angp... |
| angpined 25885 | If the angle at ABC is ` _... |
| angpieqvd 25886 | The angle ABC is ` _pi ` i... |
| chordthmlem 25887 | If ` M ` is the midpoint o... |
| chordthmlem2 25888 | If M is the midpoint of AB... |
| chordthmlem3 25889 | If M is the midpoint of AB... |
| chordthmlem4 25890 | If P is on the segment AB ... |
| chordthmlem5 25891 | If P is on the segment AB ... |
| chordthm 25892 | The intersecting chords th... |
| heron 25893 | Heron's formula gives the ... |
| quad2 25894 | The quadratic equation, wi... |
| quad 25895 | The quadratic equation. (... |
| 1cubrlem 25896 | The cube roots of unity. ... |
| 1cubr 25897 | The cube roots of unity. ... |
| dcubic1lem 25898 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 25899 | Reverse direction of ~ dcu... |
| dcubic1 25900 | Forward direction of ~ dcu... |
| dcubic 25901 | Solutions to the depressed... |
| mcubic 25902 | Solutions to a monic cubic... |
| cubic2 25903 | The solution to the genera... |
| cubic 25904 | The cubic equation, which ... |
| binom4 25905 | Work out a quartic binomia... |
| dquartlem1 25906 | Lemma for ~ dquart . (Con... |
| dquartlem2 25907 | Lemma for ~ dquart . (Con... |
| dquart 25908 | Solve a depressed quartic ... |
| quart1cl 25909 | Closure lemmas for ~ quart... |
| quart1lem 25910 | Lemma for ~ quart1 . (Con... |
| quart1 25911 | Depress a quartic equation... |
| quartlem1 25912 | Lemma for ~ quart . (Cont... |
| quartlem2 25913 | Closure lemmas for ~ quart... |
| quartlem3 25914 | Closure lemmas for ~ quart... |
| quartlem4 25915 | Closure lemmas for ~ quart... |
| quart 25916 | The quartic equation, writ... |
| asinlem 25923 | The argument to the logari... |
| asinlem2 25924 | The argument to the logari... |
| asinlem3a 25925 | Lemma for ~ asinlem3 . (C... |
| asinlem3 25926 | The argument to the logari... |
| asinf 25927 | Domain and range of the ar... |
| asincl 25928 | Closure for the arcsin fun... |
| acosf 25929 | Domain and range of the ar... |
| acoscl 25930 | Closure for the arccos fun... |
| atandm 25931 | Since the property is a li... |
| atandm2 25932 | This form of ~ atandm is a... |
| atandm3 25933 | A compact form of ~ atandm... |
| atandm4 25934 | A compact form of ~ atandm... |
| atanf 25935 | Domain and range of the ar... |
| atancl 25936 | Closure for the arctan fun... |
| asinval 25937 | Value of the arcsin functi... |
| acosval 25938 | Value of the arccos functi... |
| atanval 25939 | Value of the arctan functi... |
| atanre 25940 | A real number is in the do... |
| asinneg 25941 | The arcsine function is od... |
| acosneg 25942 | The negative symmetry rela... |
| efiasin 25943 | The exponential of the arc... |
| sinasin 25944 | The arcsine function is an... |
| cosacos 25945 | The arccosine function is ... |
| asinsinlem 25946 | Lemma for ~ asinsin . (Co... |
| asinsin 25947 | The arcsine function compo... |
| acoscos 25948 | The arccosine function is ... |
| asin1 25949 | The arcsine of ` 1 ` is ` ... |
| acos1 25950 | The arccosine of ` 1 ` is ... |
| reasinsin 25951 | The arcsine function compo... |
| asinsinb 25952 | Relationship between sine ... |
| acoscosb 25953 | Relationship between cosin... |
| asinbnd 25954 | The arcsine function has r... |
| acosbnd 25955 | The arccosine function has... |
| asinrebnd 25956 | Bounds on the arcsine func... |
| asinrecl 25957 | The arcsine function is re... |
| acosrecl 25958 | The arccosine function is ... |
| cosasin 25959 | The cosine of the arcsine ... |
| sinacos 25960 | The sine of the arccosine ... |
| atandmneg 25961 | The domain of the arctange... |
| atanneg 25962 | The arctangent function is... |
| atan0 25963 | The arctangent of zero is ... |
| atandmcj 25964 | The arctangent function di... |
| atancj 25965 | The arctangent function di... |
| atanrecl 25966 | The arctangent function is... |
| efiatan 25967 | Value of the exponential o... |
| atanlogaddlem 25968 | Lemma for ~ atanlogadd . ... |
| atanlogadd 25969 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 25970 | Lemma for ~ atanlogsub . ... |
| atanlogsub 25971 | A variation on ~ atanlogad... |
| efiatan2 25972 | Value of the exponential o... |
| 2efiatan 25973 | Value of the exponential o... |
| tanatan 25974 | The arctangent function is... |
| atandmtan 25975 | The tangent function has r... |
| cosatan 25976 | The cosine of an arctangen... |
| cosatanne0 25977 | The arctangent function ha... |
| atantan 25978 | The arctangent function is... |
| atantanb 25979 | Relationship between tange... |
| atanbndlem 25980 | Lemma for ~ atanbnd . (Co... |
| atanbnd 25981 | The arctangent function is... |
| atanord 25982 | The arctangent function is... |
| atan1 25983 | The arctangent of ` 1 ` is... |
| bndatandm 25984 | A point in the open unit d... |
| atans 25985 | The "domain of continuity"... |
| atans2 25986 | It suffices to show that `... |
| atansopn 25987 | The domain of continuity o... |
| atansssdm 25988 | The domain of continuity o... |
| ressatans 25989 | The real number line is a ... |
| dvatan 25990 | The derivative of the arct... |
| atancn 25991 | The arctangent is a contin... |
| atantayl 25992 | The Taylor series for ` ar... |
| atantayl2 25993 | The Taylor series for ` ar... |
| atantayl3 25994 | The Taylor series for ` ar... |
| leibpilem1 25995 | Lemma for ~ leibpi . (Con... |
| leibpilem2 25996 | The Leibniz formula for ` ... |
| leibpi 25997 | The Leibniz formula for ` ... |
| leibpisum 25998 | The Leibniz formula for ` ... |
| log2cnv 25999 | Using the Taylor series fo... |
| log2tlbnd 26000 | Bound the error term in th... |
| log2ublem1 26001 | Lemma for ~ log2ub . The ... |
| log2ublem2 26002 | Lemma for ~ log2ub . (Con... |
| log2ublem3 26003 | Lemma for ~ log2ub . In d... |
| log2ub 26004 | ` log 2 ` is less than ` 2... |
| log2le1 26005 | ` log 2 ` is less than ` 1... |
| birthdaylem1 26006 | Lemma for ~ birthday . (C... |
| birthdaylem2 26007 | For general ` N ` and ` K ... |
| birthdaylem3 26008 | For general ` N ` and ` K ... |
| birthday 26009 | The Birthday Problem. The... |
| dmarea 26012 | The domain of the area fun... |
| areambl 26013 | The fibers of a measurable... |
| areass 26014 | A measurable region is a s... |
| dfarea 26015 | Rewrite ~ df-area self-ref... |
| areaf 26016 | Area measurement is a func... |
| areacl 26017 | The area of a measurable r... |
| areage0 26018 | The area of a measurable r... |
| areaval 26019 | The area of a measurable r... |
| rlimcnp 26020 | Relate a limit of a real-v... |
| rlimcnp2 26021 | Relate a limit of a real-v... |
| rlimcnp3 26022 | Relate a limit of a real-v... |
| xrlimcnp 26023 | Relate a limit of a real-v... |
| efrlim 26024 | The limit of the sequence ... |
| dfef2 26025 | The limit of the sequence ... |
| cxplim 26026 | A power to a negative expo... |
| sqrtlim 26027 | The inverse square root fu... |
| rlimcxp 26028 | Any power to a positive ex... |
| o1cxp 26029 | An eventually bounded func... |
| cxp2limlem 26030 | A linear factor grows slow... |
| cxp2lim 26031 | Any power grows slower tha... |
| cxploglim 26032 | The logarithm grows slower... |
| cxploglim2 26033 | Every power of the logarit... |
| divsqrtsumlem 26034 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 26035 | The function ` F ` used in... |
| divsqrsum 26036 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 26037 | A bound on the distance of... |
| divsqrtsumo1 26038 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 26039 | Closure of a 0-1 linear co... |
| scvxcvx 26040 | A strictly convex function... |
| jensenlem1 26041 | Lemma for ~ jensen . (Con... |
| jensenlem2 26042 | Lemma for ~ jensen . (Con... |
| jensen 26043 | Jensen's inequality, a fin... |
| amgmlem 26044 | Lemma for ~ amgm . (Contr... |
| amgm 26045 | Inequality of arithmetic a... |
| logdifbnd 26048 | Bound on the difference of... |
| logdiflbnd 26049 | Lower bound on the differe... |
| emcllem1 26050 | Lemma for ~ emcl . The se... |
| emcllem2 26051 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 26052 | Lemma for ~ emcl . The fu... |
| emcllem4 26053 | Lemma for ~ emcl . The di... |
| emcllem5 26054 | Lemma for ~ emcl . The pa... |
| emcllem6 26055 | Lemma for ~ emcl . By the... |
| emcllem7 26056 | Lemma for ~ emcl and ~ har... |
| emcl 26057 | Closure and bounds for the... |
| harmonicbnd 26058 | A bound on the harmonic se... |
| harmonicbnd2 26059 | A bound on the harmonic se... |
| emre 26060 | The Euler-Mascheroni const... |
| emgt0 26061 | The Euler-Mascheroni const... |
| harmonicbnd3 26062 | A bound on the harmonic se... |
| harmoniclbnd 26063 | A bound on the harmonic se... |
| harmonicubnd 26064 | A bound on the harmonic se... |
| harmonicbnd4 26065 | The asymptotic behavior of... |
| fsumharmonic 26066 | Bound a finite sum based o... |
| zetacvg 26069 | The zeta series is converg... |
| eldmgm 26076 | Elementhood in the set of ... |
| dmgmaddn0 26077 | If ` A ` is not a nonposit... |
| dmlogdmgm 26078 | If ` A ` is in the continu... |
| rpdmgm 26079 | A positive real number is ... |
| dmgmn0 26080 | If ` A ` is not a nonposit... |
| dmgmaddnn0 26081 | If ` A ` is not a nonposit... |
| dmgmdivn0 26082 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 26083 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 26084 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 26085 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 26086 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 26087 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 26088 | The series ` G ` is unifor... |
| lgamgulm 26089 | The series ` G ` is unifor... |
| lgamgulm2 26090 | Rewrite the limit of the s... |
| lgambdd 26091 | The log-Gamma function is ... |
| lgamucov 26092 | The ` U ` regions used in ... |
| lgamucov2 26093 | The ` U ` regions used in ... |
| lgamcvglem 26094 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 26095 | The log-Gamma function is ... |
| lgamf 26096 | The log-Gamma function is ... |
| gamf 26097 | The Gamma function is a co... |
| gamcl 26098 | The exponential of the log... |
| eflgam 26099 | The exponential of the log... |
| gamne0 26100 | The Gamma function is neve... |
| igamval 26101 | Value of the inverse Gamma... |
| igamz 26102 | Value of the inverse Gamma... |
| igamgam 26103 | Value of the inverse Gamma... |
| igamlgam 26104 | Value of the inverse Gamma... |
| igamf 26105 | Closure of the inverse Gam... |
| igamcl 26106 | Closure of the inverse Gam... |
| gamigam 26107 | The Gamma function is the ... |
| lgamcvg 26108 | The series ` G ` converges... |
| lgamcvg2 26109 | The series ` G ` converges... |
| gamcvg 26110 | The pointwise exponential ... |
| lgamp1 26111 | The functional equation of... |
| gamp1 26112 | The functional equation of... |
| gamcvg2lem 26113 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 26114 | An infinite product expres... |
| regamcl 26115 | The Gamma function is real... |
| relgamcl 26116 | The log-Gamma function is ... |
| rpgamcl 26117 | The log-Gamma function is ... |
| lgam1 26118 | The log-Gamma function at ... |
| gam1 26119 | The log-Gamma function at ... |
| facgam 26120 | The Gamma function general... |
| gamfac 26121 | The Gamma function general... |
| wilthlem1 26122 | The only elements that are... |
| wilthlem2 26123 | Lemma for ~ wilth : induct... |
| wilthlem3 26124 | Lemma for ~ wilth . Here ... |
| wilth 26125 | Wilson's theorem. A numbe... |
| wilthimp 26126 | The forward implication of... |
| ftalem1 26127 | Lemma for ~ fta : "growth... |
| ftalem2 26128 | Lemma for ~ fta . There e... |
| ftalem3 26129 | Lemma for ~ fta . There e... |
| ftalem4 26130 | Lemma for ~ fta : Closure... |
| ftalem5 26131 | Lemma for ~ fta : Main pr... |
| ftalem6 26132 | Lemma for ~ fta : Dischar... |
| ftalem7 26133 | Lemma for ~ fta . Shift t... |
| fta 26134 | The Fundamental Theorem of... |
| basellem1 26135 | Lemma for ~ basel . Closu... |
| basellem2 26136 | Lemma for ~ basel . Show ... |
| basellem3 26137 | Lemma for ~ basel . Using... |
| basellem4 26138 | Lemma for ~ basel . By ~ ... |
| basellem5 26139 | Lemma for ~ basel . Using... |
| basellem6 26140 | Lemma for ~ basel . The f... |
| basellem7 26141 | Lemma for ~ basel . The f... |
| basellem8 26142 | Lemma for ~ basel . The f... |
| basellem9 26143 | Lemma for ~ basel . Since... |
| basel 26144 | The sum of the inverse squ... |
| efnnfsumcl 26157 | Finite sum closure in the ... |
| ppisval 26158 | The set of primes less tha... |
| ppisval2 26159 | The set of primes less tha... |
| ppifi 26160 | The set of primes less tha... |
| prmdvdsfi 26161 | The set of prime divisors ... |
| chtf 26162 | Domain and range of the Ch... |
| chtcl 26163 | Real closure of the Chebys... |
| chtval 26164 | Value of the Chebyshev fun... |
| efchtcl 26165 | The Chebyshev function is ... |
| chtge0 26166 | The Chebyshev function is ... |
| vmaval 26167 | Value of the von Mangoldt ... |
| isppw 26168 | Two ways to say that ` A `... |
| isppw2 26169 | Two ways to say that ` A `... |
| vmappw 26170 | Value of the von Mangoldt ... |
| vmaprm 26171 | Value of the von Mangoldt ... |
| vmacl 26172 | Closure for the von Mangol... |
| vmaf 26173 | Functionality of the von M... |
| efvmacl 26174 | The von Mangoldt is closed... |
| vmage0 26175 | The von Mangoldt function ... |
| chpval 26176 | Value of the second Chebys... |
| chpf 26177 | Functionality of the secon... |
| chpcl 26178 | Closure for the second Che... |
| efchpcl 26179 | The second Chebyshev funct... |
| chpge0 26180 | The second Chebyshev funct... |
| ppival 26181 | Value of the prime-countin... |
| ppival2 26182 | Value of the prime-countin... |
| ppival2g 26183 | Value of the prime-countin... |
| ppif 26184 | Domain and range of the pr... |
| ppicl 26185 | Real closure of the prime-... |
| muval 26186 | The value of the Möbi... |
| muval1 26187 | The value of the Möbi... |
| muval2 26188 | The value of the Möbi... |
| isnsqf 26189 | Two ways to say that a num... |
| issqf 26190 | Two ways to say that a num... |
| sqfpc 26191 | The prime count of a squar... |
| dvdssqf 26192 | A divisor of a squarefree ... |
| sqf11 26193 | A squarefree number is com... |
| muf 26194 | The Möbius function i... |
| mucl 26195 | Closure of the Möbius... |
| sgmval 26196 | The value of the divisor f... |
| sgmval2 26197 | The value of the divisor f... |
| 0sgm 26198 | The value of the sum-of-di... |
| sgmf 26199 | The divisor function is a ... |
| sgmcl 26200 | Closure of the divisor fun... |
| sgmnncl 26201 | Closure of the divisor fun... |
| mule1 26202 | The Möbius function t... |
| chtfl 26203 | The Chebyshev function doe... |
| chpfl 26204 | The second Chebyshev funct... |
| ppiprm 26205 | The prime-counting functio... |
| ppinprm 26206 | The prime-counting functio... |
| chtprm 26207 | The Chebyshev function at ... |
| chtnprm 26208 | The Chebyshev function at ... |
| chpp1 26209 | The second Chebyshev funct... |
| chtwordi 26210 | The Chebyshev function is ... |
| chpwordi 26211 | The second Chebyshev funct... |
| chtdif 26212 | The difference of the Cheb... |
| efchtdvds 26213 | The exponentiated Chebyshe... |
| ppifl 26214 | The prime-counting functio... |
| ppip1le 26215 | The prime-counting functio... |
| ppiwordi 26216 | The prime-counting functio... |
| ppidif 26217 | The difference of the prim... |
| ppi1 26218 | The prime-counting functio... |
| cht1 26219 | The Chebyshev function at ... |
| vma1 26220 | The von Mangoldt function ... |
| chp1 26221 | The second Chebyshev funct... |
| ppi1i 26222 | Inference form of ~ ppiprm... |
| ppi2i 26223 | Inference form of ~ ppinpr... |
| ppi2 26224 | The prime-counting functio... |
| ppi3 26225 | The prime-counting functio... |
| cht2 26226 | The Chebyshev function at ... |
| cht3 26227 | The Chebyshev function at ... |
| ppinncl 26228 | Closure of the prime-count... |
| chtrpcl 26229 | Closure of the Chebyshev f... |
| ppieq0 26230 | The prime-counting functio... |
| ppiltx 26231 | The prime-counting functio... |
| prmorcht 26232 | Relate the primorial (prod... |
| mumullem1 26233 | Lemma for ~ mumul . A mul... |
| mumullem2 26234 | Lemma for ~ mumul . The p... |
| mumul 26235 | The Möbius function i... |
| sqff1o 26236 | There is a bijection from ... |
| fsumdvdsdiaglem 26237 | A "diagonal commutation" o... |
| fsumdvdsdiag 26238 | A "diagonal commutation" o... |
| fsumdvdscom 26239 | A double commutation of di... |
| dvdsppwf1o 26240 | A bijection from the divis... |
| dvdsflf1o 26241 | A bijection from the numbe... |
| dvdsflsumcom 26242 | A sum commutation from ` s... |
| fsumfldivdiaglem 26243 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 26244 | The right-hand side of ~ d... |
| musum 26245 | The sum of the Möbius... |
| musumsum 26246 | Evaluate a collapsing sum ... |
| muinv 26247 | The Möbius inversion ... |
| dvdsmulf1o 26248 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 26249 | Product of two divisor sum... |
| sgmppw 26250 | The value of the divisor f... |
| 0sgmppw 26251 | A prime power ` P ^ K ` ha... |
| 1sgmprm 26252 | The sum of divisors for a ... |
| 1sgm2ppw 26253 | The sum of the divisors of... |
| sgmmul 26254 | The divisor function for f... |
| ppiublem1 26255 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 26256 | A prime greater than ` 3 `... |
| ppiub 26257 | An upper bound on the prim... |
| vmalelog 26258 | The von Mangoldt function ... |
| chtlepsi 26259 | The first Chebyshev functi... |
| chprpcl 26260 | Closure of the second Cheb... |
| chpeq0 26261 | The second Chebyshev funct... |
| chteq0 26262 | The first Chebyshev functi... |
| chtleppi 26263 | Upper bound on the ` theta... |
| chtublem 26264 | Lemma for ~ chtub . (Cont... |
| chtub 26265 | An upper bound on the Cheb... |
| fsumvma 26266 | Rewrite a sum over the von... |
| fsumvma2 26267 | Apply ~ fsumvma for the co... |
| pclogsum 26268 | The logarithmic analogue o... |
| vmasum 26269 | The sum of the von Mangold... |
| logfac2 26270 | Another expression for the... |
| chpval2 26271 | Express the second Chebysh... |
| chpchtsum 26272 | The second Chebyshev funct... |
| chpub 26273 | An upper bound on the seco... |
| logfacubnd 26274 | A simple upper bound on th... |
| logfaclbnd 26275 | A lower bound on the logar... |
| logfacbnd3 26276 | Show the stronger statemen... |
| logfacrlim 26277 | Combine the estimates ~ lo... |
| logexprlim 26278 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 26279 | Write out ~ logfacrlim as ... |
| mersenne 26280 | A Mersenne prime is a prim... |
| perfect1 26281 | Euclid's contribution to t... |
| perfectlem1 26282 | Lemma for ~ perfect . (Co... |
| perfectlem2 26283 | Lemma for ~ perfect . (Co... |
| perfect 26284 | The Euclid-Euler theorem, ... |
| dchrval 26287 | Value of the group of Diri... |
| dchrbas 26288 | Base set of the group of D... |
| dchrelbas 26289 | A Dirichlet character is a... |
| dchrelbas2 26290 | A Dirichlet character is a... |
| dchrelbas3 26291 | A Dirichlet character is a... |
| dchrelbasd 26292 | A Dirichlet character is a... |
| dchrrcl 26293 | Reverse closure for a Diri... |
| dchrmhm 26294 | A Dirichlet character is a... |
| dchrf 26295 | A Dirichlet character is a... |
| dchrelbas4 26296 | A Dirichlet character is a... |
| dchrzrh1 26297 | Value of a Dirichlet chara... |
| dchrzrhcl 26298 | A Dirichlet character take... |
| dchrzrhmul 26299 | A Dirichlet character is c... |
| dchrplusg 26300 | Group operation on the gro... |
| dchrmul 26301 | Group operation on the gro... |
| dchrmulcl 26302 | Closure of the group opera... |
| dchrn0 26303 | A Dirichlet character is n... |
| dchr1cl 26304 | Closure of the principal D... |
| dchrmulid2 26305 | Left identity for the prin... |
| dchrinvcl 26306 | Closure of the group inver... |
| dchrabl 26307 | The set of Dirichlet chara... |
| dchrfi 26308 | The group of Dirichlet cha... |
| dchrghm 26309 | A Dirichlet character rest... |
| dchr1 26310 | Value of the principal Dir... |
| dchreq 26311 | A Dirichlet character is d... |
| dchrresb 26312 | A Dirichlet character is d... |
| dchrabs 26313 | A Dirichlet character take... |
| dchrinv 26314 | The inverse of a Dirichlet... |
| dchrabs2 26315 | A Dirichlet character take... |
| dchr1re 26316 | The principal Dirichlet ch... |
| dchrptlem1 26317 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 26318 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 26319 | Lemma for ~ dchrpt . (Con... |
| dchrpt 26320 | For any element other than... |
| dchrsum2 26321 | An orthogonality relation ... |
| dchrsum 26322 | An orthogonality relation ... |
| sumdchr2 26323 | Lemma for ~ sumdchr . (Co... |
| dchrhash 26324 | There are exactly ` phi ( ... |
| sumdchr 26325 | An orthogonality relation ... |
| dchr2sum 26326 | An orthogonality relation ... |
| sum2dchr 26327 | An orthogonality relation ... |
| bcctr 26328 | Value of the central binom... |
| pcbcctr 26329 | Prime count of a central b... |
| bcmono 26330 | The binomial coefficient i... |
| bcmax 26331 | The binomial coefficient t... |
| bcp1ctr 26332 | Ratio of two central binom... |
| bclbnd 26333 | A bound on the binomial co... |
| efexple 26334 | Convert a bound on a power... |
| bpos1lem 26335 | Lemma for ~ bpos1 . (Cont... |
| bpos1 26336 | Bertrand's postulate, chec... |
| bposlem1 26337 | An upper bound on the prim... |
| bposlem2 26338 | There are no odd primes in... |
| bposlem3 26339 | Lemma for ~ bpos . Since ... |
| bposlem4 26340 | Lemma for ~ bpos . (Contr... |
| bposlem5 26341 | Lemma for ~ bpos . Bound ... |
| bposlem6 26342 | Lemma for ~ bpos . By usi... |
| bposlem7 26343 | Lemma for ~ bpos . The fu... |
| bposlem8 26344 | Lemma for ~ bpos . Evalua... |
| bposlem9 26345 | Lemma for ~ bpos . Derive... |
| bpos 26346 | Bertrand's postulate: ther... |
| zabsle1 26349 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 26350 | When ` a ` is coprime to t... |
| lgslem2 26351 | The set ` Z ` of all integ... |
| lgslem3 26352 | The set ` Z ` of all integ... |
| lgslem4 26353 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 26354 | Value of the Legendre symb... |
| lgsfval 26355 | Value of the function ` F ... |
| lgsfcl2 26356 | The function ` F ` is clos... |
| lgscllem 26357 | The Legendre symbol is an ... |
| lgsfcl 26358 | Closure of the function ` ... |
| lgsfle1 26359 | The function ` F ` has mag... |
| lgsval2lem 26360 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 26361 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 26362 | The Legendre symbol is an ... |
| lgs0 26363 | The Legendre symbol when t... |
| lgscl 26364 | The Legendre symbol is an ... |
| lgsle1 26365 | The Legendre symbol has ab... |
| lgsval2 26366 | The Legendre symbol at a p... |
| lgs2 26367 | The Legendre symbol at ` 2... |
| lgsval3 26368 | The Legendre symbol at an ... |
| lgsvalmod 26369 | The Legendre symbol is equ... |
| lgsval4 26370 | Restate ~ lgsval for nonze... |
| lgsfcl3 26371 | Closure of the function ` ... |
| lgsval4a 26372 | Same as ~ lgsval4 for posi... |
| lgscl1 26373 | The value of the Legendre ... |
| lgsneg 26374 | The Legendre symbol is eit... |
| lgsneg1 26375 | The Legendre symbol for no... |
| lgsmod 26376 | The Legendre (Jacobi) symb... |
| lgsdilem 26377 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 26378 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 26379 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 26380 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 26381 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 26382 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 26383 | The Legendre symbol is com... |
| lgsdirprm 26384 | The Legendre symbol is com... |
| lgsdir 26385 | The Legendre symbol is com... |
| lgsdilem2 26386 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 26387 | The Legendre symbol is com... |
| lgsne0 26388 | The Legendre symbol is non... |
| lgsabs1 26389 | The Legendre symbol is non... |
| lgssq 26390 | The Legendre symbol at a s... |
| lgssq2 26391 | The Legendre symbol at a s... |
| lgsprme0 26392 | The Legendre symbol at any... |
| 1lgs 26393 | The Legendre symbol at ` 1... |
| lgs1 26394 | The Legendre symbol at ` 1... |
| lgsmodeq 26395 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 26396 | The Legendre (Jacobi) symb... |
| lgsdirnn0 26397 | Variation on ~ lgsdir vali... |
| lgsdinn0 26398 | Variation on ~ lgsdi valid... |
| lgsqrlem1 26399 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 26400 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 26401 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 26402 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 26403 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 26404 | The Legendre symbol for od... |
| lgsqrmod 26405 | If the Legendre symbol of ... |
| lgsqrmodndvds 26406 | If the Legendre symbol of ... |
| lgsdchrval 26407 | The Legendre symbol functi... |
| lgsdchr 26408 | The Legendre symbol functi... |
| gausslemma2dlem0a 26409 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 26410 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 26411 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 26412 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 26413 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 26414 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 26415 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 26416 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 26417 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 26418 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 26419 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 26420 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 26421 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 26422 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 26423 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 26424 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 26425 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 26426 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 26427 | Gauss' Lemma (see also the... |
| lgseisenlem1 26428 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 26429 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 26430 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 26431 | Lemma for ~ lgseisen . Th... |
| lgseisen 26432 | Eisenstein's lemma, an exp... |
| lgsquadlem1 26433 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 26434 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 26435 | Lemma for ~ lgsquad . (Co... |
| lgsquad 26436 | The Law of Quadratic Recip... |
| lgsquad2lem1 26437 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 26438 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 26439 | Extend ~ lgsquad to coprim... |
| lgsquad3 26440 | Extend ~ lgsquad2 to integ... |
| m1lgs 26441 | The first supplement to th... |
| 2lgslem1a1 26442 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 26443 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 26444 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 26445 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 26446 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 26447 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 26448 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 26449 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 26450 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 26451 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 26452 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 26453 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 26454 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 26455 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 26456 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 26457 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 26458 | The Legendre symbol for ` ... |
| 2lgslem4 26459 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 26460 | The second supplement to t... |
| 2lgsoddprmlem1 26461 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 26462 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 26463 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 26464 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 26465 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 26466 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 26467 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 26468 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 26469 | The second supplement to t... |
| 2sqlem1 26470 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 26471 | Lemma for ~ 2sq . (Contri... |
| mul2sq 26472 | Fibonacci's identity (actu... |
| 2sqlem3 26473 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 26474 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 26475 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 26476 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 26477 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 26478 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 26479 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 26480 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 26481 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 26482 | Lemma for ~ 2sq . (Contri... |
| 2sq 26483 | All primes of the form ` 4... |
| 2sqblem 26484 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 26485 | The converse to ~ 2sq . (... |
| 2sq2 26486 | ` 2 ` is the sum of square... |
| 2sqn0 26487 | If the sum of two squares ... |
| 2sqcoprm 26488 | If the sum of two squares ... |
| 2sqmod 26489 | Given two decompositions o... |
| 2sqmo 26490 | There exists at most one d... |
| 2sqnn0 26491 | All primes of the form ` 4... |
| 2sqnn 26492 | All primes of the form ` 4... |
| addsq2reu 26493 | For each complex number ` ... |
| addsqn2reu 26494 | For each complex number ` ... |
| addsqrexnreu 26495 | For each complex number, t... |
| addsqnreup 26496 | There is no unique decompo... |
| addsq2nreurex 26497 | For each complex number ` ... |
| addsqn2reurex2 26498 | For each complex number ` ... |
| 2sqreulem1 26499 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 26500 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 26501 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 26502 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 26503 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 26504 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 26505 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 26506 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 26507 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 26508 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 26509 | There exists a unique deco... |
| 2sqreunn 26510 | There exists a unique deco... |
| 2sqreult 26511 | There exists a unique deco... |
| 2sqreultb 26512 | There exists a unique deco... |
| 2sqreunnlt 26513 | There exists a unique deco... |
| 2sqreunnltb 26514 | There exists a unique deco... |
| 2sqreuop 26515 | There exists a unique deco... |
| 2sqreuopnn 26516 | There exists a unique deco... |
| 2sqreuoplt 26517 | There exists a unique deco... |
| 2sqreuopltb 26518 | There exists a unique deco... |
| 2sqreuopnnlt 26519 | There exists a unique deco... |
| 2sqreuopnnltb 26520 | There exists a unique deco... |
| 2sqreuopb 26521 | There exists a unique deco... |
| chebbnd1lem1 26522 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 26523 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 26524 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 26525 | The Chebyshev bound: The ... |
| chtppilimlem1 26526 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 26527 | Lemma for ~ chtppilim . (... |
| chtppilim 26528 | The ` theta ` function is ... |
| chto1ub 26529 | The ` theta ` function is ... |
| chebbnd2 26530 | The Chebyshev bound, part ... |
| chto1lb 26531 | The ` theta ` function is ... |
| chpchtlim 26532 | The ` psi ` and ` theta ` ... |
| chpo1ub 26533 | The ` psi ` function is up... |
| chpo1ubb 26534 | The ` psi ` function is up... |
| vmadivsum 26535 | The sum of the von Mangold... |
| vmadivsumb 26536 | Give a total bound on the ... |
| rplogsumlem1 26537 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 26538 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 26539 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 26540 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 26541 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 26542 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 26543 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 26544 | Lemma for ~ dchrisum . Le... |
| dchrisum 26545 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 26546 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 26547 | The sum of the Möbius... |
| dchrvmasumlem1 26548 | An alternative expression ... |
| dchrvmasum2lem 26549 | Give an expression for ` l... |
| dchrvmasum2if 26550 | Combine the results of ~ d... |
| dchrvmasumlem2 26551 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 26552 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 26553 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 26554 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 26555 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 26556 | An asymptotic approximatio... |
| dchrvmaeq0 26557 | The set ` W ` is the colle... |
| dchrisum0fval 26558 | Value of the function ` F ... |
| dchrisum0fmul 26559 | The function ` F ` , the d... |
| dchrisum0ff 26560 | The function ` F ` is a re... |
| dchrisum0flblem1 26561 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 26562 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 26563 | The divisor sum of a real ... |
| dchrisum0fno1 26564 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 26565 | A partial result along the... |
| dchrisum0re 26566 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 26567 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 26568 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 26569 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 26570 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 26571 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 26572 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 26573 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 26574 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 26575 | The sum of the Möbius... |
| dchrvmasumlem 26576 | The sum of the Möbius... |
| dchrmusum 26577 | The sum of the Möbius... |
| dchrvmasum 26578 | The sum of the von Mangold... |
| rpvmasum 26579 | The sum of the von Mangold... |
| rplogsum 26580 | The sum of ` log p / p ` o... |
| dirith2 26581 | Dirichlet's theorem: there... |
| dirith 26582 | Dirichlet's theorem: there... |
| mudivsum 26583 | Asymptotic formula for ` s... |
| mulogsumlem 26584 | Lemma for ~ mulogsum . (C... |
| mulogsum 26585 | Asymptotic formula for ... |
| logdivsum 26586 | Asymptotic analysis of ... |
| mulog2sumlem1 26587 | Asymptotic formula for ... |
| mulog2sumlem2 26588 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 26589 | Lemma for ~ mulog2sum . (... |
| mulog2sum 26590 | Asymptotic formula for ... |
| vmalogdivsum2 26591 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 26592 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 26593 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 26594 | The sum ` sum_ m n <_ x , ... |
| logsqvma 26595 | A formula for ` log ^ 2 ( ... |
| logsqvma2 26596 | The Möbius inverse of... |
| log2sumbnd 26597 | Bound on the difference be... |
| selberglem1 26598 | Lemma for ~ selberg . Est... |
| selberglem2 26599 | Lemma for ~ selberg . (Co... |
| selberglem3 26600 | Lemma for ~ selberg . Est... |
| selberg 26601 | Selberg's symmetry formula... |
| selbergb 26602 | Convert eventual boundedne... |
| selberg2lem 26603 | Lemma for ~ selberg2 . Eq... |
| selberg2 26604 | Selberg's symmetry formula... |
| selberg2b 26605 | Convert eventual boundedne... |
| chpdifbndlem1 26606 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 26607 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 26608 | A bound on the difference ... |
| logdivbnd 26609 | A bound on a sum of logs, ... |
| selberg3lem1 26610 | Introduce a log weighting ... |
| selberg3lem2 26611 | Lemma for ~ selberg3 . Eq... |
| selberg3 26612 | Introduce a log weighting ... |
| selberg4lem1 26613 | Lemma for ~ selberg4 . Eq... |
| selberg4 26614 | The Selberg symmetry formu... |
| pntrval 26615 | Define the residual of the... |
| pntrf 26616 | Functionality of the resid... |
| pntrmax 26617 | There is a bound on the re... |
| pntrsumo1 26618 | A bound on a sum over ` R ... |
| pntrsumbnd 26619 | A bound on a sum over ` R ... |
| pntrsumbnd2 26620 | A bound on a sum over ` R ... |
| selbergr 26621 | Selberg's symmetry formula... |
| selberg3r 26622 | Selberg's symmetry formula... |
| selberg4r 26623 | Selberg's symmetry formula... |
| selberg34r 26624 | The sum of ~ selberg3r and... |
| pntsval 26625 | Define the "Selberg functi... |
| pntsf 26626 | Functionality of the Selbe... |
| selbergs 26627 | Selberg's symmetry formula... |
| selbergsb 26628 | Selberg's symmetry formula... |
| pntsval2 26629 | The Selberg function can b... |
| pntrlog2bndlem1 26630 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 26631 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 26632 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 26633 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 26634 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 26635 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 26636 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 26637 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 26638 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 26639 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 26640 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 26641 | Lemma for ~ pnt . Establi... |
| pntibndlem1 26642 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 26643 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 26644 | Lemma for ~ pntibnd . The... |
| pntibndlem3 26645 | Lemma for ~ pntibnd . Pac... |
| pntibnd 26646 | Lemma for ~ pnt . Establi... |
| pntlemd 26647 | Lemma for ~ pnt . Closure... |
| pntlemc 26648 | Lemma for ~ pnt . Closure... |
| pntlema 26649 | Lemma for ~ pnt . Closure... |
| pntlemb 26650 | Lemma for ~ pnt . Unpack ... |
| pntlemg 26651 | Lemma for ~ pnt . Closure... |
| pntlemh 26652 | Lemma for ~ pnt . Bounds ... |
| pntlemn 26653 | Lemma for ~ pnt . The "na... |
| pntlemq 26654 | Lemma for ~ pntlemj . (Co... |
| pntlemr 26655 | Lemma for ~ pntlemj . (Co... |
| pntlemj 26656 | Lemma for ~ pnt . The ind... |
| pntlemi 26657 | Lemma for ~ pnt . Elimina... |
| pntlemf 26658 | Lemma for ~ pnt . Add up ... |
| pntlemk 26659 | Lemma for ~ pnt . Evaluat... |
| pntlemo 26660 | Lemma for ~ pnt . Combine... |
| pntleme 26661 | Lemma for ~ pnt . Package... |
| pntlem3 26662 | Lemma for ~ pnt . Equatio... |
| pntlemp 26663 | Lemma for ~ pnt . Wrappin... |
| pntleml 26664 | Lemma for ~ pnt . Equatio... |
| pnt3 26665 | The Prime Number Theorem, ... |
| pnt2 26666 | The Prime Number Theorem, ... |
| pnt 26667 | The Prime Number Theorem: ... |
| abvcxp 26668 | Raising an absolute value ... |
| padicfval 26669 | Value of the p-adic absolu... |
| padicval 26670 | Value of the p-adic absolu... |
| ostth2lem1 26671 | Lemma for ~ ostth2 , altho... |
| qrngbas 26672 | The base set of the field ... |
| qdrng 26673 | The rationals form a divis... |
| qrng0 26674 | The zero element of the fi... |
| qrng1 26675 | The unit element of the fi... |
| qrngneg 26676 | The additive inverse in th... |
| qrngdiv 26677 | The division operation in ... |
| qabvle 26678 | By using induction on ` N ... |
| qabvexp 26679 | Induct the product rule ~ ... |
| ostthlem1 26680 | Lemma for ~ ostth . If tw... |
| ostthlem2 26681 | Lemma for ~ ostth . Refin... |
| qabsabv 26682 | The regular absolute value... |
| padicabv 26683 | The p-adic absolute value ... |
| padicabvf 26684 | The p-adic absolute value ... |
| padicabvcxp 26685 | All positive powers of the... |
| ostth1 26686 | - Lemma for ~ ostth : triv... |
| ostth2lem2 26687 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 26688 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 26689 | Lemma for ~ ostth2 . (Con... |
| ostth2 26690 | - Lemma for ~ ostth : regu... |
| ostth3 26691 | - Lemma for ~ ostth : p-ad... |
| ostth 26692 | Ostrowski's theorem, which... |
| itvndx 26703 | Index value of the Interva... |
| lngndx 26704 | Index value of the "line" ... |
| itvid 26705 | Utility theorem: index-ind... |
| lngid 26706 | Utility theorem: index-ind... |
| slotsinbpsd 26707 | The slots ` Base ` , ` +g ... |
| slotslnbpsd 26708 | The slots ` Base ` , ` +g ... |
| trkgstr 26709 | Functionality of a Tarski ... |
| trkgbas 26710 | The base set of a Tarski g... |
| trkgdist 26711 | The measure of a distance ... |
| trkgitv 26712 | The congruence relation in... |
| istrkgc 26719 | Property of being a Tarski... |
| istrkgb 26720 | Property of being a Tarski... |
| istrkgcb 26721 | Property of being a Tarski... |
| istrkge 26722 | Property of fulfilling Euc... |
| istrkgl 26723 | Building lines from the se... |
| istrkgld 26724 | Property of fulfilling the... |
| istrkg2ld 26725 | Property of fulfilling the... |
| istrkg3ld 26726 | Property of fulfilling the... |
| axtgcgrrflx 26727 | Axiom of reflexivity of co... |
| axtgcgrid 26728 | Axiom of identity of congr... |
| axtgsegcon 26729 | Axiom of segment construct... |
| axtg5seg 26730 | Five segments axiom, Axiom... |
| axtgbtwnid 26731 | Identity of Betweenness. ... |
| axtgpasch 26732 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 26733 | Axiom of Continuity. Axio... |
| axtgcont 26734 | Axiom of Continuity. Axio... |
| axtglowdim2 26735 | Lower dimension axiom for ... |
| axtgupdim2 26736 | Upper dimension axiom for ... |
| axtgeucl 26737 | Euclid's Axiom. Axiom A10... |
| tgjustf 26738 | Given any function ` F ` ,... |
| tgjustr 26739 | Given any equivalence rela... |
| tgjustc1 26740 | A justification for using ... |
| tgjustc2 26741 | A justification for using ... |
| tgcgrcomimp 26742 | Congruence commutes on the... |
| tgcgrcomr 26743 | Congruence commutes on the... |
| tgcgrcoml 26744 | Congruence commutes on the... |
| tgcgrcomlr 26745 | Congruence commutes on bot... |
| tgcgreqb 26746 | Congruence and equality. ... |
| tgcgreq 26747 | Congruence and equality. ... |
| tgcgrneq 26748 | Congruence and equality. ... |
| tgcgrtriv 26749 | Degenerate segments are co... |
| tgcgrextend 26750 | Link congruence over a pai... |
| tgsegconeq 26751 | Two points that satisfy th... |
| tgbtwntriv2 26752 | Betweenness always holds f... |
| tgbtwncom 26753 | Betweenness commutes. The... |
| tgbtwncomb 26754 | Betweenness commutes, bico... |
| tgbtwnne 26755 | Betweenness and inequality... |
| tgbtwntriv1 26756 | Betweenness always holds f... |
| tgbtwnswapid 26757 | If you can swap the first ... |
| tgbtwnintr 26758 | Inner transitivity law for... |
| tgbtwnexch3 26759 | Exchange the first endpoin... |
| tgbtwnouttr2 26760 | Outer transitivity law for... |
| tgbtwnexch2 26761 | Exchange the outer point o... |
| tgbtwnouttr 26762 | Outer transitivity law for... |
| tgbtwnexch 26763 | Outer transitivity law for... |
| tgtrisegint 26764 | A line segment between two... |
| tglowdim1 26765 | Lower dimension axiom for ... |
| tglowdim1i 26766 | Lower dimension axiom for ... |
| tgldimor 26767 | Excluded-middle like state... |
| tgldim0eq 26768 | In dimension zero, any two... |
| tgldim0itv 26769 | In dimension zero, any two... |
| tgldim0cgr 26770 | In dimension zero, any two... |
| tgbtwndiff 26771 | There is always a ` c ` di... |
| tgdim01 26772 | In geometries of dimension... |
| tgifscgr 26773 | Inner five segment congrue... |
| tgcgrsub 26774 | Removing identical parts f... |
| iscgrg 26777 | The congruence property fo... |
| iscgrgd 26778 | The property for two seque... |
| iscgrglt 26779 | The property for two seque... |
| trgcgrg 26780 | The property for two trian... |
| trgcgr 26781 | Triangle congruence. (Con... |
| ercgrg 26782 | The shape congruence relat... |
| tgcgrxfr 26783 | A line segment can be divi... |
| cgr3id 26784 | Reflexivity law for three-... |
| cgr3simp1 26785 | Deduce segment congruence ... |
| cgr3simp2 26786 | Deduce segment congruence ... |
| cgr3simp3 26787 | Deduce segment congruence ... |
| cgr3swap12 26788 | Permutation law for three-... |
| cgr3swap23 26789 | Permutation law for three-... |
| cgr3swap13 26790 | Permutation law for three-... |
| cgr3rotr 26791 | Permutation law for three-... |
| cgr3rotl 26792 | Permutation law for three-... |
| trgcgrcom 26793 | Commutative law for three-... |
| cgr3tr 26794 | Transitivity law for three... |
| tgbtwnxfr 26795 | A condition for extending ... |
| tgcgr4 26796 | Two quadrilaterals to be c... |
| isismt 26799 | Property of being an isome... |
| ismot 26800 | Property of being an isome... |
| motcgr 26801 | Property of a motion: dist... |
| idmot 26802 | The identity is a motion. ... |
| motf1o 26803 | Motions are bijections. (... |
| motcl 26804 | Closure of motions. (Cont... |
| motco 26805 | The composition of two mot... |
| cnvmot 26806 | The converse of a motion i... |
| motplusg 26807 | The operation for motions ... |
| motgrp 26808 | The motions of a geometry ... |
| motcgrg 26809 | Property of a motion: dist... |
| motcgr3 26810 | Property of a motion: dist... |
| tglng 26811 | Lines of a Tarski Geometry... |
| tglnfn 26812 | Lines as functions. (Cont... |
| tglnunirn 26813 | Lines are sets of points. ... |
| tglnpt 26814 | Lines are sets of points. ... |
| tglngne 26815 | It takes two different poi... |
| tglngval 26816 | The line going through poi... |
| tglnssp 26817 | Lines are subset of the ge... |
| tgellng 26818 | Property of lying on the l... |
| tgcolg 26819 | We choose the notation ` (... |
| btwncolg1 26820 | Betweenness implies coline... |
| btwncolg2 26821 | Betweenness implies coline... |
| btwncolg3 26822 | Betweenness implies coline... |
| colcom 26823 | Swapping the points defini... |
| colrot1 26824 | Rotating the points defini... |
| colrot2 26825 | Rotating the points defini... |
| ncolcom 26826 | Swapping non-colinear poin... |
| ncolrot1 26827 | Rotating non-colinear poin... |
| ncolrot2 26828 | Rotating non-colinear poin... |
| tgdim01ln 26829 | In geometries of dimension... |
| ncoltgdim2 26830 | If there are three non-col... |
| lnxfr 26831 | Transfer law for colineari... |
| lnext 26832 | Extend a line with a missi... |
| tgfscgr 26833 | Congruence law for the gen... |
| lncgr 26834 | Congruence rule for lines.... |
| lnid 26835 | Identity law for points on... |
| tgidinside 26836 | Law for finding a point in... |
| tgbtwnconn1lem1 26837 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 26838 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 26839 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 26840 | Connectivity law for betwe... |
| tgbtwnconn2 26841 | Another connectivity law f... |
| tgbtwnconn3 26842 | Inner connectivity law for... |
| tgbtwnconnln3 26843 | Derive colinearity from be... |
| tgbtwnconn22 26844 | Double connectivity law fo... |
| tgbtwnconnln1 26845 | Derive colinearity from be... |
| tgbtwnconnln2 26846 | Derive colinearity from be... |
| legval 26849 | Value of the less-than rel... |
| legov 26850 | Value of the less-than rel... |
| legov2 26851 | An equivalent definition o... |
| legid 26852 | Reflexivity of the less-th... |
| btwnleg 26853 | Betweenness implies less-t... |
| legtrd 26854 | Transitivity of the less-t... |
| legtri3 26855 | Equality from the less-tha... |
| legtrid 26856 | Trichotomy law for the les... |
| leg0 26857 | Degenerated (zero-length) ... |
| legeq 26858 | Deduce equality from "less... |
| legbtwn 26859 | Deduce betweenness from "l... |
| tgcgrsub2 26860 | Removing identical parts f... |
| ltgseg 26861 | The set ` E ` denotes the ... |
| ltgov 26862 | Strict "shorter than" geom... |
| legov3 26863 | An equivalent definition o... |
| legso 26864 | The "shorter than" relatio... |
| ishlg 26867 | Rays : Definition 6.1 of ... |
| hlcomb 26868 | The half-line relation com... |
| hlcomd 26869 | The half-line relation com... |
| hlne1 26870 | The half-line relation imp... |
| hlne2 26871 | The half-line relation imp... |
| hlln 26872 | The half-line relation imp... |
| hleqnid 26873 | The endpoint does not belo... |
| hlid 26874 | The half-line relation is ... |
| hltr 26875 | The half-line relation is ... |
| hlbtwn 26876 | Betweenness is a sufficien... |
| btwnhl1 26877 | Deduce half-line from betw... |
| btwnhl2 26878 | Deduce half-line from betw... |
| btwnhl 26879 | Swap betweenness for a hal... |
| lnhl 26880 | Either a point ` C ` on th... |
| hlcgrex 26881 | Construct a point on a hal... |
| hlcgreulem 26882 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 26883 | The point constructed in ~... |
| btwnlng1 26884 | Betweenness implies coline... |
| btwnlng2 26885 | Betweenness implies coline... |
| btwnlng3 26886 | Betweenness implies coline... |
| lncom 26887 | Swapping the points defini... |
| lnrot1 26888 | Rotating the points defini... |
| lnrot2 26889 | Rotating the points defini... |
| ncolne1 26890 | Non-colinear points are di... |
| ncolne2 26891 | Non-colinear points are di... |
| tgisline 26892 | The property of being a pr... |
| tglnne 26893 | It takes two different poi... |
| tglndim0 26894 | There are no lines in dime... |
| tgelrnln 26895 | The property of being a pr... |
| tglineeltr 26896 | Transitivity law for lines... |
| tglineelsb2 26897 | If ` S ` lies on PQ , then... |
| tglinerflx1 26898 | Reflexivity law for line m... |
| tglinerflx2 26899 | Reflexivity law for line m... |
| tglinecom 26900 | Commutativity law for line... |
| tglinethru 26901 | If ` A ` is a line contain... |
| tghilberti1 26902 | There is a line through an... |
| tghilberti2 26903 | There is at most one line ... |
| tglinethrueu 26904 | There is a unique line goi... |
| tglnne0 26905 | A line ` A ` has at least ... |
| tglnpt2 26906 | Find a second point on a l... |
| tglineintmo 26907 | Two distinct lines interse... |
| tglineineq 26908 | Two distinct lines interse... |
| tglineneq 26909 | Given three non-colinear p... |
| tglineinteq 26910 | Two distinct lines interse... |
| ncolncol 26911 | Deduce non-colinearity fro... |
| coltr 26912 | A transitivity law for col... |
| coltr3 26913 | A transitivity law for col... |
| colline 26914 | Three points are colinear ... |
| tglowdim2l 26915 | Reformulation of the lower... |
| tglowdim2ln 26916 | There is always one point ... |
| mirreu3 26919 | Existential uniqueness of ... |
| mirval 26920 | Value of the point inversi... |
| mirfv 26921 | Value of the point inversi... |
| mircgr 26922 | Property of the image by t... |
| mirbtwn 26923 | Property of the image by t... |
| ismir 26924 | Property of the image by t... |
| mirf 26925 | Point inversion as functio... |
| mircl 26926 | Closure of the point inver... |
| mirmir 26927 | The point inversion functi... |
| mircom 26928 | Variation on ~ mirmir . (... |
| mirreu 26929 | Any point has a unique ant... |
| mireq 26930 | Equality deduction for poi... |
| mirinv 26931 | The only invariant point o... |
| mirne 26932 | Mirror of non-center point... |
| mircinv 26933 | The center point is invari... |
| mirf1o 26934 | The point inversion functi... |
| miriso 26935 | The point inversion functi... |
| mirbtwni 26936 | Point inversion preserves ... |
| mirbtwnb 26937 | Point inversion preserves ... |
| mircgrs 26938 | Point inversion preserves ... |
| mirmir2 26939 | Point inversion of a point... |
| mirmot 26940 | Point investion is a motio... |
| mirln 26941 | If two points are on the s... |
| mirln2 26942 | If a point and its mirror ... |
| mirconn 26943 | Point inversion of connect... |
| mirhl 26944 | If two points ` X ` and ` ... |
| mirbtwnhl 26945 | If the center of the point... |
| mirhl2 26946 | Deduce half-line relation ... |
| mircgrextend 26947 | Link congruence over a pai... |
| mirtrcgr 26948 | Point inversion of one poi... |
| mirauto 26949 | Point inversion preserves ... |
| miduniq 26950 | Uniqueness of the middle p... |
| miduniq1 26951 | Uniqueness of the middle p... |
| miduniq2 26952 | If two point inversions co... |
| colmid 26953 | Colinearity and equidistan... |
| symquadlem 26954 | Lemma of the symetrial qua... |
| krippenlem 26955 | Lemma for ~ krippen . We ... |
| krippen 26956 | Krippenlemma (German for c... |
| midexlem 26957 | Lemma for the existence of... |
| israg 26962 | Property for 3 points A, B... |
| ragcom 26963 | Commutative rule for right... |
| ragcol 26964 | The right angle property i... |
| ragmir 26965 | Right angle property is pr... |
| mirrag 26966 | Right angle is conserved b... |
| ragtrivb 26967 | Trivial right angle. Theo... |
| ragflat2 26968 | Deduce equality from two r... |
| ragflat 26969 | Deduce equality from two r... |
| ragtriva 26970 | Trivial right angle. Theo... |
| ragflat3 26971 | Right angle and colinearit... |
| ragcgr 26972 | Right angle and colinearit... |
| motrag 26973 | Right angles are preserved... |
| ragncol 26974 | Right angle implies non-co... |
| perpln1 26975 | Derive a line from perpend... |
| perpln2 26976 | Derive a line from perpend... |
| isperp 26977 | Property for 2 lines A, B ... |
| perpcom 26978 | The "perpendicular" relati... |
| perpneq 26979 | Two perpendicular lines ar... |
| isperp2 26980 | Property for 2 lines A, B,... |
| isperp2d 26981 | One direction of ~ isperp2... |
| ragperp 26982 | Deduce that two lines are ... |
| footexALT 26983 | Alternative version of ~ f... |
| footexlem1 26984 | Lemma for ~ footex . (Con... |
| footexlem2 26985 | Lemma for ~ footex . (Con... |
| footex 26986 | From a point ` C ` outside... |
| foot 26987 | From a point ` C ` outside... |
| footne 26988 | Uniqueness of the foot poi... |
| footeq 26989 | Uniqueness of the foot poi... |
| hlperpnel 26990 | A point on a half-line whi... |
| perprag 26991 | Deduce a right angle from ... |
| perpdragALT 26992 | Deduce a right angle from ... |
| perpdrag 26993 | Deduce a right angle from ... |
| colperp 26994 | Deduce a perpendicularity ... |
| colperpexlem1 26995 | Lemma for ~ colperp . Fir... |
| colperpexlem2 26996 | Lemma for ~ colperpex . S... |
| colperpexlem3 26997 | Lemma for ~ colperpex . C... |
| colperpex 26998 | In dimension 2 and above, ... |
| mideulem2 26999 | Lemma for ~ opphllem , whi... |
| opphllem 27000 | Lemma 8.24 of [Schwabhause... |
| mideulem 27001 | Lemma for ~ mideu . We ca... |
| midex 27002 | Existence of the midpoint,... |
| mideu 27003 | Existence and uniqueness o... |
| islnopp 27004 | The property for two point... |
| islnoppd 27005 | Deduce that ` A ` and ` B ... |
| oppne1 27006 | Points lying on opposite s... |
| oppne2 27007 | Points lying on opposite s... |
| oppne3 27008 | Points lying on opposite s... |
| oppcom 27009 | Commutativity rule for "op... |
| opptgdim2 27010 | If two points opposite to ... |
| oppnid 27011 | The "opposite to a line" r... |
| opphllem1 27012 | Lemma for ~ opphl . (Cont... |
| opphllem2 27013 | Lemma for ~ opphl . Lemma... |
| opphllem3 27014 | Lemma for ~ opphl : We as... |
| opphllem4 27015 | Lemma for ~ opphl . (Cont... |
| opphllem5 27016 | Second part of Lemma 9.4 o... |
| opphllem6 27017 | First part of Lemma 9.4 of... |
| oppperpex 27018 | Restating ~ colperpex usin... |
| opphl 27019 | If two points ` A ` and ` ... |
| outpasch 27020 | Axiom of Pasch, outer form... |
| hlpasch 27021 | An application of the axio... |
| ishpg 27024 | Value of the half-plane re... |
| hpgbr 27025 | Half-planes : property for... |
| hpgne1 27026 | Points on the open half pl... |
| hpgne2 27027 | Points on the open half pl... |
| lnopp2hpgb 27028 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 27029 | If two points lie on the o... |
| hpgerlem 27030 | Lemma for the proof that t... |
| hpgid 27031 | The half-plane relation is... |
| hpgcom 27032 | The half-plane relation co... |
| hpgtr 27033 | The half-plane relation is... |
| colopp 27034 | Opposite sides of a line f... |
| colhp 27035 | Half-plane relation for co... |
| hphl 27036 | If two points are on the s... |
| midf 27041 | Midpoint as a function. (... |
| midcl 27042 | Closure of the midpoint. ... |
| ismidb 27043 | Property of the midpoint. ... |
| midbtwn 27044 | Betweenness of midpoint. ... |
| midcgr 27045 | Congruence of midpoint. (... |
| midid 27046 | Midpoint of a null segment... |
| midcom 27047 | Commutativity rule for the... |
| mirmid 27048 | Point inversion preserves ... |
| lmieu 27049 | Uniqueness of the line mir... |
| lmif 27050 | Line mirror as a function.... |
| lmicl 27051 | Closure of the line mirror... |
| islmib 27052 | Property of the line mirro... |
| lmicom 27053 | The line mirroring functio... |
| lmilmi 27054 | Line mirroring is an invol... |
| lmireu 27055 | Any point has a unique ant... |
| lmieq 27056 | Equality deduction for lin... |
| lmiinv 27057 | The invariants of the line... |
| lmicinv 27058 | The mirroring line is an i... |
| lmimid 27059 | If we have a right angle, ... |
| lmif1o 27060 | The line mirroring functio... |
| lmiisolem 27061 | Lemma for ~ lmiiso . (Con... |
| lmiiso 27062 | The line mirroring functio... |
| lmimot 27063 | Line mirroring is a motion... |
| hypcgrlem1 27064 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 27065 | Lemma for ~ hypcgr , case ... |
| hypcgr 27066 | If the catheti of two righ... |
| lmiopp 27067 | Line mirroring produces po... |
| lnperpex 27068 | Existence of a perpendicul... |
| trgcopy 27069 | Triangle construction: a c... |
| trgcopyeulem 27070 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 27071 | Triangle construction: a c... |
| iscgra 27074 | Property for two angles AB... |
| iscgra1 27075 | A special version of ~ isc... |
| iscgrad 27076 | Sufficient conditions for ... |
| cgrane1 27077 | Angles imply inequality. ... |
| cgrane2 27078 | Angles imply inequality. ... |
| cgrane3 27079 | Angles imply inequality. ... |
| cgrane4 27080 | Angles imply inequality. ... |
| cgrahl1 27081 | Angle congruence is indepe... |
| cgrahl2 27082 | Angle congruence is indepe... |
| cgracgr 27083 | First direction of proposi... |
| cgraid 27084 | Angle congruence is reflex... |
| cgraswap 27085 | Swap rays in a congruence ... |
| cgrcgra 27086 | Triangle congruence implie... |
| cgracom 27087 | Angle congruence commutes.... |
| cgratr 27088 | Angle congruence is transi... |
| flatcgra 27089 | Flat angles are congruent.... |
| cgraswaplr 27090 | Swap both side of angle co... |
| cgrabtwn 27091 | Angle congruence preserves... |
| cgrahl 27092 | Angle congruence preserves... |
| cgracol 27093 | Angle congruence preserves... |
| cgrancol 27094 | Angle congruence preserves... |
| dfcgra2 27095 | This is the full statement... |
| sacgr 27096 | Supplementary angles of co... |
| oacgr 27097 | Vertical angle theorem. V... |
| acopy 27098 | Angle construction. Theor... |
| acopyeu 27099 | Angle construction. Theor... |
| isinag 27103 | Property for point ` X ` t... |
| isinagd 27104 | Sufficient conditions for ... |
| inagflat 27105 | Any point lies in a flat a... |
| inagswap 27106 | Swap the order of the half... |
| inagne1 27107 | Deduce inequality from the... |
| inagne2 27108 | Deduce inequality from the... |
| inagne3 27109 | Deduce inequality from the... |
| inaghl 27110 | The "point lie in angle" r... |
| isleag 27112 | Geometrical "less than" pr... |
| isleagd 27113 | Sufficient condition for "... |
| leagne1 27114 | Deduce inequality from the... |
| leagne2 27115 | Deduce inequality from the... |
| leagne3 27116 | Deduce inequality from the... |
| leagne4 27117 | Deduce inequality from the... |
| cgrg3col4 27118 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 27119 | First congruence theorem: ... |
| tgsas 27120 | First congruence theorem: ... |
| tgsas2 27121 | First congruence theorem: ... |
| tgsas3 27122 | First congruence theorem: ... |
| tgasa1 27123 | Second congruence theorem:... |
| tgasa 27124 | Second congruence theorem:... |
| tgsss1 27125 | Third congruence theorem: ... |
| tgsss2 27126 | Third congruence theorem: ... |
| tgsss3 27127 | Third congruence theorem: ... |
| dfcgrg2 27128 | Congruence for two triangl... |
| isoas 27129 | Congruence theorem for iso... |
| iseqlg 27132 | Property of a triangle bei... |
| iseqlgd 27133 | Condition for a triangle t... |
| f1otrgds 27134 | Convenient lemma for ~ f1o... |
| f1otrgitv 27135 | Convenient lemma for ~ f1o... |
| f1otrg 27136 | A bijection between bases ... |
| f1otrge 27137 | A bijection between bases ... |
| ttgval 27140 | Define a function to augme... |
| ttglem 27141 | Lemma for ~ ttgbas , ~ ttg... |
| ttglemOLD 27142 | Obsolete version of ~ ttgl... |
| ttgbas 27143 | The base set of a subcompl... |
| ttgbasOLD 27144 | Obsolete proof of ~ ttgbas... |
| ttgplusg 27145 | The addition operation of ... |
| ttgplusgOLD 27146 | Obsolete proof of ~ ttgplu... |
| ttgsub 27147 | The subtraction operation ... |
| ttgvsca 27148 | The scalar product of a su... |
| ttgvscaOLD 27149 | Obsolete proof of ~ ttgvsc... |
| ttgds 27150 | The metric of a subcomplex... |
| ttgdsOLD 27151 | Obsolete proof of ~ ttgds ... |
| ttgitvval 27152 | Betweenness for a subcompl... |
| ttgelitv 27153 | Betweenness for a subcompl... |
| ttgbtwnid 27154 | Any subcomplex module equi... |
| ttgcontlem1 27155 | Lemma for % ttgcont . (Co... |
| xmstrkgc 27156 | Any metric space fulfills ... |
| cchhllem 27157 | Lemma for chlbas and chlvs... |
| cchhllemOLD 27158 | Obsolete version of ~ cchh... |
| elee 27165 | Membership in a Euclidean ... |
| mptelee 27166 | A condition for a mapping ... |
| eleenn 27167 | If ` A ` is in ` ( EE `` N... |
| eleei 27168 | The forward direction of ~... |
| eedimeq 27169 | A point belongs to at most... |
| brbtwn 27170 | The binary relation form o... |
| brcgr 27171 | The binary relation form o... |
| fveere 27172 | The function value of a po... |
| fveecn 27173 | The function value of a po... |
| eqeefv 27174 | Two points are equal iff t... |
| eqeelen 27175 | Two points are equal iff t... |
| brbtwn2 27176 | Alternate characterization... |
| colinearalglem1 27177 | Lemma for ~ colinearalg . ... |
| colinearalglem2 27178 | Lemma for ~ colinearalg . ... |
| colinearalglem3 27179 | Lemma for ~ colinearalg . ... |
| colinearalglem4 27180 | Lemma for ~ colinearalg . ... |
| colinearalg 27181 | An algebraic characterizat... |
| eleesub 27182 | Membership of a subtractio... |
| eleesubd 27183 | Membership of a subtractio... |
| axdimuniq 27184 | The unique dimension axiom... |
| axcgrrflx 27185 | ` A ` is as far from ` B `... |
| axcgrtr 27186 | Congruence is transitive. ... |
| axcgrid 27187 | If there is no distance be... |
| axsegconlem1 27188 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 27189 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 27190 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 27191 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 27192 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 27193 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 27194 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 27195 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 27196 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 27197 | Lemma for ~ axsegcon . Sh... |
| axsegcon 27198 | Any segment ` A B ` can be... |
| ax5seglem1 27199 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 27200 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 27201 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 27202 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 27203 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 27204 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 27205 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 27206 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 27207 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 27208 | Lemma for ~ ax5seg . Take... |
| ax5seg 27209 | The five segment axiom. T... |
| axbtwnid 27210 | Points are indivisible. T... |
| axpaschlem 27211 | Lemma for ~ axpasch . Set... |
| axpasch 27212 | The inner Pasch axiom. Ta... |
| axlowdimlem1 27213 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 27214 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 27215 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 27216 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 27217 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 27218 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 27219 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 27220 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 27221 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 27222 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 27223 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 27224 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 27225 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 27226 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 27227 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 27228 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 27229 | Lemma for ~ axlowdim . Es... |
| axlowdim1 27230 | The lower dimension axiom ... |
| axlowdim2 27231 | The lower two-dimensional ... |
| axlowdim 27232 | The general lower dimensio... |
| axeuclidlem 27233 | Lemma for ~ axeuclid . Ha... |
| axeuclid 27234 | Euclid's axiom. Take an a... |
| axcontlem1 27235 | Lemma for ~ axcont . Chan... |
| axcontlem2 27236 | Lemma for ~ axcont . The ... |
| axcontlem3 27237 | Lemma for ~ axcont . Give... |
| axcontlem4 27238 | Lemma for ~ axcont . Give... |
| axcontlem5 27239 | Lemma for ~ axcont . Comp... |
| axcontlem6 27240 | Lemma for ~ axcont . Stat... |
| axcontlem7 27241 | Lemma for ~ axcont . Give... |
| axcontlem8 27242 | Lemma for ~ axcont . A po... |
| axcontlem9 27243 | Lemma for ~ axcont . Give... |
| axcontlem10 27244 | Lemma for ~ axcont . Give... |
| axcontlem11 27245 | Lemma for ~ axcont . Elim... |
| axcontlem12 27246 | Lemma for ~ axcont . Elim... |
| axcont 27247 | The axiom of continuity. ... |
| eengv 27250 | The value of the Euclidean... |
| eengstr 27251 | The Euclidean geometry as ... |
| eengbas 27252 | The Base of the Euclidean ... |
| ebtwntg 27253 | The betweenness relation u... |
| ecgrtg 27254 | The congruence relation us... |
| elntg 27255 | The line definition in the... |
| elntg2 27256 | The line definition in the... |
| eengtrkg 27257 | The geometry structure for... |
| eengtrkge 27258 | The geometry structure for... |
| edgfid 27261 | Utility theorem: index-ind... |
| edgfndx 27262 | Index value of the ~ df-ed... |
| edgfndxnn 27263 | The index value of the edg... |
| edgfndxid 27264 | The value of the edge func... |
| edgfndxidOLD 27265 | Obsolete version of ~ edgf... |
| baseltedgf 27266 | The index value of the ` B... |
| baseltedgfOLD 27267 | Obsolete proof of ~ baselt... |
| basendxnedgfndx 27268 | The slots ` Base ` and ` .... |
| vtxval 27273 | The set of vertices of a g... |
| iedgval 27274 | The set of indexed edges o... |
| 1vgrex 27275 | A graph with at least one ... |
| opvtxval 27276 | The set of vertices of a g... |
| opvtxfv 27277 | The set of vertices of a g... |
| opvtxov 27278 | The set of vertices of a g... |
| opiedgval 27279 | The set of indexed edges o... |
| opiedgfv 27280 | The set of indexed edges o... |
| opiedgov 27281 | The set of indexed edges o... |
| opvtxfvi 27282 | The set of vertices of a g... |
| opiedgfvi 27283 | The set of indexed edges o... |
| funvtxdmge2val 27284 | The set of vertices of an ... |
| funiedgdmge2val 27285 | The set of indexed edges o... |
| funvtxdm2val 27286 | The set of vertices of an ... |
| funiedgdm2val 27287 | The set of indexed edges o... |
| funvtxval0 27288 | The set of vertices of an ... |
| basvtxval 27289 | The set of vertices of a g... |
| edgfiedgval 27290 | The set of indexed edges o... |
| funvtxval 27291 | The set of vertices of a g... |
| funiedgval 27292 | The set of indexed edges o... |
| structvtxvallem 27293 | Lemma for ~ structvtxval a... |
| structvtxval 27294 | The set of vertices of an ... |
| structiedg0val 27295 | The set of indexed edges o... |
| structgrssvtxlem 27296 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 27297 | The set of vertices of a g... |
| structgrssiedg 27298 | The set of indexed edges o... |
| struct2grstr 27299 | A graph represented as an ... |
| struct2grvtx 27300 | The set of vertices of a g... |
| struct2griedg 27301 | The set of indexed edges o... |
| graop 27302 | Any representation of a gr... |
| grastruct 27303 | Any representation of a gr... |
| gropd 27304 | If any representation of a... |
| grstructd 27305 | If any representation of a... |
| gropeld 27306 | If any representation of a... |
| grstructeld 27307 | If any representation of a... |
| setsvtx 27308 | The vertices of a structur... |
| setsiedg 27309 | The (indexed) edges of a s... |
| snstrvtxval 27310 | The set of vertices of a g... |
| snstriedgval 27311 | The set of indexed edges o... |
| vtxval0 27312 | Degenerated case 1 for ver... |
| iedgval0 27313 | Degenerated case 1 for edg... |
| vtxvalsnop 27314 | Degenerated case 2 for ver... |
| iedgvalsnop 27315 | Degenerated case 2 for edg... |
| vtxval3sn 27316 | Degenerated case 3 for ver... |
| iedgval3sn 27317 | Degenerated case 3 for edg... |
| vtxvalprc 27318 | Degenerated case 4 for ver... |
| iedgvalprc 27319 | Degenerated case 4 for edg... |
| edgval 27322 | The edges of a graph. (Co... |
| iedgedg 27323 | An indexed edge is an edge... |
| edgopval 27324 | The edges of a graph repre... |
| edgov 27325 | The edges of a graph repre... |
| edgstruct 27326 | The edges of a graph repre... |
| edgiedgb 27327 | A set is an edge iff it is... |
| edg0iedg0 27328 | There is no edge in a grap... |
| isuhgr 27333 | The predicate "is an undir... |
| isushgr 27334 | The predicate "is an undir... |
| uhgrf 27335 | The edge function of an un... |
| ushgrf 27336 | The edge function of an un... |
| uhgrss 27337 | An edge is a subset of ver... |
| uhgreq12g 27338 | If two sets have the same ... |
| uhgrfun 27339 | The edge function of an un... |
| uhgrn0 27340 | An edge is a nonempty subs... |
| lpvtx 27341 | The endpoints of a loop (w... |
| ushgruhgr 27342 | An undirected simple hyper... |
| isuhgrop 27343 | The property of being an u... |
| uhgr0e 27344 | The empty graph, with vert... |
| uhgr0vb 27345 | The null graph, with no ve... |
| uhgr0 27346 | The null graph represented... |
| uhgrun 27347 | The union ` U ` of two (un... |
| uhgrunop 27348 | The union of two (undirect... |
| ushgrun 27349 | The union ` U ` of two (un... |
| ushgrunop 27350 | The union of two (undirect... |
| uhgrstrrepe 27351 | Replacing (or adding) the ... |
| incistruhgr 27352 | An _incidence structure_ `... |
| isupgr 27357 | The property of being an u... |
| wrdupgr 27358 | The property of being an u... |
| upgrf 27359 | The edge function of an un... |
| upgrfn 27360 | The edge function of an un... |
| upgrss 27361 | An edge is a subset of ver... |
| upgrn0 27362 | An edge is a nonempty subs... |
| upgrle 27363 | An edge of an undirected p... |
| upgrfi 27364 | An edge is a finite subset... |
| upgrex 27365 | An edge is an unordered pa... |
| upgrbi 27366 | Show that an unordered pai... |
| upgrop 27367 | A pseudograph represented ... |
| isumgr 27368 | The property of being an u... |
| isumgrs 27369 | The simplified property of... |
| wrdumgr 27370 | The property of being an u... |
| umgrf 27371 | The edge function of an un... |
| umgrfn 27372 | The edge function of an un... |
| umgredg2 27373 | An edge of a multigraph ha... |
| umgrbi 27374 | Show that an unordered pai... |
| upgruhgr 27375 | An undirected pseudograph ... |
| umgrupgr 27376 | An undirected multigraph i... |
| umgruhgr 27377 | An undirected multigraph i... |
| upgrle2 27378 | An edge of an undirected p... |
| umgrnloopv 27379 | In a multigraph, there is ... |
| umgredgprv 27380 | In a multigraph, an edge i... |
| umgrnloop 27381 | In a multigraph, there is ... |
| umgrnloop0 27382 | A multigraph has no loops.... |
| umgr0e 27383 | The empty graph, with vert... |
| upgr0e 27384 | The empty graph, with vert... |
| upgr1elem 27385 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 27386 | A pseudograph with one edg... |
| upgr0eop 27387 | The empty graph, with vert... |
| upgr1eop 27388 | A pseudograph with one edg... |
| upgr0eopALT 27389 | Alternate proof of ~ upgr0... |
| upgr1eopALT 27390 | Alternate proof of ~ upgr1... |
| upgrun 27391 | The union ` U ` of two pse... |
| upgrunop 27392 | The union of two pseudogra... |
| umgrun 27393 | The union ` U ` of two mul... |
| umgrunop 27394 | The union of two multigrap... |
| umgrislfupgrlem 27395 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 27396 | A multigraph is a loop-fre... |
| lfgredgge2 27397 | An edge of a loop-free gra... |
| lfgrnloop 27398 | A loop-free graph has no l... |
| uhgredgiedgb 27399 | In a hypergraph, a set is ... |
| uhgriedg0edg0 27400 | A hypergraph has no edges ... |
| uhgredgn0 27401 | An edge of a hypergraph is... |
| edguhgr 27402 | An edge of a hypergraph is... |
| uhgredgrnv 27403 | An edge of a hypergraph co... |
| uhgredgss 27404 | The set of edges of a hype... |
| upgredgss 27405 | The set of edges of a pseu... |
| umgredgss 27406 | The set of edges of a mult... |
| edgupgr 27407 | Properties of an edge of a... |
| edgumgr 27408 | Properties of an edge of a... |
| uhgrvtxedgiedgb 27409 | In a hypergraph, a vertex ... |
| upgredg 27410 | For each edge in a pseudog... |
| umgredg 27411 | For each edge in a multigr... |
| upgrpredgv 27412 | An edge of a pseudograph a... |
| umgrpredgv 27413 | An edge of a multigraph al... |
| upgredg2vtx 27414 | For a vertex incident to a... |
| upgredgpr 27415 | If a proper pair (of verti... |
| edglnl 27416 | The edges incident with a ... |
| numedglnl 27417 | The number of edges incide... |
| umgredgne 27418 | An edge of a multigraph al... |
| umgrnloop2 27419 | A multigraph has no loops.... |
| umgredgnlp 27420 | An edge of a multigraph is... |
| isuspgr 27425 | The property of being a si... |
| isusgr 27426 | The property of being a si... |
| uspgrf 27427 | The edge function of a sim... |
| usgrf 27428 | The edge function of a sim... |
| isusgrs 27429 | The property of being a si... |
| usgrfs 27430 | The edge function of a sim... |
| usgrfun 27431 | The edge function of a sim... |
| usgredgss 27432 | The set of edges of a simp... |
| edgusgr 27433 | An edge of a simple graph ... |
| isuspgrop 27434 | The property of being an u... |
| isusgrop 27435 | The property of being an u... |
| usgrop 27436 | A simple graph represented... |
| isausgr 27437 | The property of an unorder... |
| ausgrusgrb 27438 | The equivalence of the def... |
| usgrausgri 27439 | A simple graph represented... |
| ausgrumgri 27440 | If an alternatively define... |
| ausgrusgri 27441 | The equivalence of the def... |
| usgrausgrb 27442 | The equivalence of the def... |
| usgredgop 27443 | An edge of a simple graph ... |
| usgrf1o 27444 | The edge function of a sim... |
| usgrf1 27445 | The edge function of a sim... |
| uspgrf1oedg 27446 | The edge function of a sim... |
| usgrss 27447 | An edge is a subset of ver... |
| uspgrushgr 27448 | A simple pseudograph is an... |
| uspgrupgr 27449 | A simple pseudograph is an... |
| uspgrupgrushgr 27450 | A graph is a simple pseudo... |
| usgruspgr 27451 | A simple graph is a simple... |
| usgrumgr 27452 | A simple graph is an undir... |
| usgrumgruspgr 27453 | A graph is a simple graph ... |
| usgruspgrb 27454 | A class is a simple graph ... |
| usgrupgr 27455 | A simple graph is an undir... |
| usgruhgr 27456 | A simple graph is an undir... |
| usgrislfuspgr 27457 | A simple graph is a loop-f... |
| uspgrun 27458 | The union ` U ` of two sim... |
| uspgrunop 27459 | The union of two simple ps... |
| usgrun 27460 | The union ` U ` of two sim... |
| usgrunop 27461 | The union of two simple gr... |
| usgredg2 27462 | The value of the "edge fun... |
| usgredg2ALT 27463 | Alternate proof of ~ usgre... |
| usgredgprv 27464 | In a simple graph, an edge... |
| usgredgprvALT 27465 | Alternate proof of ~ usgre... |
| usgredgppr 27466 | An edge of a simple graph ... |
| usgrpredgv 27467 | An edge of a simple graph ... |
| edgssv2 27468 | An edge of a simple graph ... |
| usgredg 27469 | For each edge in a simple ... |
| usgrnloopv 27470 | In a simple graph, there i... |
| usgrnloopvALT 27471 | Alternate proof of ~ usgrn... |
| usgrnloop 27472 | In a simple graph, there i... |
| usgrnloopALT 27473 | Alternate proof of ~ usgrn... |
| usgrnloop0 27474 | A simple graph has no loop... |
| usgrnloop0ALT 27475 | Alternate proof of ~ usgrn... |
| usgredgne 27476 | An edge of a simple graph ... |
| usgrf1oedg 27477 | The edge function of a sim... |
| uhgr2edg 27478 | If a vertex is adjacent to... |
| umgr2edg 27479 | If a vertex is adjacent to... |
| usgr2edg 27480 | If a vertex is adjacent to... |
| umgr2edg1 27481 | If a vertex is adjacent to... |
| usgr2edg1 27482 | If a vertex is adjacent to... |
| umgrvad2edg 27483 | If a vertex is adjacent to... |
| umgr2edgneu 27484 | If a vertex is adjacent to... |
| usgrsizedg 27485 | In a simple graph, the siz... |
| usgredg3 27486 | The value of the "edge fun... |
| usgredg4 27487 | For a vertex incident to a... |
| usgredgreu 27488 | For a vertex incident to a... |
| usgredg2vtx 27489 | For a vertex incident to a... |
| uspgredg2vtxeu 27490 | For a vertex incident to a... |
| usgredg2vtxeu 27491 | For a vertex incident to a... |
| usgredg2vtxeuALT 27492 | Alternate proof of ~ usgre... |
| uspgredg2vlem 27493 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 27494 | In a simple pseudograph, t... |
| usgredg2vlem1 27495 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 27496 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 27497 | In a simple graph, the map... |
| usgriedgleord 27498 | Alternate version of ~ usg... |
| ushgredgedg 27499 | In a simple hypergraph the... |
| usgredgedg 27500 | In a simple graph there is... |
| ushgredgedgloop 27501 | In a simple hypergraph the... |
| uspgredgleord 27502 | In a simple pseudograph th... |
| usgredgleord 27503 | In a simple graph the numb... |
| usgredgleordALT 27504 | Alternate proof for ~ usgr... |
| usgrstrrepe 27505 | Replacing (or adding) the ... |
| usgr0e 27506 | The empty graph, with vert... |
| usgr0vb 27507 | The null graph, with no ve... |
| uhgr0v0e 27508 | The null graph, with no ve... |
| uhgr0vsize0 27509 | The size of a hypergraph w... |
| uhgr0edgfi 27510 | A graph of order 0 (i.e. w... |
| usgr0v 27511 | The null graph, with no ve... |
| uhgr0vusgr 27512 | The null graph, with no ve... |
| usgr0 27513 | The null graph represented... |
| uspgr1e 27514 | A simple pseudograph with ... |
| usgr1e 27515 | A simple graph with one ed... |
| usgr0eop 27516 | The empty graph, with vert... |
| uspgr1eop 27517 | A simple pseudograph with ... |
| uspgr1ewop 27518 | A simple pseudograph with ... |
| uspgr1v1eop 27519 | A simple pseudograph with ... |
| usgr1eop 27520 | A simple graph with (at le... |
| uspgr2v1e2w 27521 | A simple pseudograph with ... |
| usgr2v1e2w 27522 | A simple graph with two ve... |
| edg0usgr 27523 | A class without edges is a... |
| lfuhgr1v0e 27524 | A loop-free hypergraph wit... |
| usgr1vr 27525 | A simple graph with one ve... |
| usgr1v 27526 | A class with one (or no) v... |
| usgr1v0edg 27527 | A class with one (or no) v... |
| usgrexmpldifpr 27528 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 27529 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 27530 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 27531 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 27532 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 27533 | ` G ` is a simple graph of... |
| griedg0prc 27534 | The class of empty graphs ... |
| griedg0ssusgr 27535 | The class of all simple gr... |
| usgrprc 27536 | The class of simple graphs... |
| relsubgr 27539 | The class of the subgraph ... |
| subgrv 27540 | If a class is a subgraph o... |
| issubgr 27541 | The property of a set to b... |
| issubgr2 27542 | The property of a set to b... |
| subgrprop 27543 | The properties of a subgra... |
| subgrprop2 27544 | The properties of a subgra... |
| uhgrissubgr 27545 | The property of a hypergra... |
| subgrprop3 27546 | The properties of a subgra... |
| egrsubgr 27547 | An empty graph consisting ... |
| 0grsubgr 27548 | The null graph (represente... |
| 0uhgrsubgr 27549 | The null graph (as hypergr... |
| uhgrsubgrself 27550 | A hypergraph is a subgraph... |
| subgrfun 27551 | The edge function of a sub... |
| subgruhgrfun 27552 | The edge function of a sub... |
| subgreldmiedg 27553 | An element of the domain o... |
| subgruhgredgd 27554 | An edge of a subgraph of a... |
| subumgredg2 27555 | An edge of a subgraph of a... |
| subuhgr 27556 | A subgraph of a hypergraph... |
| subupgr 27557 | A subgraph of a pseudograp... |
| subumgr 27558 | A subgraph of a multigraph... |
| subusgr 27559 | A subgraph of a simple gra... |
| uhgrspansubgrlem 27560 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 27561 | A spanning subgraph ` S ` ... |
| uhgrspan 27562 | A spanning subgraph ` S ` ... |
| upgrspan 27563 | A spanning subgraph ` S ` ... |
| umgrspan 27564 | A spanning subgraph ` S ` ... |
| usgrspan 27565 | A spanning subgraph ` S ` ... |
| uhgrspanop 27566 | A spanning subgraph of a h... |
| upgrspanop 27567 | A spanning subgraph of a p... |
| umgrspanop 27568 | A spanning subgraph of a m... |
| usgrspanop 27569 | A spanning subgraph of a s... |
| uhgrspan1lem1 27570 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 27571 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 27572 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 27573 | The induced subgraph ` S `... |
| upgrreslem 27574 | Lemma for ~ upgrres . (Co... |
| umgrreslem 27575 | Lemma for ~ umgrres and ~ ... |
| upgrres 27576 | A subgraph obtained by rem... |
| umgrres 27577 | A subgraph obtained by rem... |
| usgrres 27578 | A subgraph obtained by rem... |
| upgrres1lem1 27579 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 27580 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 27581 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 27582 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 27583 | A pseudograph obtained by ... |
| umgrres1 27584 | A multigraph obtained by r... |
| usgrres1 27585 | Restricting a simple graph... |
| isfusgr 27588 | The property of being a fi... |
| fusgrvtxfi 27589 | A finite simple graph has ... |
| isfusgrf1 27590 | The property of being a fi... |
| isfusgrcl 27591 | The property of being a fi... |
| fusgrusgr 27592 | A finite simple graph is a... |
| opfusgr 27593 | A finite simple graph repr... |
| usgredgffibi 27594 | The number of edges in a s... |
| fusgredgfi 27595 | In a finite simple graph t... |
| usgr1v0e 27596 | The size of a (finite) sim... |
| usgrfilem 27597 | In a finite simple graph, ... |
| fusgrfisbase 27598 | Induction base for ~ fusgr... |
| fusgrfisstep 27599 | Induction step in ~ fusgrf... |
| fusgrfis 27600 | A finite simple graph is o... |
| fusgrfupgrfs 27601 | A finite simple graph is a... |
| nbgrprc0 27604 | The set of neighbors is em... |
| nbgrcl 27605 | If a class ` X ` has at le... |
| nbgrval 27606 | The set of neighbors of a ... |
| dfnbgr2 27607 | Alternate definition of th... |
| dfnbgr3 27608 | Alternate definition of th... |
| nbgrnvtx0 27609 | If a class ` X ` is not a ... |
| nbgrel 27610 | Characterization of a neig... |
| nbgrisvtx 27611 | Every neighbor ` N ` of a ... |
| nbgrssvtx 27612 | The neighbors of a vertex ... |
| nbuhgr 27613 | The set of neighbors of a ... |
| nbupgr 27614 | The set of neighbors of a ... |
| nbupgrel 27615 | A neighbor of a vertex in ... |
| nbumgrvtx 27616 | The set of neighbors of a ... |
| nbumgr 27617 | The set of neighbors of an... |
| nbusgrvtx 27618 | The set of neighbors of a ... |
| nbusgr 27619 | The set of neighbors of an... |
| nbgr2vtx1edg 27620 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 27621 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 27622 | If a hypergraph has two ve... |
| nbusgreledg 27623 | A class/vertex is a neighb... |
| uhgrnbgr0nb 27624 | A vertex which is not endp... |
| nbgr0vtxlem 27625 | Lemma for ~ nbgr0vtx and ~... |
| nbgr0vtx 27626 | In a null graph (with no v... |
| nbgr0edg 27627 | In an empty graph (with no... |
| nbgr1vtx 27628 | In a graph with one vertex... |
| nbgrnself 27629 | A vertex in a graph is not... |
| nbgrnself2 27630 | A class ` X ` is not a nei... |
| nbgrssovtx 27631 | The neighbors of a vertex ... |
| nbgrssvwo2 27632 | The neighbors of a vertex ... |
| nbgrsym 27633 | In a graph, the neighborho... |
| nbupgrres 27634 | The neighborhood of a vert... |
| usgrnbcnvfv 27635 | Applying the edge function... |
| nbusgredgeu 27636 | For each neighbor of a ver... |
| edgnbusgreu 27637 | For each edge incident to ... |
| nbusgredgeu0 27638 | For each neighbor of a ver... |
| nbusgrf1o0 27639 | The mapping of neighbors o... |
| nbusgrf1o1 27640 | The set of neighbors of a ... |
| nbusgrf1o 27641 | The set of neighbors of a ... |
| nbedgusgr 27642 | The number of neighbors of... |
| edgusgrnbfin 27643 | The number of neighbors of... |
| nbusgrfi 27644 | The class of neighbors of ... |
| nbfiusgrfi 27645 | The class of neighbors of ... |
| hashnbusgrnn0 27646 | The number of neighbors of... |
| nbfusgrlevtxm1 27647 | The number of neighbors of... |
| nbfusgrlevtxm2 27648 | If there is a vertex which... |
| nbusgrvtxm1 27649 | If the number of neighbors... |
| nb3grprlem1 27650 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 27651 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 27652 | The neighbors of a vertex ... |
| nb3grpr2 27653 | The neighbors of a vertex ... |
| nb3gr2nb 27654 | If the neighbors of two ve... |
| uvtxval 27657 | The set of all universal v... |
| uvtxel 27658 | A universal vertex, i.e. a... |
| uvtxisvtx 27659 | A universal vertex is a ve... |
| uvtxssvtx 27660 | The set of the universal v... |
| vtxnbuvtx 27661 | A universal vertex has all... |
| uvtxnbgrss 27662 | A universal vertex has all... |
| uvtxnbgrvtx 27663 | A universal vertex is neig... |
| uvtx0 27664 | There is no universal vert... |
| isuvtx 27665 | The set of all universal v... |
| uvtxel1 27666 | Characterization of a univ... |
| uvtx01vtx 27667 | If a graph/class has no ed... |
| uvtx2vtx1edg 27668 | If a graph has two vertice... |
| uvtx2vtx1edgb 27669 | If a hypergraph has two ve... |
| uvtxnbgr 27670 | A universal vertex has all... |
| uvtxnbgrb 27671 | A vertex is universal iff ... |
| uvtxusgr 27672 | The set of all universal v... |
| uvtxusgrel 27673 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 27674 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 27675 | If the number of neighbors... |
| uvtxnbvtxm1 27676 | A universal vertex has ` n... |
| nbupgruvtxres 27677 | The neighborhood of a univ... |
| uvtxupgrres 27678 | A universal vertex is univ... |
| cplgruvtxb 27683 | A graph ` G ` is complete ... |
| prcliscplgr 27684 | A proper class (representi... |
| iscplgr 27685 | The property of being a co... |
| iscplgrnb 27686 | A graph is complete iff al... |
| iscplgredg 27687 | A graph ` G ` is complete ... |
| iscusgr 27688 | The property of being a co... |
| cusgrusgr 27689 | A complete simple graph is... |
| cusgrcplgr 27690 | A complete simple graph is... |
| iscusgrvtx 27691 | A simple graph is complete... |
| cusgruvtxb 27692 | A simple graph is complete... |
| iscusgredg 27693 | A simple graph is complete... |
| cusgredg 27694 | In a complete simple graph... |
| cplgr0 27695 | The null graph (with no ve... |
| cusgr0 27696 | The null graph (with no ve... |
| cplgr0v 27697 | A null graph (with no vert... |
| cusgr0v 27698 | A graph with no vertices a... |
| cplgr1vlem 27699 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 27700 | A graph with one vertex is... |
| cusgr1v 27701 | A graph with one vertex an... |
| cplgr2v 27702 | An undirected hypergraph w... |
| cplgr2vpr 27703 | An undirected hypergraph w... |
| nbcplgr 27704 | In a complete graph, each ... |
| cplgr3v 27705 | A pseudograph with three (... |
| cusgr3vnbpr 27706 | The neighbors of a vertex ... |
| cplgrop 27707 | A complete graph represent... |
| cusgrop 27708 | A complete simple graph re... |
| cusgrexilem1 27709 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 27710 | Lemma for ~ usgrexi . (Co... |
| usgrexi 27711 | An arbitrary set regarded ... |
| cusgrexilem2 27712 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 27713 | An arbitrary set ` V ` reg... |
| cusgrexg 27714 | For each set there is a se... |
| structtousgr 27715 | Any (extensible) structure... |
| structtocusgr 27716 | Any (extensible) structure... |
| cffldtocusgr 27717 | The field of complex numbe... |
| cusgrres 27718 | Restricting a complete sim... |
| cusgrsizeindb0 27719 | Base case of the induction... |
| cusgrsizeindb1 27720 | Base case of the induction... |
| cusgrsizeindslem 27721 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 27722 | Part 1 of induction step i... |
| cusgrsize2inds 27723 | Induction step in ~ cusgrs... |
| cusgrsize 27724 | The size of a finite compl... |
| cusgrfilem1 27725 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 27726 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 27727 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 27728 | If the size of a complete ... |
| usgredgsscusgredg 27729 | A simple graph is a subgra... |
| usgrsscusgr 27730 | A simple graph is a subgra... |
| sizusglecusglem1 27731 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 27732 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 27733 | The size of a simple graph... |
| fusgrmaxsize 27734 | The maximum size of a fini... |
| vtxdgfval 27737 | The value of the vertex de... |
| vtxdgval 27738 | The degree of a vertex. (... |
| vtxdgfival 27739 | The degree of a vertex for... |
| vtxdgop 27740 | The vertex degree expresse... |
| vtxdgf 27741 | The vertex degree function... |
| vtxdgelxnn0 27742 | The degree of a vertex is ... |
| vtxdg0v 27743 | The degree of a vertex in ... |
| vtxdg0e 27744 | The degree of a vertex in ... |
| vtxdgfisnn0 27745 | The degree of a vertex in ... |
| vtxdgfisf 27746 | The vertex degree function... |
| vtxdeqd 27747 | Equality theorem for the v... |
| vtxduhgr0e 27748 | The degree of a vertex in ... |
| vtxdlfuhgr1v 27749 | The degree of the vertex i... |
| vdumgr0 27750 | A vertex in a multigraph h... |
| vtxdun 27751 | The degree of a vertex in ... |
| vtxdfiun 27752 | The degree of a vertex in ... |
| vtxduhgrun 27753 | The degree of a vertex in ... |
| vtxduhgrfiun 27754 | The degree of a vertex in ... |
| vtxdlfgrval 27755 | The value of the vertex de... |
| vtxdumgrval 27756 | The value of the vertex de... |
| vtxdusgrval 27757 | The value of the vertex de... |
| vtxd0nedgb 27758 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 27759 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 27760 | The value of the vertex de... |
| vtxdusgrfvedg 27761 | The value of the vertex de... |
| vtxduhgr0nedg 27762 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 27763 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 27764 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 27765 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 27766 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 27767 | The vertex degree function... |
| vtxdgfusgr 27768 | In a finite simple graph, ... |
| fusgrn0degnn0 27769 | In a nonempty, finite grap... |
| 1loopgruspgr 27770 | A graph with one edge whic... |
| 1loopgredg 27771 | The set of edges in a grap... |
| 1loopgrnb0 27772 | In a graph (simple pseudog... |
| 1loopgrvd2 27773 | The vertex degree of a one... |
| 1loopgrvd0 27774 | The vertex degree of a one... |
| 1hevtxdg0 27775 | The vertex degree of verte... |
| 1hevtxdg1 27776 | The vertex degree of verte... |
| 1hegrvtxdg1 27777 | The vertex degree of a gra... |
| 1hegrvtxdg1r 27778 | The vertex degree of a gra... |
| 1egrvtxdg1 27779 | The vertex degree of a one... |
| 1egrvtxdg1r 27780 | The vertex degree of a one... |
| 1egrvtxdg0 27781 | The vertex degree of a one... |
| p1evtxdeqlem 27782 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 27783 | If an edge ` E ` which doe... |
| p1evtxdp1 27784 | If an edge ` E ` (not bein... |
| uspgrloopvtx 27785 | The set of vertices in a g... |
| uspgrloopvtxel 27786 | A vertex in a graph (simpl... |
| uspgrloopiedg 27787 | The set of edges in a grap... |
| uspgrloopedg 27788 | The set of edges in a grap... |
| uspgrloopnb0 27789 | In a graph (simple pseudog... |
| uspgrloopvd2 27790 | The vertex degree of a one... |
| umgr2v2evtx 27791 | The set of vertices in a m... |
| umgr2v2evtxel 27792 | A vertex in a multigraph w... |
| umgr2v2eiedg 27793 | The edge function in a mul... |
| umgr2v2eedg 27794 | The set of edges in a mult... |
| umgr2v2e 27795 | A multigraph with two edge... |
| umgr2v2enb1 27796 | In a multigraph with two e... |
| umgr2v2evd2 27797 | In a multigraph with two e... |
| hashnbusgrvd 27798 | In a simple graph, the num... |
| usgruvtxvdb 27799 | In a finite simple graph w... |
| vdiscusgrb 27800 | A finite simple graph with... |
| vdiscusgr 27801 | In a finite complete simpl... |
| vtxdusgradjvtx 27802 | The degree of a vertex in ... |
| usgrvd0nedg 27803 | If a vertex in a simple gr... |
| uhgrvd00 27804 | If every vertex in a hyper... |
| usgrvd00 27805 | If every vertex in a simpl... |
| vdegp1ai 27806 | The induction step for a v... |
| vdegp1bi 27807 | The induction step for a v... |
| vdegp1ci 27808 | The induction step for a v... |
| vtxdginducedm1lem1 27809 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 27810 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 27811 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 27812 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 27813 | The degree of a vertex ` v... |
| vtxdginducedm1fi 27814 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 27815 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 27816 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 27817 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 27818 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 27819 | Induction step of ~ finsum... |
| finsumvtxdg2size 27820 | The sum of the degrees of ... |
| fusgr1th 27821 | The sum of the degrees of ... |
| finsumvtxdgeven 27822 | The sum of the degrees of ... |
| vtxdgoddnumeven 27823 | The number of vertices of ... |
| fusgrvtxdgonume 27824 | The number of vertices of ... |
| isrgr 27829 | The property of a class be... |
| rgrprop 27830 | The properties of a k-regu... |
| isrusgr 27831 | The property of being a k-... |
| rusgrprop 27832 | The properties of a k-regu... |
| rusgrrgr 27833 | A k-regular simple graph i... |
| rusgrusgr 27834 | A k-regular simple graph i... |
| finrusgrfusgr 27835 | A finite regular simple gr... |
| isrusgr0 27836 | The property of being a k-... |
| rusgrprop0 27837 | The properties of a k-regu... |
| usgreqdrusgr 27838 | If all vertices in a simpl... |
| fusgrregdegfi 27839 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 27840 | If all vertices in a nonem... |
| frusgrnn0 27841 | In a nonempty finite k-reg... |
| 0edg0rgr 27842 | A graph is 0-regular if it... |
| uhgr0edg0rgr 27843 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 27844 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 27845 | A simple graph is 0-regula... |
| 0vtxrgr 27846 | A null graph (with no vert... |
| 0vtxrusgr 27847 | A graph with no vertices a... |
| 0uhgrrusgr 27848 | The null graph as hypergra... |
| 0grrusgr 27849 | The null graph represented... |
| 0grrgr 27850 | The null graph represented... |
| cusgrrusgr 27851 | A complete simple graph wi... |
| cusgrm1rusgr 27852 | A finite simple graph with... |
| rusgrpropnb 27853 | The properties of a k-regu... |
| rusgrpropedg 27854 | The properties of a k-regu... |
| rusgrpropadjvtx 27855 | The properties of a k-regu... |
| rusgrnumwrdl2 27856 | In a k-regular simple grap... |
| rusgr1vtxlem 27857 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 27858 | If a k-regular simple grap... |
| rgrusgrprc 27859 | The class of 0-regular sim... |
| rusgrprc 27860 | The class of 0-regular sim... |
| rgrprc 27861 | The class of 0-regular gra... |
| rgrprcx 27862 | The class of 0-regular gra... |
| rgrx0ndm 27863 | 0 is not in the domain of ... |
| rgrx0nd 27864 | The potentially alternativ... |
| ewlksfval 27871 | The set of s-walks of edge... |
| isewlk 27872 | Conditions for a function ... |
| ewlkprop 27873 | Properties of an s-walk of... |
| ewlkinedg 27874 | The intersection (common v... |
| ewlkle 27875 | An s-walk of edges is also... |
| upgrewlkle2 27876 | In a pseudograph, there is... |
| wkslem1 27877 | Lemma 1 for walks to subst... |
| wkslem2 27878 | Lemma 2 for walks to subst... |
| wksfval 27879 | The set of walks (in an un... |
| iswlk 27880 | Properties of a pair of fu... |
| wlkprop 27881 | Properties of a walk. (Co... |
| wlkv 27882 | The classes involved in a ... |
| iswlkg 27883 | Generalization of ~ iswlk ... |
| wlkf 27884 | The mapping enumerating th... |
| wlkcl 27885 | A walk has length ` # ( F ... |
| wlkp 27886 | The mapping enumerating th... |
| wlkpwrd 27887 | The sequence of vertices o... |
| wlklenvp1 27888 | The number of vertices of ... |
| wksv 27889 | The class of walks is a se... |
| wlkn0 27890 | The sequence of vertices o... |
| wlklenvm1 27891 | The number of edges of a w... |
| ifpsnprss 27892 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 27893 | Each pair of adjacent vert... |
| wlkvtxiedg 27894 | The vertices of a walk are... |
| relwlk 27895 | The set ` ( Walks `` G ) `... |
| wlkvv 27896 | If there is at least one w... |
| wlkop 27897 | A walk is an ordered pair.... |
| wlkcpr 27898 | A walk as class with two c... |
| wlk2f 27899 | If there is a walk ` W ` t... |
| wlkcomp 27900 | A walk expressed by proper... |
| wlkcompim 27901 | Implications for the prope... |
| wlkelwrd 27902 | The components of a walk a... |
| wlkeq 27903 | Conditions for two walks (... |
| edginwlk 27904 | The value of the edge func... |
| upgredginwlk 27905 | The value of the edge func... |
| iedginwlk 27906 | The value of the edge func... |
| wlkl1loop 27907 | A walk of length 1 from a ... |
| wlk1walk 27908 | A walk is a 1-walk "on the... |
| wlk1ewlk 27909 | A walk is an s-walk "on th... |
| upgriswlk 27910 | Properties of a pair of fu... |
| upgrwlkedg 27911 | The edges of a walk in a p... |
| upgrwlkcompim 27912 | Implications for the prope... |
| wlkvtxedg 27913 | The vertices of a walk are... |
| upgrwlkvtxedg 27914 | The pairs of connected ver... |
| uspgr2wlkeq 27915 | Conditions for two walks w... |
| uspgr2wlkeq2 27916 | Conditions for two walks w... |
| uspgr2wlkeqi 27917 | Conditions for two walks w... |
| umgrwlknloop 27918 | In a multigraph, each walk... |
| wlkRes 27919 | Restrictions of walks (i.e... |
| wlkv0 27920 | If there is a walk in the ... |
| g0wlk0 27921 | There is no walk in a null... |
| 0wlk0 27922 | There is no walk for the e... |
| wlk0prc 27923 | There is no walk in a null... |
| wlklenvclwlk 27924 | The number of vertices in ... |
| wlklenvclwlkOLD 27925 | Obsolete version of ~ wlkl... |
| wlkson 27926 | The set of walks between t... |
| iswlkon 27927 | Properties of a pair of fu... |
| wlkonprop 27928 | Properties of a walk betwe... |
| wlkpvtx 27929 | A walk connects vertices. ... |
| wlkepvtx 27930 | The endpoints of a walk ar... |
| wlkoniswlk 27931 | A walk between two vertice... |
| wlkonwlk 27932 | A walk is a walk between i... |
| wlkonwlk1l 27933 | A walk is a walk from its ... |
| wlksoneq1eq2 27934 | Two walks with identical s... |
| wlkonl1iedg 27935 | If there is a walk between... |
| wlkon2n0 27936 | The length of a walk betwe... |
| 2wlklem 27937 | Lemma for theorems for wal... |
| upgr2wlk 27938 | Properties of a pair of fu... |
| wlkreslem 27939 | Lemma for ~ wlkres . (Con... |
| wlkres 27940 | The restriction ` <. H , Q... |
| redwlklem 27941 | Lemma for ~ redwlk . (Con... |
| redwlk 27942 | A walk ending at the last ... |
| wlkp1lem1 27943 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 27944 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 27945 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 27946 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 27947 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 27948 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 27949 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 27950 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 27951 | Append one path segment (e... |
| wlkdlem1 27952 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 27953 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 27954 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 27955 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 27956 | Two words representing a w... |
| lfgrwlkprop 27957 | Two adjacent vertices in a... |
| lfgriswlk 27958 | Conditions for a pair of f... |
| lfgrwlknloop 27959 | In a loop-free graph, each... |
| reltrls 27964 | The set ` ( Trails `` G ) ... |
| trlsfval 27965 | The set of trails (in an u... |
| istrl 27966 | Conditions for a pair of c... |
| trliswlk 27967 | A trail is a walk. (Contr... |
| trlf1 27968 | The enumeration ` F ` of a... |
| trlreslem 27969 | Lemma for ~ trlres . Form... |
| trlres 27970 | The restriction ` <. H , Q... |
| upgrtrls 27971 | The set of trails in a pse... |
| upgristrl 27972 | Properties of a pair of fu... |
| upgrf1istrl 27973 | Properties of a pair of a ... |
| wksonproplem 27974 | Lemma for theorems for pro... |
| trlsonfval 27975 | The set of trails between ... |
| istrlson 27976 | Properties of a pair of fu... |
| trlsonprop 27977 | Properties of a trail betw... |
| trlsonistrl 27978 | A trail between two vertic... |
| trlsonwlkon 27979 | A trail between two vertic... |
| trlontrl 27980 | A trail is a trail between... |
| relpths 27989 | The set ` ( Paths `` G ) `... |
| pthsfval 27990 | The set of paths (in an un... |
| spthsfval 27991 | The set of simple paths (i... |
| ispth 27992 | Conditions for a pair of c... |
| isspth 27993 | Conditions for a pair of c... |
| pthistrl 27994 | A path is a trail (in an u... |
| spthispth 27995 | A simple path is a path (i... |
| pthiswlk 27996 | A path is a walk (in an un... |
| spthiswlk 27997 | A simple path is a walk (i... |
| pthdivtx 27998 | The inner vertices of a pa... |
| pthdadjvtx 27999 | The adjacent vertices of a... |
| 2pthnloop 28000 | A path of length at least ... |
| upgr2pthnlp 28001 | A path of length at least ... |
| spthdifv 28002 | The vertices of a simple p... |
| spthdep 28003 | A simple path (at least of... |
| pthdepisspth 28004 | A path with different star... |
| upgrwlkdvdelem 28005 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 28006 | In a pseudograph, all edge... |
| upgrspthswlk 28007 | The set of simple paths in... |
| upgrwlkdvspth 28008 | A walk consisting of diffe... |
| pthsonfval 28009 | The set of paths between t... |
| spthson 28010 | The set of simple paths be... |
| ispthson 28011 | Properties of a pair of fu... |
| isspthson 28012 | Properties of a pair of fu... |
| pthsonprop 28013 | Properties of a path betwe... |
| spthonprop 28014 | Properties of a simple pat... |
| pthonispth 28015 | A path between two vertice... |
| pthontrlon 28016 | A path between two vertice... |
| pthonpth 28017 | A path is a path between i... |
| isspthonpth 28018 | A pair of functions is a s... |
| spthonisspth 28019 | A simple path between to v... |
| spthonpthon 28020 | A simple path between two ... |
| spthonepeq 28021 | The endpoints of a simple ... |
| uhgrwkspthlem1 28022 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 28023 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 28024 | Any walk of length 1 betwe... |
| usgr2wlkneq 28025 | The vertices and edges are... |
| usgr2wlkspthlem1 28026 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 28027 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 28028 | In a simple graph, any wal... |
| usgr2trlncl 28029 | In a simple graph, any tra... |
| usgr2trlspth 28030 | In a simple graph, any tra... |
| usgr2pthspth 28031 | In a simple graph, any pat... |
| usgr2pthlem 28032 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 28033 | In a simple graph, there i... |
| usgr2pth0 28034 | In a simply graph, there i... |
| pthdlem1 28035 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 28036 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 28037 | Lemma 2 for ~ pthd . (Con... |
| pthd 28038 | Two words representing a t... |
| clwlks 28041 | The set of closed walks (i... |
| isclwlk 28042 | A pair of functions repres... |
| clwlkiswlk 28043 | A closed walk is a walk (i... |
| clwlkwlk 28044 | Closed walks are walks (in... |
| clwlkswks 28045 | Closed walks are walks (in... |
| isclwlke 28046 | Properties of a pair of fu... |
| isclwlkupgr 28047 | Properties of a pair of fu... |
| clwlkcomp 28048 | A closed walk expressed by... |
| clwlkcompim 28049 | Implications for the prope... |
| upgrclwlkcompim 28050 | Implications for the prope... |
| clwlkcompbp 28051 | Basic properties of the co... |
| clwlkl1loop 28052 | A closed walk of length 1 ... |
| crcts 28057 | The set of circuits (in an... |
| cycls 28058 | The set of cycles (in an u... |
| iscrct 28059 | Sufficient and necessary c... |
| iscycl 28060 | Sufficient and necessary c... |
| crctprop 28061 | The properties of a circui... |
| cyclprop 28062 | The properties of a cycle:... |
| crctisclwlk 28063 | A circuit is a closed walk... |
| crctistrl 28064 | A circuit is a trail. (Co... |
| crctiswlk 28065 | A circuit is a walk. (Con... |
| cyclispth 28066 | A cycle is a path. (Contr... |
| cycliswlk 28067 | A cycle is a walk. (Contr... |
| cycliscrct 28068 | A cycle is a circuit. (Co... |
| cyclnspth 28069 | A (non-trivial) cycle is n... |
| cyclispthon 28070 | A cycle is a path starting... |
| lfgrn1cycl 28071 | In a loop-free graph there... |
| usgr2trlncrct 28072 | In a simple graph, any tra... |
| umgrn1cycl 28073 | In a multigraph graph (wit... |
| uspgrn2crct 28074 | In a simple pseudograph th... |
| usgrn2cycl 28075 | In a simple graph there ar... |
| crctcshwlkn0lem1 28076 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 28077 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 28078 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 28079 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 28080 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 28081 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 28082 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 28083 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 28084 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 28085 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 28086 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 28087 | Cyclically shifting the in... |
| crctcshwlk 28088 | Cyclically shifting the in... |
| crctcshtrl 28089 | Cyclically shifting the in... |
| crctcsh 28090 | Cyclically shifting the in... |
| wwlks 28101 | The set of walks (in an un... |
| iswwlks 28102 | A word over the set of ver... |
| wwlksn 28103 | The set of walks (in an un... |
| iswwlksn 28104 | A word over the set of ver... |
| wwlksnprcl 28105 | Derivation of the length o... |
| iswwlksnx 28106 | Properties of a word to re... |
| wwlkbp 28107 | Basic properties of a walk... |
| wwlknbp 28108 | Basic properties of a walk... |
| wwlknp 28109 | Properties of a set being ... |
| wwlknbp1 28110 | Other basic properties of ... |
| wwlknvtx 28111 | The symbols of a word ` W ... |
| wwlknllvtx 28112 | If a word ` W ` represents... |
| wwlknlsw 28113 | If a word represents a wal... |
| wspthsn 28114 | The set of simple paths of... |
| iswspthn 28115 | An element of the set of s... |
| wspthnp 28116 | Properties of a set being ... |
| wwlksnon 28117 | The set of walks of a fixe... |
| wspthsnon 28118 | The set of simple paths of... |
| iswwlksnon 28119 | The set of walks of a fixe... |
| wwlksnon0 28120 | Sufficient conditions for ... |
| wwlksonvtx 28121 | If a word ` W ` represents... |
| iswspthsnon 28122 | The set of simple paths of... |
| wwlknon 28123 | An element of the set of w... |
| wspthnon 28124 | An element of the set of s... |
| wspthnonp 28125 | Properties of a set being ... |
| wspthneq1eq2 28126 | Two simple paths with iden... |
| wwlksn0s 28127 | The set of all walks as wo... |
| wwlkssswrd 28128 | Walks (represented by word... |
| wwlksn0 28129 | A walk of length 0 is repr... |
| 0enwwlksnge1 28130 | In graphs without edges, t... |
| wwlkswwlksn 28131 | A walk of a fixed length a... |
| wwlkssswwlksn 28132 | The walks of a fixed lengt... |
| wlkiswwlks1 28133 | The sequence of vertices i... |
| wlklnwwlkln1 28134 | The sequence of vertices i... |
| wlkiswwlks2lem1 28135 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 28136 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 28137 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 28138 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 28139 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 28140 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 28141 | A walk as word corresponds... |
| wlkiswwlks 28142 | A walk as word corresponds... |
| wlkiswwlksupgr2 28143 | A walk as word corresponds... |
| wlkiswwlkupgr 28144 | A walk as word corresponds... |
| wlkswwlksf1o 28145 | The mapping of (ordinary) ... |
| wlkswwlksen 28146 | The set of walks as words ... |
| wwlksm1edg 28147 | Removing the trailing edge... |
| wlklnwwlkln2lem 28148 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 28149 | A walk of length ` N ` as ... |
| wlklnwwlkn 28150 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 28151 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 28152 | A walk of length ` N ` as ... |
| wlknewwlksn 28153 | If a walk in a pseudograph... |
| wlknwwlksnbij 28154 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 28155 | In a simple pseudograph, t... |
| wlknwwlksneqs 28156 | The set of walks of a fixe... |
| wwlkseq 28157 | Equality of two walks (as ... |
| wwlksnred 28158 | Reduction of a walk (as wo... |
| wwlksnext 28159 | Extension of a walk (as wo... |
| wwlksnextbi 28160 | Extension of a walk (as wo... |
| wwlksnredwwlkn 28161 | For each walk (as word) of... |
| wwlksnredwwlkn0 28162 | For each walk (as word) of... |
| wwlksnextwrd 28163 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 28164 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 28165 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 28166 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 28167 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 28168 | There is a bijection betwe... |
| wwlksnexthasheq 28169 | The number of the extensio... |
| disjxwwlksn 28170 | Sets of walks (as words) e... |
| wwlksnndef 28171 | Conditions for ` WWalksN `... |
| wwlksnfi 28172 | The number of walks repres... |
| wlksnfi 28173 | The number of walks of fix... |
| wlksnwwlknvbij 28174 | There is a bijection betwe... |
| wwlksnextproplem1 28175 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 28176 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 28177 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 28178 | Adding additional properti... |
| disjxwwlkn 28179 | Sets of walks (as words) e... |
| hashwwlksnext 28180 | Number of walks (as words)... |
| wwlksnwwlksnon 28181 | A walk of fixed length is ... |
| wspthsnwspthsnon 28182 | A simple path of fixed len... |
| wspthsnonn0vne 28183 | If the set of simple paths... |
| wspthsswwlkn 28184 | The set of simple paths of... |
| wspthnfi 28185 | In a finite graph, the set... |
| wwlksnonfi 28186 | In a finite graph, the set... |
| wspthsswwlknon 28187 | The set of simple paths of... |
| wspthnonfi 28188 | In a finite graph, the set... |
| wspniunwspnon 28189 | The set of nonempty simple... |
| wspn0 28190 | If there are no vertices, ... |
| 2wlkdlem1 28191 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 28192 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 28193 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 28194 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 28195 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 28196 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 28197 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 28198 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 28199 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 28200 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 28201 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 28202 | Construction of a walk fro... |
| 2wlkond 28203 | A walk of length 2 from on... |
| 2trld 28204 | Construction of a trail fr... |
| 2trlond 28205 | A trail of length 2 from o... |
| 2pthd 28206 | A path of length 2 from on... |
| 2spthd 28207 | A simple path of length 2 ... |
| 2pthond 28208 | A simple path of length 2 ... |
| 2pthon3v 28209 | For a vertex adjacent to t... |
| umgr2adedgwlklem 28210 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 28211 | In a multigraph, two adjac... |
| umgr2adedgwlkon 28212 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 28213 | Alternate proof for ~ umgr... |
| umgr2adedgspth 28214 | In a multigraph, two adjac... |
| umgr2wlk 28215 | In a multigraph, there is ... |
| umgr2wlkon 28216 | For each pair of adjacent ... |
| elwwlks2s3 28217 | A walk of length 2 as word... |
| midwwlks2s3 28218 | There is a vertex between ... |
| wwlks2onv 28219 | If a length 3 string repre... |
| elwwlks2ons3im 28220 | A walk as word of length 2... |
| elwwlks2ons3 28221 | For each walk of length 2 ... |
| s3wwlks2on 28222 | A length 3 string which re... |
| umgrwwlks2on 28223 | A walk of length 2 between... |
| wwlks2onsym 28224 | There is a walk of length ... |
| elwwlks2on 28225 | A walk of length 2 between... |
| elwspths2on 28226 | A simple path of length 2 ... |
| wpthswwlks2on 28227 | For two different vertices... |
| 2wspdisj 28228 | All simple paths of length... |
| 2wspiundisj 28229 | All simple paths of length... |
| usgr2wspthons3 28230 | A simple path of length 2 ... |
| usgr2wspthon 28231 | A simple path of length 2 ... |
| elwwlks2 28232 | A walk of length 2 between... |
| elwspths2spth 28233 | A simple path of length 2 ... |
| rusgrnumwwlkl1 28234 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 28235 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 28236 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 28237 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 28238 | Induction base 1 for ~ rus... |
| rusgr0edg 28239 | Special case for graphs wi... |
| rusgrnumwwlks 28240 | Induction step for ~ rusgr... |
| rusgrnumwwlk 28241 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 28242 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 28243 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 28244 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 28245 | In a ` K `-regular graph, ... |
| clwwlk 28248 | The set of closed walks (i... |
| isclwwlk 28249 | Properties of a word to re... |
| clwwlkbp 28250 | Basic properties of a clos... |
| clwwlkgt0 28251 | There is no empty closed w... |
| clwwlksswrd 28252 | Closed walks (represented ... |
| clwwlk1loop 28253 | A closed walk of length 1 ... |
| clwwlkccatlem 28254 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 28255 | The concatenation of two w... |
| umgrclwwlkge2 28256 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 28257 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 28258 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 28259 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 28260 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 28261 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 28262 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 28263 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 28264 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 28265 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 28266 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 28267 | A closed walk as word of l... |
| clwlkclwwlk2 28268 | A closed walk corresponds ... |
| clwlkclwwlkflem 28269 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 28270 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 28271 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 28272 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 28273 | ` F ` is a function from t... |
| clwlkclwwlkfo 28274 | ` F ` is a function from t... |
| clwlkclwwlkf1 28275 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 28276 | ` F ` is a bijection betwe... |
| clwlkclwwlken 28277 | The set of the nonempty cl... |
| clwwisshclwwslemlem 28278 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 28279 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 28280 | Cyclically shifting a clos... |
| clwwisshclwwsn 28281 | Cyclically shifting a clos... |
| erclwwlkrel 28282 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 28283 | Two classes are equivalent... |
| erclwwlkeqlen 28284 | If two classes are equival... |
| erclwwlkref 28285 | ` .~ ` is a reflexive rela... |
| erclwwlksym 28286 | ` .~ ` is a symmetric rela... |
| erclwwlktr 28287 | ` .~ ` is a transitive rel... |
| erclwwlk 28288 | ` .~ ` is an equivalence r... |
| clwwlkn 28291 | The set of closed walks of... |
| isclwwlkn 28292 | A word over the set of ver... |
| clwwlkn0 28293 | There is no closed walk of... |
| clwwlkneq0 28294 | Sufficient conditions for ... |
| clwwlkclwwlkn 28295 | A closed walk of a fixed l... |
| clwwlksclwwlkn 28296 | The closed walks of a fixe... |
| clwwlknlen 28297 | The length of a word repre... |
| clwwlknnn 28298 | The length of a closed wal... |
| clwwlknwrd 28299 | A closed walk of a fixed l... |
| clwwlknbp 28300 | Basic properties of a clos... |
| isclwwlknx 28301 | Characterization of a word... |
| clwwlknp 28302 | Properties of a set being ... |
| clwwlknwwlksn 28303 | A word representing a clos... |
| clwwlknlbonbgr1 28304 | The last but one vertex in... |
| clwwlkinwwlk 28305 | If the initial vertex of a... |
| clwwlkn1 28306 | A closed walk of length 1 ... |
| loopclwwlkn1b 28307 | The singleton word consist... |
| clwwlkn1loopb 28308 | A word represents a closed... |
| clwwlkn2 28309 | A closed walk of length 2 ... |
| clwwlknfi 28310 | If there is only a finite ... |
| clwwlkel 28311 | Obtaining a closed walk (a... |
| clwwlkf 28312 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 28313 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 28314 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 28315 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 28316 | F is a 1-1 onto function, ... |
| clwwlken 28317 | The set of closed walks of... |
| clwwlknwwlkncl 28318 | Obtaining a closed walk (a... |
| clwwlkwwlksb 28319 | A nonempty word over verti... |
| clwwlknwwlksnb 28320 | A word over vertices repre... |
| clwwlkext2edg 28321 | If a word concatenated wit... |
| wwlksext2clwwlk 28322 | If a word represents a wal... |
| wwlksubclwwlk 28323 | Any prefix of a word repre... |
| clwwnisshclwwsn 28324 | Cyclically shifting a clos... |
| eleclclwwlknlem1 28325 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 28326 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 28327 | The set of cyclical shifts... |
| clwwlknccat 28328 | The concatenation of two w... |
| umgr2cwwk2dif 28329 | If a word represents a clo... |
| umgr2cwwkdifex 28330 | If a word represents a clo... |
| erclwwlknrel 28331 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 28332 | Two classes are equivalent... |
| erclwwlkneqlen 28333 | If two classes are equival... |
| erclwwlknref 28334 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 28335 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 28336 | ` .~ ` is a transitive rel... |
| erclwwlkn 28337 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 28338 | The quotient set of the se... |
| hashclwwlkn0 28339 | The number of closed walks... |
| eclclwwlkn1 28340 | An equivalence class accor... |
| eleclclwwlkn 28341 | A member of an equivalence... |
| hashecclwwlkn1 28342 | The size of every equivale... |
| umgrhashecclwwlk 28343 | The size of every equivale... |
| fusgrhashclwwlkn 28344 | The size of the set of clo... |
| clwwlkndivn 28345 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 28346 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 28347 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 28348 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 28349 | There is a one-to-one onto... |
| clwlkssizeeq 28350 | The size of the set of clo... |
| clwlksndivn 28351 | The size of the set of clo... |
| clwwlknonmpo 28354 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 28355 | The set of closed walks on... |
| isclwwlknon 28356 | A word over the set of ver... |
| clwwlk0on0 28357 | There is no word over the ... |
| clwwlknon0 28358 | Sufficient conditions for ... |
| clwwlknonfin 28359 | In a finite graph ` G ` , ... |
| clwwlknonel 28360 | Characterization of a word... |
| clwwlknonccat 28361 | The concatenation of two w... |
| clwwlknon1 28362 | The set of closed walks on... |
| clwwlknon1loop 28363 | If there is a loop at vert... |
| clwwlknon1nloop 28364 | If there is no loop at ver... |
| clwwlknon1sn 28365 | The set of (closed) walks ... |
| clwwlknon1le1 28366 | There is at most one (clos... |
| clwwlknon2 28367 | The set of closed walks on... |
| clwwlknon2x 28368 | The set of closed walks on... |
| s2elclwwlknon2 28369 | Sufficient conditions of a... |
| clwwlknon2num 28370 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 28371 | A word over vertices repre... |
| clwwlknonex2lem1 28372 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 28373 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 28374 | Extending a closed walk ` ... |
| clwwlknonex2e 28375 | Extending a closed walk ` ... |
| clwwlknondisj 28376 | The sets of closed walks o... |
| clwwlknun 28377 | The set of closed walks of... |
| clwwlkvbij 28378 | There is a bijection betwe... |
| 0ewlk 28379 | The empty set (empty seque... |
| 1ewlk 28380 | A sequence of 1 edge is an... |
| 0wlk 28381 | A pair of an empty set (of... |
| is0wlk 28382 | A pair of an empty set (of... |
| 0wlkonlem1 28383 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 28384 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 28385 | A walk of length 0 from a ... |
| 0wlkons1 28386 | A walk of length 0 from a ... |
| 0trl 28387 | A pair of an empty set (of... |
| is0trl 28388 | A pair of an empty set (of... |
| 0trlon 28389 | A trail of length 0 from a... |
| 0pth 28390 | A pair of an empty set (of... |
| 0spth 28391 | A pair of an empty set (of... |
| 0pthon 28392 | A path of length 0 from a ... |
| 0pthon1 28393 | A path of length 0 from a ... |
| 0pthonv 28394 | For each vertex there is a... |
| 0clwlk 28395 | A pair of an empty set (of... |
| 0clwlkv 28396 | Any vertex (more precisely... |
| 0clwlk0 28397 | There is no closed walk in... |
| 0crct 28398 | A pair of an empty set (of... |
| 0cycl 28399 | A pair of an empty set (of... |
| 1pthdlem1 28400 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 28401 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 28402 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 28403 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 28404 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 28405 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 28406 | In a graph with two vertic... |
| 1trld 28407 | In a graph with two vertic... |
| 1pthd 28408 | In a graph with two vertic... |
| 1pthond 28409 | In a graph with two vertic... |
| upgr1wlkdlem1 28410 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 28411 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 28412 | In a pseudograph with two ... |
| upgr1trld 28413 | In a pseudograph with two ... |
| upgr1pthd 28414 | In a pseudograph with two ... |
| upgr1pthond 28415 | In a pseudograph with two ... |
| lppthon 28416 | A loop (which is an edge a... |
| lp1cycl 28417 | A loop (which is an edge a... |
| 1pthon2v 28418 | For each pair of adjacent ... |
| 1pthon2ve 28419 | For each pair of adjacent ... |
| wlk2v2elem1 28420 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 28421 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 28422 | In a graph with two vertic... |
| ntrl2v2e 28423 | A walk which is not a trai... |
| 3wlkdlem1 28424 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 28425 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 28426 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 28427 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 28428 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 28429 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 28430 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 28431 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 28432 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 28433 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 28434 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 28435 | Construction of a walk fro... |
| 3wlkond 28436 | A walk of length 3 from on... |
| 3trld 28437 | Construction of a trail fr... |
| 3trlond 28438 | A trail of length 3 from o... |
| 3pthd 28439 | A path of length 3 from on... |
| 3pthond 28440 | A path of length 3 from on... |
| 3spthd 28441 | A simple path of length 3 ... |
| 3spthond 28442 | A simple path of length 3 ... |
| 3cycld 28443 | Construction of a 3-cycle ... |
| 3cyclpd 28444 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 28445 | If there is a cycle of len... |
| uhgr3cyclexlem 28446 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 28447 | If there are three differe... |
| umgr3cyclex 28448 | If there are three (differ... |
| umgr3v3e3cycl 28449 | If and only if there is a ... |
| upgr4cycl4dv4e 28450 | If there is a cycle of len... |
| dfconngr1 28453 | Alternative definition of ... |
| isconngr 28454 | The property of being a co... |
| isconngr1 28455 | The property of being a co... |
| cusconngr 28456 | A complete hypergraph is c... |
| 0conngr 28457 | A graph without vertices i... |
| 0vconngr 28458 | A graph without vertices i... |
| 1conngr 28459 | A graph with (at most) one... |
| conngrv2edg 28460 | A vertex in a connected gr... |
| vdn0conngrumgrv2 28461 | A vertex in a connected mu... |
| releupth 28464 | The set ` ( EulerPaths `` ... |
| eupths 28465 | The Eulerian paths on the ... |
| iseupth 28466 | The property " ` <. F , P ... |
| iseupthf1o 28467 | The property " ` <. F , P ... |
| eupthi 28468 | Properties of an Eulerian ... |
| eupthf1o 28469 | The ` F ` function in an E... |
| eupthfi 28470 | Any graph with an Eulerian... |
| eupthseg 28471 | The ` N ` -th edge in an e... |
| upgriseupth 28472 | The property " ` <. F , P ... |
| upgreupthi 28473 | Properties of an Eulerian ... |
| upgreupthseg 28474 | The ` N ` -th edge in an e... |
| eupthcl 28475 | An Eulerian path has lengt... |
| eupthistrl 28476 | An Eulerian path is a trai... |
| eupthiswlk 28477 | An Eulerian path is a walk... |
| eupthpf 28478 | The ` P ` function in an E... |
| eupth0 28479 | There is an Eulerian path ... |
| eupthres 28480 | The restriction ` <. H , Q... |
| eupthp1 28481 | Append one path segment to... |
| eupth2eucrct 28482 | Append one path segment to... |
| eupth2lem1 28483 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 28484 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 28485 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 28486 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 28487 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 28488 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 28489 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 28490 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 28491 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 28492 | Formerly part of proof of ... |
| eupth2lem3lem1 28493 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 28494 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 28495 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 28496 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 28497 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 28498 | Formerly part of proof of ... |
| eupth2lem3lem7 28499 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 28500 | Formerly part of proof of ... |
| eupth2lem3 28501 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 28502 | Lemma for ~ eupth2 (induct... |
| eupth2lems 28503 | Lemma for ~ eupth2 (induct... |
| eupth2 28504 | The only vertices of odd d... |
| eulerpathpr 28505 | A graph with an Eulerian p... |
| eulerpath 28506 | A pseudograph with an Eule... |
| eulercrct 28507 | A pseudograph with an Eule... |
| eucrctshift 28508 | Cyclically shifting the in... |
| eucrct2eupth1 28509 | Removing one edge ` ( I ``... |
| eucrct2eupth 28510 | Removing one edge ` ( I ``... |
| konigsbergvtx 28511 | The set of vertices of the... |
| konigsbergiedg 28512 | The indexed edges of the K... |
| konigsbergiedgw 28513 | The indexed edges of the K... |
| konigsbergssiedgwpr 28514 | Each subset of the indexed... |
| konigsbergssiedgw 28515 | Each subset of the indexed... |
| konigsbergumgr 28516 | The Königsberg graph ... |
| konigsberglem1 28517 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 28518 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 28519 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 28520 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 28521 | Lemma 5 for ~ konigsberg :... |
| konigsberg 28522 | The Königsberg Bridge... |
| isfrgr 28525 | The property of being a fr... |
| frgrusgr 28526 | A friendship graph is a si... |
| frgr0v 28527 | Any null graph (set with n... |
| frgr0vb 28528 | Any null graph (without ve... |
| frgruhgr0v 28529 | Any null graph (without ve... |
| frgr0 28530 | The null graph (graph with... |
| frcond1 28531 | The friendship condition: ... |
| frcond2 28532 | The friendship condition: ... |
| frgreu 28533 | Variant of ~ frcond2 : An... |
| frcond3 28534 | The friendship condition, ... |
| frcond4 28535 | The friendship condition, ... |
| frgr1v 28536 | Any graph with (at most) o... |
| nfrgr2v 28537 | Any graph with two (differ... |
| frgr3vlem1 28538 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 28539 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 28540 | Any graph with three verti... |
| 1vwmgr 28541 | Every graph with one verte... |
| 3vfriswmgrlem 28542 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 28543 | Every friendship graph wit... |
| 1to2vfriswmgr 28544 | Every friendship graph wit... |
| 1to3vfriswmgr 28545 | Every friendship graph wit... |
| 1to3vfriendship 28546 | The friendship theorem for... |
| 2pthfrgrrn 28547 | Between any two (different... |
| 2pthfrgrrn2 28548 | Between any two (different... |
| 2pthfrgr 28549 | Between any two (different... |
| 3cyclfrgrrn1 28550 | Every vertex in a friendsh... |
| 3cyclfrgrrn 28551 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 28552 | Every vertex in a friendsh... |
| 3cyclfrgr 28553 | Every vertex in a friendsh... |
| 4cycl2v2nb 28554 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 28555 | In a 4-cycle, two distinct... |
| n4cyclfrgr 28556 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 28557 | A graph with a 4-cycle is ... |
| frgrnbnb 28558 | If two neighbors ` U ` and... |
| frgrconngr 28559 | A friendship graph is conn... |
| vdgn0frgrv2 28560 | A vertex in a friendship g... |
| vdgn1frgrv2 28561 | Any vertex in a friendship... |
| vdgn1frgrv3 28562 | Any vertex in a friendship... |
| vdgfrgrgt2 28563 | Any vertex in a friendship... |
| frgrncvvdeqlem1 28564 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 28565 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 28566 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 28567 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 28568 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 28569 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 28570 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 28571 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 28572 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 28573 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 28574 | In a friendship graph, two... |
| frgrwopreglem4a 28575 | In a friendship graph any ... |
| frgrwopreglem5a 28576 | If a friendship graph has ... |
| frgrwopreglem1 28577 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 28578 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 28579 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 28580 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 28581 | According to statement 5 i... |
| frgrwopregbsn 28582 | According to statement 5 i... |
| frgrwopreg1 28583 | According to statement 5 i... |
| frgrwopreg2 28584 | According to statement 5 i... |
| frgrwopreglem5lem 28585 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 28586 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 28587 | Alternate direct proof of ... |
| frgrwopreg 28588 | In a friendship graph ther... |
| frgrregorufr0 28589 | In a friendship graph ther... |
| frgrregorufr 28590 | If there is a vertex havin... |
| frgrregorufrg 28591 | If there is a vertex havin... |
| frgr2wwlkeu 28592 | For two different vertices... |
| frgr2wwlkn0 28593 | In a friendship graph, the... |
| frgr2wwlk1 28594 | In a friendship graph, the... |
| frgr2wsp1 28595 | In a friendship graph, the... |
| frgr2wwlkeqm 28596 | If there is a (simple) pat... |
| frgrhash2wsp 28597 | The number of simple paths... |
| fusgreg2wsplem 28598 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 28599 | The set of paths of length... |
| fusgreghash2wspv 28600 | According to statement 7 i... |
| fusgreg2wsp 28601 | In a finite simple graph, ... |
| 2wspmdisj 28602 | The sets of paths of lengt... |
| fusgreghash2wsp 28603 | In a finite k-regular grap... |
| frrusgrord0lem 28604 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 28605 | If a nonempty finite frien... |
| frrusgrord 28606 | If a nonempty finite frien... |
| numclwwlk2lem1lem 28607 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 28608 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 28609 | If the initial vertex of a... |
| clwwnonrepclwwnon 28610 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 28611 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 28612 | Value of operation ` C ` ,... |
| 2clwwlk2 28613 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 28614 | Characterization of an ele... |
| 2clwwlk2clwwlk 28615 | An element of the value of... |
| numclwwlk1lem2foalem 28616 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 28617 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 28618 | Characterization of an ele... |
| numclwwlk1lem2foa 28619 | Going forth and back from ... |
| numclwwlk1lem2f 28620 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 28621 | Value of the function ` T ... |
| numclwwlk1lem2f1 28622 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 28623 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 28624 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 28625 | The set of double loops of... |
| numclwwlk1 28626 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 28627 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 28628 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 28629 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 28630 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 28631 | The sets of the two repres... |
| wlkl0 28632 | There is exactly one walk ... |
| clwlknon2num 28633 | There are k walks of lengt... |
| numclwlk1lem1 28634 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 28635 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 28636 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 28637 | Value of operation ` H ` ,... |
| numclwwlkovh 28638 | Value of operation ` H ` ,... |
| numclwwlkovq 28639 | Value of operation ` Q ` ,... |
| numclwwlkqhash 28640 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 28641 | In a friendship graph, for... |
| numclwlk2lem2f 28642 | ` R ` is a function mappin... |
| numclwlk2lem2fv 28643 | Value of the function ` R ... |
| numclwlk2lem2f1o 28644 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 28645 | In a friendship graph, the... |
| numclwwlk2 28646 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 28647 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 28648 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 28649 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 28650 | Statement 12 in [Huneke] p... |
| numclwwlk4 28651 | The total number of closed... |
| numclwwlk5lem 28652 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 28653 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 28654 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 28655 | For a prime divisor ` P ` ... |
| numclwwlk7 28656 | Statement 14 in [Huneke] p... |
| numclwwlk8 28657 | The size of the set of clo... |
| frgrreggt1 28658 | If a finite nonempty frien... |
| frgrreg 28659 | If a finite nonempty frien... |
| frgrregord013 28660 | If a finite friendship gra... |
| frgrregord13 28661 | If a nonempty finite frien... |
| frgrogt3nreg 28662 | If a finite friendship gra... |
| friendshipgt3 28663 | The friendship theorem for... |
| friendship 28664 | The friendship theorem: I... |
| conventions 28665 |
H... |
| conventions-labels 28666 |
... |
| conventions-comments 28667 |
... |
| natded 28668 | Here are typical n... |
| ex-natded5.2 28669 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 28670 | A more efficient proof of ... |
| ex-natded5.2i 28671 | The same as ~ ex-natded5.2... |
| ex-natded5.3 28672 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 28673 | A more efficient proof of ... |
| ex-natded5.3i 28674 | The same as ~ ex-natded5.3... |
| ex-natded5.5 28675 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 28676 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 28677 | A more efficient proof of ... |
| ex-natded5.8 28678 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 28679 | A more efficient proof of ... |
| ex-natded5.13 28680 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 28681 | A more efficient proof of ... |
| ex-natded9.20 28682 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 28683 | A more efficient proof of ... |
| ex-natded9.26 28684 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 28685 | A more efficient proof of ... |
| ex-or 28686 | Example for ~ df-or . Exa... |
| ex-an 28687 | Example for ~ df-an . Exa... |
| ex-dif 28688 | Example for ~ df-dif . Ex... |
| ex-un 28689 | Example for ~ df-un . Exa... |
| ex-in 28690 | Example for ~ df-in . Exa... |
| ex-uni 28691 | Example for ~ df-uni . Ex... |
| ex-ss 28692 | Example for ~ df-ss . Exa... |
| ex-pss 28693 | Example for ~ df-pss . Ex... |
| ex-pw 28694 | Example for ~ df-pw . Exa... |
| ex-pr 28695 | Example for ~ df-pr . (Co... |
| ex-br 28696 | Example for ~ df-br . Exa... |
| ex-opab 28697 | Example for ~ df-opab . E... |
| ex-eprel 28698 | Example for ~ df-eprel . ... |
| ex-id 28699 | Example for ~ df-id . Exa... |
| ex-po 28700 | Example for ~ df-po . Exa... |
| ex-xp 28701 | Example for ~ df-xp . Exa... |
| ex-cnv 28702 | Example for ~ df-cnv . Ex... |
| ex-co 28703 | Example for ~ df-co . Exa... |
| ex-dm 28704 | Example for ~ df-dm . Exa... |
| ex-rn 28705 | Example for ~ df-rn . Exa... |
| ex-res 28706 | Example for ~ df-res . Ex... |
| ex-ima 28707 | Example for ~ df-ima . Ex... |
| ex-fv 28708 | Example for ~ df-fv . Exa... |
| ex-1st 28709 | Example for ~ df-1st . Ex... |
| ex-2nd 28710 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 28711 | Example for ~ df-dec , 100... |
| ex-fl 28712 | Example for ~ df-fl . Exa... |
| ex-ceil 28713 | Example for ~ df-ceil . (... |
| ex-mod 28714 | Example for ~ df-mod . (C... |
| ex-exp 28715 | Example for ~ df-exp . (C... |
| ex-fac 28716 | Example for ~ df-fac . (C... |
| ex-bc 28717 | Example for ~ df-bc . (Co... |
| ex-hash 28718 | Example for ~ df-hash . (... |
| ex-sqrt 28719 | Example for ~ df-sqrt . (... |
| ex-abs 28720 | Example for ~ df-abs . (C... |
| ex-dvds 28721 | Example for ~ df-dvds : 3 ... |
| ex-gcd 28722 | Example for ~ df-gcd . (C... |
| ex-lcm 28723 | Example for ~ df-lcm . (C... |
| ex-prmo 28724 | Example for ~ df-prmo : ` ... |
| aevdemo 28725 | Proof illustrating the com... |
| ex-ind-dvds 28726 | Example of a proof by indu... |
| ex-fpar 28727 | Formalized example provide... |
| avril1 28728 | Poisson d'Avril's Theorem.... |
| 2bornot2b 28729 | The law of excluded middle... |
| helloworld 28730 | The classic "Hello world" ... |
| 1p1e2apr1 28731 | One plus one equals two. ... |
| eqid1 28732 | Law of identity (reflexivi... |
| 1div0apr 28733 | Division by zero is forbid... |
| topnfbey 28734 | Nothing seems to be imposs... |
| 9p10ne21 28735 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 28736 | 9 + 10 equals 21. This as... |
| isplig 28739 | The predicate "is a planar... |
| ispligb 28740 | The predicate "is a planar... |
| tncp 28741 | In any planar incidence ge... |
| l2p 28742 | For any line in a planar i... |
| lpni 28743 | For any line in a planar i... |
| nsnlplig 28744 | There is no "one-point lin... |
| nsnlpligALT 28745 | Alternate version of ~ nsn... |
| n0lplig 28746 | There is no "empty line" i... |
| n0lpligALT 28747 | Alternate version of ~ n0l... |
| eulplig 28748 | Through two distinct point... |
| pliguhgr 28749 | Any planar incidence geome... |
| dummylink 28750 | Alias for ~ a1ii that may ... |
| id1 28751 | Alias for ~ idALT that may... |
| isgrpo 28760 | The predicate "is a group ... |
| isgrpoi 28761 | Properties that determine ... |
| grpofo 28762 | A group operation maps ont... |
| grpocl 28763 | Closure law for a group op... |
| grpolidinv 28764 | A group has a left identit... |
| grpon0 28765 | The base set of a group is... |
| grpoass 28766 | A group operation is assoc... |
| grpoidinvlem1 28767 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 28768 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 28769 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 28770 | Lemma for ~ grpoidinv . (... |
| grpoidinv 28771 | A group has a left and rig... |
| grpoideu 28772 | The left identity element ... |
| grporndm 28773 | A group's range in terms o... |
| 0ngrp 28774 | The empty set is not a gro... |
| gidval 28775 | The value of the identity ... |
| grpoidval 28776 | Lemma for ~ grpoidcl and o... |
| grpoidcl 28777 | The identity element of a ... |
| grpoidinv2 28778 | A group's properties using... |
| grpolid 28779 | The identity element of a ... |
| grporid 28780 | The identity element of a ... |
| grporcan 28781 | Right cancellation law for... |
| grpoinveu 28782 | The left inverse element o... |
| grpoid 28783 | Two ways of saying that an... |
| grporn 28784 | The range of a group opera... |
| grpoinvfval 28785 | The inverse function of a ... |
| grpoinvval 28786 | The inverse of a group ele... |
| grpoinvcl 28787 | A group element's inverse ... |
| grpoinv 28788 | The properties of a group ... |
| grpolinv 28789 | The left inverse of a grou... |
| grporinv 28790 | The right inverse of a gro... |
| grpoinvid1 28791 | The inverse of a group ele... |
| grpoinvid2 28792 | The inverse of a group ele... |
| grpolcan 28793 | Left cancellation law for ... |
| grpo2inv 28794 | Double inverse law for gro... |
| grpoinvf 28795 | Mapping of the inverse fun... |
| grpoinvop 28796 | The inverse of the group o... |
| grpodivfval 28797 | Group division (or subtrac... |
| grpodivval 28798 | Group division (or subtrac... |
| grpodivinv 28799 | Group division by an inver... |
| grpoinvdiv 28800 | Inverse of a group divisio... |
| grpodivf 28801 | Mapping for group division... |
| grpodivcl 28802 | Closure of group division ... |
| grpodivdiv 28803 | Double group division. (C... |
| grpomuldivass 28804 | Associative-type law for m... |
| grpodivid 28805 | Division of a group member... |
| grponpcan 28806 | Cancellation law for group... |
| isablo 28809 | The predicate "is an Abeli... |
| ablogrpo 28810 | An Abelian group operation... |
| ablocom 28811 | An Abelian group operation... |
| ablo32 28812 | Commutative/associative la... |
| ablo4 28813 | Commutative/associative la... |
| isabloi 28814 | Properties that determine ... |
| ablomuldiv 28815 | Law for group multiplicati... |
| ablodivdiv 28816 | Law for double group divis... |
| ablodivdiv4 28817 | Law for double group divis... |
| ablodiv32 28818 | Swap the second and third ... |
| ablonncan 28819 | Cancellation law for group... |
| ablonnncan1 28820 | Cancellation law for group... |
| vcrel 28823 | The class of all complex v... |
| vciOLD 28824 | Obsolete version of ~ cvsi... |
| vcsm 28825 | Functionality of th scalar... |
| vccl 28826 | Closure of the scalar prod... |
| vcidOLD 28827 | Identity element for the s... |
| vcdi 28828 | Distributive law for the s... |
| vcdir 28829 | Distributive law for the s... |
| vcass 28830 | Associative law for the sc... |
| vc2OLD 28831 | A vector plus itself is tw... |
| vcablo 28832 | Vector addition is an Abel... |
| vcgrp 28833 | Vector addition is a group... |
| vclcan 28834 | Left cancellation law for ... |
| vczcl 28835 | The zero vector is a vecto... |
| vc0rid 28836 | The zero vector is a right... |
| vc0 28837 | Zero times a vector is the... |
| vcz 28838 | Anything times the zero ve... |
| vcm 28839 | Minus 1 times a vector is ... |
| isvclem 28840 | Lemma for ~ isvcOLD . (Co... |
| vcex 28841 | The components of a comple... |
| isvcOLD 28842 | The predicate "is a comple... |
| isvciOLD 28843 | Properties that determine ... |
| cnaddabloOLD 28844 | Obsolete version of ~ cnad... |
| cnidOLD 28845 | Obsolete version of ~ cnad... |
| cncvcOLD 28846 | Obsolete version of ~ cncv... |
| nvss 28856 | Structure of the class of ... |
| nvvcop 28857 | A normed complex vector sp... |
| nvrel 28865 | The class of all normed co... |
| vafval 28866 | Value of the function for ... |
| bafval 28867 | Value of the function for ... |
| smfval 28868 | Value of the function for ... |
| 0vfval 28869 | Value of the function for ... |
| nmcvfval 28870 | Value of the norm function... |
| nvop2 28871 | A normed complex vector sp... |
| nvvop 28872 | The vector space component... |
| isnvlem 28873 | Lemma for ~ isnv . (Contr... |
| nvex 28874 | The components of a normed... |
| isnv 28875 | The predicate "is a normed... |
| isnvi 28876 | Properties that determine ... |
| nvi 28877 | The properties of a normed... |
| nvvc 28878 | The vector space component... |
| nvablo 28879 | The vector addition operat... |
| nvgrp 28880 | The vector addition operat... |
| nvgf 28881 | Mapping for the vector add... |
| nvsf 28882 | Mapping for the scalar mul... |
| nvgcl 28883 | Closure law for the vector... |
| nvcom 28884 | The vector addition (group... |
| nvass 28885 | The vector addition (group... |
| nvadd32 28886 | Commutative/associative la... |
| nvrcan 28887 | Right cancellation law for... |
| nvadd4 28888 | Rearrangement of 4 terms i... |
| nvscl 28889 | Closure law for the scalar... |
| nvsid 28890 | Identity element for the s... |
| nvsass 28891 | Associative law for the sc... |
| nvscom 28892 | Commutative law for the sc... |
| nvdi 28893 | Distributive law for the s... |
| nvdir 28894 | Distributive law for the s... |
| nv2 28895 | A vector plus itself is tw... |
| vsfval 28896 | Value of the function for ... |
| nvzcl 28897 | Closure law for the zero v... |
| nv0rid 28898 | The zero vector is a right... |
| nv0lid 28899 | The zero vector is a left ... |
| nv0 28900 | Zero times a vector is the... |
| nvsz 28901 | Anything times the zero ve... |
| nvinv 28902 | Minus 1 times a vector is ... |
| nvinvfval 28903 | Function for the negative ... |
| nvm 28904 | Vector subtraction in term... |
| nvmval 28905 | Value of vector subtractio... |
| nvmval2 28906 | Value of vector subtractio... |
| nvmfval 28907 | Value of the function for ... |
| nvmf 28908 | Mapping for the vector sub... |
| nvmcl 28909 | Closure law for the vector... |
| nvnnncan1 28910 | Cancellation law for vecto... |
| nvmdi 28911 | Distributive law for scala... |
| nvnegneg 28912 | Double negative of a vecto... |
| nvmul0or 28913 | If a scalar product is zer... |
| nvrinv 28914 | A vector minus itself. (C... |
| nvlinv 28915 | Minus a vector plus itself... |
| nvpncan2 28916 | Cancellation law for vecto... |
| nvpncan 28917 | Cancellation law for vecto... |
| nvaddsub 28918 | Commutative/associative la... |
| nvnpcan 28919 | Cancellation law for a nor... |
| nvaddsub4 28920 | Rearrangement of 4 terms i... |
| nvmeq0 28921 | The difference between two... |
| nvmid 28922 | A vector minus itself is t... |
| nvf 28923 | Mapping for the norm funct... |
| nvcl 28924 | The norm of a normed compl... |
| nvcli 28925 | The norm of a normed compl... |
| nvs 28926 | Proportionality property o... |
| nvsge0 28927 | The norm of a scalar produ... |
| nvm1 28928 | The norm of the negative o... |
| nvdif 28929 | The norm of the difference... |
| nvpi 28930 | The norm of a vector plus ... |
| nvz0 28931 | The norm of a zero vector ... |
| nvz 28932 | The norm of a vector is ze... |
| nvtri 28933 | Triangle inequality for th... |
| nvmtri 28934 | Triangle inequality for th... |
| nvabs 28935 | Norm difference property o... |
| nvge0 28936 | The norm of a normed compl... |
| nvgt0 28937 | A nonzero norm is positive... |
| nv1 28938 | From any nonzero vector, c... |
| nvop 28939 | A complex inner product sp... |
| cnnv 28940 | The set of complex numbers... |
| cnnvg 28941 | The vector addition (group... |
| cnnvba 28942 | The base set of the normed... |
| cnnvs 28943 | The scalar product operati... |
| cnnvnm 28944 | The norm operation of the ... |
| cnnvm 28945 | The vector subtraction ope... |
| elimnv 28946 | Hypothesis elimination lem... |
| elimnvu 28947 | Hypothesis elimination lem... |
| imsval 28948 | Value of the induced metri... |
| imsdval 28949 | Value of the induced metri... |
| imsdval2 28950 | Value of the distance func... |
| nvnd 28951 | The norm of a normed compl... |
| imsdf 28952 | Mapping for the induced me... |
| imsmetlem 28953 | Lemma for ~ imsmet . (Con... |
| imsmet 28954 | The induced metric of a no... |
| imsxmet 28955 | The induced metric of a no... |
| cnims 28956 | The metric induced on the ... |
| vacn 28957 | Vector addition is jointly... |
| nmcvcn 28958 | The norm of a normed compl... |
| nmcnc 28959 | The norm of a normed compl... |
| smcnlem 28960 | Lemma for ~ smcn . (Contr... |
| smcn 28961 | Scalar multiplication is j... |
| vmcn 28962 | Vector subtraction is join... |
| dipfval 28965 | The inner product function... |
| ipval 28966 | Value of the inner product... |
| ipval2lem2 28967 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 28968 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 28969 | Lemma for ~ ipval3 . (Con... |
| ipval2 28970 | Expansion of the inner pro... |
| 4ipval2 28971 | Four times the inner produ... |
| ipval3 28972 | Expansion of the inner pro... |
| ipidsq 28973 | The inner product of a vec... |
| ipnm 28974 | Norm expressed in terms of... |
| dipcl 28975 | An inner product is a comp... |
| ipf 28976 | Mapping for the inner prod... |
| dipcj 28977 | The complex conjugate of a... |
| ipipcj 28978 | An inner product times its... |
| diporthcom 28979 | Orthogonality (meaning inn... |
| dip0r 28980 | Inner product with a zero ... |
| dip0l 28981 | Inner product with a zero ... |
| ipz 28982 | The inner product of a vec... |
| dipcn 28983 | Inner product is jointly c... |
| sspval 28986 | The set of all subspaces o... |
| isssp 28987 | The predicate "is a subspa... |
| sspid 28988 | A normed complex vector sp... |
| sspnv 28989 | A subspace is a normed com... |
| sspba 28990 | The base set of a subspace... |
| sspg 28991 | Vector addition on a subsp... |
| sspgval 28992 | Vector addition on a subsp... |
| ssps 28993 | Scalar multiplication on a... |
| sspsval 28994 | Scalar multiplication on a... |
| sspmlem 28995 | Lemma for ~ sspm and other... |
| sspmval 28996 | Vector addition on a subsp... |
| sspm 28997 | Vector subtraction on a su... |
| sspz 28998 | The zero vector of a subsp... |
| sspn 28999 | The norm on a subspace is ... |
| sspnval 29000 | The norm on a subspace in ... |
| sspimsval 29001 | The induced metric on a su... |
| sspims 29002 | The induced metric on a su... |
| lnoval 29015 | The set of linear operator... |
| islno 29016 | The predicate "is a linear... |
| lnolin 29017 | Basic linearity property o... |
| lnof 29018 | A linear operator is a map... |
| lno0 29019 | The value of a linear oper... |
| lnocoi 29020 | The composition of two lin... |
| lnoadd 29021 | Addition property of a lin... |
| lnosub 29022 | Subtraction property of a ... |
| lnomul 29023 | Scalar multiplication prop... |
| nvo00 29024 | Two ways to express a zero... |
| nmoofval 29025 | The operator norm function... |
| nmooval 29026 | The operator norm function... |
| nmosetre 29027 | The set in the supremum of... |
| nmosetn0 29028 | The set in the supremum of... |
| nmoxr 29029 | The norm of an operator is... |
| nmooge0 29030 | The norm of an operator is... |
| nmorepnf 29031 | The norm of an operator is... |
| nmoreltpnf 29032 | The norm of any operator i... |
| nmogtmnf 29033 | The norm of an operator is... |
| nmoolb 29034 | A lower bound for an opera... |
| nmoubi 29035 | An upper bound for an oper... |
| nmoub3i 29036 | An upper bound for an oper... |
| nmoub2i 29037 | An upper bound for an oper... |
| nmobndi 29038 | Two ways to express that a... |
| nmounbi 29039 | Two ways two express that ... |
| nmounbseqi 29040 | An unbounded operator dete... |
| nmounbseqiALT 29041 | Alternate shorter proof of... |
| nmobndseqi 29042 | A bounded sequence determi... |
| nmobndseqiALT 29043 | Alternate shorter proof of... |
| bloval 29044 | The class of bounded linea... |
| isblo 29045 | The predicate "is a bounde... |
| isblo2 29046 | The predicate "is a bounde... |
| bloln 29047 | A bounded operator is a li... |
| blof 29048 | A bounded operator is an o... |
| nmblore 29049 | The norm of a bounded oper... |
| 0ofval 29050 | The zero operator between ... |
| 0oval 29051 | Value of the zero operator... |
| 0oo 29052 | The zero operator is an op... |
| 0lno 29053 | The zero operator is linea... |
| nmoo0 29054 | The operator norm of the z... |
| 0blo 29055 | The zero operator is a bou... |
| nmlno0lem 29056 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 29057 | The norm of a linear opera... |
| nmlno0 29058 | The norm of a linear opera... |
| nmlnoubi 29059 | An upper bound for the ope... |
| nmlnogt0 29060 | The norm of a nonzero line... |
| lnon0 29061 | The domain of a nonzero li... |
| nmblolbii 29062 | A lower bound for the norm... |
| nmblolbi 29063 | A lower bound for the norm... |
| isblo3i 29064 | The predicate "is a bounde... |
| blo3i 29065 | Properties that determine ... |
| blometi 29066 | Upper bound for the distan... |
| blocnilem 29067 | Lemma for ~ blocni and ~ l... |
| blocni 29068 | A linear operator is conti... |
| lnocni 29069 | If a linear operator is co... |
| blocn 29070 | A linear operator is conti... |
| blocn2 29071 | A bounded linear operator ... |
| ajfval 29072 | The adjoint function. (Co... |
| hmoval 29073 | The set of Hermitian (self... |
| ishmo 29074 | The predicate "is a hermit... |
| phnv 29077 | Every complex inner produc... |
| phrel 29078 | The class of all complex i... |
| phnvi 29079 | Every complex inner produc... |
| isphg 29080 | The predicate "is a comple... |
| phop 29081 | A complex inner product sp... |
| cncph 29082 | The set of complex numbers... |
| elimph 29083 | Hypothesis elimination lem... |
| elimphu 29084 | Hypothesis elimination lem... |
| isph 29085 | The predicate "is an inner... |
| phpar2 29086 | The parallelogram law for ... |
| phpar 29087 | The parallelogram law for ... |
| ip0i 29088 | A slight variant of Equati... |
| ip1ilem 29089 | Lemma for ~ ip1i . (Contr... |
| ip1i 29090 | Equation 6.47 of [Ponnusam... |
| ip2i 29091 | Equation 6.48 of [Ponnusam... |
| ipdirilem 29092 | Lemma for ~ ipdiri . (Con... |
| ipdiri 29093 | Distributive law for inner... |
| ipasslem1 29094 | Lemma for ~ ipassi . Show... |
| ipasslem2 29095 | Lemma for ~ ipassi . Show... |
| ipasslem3 29096 | Lemma for ~ ipassi . Show... |
| ipasslem4 29097 | Lemma for ~ ipassi . Show... |
| ipasslem5 29098 | Lemma for ~ ipassi . Show... |
| ipasslem7 29099 | Lemma for ~ ipassi . Show... |
| ipasslem8 29100 | Lemma for ~ ipassi . By ~... |
| ipasslem9 29101 | Lemma for ~ ipassi . Conc... |
| ipasslem10 29102 | Lemma for ~ ipassi . Show... |
| ipasslem11 29103 | Lemma for ~ ipassi . Show... |
| ipassi 29104 | Associative law for inner ... |
| dipdir 29105 | Distributive law for inner... |
| dipdi 29106 | Distributive law for inner... |
| ip2dii 29107 | Inner product of two sums.... |
| dipass 29108 | Associative law for inner ... |
| dipassr 29109 | "Associative" law for seco... |
| dipassr2 29110 | "Associative" law for inne... |
| dipsubdir 29111 | Distributive law for inner... |
| dipsubdi 29112 | Distributive law for inner... |
| pythi 29113 | The Pythagorean theorem fo... |
| siilem1 29114 | Lemma for ~ sii . (Contri... |
| siilem2 29115 | Lemma for ~ sii . (Contri... |
| siii 29116 | Inference from ~ sii . (C... |
| sii 29117 | Obsolete version of ~ ipca... |
| ipblnfi 29118 | A function ` F ` generated... |
| ip2eqi 29119 | Two vectors are equal iff ... |
| phoeqi 29120 | A condition implying that ... |
| ajmoi 29121 | Every operator has at most... |
| ajfuni 29122 | The adjoint function is a ... |
| ajfun 29123 | The adjoint function is a ... |
| ajval 29124 | Value of the adjoint funct... |
| iscbn 29127 | A complex Banach space is ... |
| cbncms 29128 | The induced metric on comp... |
| bnnv 29129 | Every complex Banach space... |
| bnrel 29130 | The class of all complex B... |
| bnsscmcl 29131 | A subspace of a Banach spa... |
| cnbn 29132 | The set of complex numbers... |
| ubthlem1 29133 | Lemma for ~ ubth . The fu... |
| ubthlem2 29134 | Lemma for ~ ubth . Given ... |
| ubthlem3 29135 | Lemma for ~ ubth . Prove ... |
| ubth 29136 | Uniform Boundedness Theore... |
| minvecolem1 29137 | Lemma for ~ minveco . The... |
| minvecolem2 29138 | Lemma for ~ minveco . Any... |
| minvecolem3 29139 | Lemma for ~ minveco . The... |
| minvecolem4a 29140 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 29141 | Lemma for ~ minveco . The... |
| minvecolem4c 29142 | Lemma for ~ minveco . The... |
| minvecolem4 29143 | Lemma for ~ minveco . The... |
| minvecolem5 29144 | Lemma for ~ minveco . Dis... |
| minvecolem6 29145 | Lemma for ~ minveco . Any... |
| minvecolem7 29146 | Lemma for ~ minveco . Sin... |
| minveco 29147 | Minimizing vector theorem,... |
| ishlo 29150 | The predicate "is a comple... |
| hlobn 29151 | Every complex Hilbert spac... |
| hlph 29152 | Every complex Hilbert spac... |
| hlrel 29153 | The class of all complex H... |
| hlnv 29154 | Every complex Hilbert spac... |
| hlnvi 29155 | Every complex Hilbert spac... |
| hlvc 29156 | Every complex Hilbert spac... |
| hlcmet 29157 | The induced metric on a co... |
| hlmet 29158 | The induced metric on a co... |
| hlpar2 29159 | The parallelogram law sati... |
| hlpar 29160 | The parallelogram law sati... |
| hlex 29161 | The base set of a Hilbert ... |
| hladdf 29162 | Mapping for Hilbert space ... |
| hlcom 29163 | Hilbert space vector addit... |
| hlass 29164 | Hilbert space vector addit... |
| hl0cl 29165 | The Hilbert space zero vec... |
| hladdid 29166 | Hilbert space addition wit... |
| hlmulf 29167 | Mapping for Hilbert space ... |
| hlmulid 29168 | Hilbert space scalar multi... |
| hlmulass 29169 | Hilbert space scalar multi... |
| hldi 29170 | Hilbert space scalar multi... |
| hldir 29171 | Hilbert space scalar multi... |
| hlmul0 29172 | Hilbert space scalar multi... |
| hlipf 29173 | Mapping for Hilbert space ... |
| hlipcj 29174 | Conjugate law for Hilbert ... |
| hlipdir 29175 | Distributive law for Hilbe... |
| hlipass 29176 | Associative law for Hilber... |
| hlipgt0 29177 | The inner product of a Hil... |
| hlcompl 29178 | Completeness of a Hilbert ... |
| cnchl 29179 | The set of complex numbers... |
| htthlem 29180 | Lemma for ~ htth . The co... |
| htth 29181 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 29237 | The group (addition) opera... |
| h2hsm 29238 | The scalar product operati... |
| h2hnm 29239 | The norm function of Hilbe... |
| h2hvs 29240 | The vector subtraction ope... |
| h2hmetdval 29241 | Value of the distance func... |
| h2hcau 29242 | The Cauchy sequences of Hi... |
| h2hlm 29243 | The limit sequences of Hil... |
| axhilex-zf 29244 | Derive Axiom ~ ax-hilex fr... |
| axhfvadd-zf 29245 | Derive Axiom ~ ax-hfvadd f... |
| axhvcom-zf 29246 | Derive Axiom ~ ax-hvcom fr... |
| axhvass-zf 29247 | Derive Axiom ~ ax-hvass fr... |
| axhv0cl-zf 29248 | Derive Axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 29249 | Derive Axiom ~ ax-hvaddid ... |
| axhfvmul-zf 29250 | Derive Axiom ~ ax-hfvmul f... |
| axhvmulid-zf 29251 | Derive Axiom ~ ax-hvmulid ... |
| axhvmulass-zf 29252 | Derive Axiom ~ ax-hvmulass... |
| axhvdistr1-zf 29253 | Derive Axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 29254 | Derive Axiom ~ ax-hvdistr2... |
| axhvmul0-zf 29255 | Derive Axiom ~ ax-hvmul0 f... |
| axhfi-zf 29256 | Derive Axiom ~ ax-hfi from... |
| axhis1-zf 29257 | Derive Axiom ~ ax-his1 fro... |
| axhis2-zf 29258 | Derive Axiom ~ ax-his2 fro... |
| axhis3-zf 29259 | Derive Axiom ~ ax-his3 fro... |
| axhis4-zf 29260 | Derive Axiom ~ ax-his4 fro... |
| axhcompl-zf 29261 | Derive Axiom ~ ax-hcompl f... |
| hvmulex 29274 | The Hilbert space scalar p... |
| hvaddcl 29275 | Closure of vector addition... |
| hvmulcl 29276 | Closure of scalar multipli... |
| hvmulcli 29277 | Closure inference for scal... |
| hvsubf 29278 | Mapping domain and codomai... |
| hvsubval 29279 | Value of vector subtractio... |
| hvsubcl 29280 | Closure of vector subtract... |
| hvaddcli 29281 | Closure of vector addition... |
| hvcomi 29282 | Commutation of vector addi... |
| hvsubvali 29283 | Value of vector subtractio... |
| hvsubcli 29284 | Closure of vector subtract... |
| ifhvhv0 29285 | Prove ` if ( A e. ~H , A ,... |
| hvaddid2 29286 | Addition with the zero vec... |
| hvmul0 29287 | Scalar multiplication with... |
| hvmul0or 29288 | If a scalar product is zer... |
| hvsubid 29289 | Subtraction of a vector fr... |
| hvnegid 29290 | Addition of negative of a ... |
| hv2neg 29291 | Two ways to express the ne... |
| hvaddid2i 29292 | Addition with the zero vec... |
| hvnegidi 29293 | Addition of negative of a ... |
| hv2negi 29294 | Two ways to express the ne... |
| hvm1neg 29295 | Convert minus one times a ... |
| hvaddsubval 29296 | Value of vector addition i... |
| hvadd32 29297 | Commutative/associative la... |
| hvadd12 29298 | Commutative/associative la... |
| hvadd4 29299 | Hilbert vector space addit... |
| hvsub4 29300 | Hilbert vector space addit... |
| hvaddsub12 29301 | Commutative/associative la... |
| hvpncan 29302 | Addition/subtraction cance... |
| hvpncan2 29303 | Addition/subtraction cance... |
| hvaddsubass 29304 | Associativity of sum and d... |
| hvpncan3 29305 | Subtraction and addition o... |
| hvmulcom 29306 | Scalar multiplication comm... |
| hvsubass 29307 | Hilbert vector space assoc... |
| hvsub32 29308 | Hilbert vector space commu... |
| hvmulassi 29309 | Scalar multiplication asso... |
| hvmulcomi 29310 | Scalar multiplication comm... |
| hvmul2negi 29311 | Double negative in scalar ... |
| hvsubdistr1 29312 | Scalar multiplication dist... |
| hvsubdistr2 29313 | Scalar multiplication dist... |
| hvdistr1i 29314 | Scalar multiplication dist... |
| hvsubdistr1i 29315 | Scalar multiplication dist... |
| hvassi 29316 | Hilbert vector space assoc... |
| hvadd32i 29317 | Hilbert vector space commu... |
| hvsubassi 29318 | Hilbert vector space assoc... |
| hvsub32i 29319 | Hilbert vector space commu... |
| hvadd12i 29320 | Hilbert vector space commu... |
| hvadd4i 29321 | Hilbert vector space addit... |
| hvsubsub4i 29322 | Hilbert vector space addit... |
| hvsubsub4 29323 | Hilbert vector space addit... |
| hv2times 29324 | Two times a vector. (Cont... |
| hvnegdii 29325 | Distribution of negative o... |
| hvsubeq0i 29326 | If the difference between ... |
| hvsubcan2i 29327 | Vector cancellation law. ... |
| hvaddcani 29328 | Cancellation law for vecto... |
| hvsubaddi 29329 | Relationship between vecto... |
| hvnegdi 29330 | Distribution of negative o... |
| hvsubeq0 29331 | If the difference between ... |
| hvaddeq0 29332 | If the sum of two vectors ... |
| hvaddcan 29333 | Cancellation law for vecto... |
| hvaddcan2 29334 | Cancellation law for vecto... |
| hvmulcan 29335 | Cancellation law for scala... |
| hvmulcan2 29336 | Cancellation law for scala... |
| hvsubcan 29337 | Cancellation law for vecto... |
| hvsubcan2 29338 | Cancellation law for vecto... |
| hvsub0 29339 | Subtraction of a zero vect... |
| hvsubadd 29340 | Relationship between vecto... |
| hvaddsub4 29341 | Hilbert vector space addit... |
| hicl 29343 | Closure of inner product. ... |
| hicli 29344 | Closure inference for inne... |
| his5 29349 | Associative law for inner ... |
| his52 29350 | Associative law for inner ... |
| his35 29351 | Move scalar multiplication... |
| his35i 29352 | Move scalar multiplication... |
| his7 29353 | Distributive law for inner... |
| hiassdi 29354 | Distributive/associative l... |
| his2sub 29355 | Distributive law for inner... |
| his2sub2 29356 | Distributive law for inner... |
| hire 29357 | A necessary and sufficient... |
| hiidrcl 29358 | Real closure of inner prod... |
| hi01 29359 | Inner product with the 0 v... |
| hi02 29360 | Inner product with the 0 v... |
| hiidge0 29361 | Inner product with self is... |
| his6 29362 | Zero inner product with se... |
| his1i 29363 | Conjugate law for inner pr... |
| abshicom 29364 | Commuted inner products ha... |
| hial0 29365 | A vector whose inner produ... |
| hial02 29366 | A vector whose inner produ... |
| hisubcomi 29367 | Two vector subtractions si... |
| hi2eq 29368 | Lemma used to prove equali... |
| hial2eq 29369 | Two vectors whose inner pr... |
| hial2eq2 29370 | Two vectors whose inner pr... |
| orthcom 29371 | Orthogonality commutes. (... |
| normlem0 29372 | Lemma used to derive prope... |
| normlem1 29373 | Lemma used to derive prope... |
| normlem2 29374 | Lemma used to derive prope... |
| normlem3 29375 | Lemma used to derive prope... |
| normlem4 29376 | Lemma used to derive prope... |
| normlem5 29377 | Lemma used to derive prope... |
| normlem6 29378 | Lemma used to derive prope... |
| normlem7 29379 | Lemma used to derive prope... |
| normlem8 29380 | Lemma used to derive prope... |
| normlem9 29381 | Lemma used to derive prope... |
| normlem7tALT 29382 | Lemma used to derive prope... |
| bcseqi 29383 | Equality case of Bunjakova... |
| normlem9at 29384 | Lemma used to derive prope... |
| dfhnorm2 29385 | Alternate definition of th... |
| normf 29386 | The norm function maps fro... |
| normval 29387 | The value of the norm of a... |
| normcl 29388 | Real closure of the norm o... |
| normge0 29389 | The norm of a vector is no... |
| normgt0 29390 | The norm of nonzero vector... |
| norm0 29391 | The norm of a zero vector.... |
| norm-i 29392 | Theorem 3.3(i) of [Beran] ... |
| normne0 29393 | A norm is nonzero iff its ... |
| normcli 29394 | Real closure of the norm o... |
| normsqi 29395 | The square of a norm. (Co... |
| norm-i-i 29396 | Theorem 3.3(i) of [Beran] ... |
| normsq 29397 | The square of a norm. (Co... |
| normsub0i 29398 | Two vectors are equal iff ... |
| normsub0 29399 | Two vectors are equal iff ... |
| norm-ii-i 29400 | Triangle inequality for no... |
| norm-ii 29401 | Triangle inequality for no... |
| norm-iii-i 29402 | Theorem 3.3(iii) of [Beran... |
| norm-iii 29403 | Theorem 3.3(iii) of [Beran... |
| normsubi 29404 | Negative doesn't change th... |
| normpythi 29405 | Analogy to Pythagorean the... |
| normsub 29406 | Swapping order of subtract... |
| normneg 29407 | The norm of a vector equal... |
| normpyth 29408 | Analogy to Pythagorean the... |
| normpyc 29409 | Corollary to Pythagorean t... |
| norm3difi 29410 | Norm of differences around... |
| norm3adifii 29411 | Norm of differences around... |
| norm3lem 29412 | Lemma involving norm of di... |
| norm3dif 29413 | Norm of differences around... |
| norm3dif2 29414 | Norm of differences around... |
| norm3lemt 29415 | Lemma involving norm of di... |
| norm3adifi 29416 | Norm of differences around... |
| normpari 29417 | Parallelogram law for norm... |
| normpar 29418 | Parallelogram law for norm... |
| normpar2i 29419 | Corollary of parallelogram... |
| polid2i 29420 | Generalized polarization i... |
| polidi 29421 | Polarization identity. Re... |
| polid 29422 | Polarization identity. Re... |
| hilablo 29423 | Hilbert space vector addit... |
| hilid 29424 | The group identity element... |
| hilvc 29425 | Hilbert space is a complex... |
| hilnormi 29426 | Hilbert space norm in term... |
| hilhhi 29427 | Deduce the structure of Hi... |
| hhnv 29428 | Hilbert space is a normed ... |
| hhva 29429 | The group (addition) opera... |
| hhba 29430 | The base set of Hilbert sp... |
| hh0v 29431 | The zero vector of Hilbert... |
| hhsm 29432 | The scalar product operati... |
| hhvs 29433 | The vector subtraction ope... |
| hhnm 29434 | The norm function of Hilbe... |
| hhims 29435 | The induced metric of Hilb... |
| hhims2 29436 | Hilbert space distance met... |
| hhmet 29437 | The induced metric of Hilb... |
| hhxmet 29438 | The induced metric of Hilb... |
| hhmetdval 29439 | Value of the distance func... |
| hhip 29440 | The inner product operatio... |
| hhph 29441 | The Hilbert space of the H... |
| bcsiALT 29442 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 29443 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 29444 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 29445 | Corollary of the Bunjakova... |
| bcs3 29446 | Corollary of the Bunjakova... |
| hcau 29447 | Member of the set of Cauch... |
| hcauseq 29448 | A Cauchy sequences on a Hi... |
| hcaucvg 29449 | A Cauchy sequence on a Hil... |
| seq1hcau 29450 | A sequence on a Hilbert sp... |
| hlimi 29451 | Express the predicate: Th... |
| hlimseqi 29452 | A sequence with a limit on... |
| hlimveci 29453 | Closure of the limit of a ... |
| hlimconvi 29454 | Convergence of a sequence ... |
| hlim2 29455 | The limit of a sequence on... |
| hlimadd 29456 | Limit of the sum of two se... |
| hilmet 29457 | The Hilbert space norm det... |
| hilxmet 29458 | The Hilbert space norm det... |
| hilmetdval 29459 | Value of the distance func... |
| hilims 29460 | Hilbert space distance met... |
| hhcau 29461 | The Cauchy sequences of Hi... |
| hhlm 29462 | The limit sequences of Hil... |
| hhcmpl 29463 | Lemma used for derivation ... |
| hilcompl 29464 | Lemma used for derivation ... |
| hhcms 29466 | The Hilbert space induced ... |
| hhhl 29467 | The Hilbert space structur... |
| hilcms 29468 | The Hilbert space norm det... |
| hilhl 29469 | The Hilbert space of the H... |
| issh 29471 | Subspace ` H ` of a Hilber... |
| issh2 29472 | Subspace ` H ` of a Hilber... |
| shss 29473 | A subspace is a subset of ... |
| shel 29474 | A member of a subspace of ... |
| shex 29475 | The set of subspaces of a ... |
| shssii 29476 | A closed subspace of a Hil... |
| sheli 29477 | A member of a subspace of ... |
| shelii 29478 | A member of a subspace of ... |
| sh0 29479 | The zero vector belongs to... |
| shaddcl 29480 | Closure of vector addition... |
| shmulcl 29481 | Closure of vector scalar m... |
| issh3 29482 | Subspace ` H ` of a Hilber... |
| shsubcl 29483 | Closure of vector subtract... |
| isch 29485 | Closed subspace ` H ` of a... |
| isch2 29486 | Closed subspace ` H ` of a... |
| chsh 29487 | A closed subspace is a sub... |
| chsssh 29488 | Closed subspaces are subsp... |
| chex 29489 | The set of closed subspace... |
| chshii 29490 | A closed subspace is a sub... |
| ch0 29491 | The zero vector belongs to... |
| chss 29492 | A closed subspace of a Hil... |
| chel 29493 | A member of a closed subsp... |
| chssii 29494 | A closed subspace of a Hil... |
| cheli 29495 | A member of a closed subsp... |
| chelii 29496 | A member of a closed subsp... |
| chlimi 29497 | The limit property of a cl... |
| hlim0 29498 | The zero sequence in Hilbe... |
| hlimcaui 29499 | If a sequence in Hilbert s... |
| hlimf 29500 | Function-like behavior of ... |
| hlimuni 29501 | A Hilbert space sequence c... |
| hlimreui 29502 | The limit of a Hilbert spa... |
| hlimeui 29503 | The limit of a Hilbert spa... |
| isch3 29504 | A Hilbert subspace is clos... |
| chcompl 29505 | Completeness of a closed s... |
| helch 29506 | The unit Hilbert lattice e... |
| ifchhv 29507 | Prove ` if ( A e. CH , A ,... |
| helsh 29508 | Hilbert space is a subspac... |
| shsspwh 29509 | Subspaces are subsets of H... |
| chsspwh 29510 | Closed subspaces are subse... |
| hsn0elch 29511 | The zero subspace belongs ... |
| norm1 29512 | From any nonzero Hilbert s... |
| norm1exi 29513 | A normalized vector exists... |
| norm1hex 29514 | A normalized vector can ex... |
| elch0 29517 | Membership in zero for clo... |
| h0elch 29518 | The zero subspace is a clo... |
| h0elsh 29519 | The zero subspace is a sub... |
| hhssva 29520 | The vector addition operat... |
| hhsssm 29521 | The scalar multiplication ... |
| hhssnm 29522 | The norm operation on a su... |
| issubgoilem 29523 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 29524 | Lemma for ~ hhssabloi . F... |
| hhssabloi 29525 | Abelian group property of ... |
| hhssablo 29526 | Abelian group property of ... |
| hhssnv 29527 | Normed complex vector spac... |
| hhssnvt 29528 | Normed complex vector spac... |
| hhsst 29529 | A member of ` SH ` is a su... |
| hhshsslem1 29530 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 29531 | Lemma for ~ hhsssh . (Con... |
| hhsssh 29532 | The predicate " ` H ` is a... |
| hhsssh2 29533 | The predicate " ` H ` is a... |
| hhssba 29534 | The base set of a subspace... |
| hhssvs 29535 | The vector subtraction ope... |
| hhssvsf 29536 | Mapping of the vector subt... |
| hhssims 29537 | Induced metric of a subspa... |
| hhssims2 29538 | Induced metric of a subspa... |
| hhssmet 29539 | Induced metric of a subspa... |
| hhssmetdval 29540 | Value of the distance func... |
| hhsscms 29541 | The induced metric of a cl... |
| hhssbnOLD 29542 | Obsolete version of ~ cssb... |
| ocval 29543 | Value of orthogonal comple... |
| ocel 29544 | Membership in orthogonal c... |
| shocel 29545 | Membership in orthogonal c... |
| ocsh 29546 | The orthogonal complement ... |
| shocsh 29547 | The orthogonal complement ... |
| ocss 29548 | An orthogonal complement i... |
| shocss 29549 | An orthogonal complement i... |
| occon 29550 | Contraposition law for ort... |
| occon2 29551 | Double contraposition for ... |
| occon2i 29552 | Double contraposition for ... |
| oc0 29553 | The zero vector belongs to... |
| ocorth 29554 | Members of a subset and it... |
| shocorth 29555 | Members of a subspace and ... |
| ococss 29556 | Inclusion in complement of... |
| shococss 29557 | Inclusion in complement of... |
| shorth 29558 | Members of orthogonal subs... |
| ocin 29559 | Intersection of a Hilbert ... |
| occon3 29560 | Hilbert lattice contraposi... |
| ocnel 29561 | A nonzero vector in the co... |
| chocvali 29562 | Value of the orthogonal co... |
| shuni 29563 | Two subspaces with trivial... |
| chocunii 29564 | Lemma for uniqueness part ... |
| pjhthmo 29565 | Projection Theorem, unique... |
| occllem 29566 | Lemma for ~ occl . (Contr... |
| occl 29567 | Closure of complement of H... |
| shoccl 29568 | Closure of complement of H... |
| choccl 29569 | Closure of complement of H... |
| choccli 29570 | Closure of ` CH ` orthocom... |
| shsval 29575 | Value of subspace sum of t... |
| shsss 29576 | The subspace sum is a subs... |
| shsel 29577 | Membership in the subspace... |
| shsel3 29578 | Membership in the subspace... |
| shseli 29579 | Membership in subspace sum... |
| shscli 29580 | Closure of subspace sum. ... |
| shscl 29581 | Closure of subspace sum. ... |
| shscom 29582 | Commutative law for subspa... |
| shsva 29583 | Vector sum belongs to subs... |
| shsel1 29584 | A subspace sum contains a ... |
| shsel2 29585 | A subspace sum contains a ... |
| shsvs 29586 | Vector subtraction belongs... |
| shsub1 29587 | Subspace sum is an upper b... |
| shsub2 29588 | Subspace sum is an upper b... |
| choc0 29589 | The orthocomplement of the... |
| choc1 29590 | The orthocomplement of the... |
| chocnul 29591 | Orthogonal complement of t... |
| shintcli 29592 | Closure of intersection of... |
| shintcl 29593 | The intersection of a none... |
| chintcli 29594 | The intersection of a none... |
| chintcl 29595 | The intersection (infimum)... |
| spanval 29596 | Value of the linear span o... |
| hsupval 29597 | Value of supremum of set o... |
| chsupval 29598 | The value of the supremum ... |
| spancl 29599 | The span of a subset of Hi... |
| elspancl 29600 | A member of a span is a ve... |
| shsupcl 29601 | Closure of the subspace su... |
| hsupcl 29602 | Closure of supremum of set... |
| chsupcl 29603 | Closure of supremum of sub... |
| hsupss 29604 | Subset relation for suprem... |
| chsupss 29605 | Subset relation for suprem... |
| hsupunss 29606 | The union of a set of Hilb... |
| chsupunss 29607 | The union of a set of clos... |
| spanss2 29608 | A subset of Hilbert space ... |
| shsupunss 29609 | The union of a set of subs... |
| spanid 29610 | A subspace of Hilbert spac... |
| spanss 29611 | Ordering relationship for ... |
| spanssoc 29612 | The span of a subset of Hi... |
| sshjval 29613 | Value of join for subsets ... |
| shjval 29614 | Value of join in ` SH ` . ... |
| chjval 29615 | Value of join in ` CH ` . ... |
| chjvali 29616 | Value of join in ` CH ` . ... |
| sshjval3 29617 | Value of join for subsets ... |
| sshjcl 29618 | Closure of join for subset... |
| shjcl 29619 | Closure of join in ` SH ` ... |
| chjcl 29620 | Closure of join in ` CH ` ... |
| shjcom 29621 | Commutative law for Hilber... |
| shless 29622 | Subset implies subset of s... |
| shlej1 29623 | Add disjunct to both sides... |
| shlej2 29624 | Add disjunct to both sides... |
| shincli 29625 | Closure of intersection of... |
| shscomi 29626 | Commutative law for subspa... |
| shsvai 29627 | Vector sum belongs to subs... |
| shsel1i 29628 | A subspace sum contains a ... |
| shsel2i 29629 | A subspace sum contains a ... |
| shsvsi 29630 | Vector subtraction belongs... |
| shunssi 29631 | Union is smaller than subs... |
| shunssji 29632 | Union is smaller than Hilb... |
| shsleji 29633 | Subspace sum is smaller th... |
| shjcomi 29634 | Commutative law for join i... |
| shsub1i 29635 | Subspace sum is an upper b... |
| shsub2i 29636 | Subspace sum is an upper b... |
| shub1i 29637 | Hilbert lattice join is an... |
| shjcli 29638 | Closure of ` CH ` join. (... |
| shjshcli 29639 | ` SH ` closure of join. (... |
| shlessi 29640 | Subset implies subset of s... |
| shlej1i 29641 | Add disjunct to both sides... |
| shlej2i 29642 | Add disjunct to both sides... |
| shslej 29643 | Subspace sum is smaller th... |
| shincl 29644 | Closure of intersection of... |
| shub1 29645 | Hilbert lattice join is an... |
| shub2 29646 | A subspace is a subset of ... |
| shsidmi 29647 | Idempotent law for Hilbert... |
| shslubi 29648 | The least upper bound law ... |
| shlesb1i 29649 | Hilbert lattice ordering i... |
| shsval2i 29650 | An alternate way to expres... |
| shsval3i 29651 | An alternate way to expres... |
| shmodsi 29652 | The modular law holds for ... |
| shmodi 29653 | The modular law is implied... |
| pjhthlem1 29654 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 29655 | Lemma for ~ pjhth . (Cont... |
| pjhth 29656 | Projection Theorem: Any H... |
| pjhtheu 29657 | Projection Theorem: Any H... |
| pjhfval 29659 | The value of the projectio... |
| pjhval 29660 | Value of a projection. (C... |
| pjpreeq 29661 | Equality with a projection... |
| pjeq 29662 | Equality with a projection... |
| axpjcl 29663 | Closure of a projection in... |
| pjhcl 29664 | Closure of a projection in... |
| omlsilem 29665 | Lemma for orthomodular law... |
| omlsii 29666 | Subspace inference form of... |
| omlsi 29667 | Subspace form of orthomodu... |
| ococi 29668 | Complement of complement o... |
| ococ 29669 | Complement of complement o... |
| dfch2 29670 | Alternate definition of th... |
| ococin 29671 | The double complement is t... |
| hsupval2 29672 | Alternate definition of su... |
| chsupval2 29673 | The value of the supremum ... |
| sshjval2 29674 | Value of join in the set o... |
| chsupid 29675 | A subspace is the supremum... |
| chsupsn 29676 | Value of supremum of subse... |
| shlub 29677 | Hilbert lattice join is th... |
| shlubi 29678 | Hilbert lattice join is th... |
| pjhtheu2 29679 | Uniqueness of ` y ` for th... |
| pjcli 29680 | Closure of a projection in... |
| pjhcli 29681 | Closure of a projection in... |
| pjpjpre 29682 | Decomposition of a vector ... |
| axpjpj 29683 | Decomposition of a vector ... |
| pjclii 29684 | Closure of a projection in... |
| pjhclii 29685 | Closure of a projection in... |
| pjpj0i 29686 | Decomposition of a vector ... |
| pjpji 29687 | Decomposition of a vector ... |
| pjpjhth 29688 | Projection Theorem: Any H... |
| pjpjhthi 29689 | Projection Theorem: Any H... |
| pjop 29690 | Orthocomplement projection... |
| pjpo 29691 | Projection in terms of ort... |
| pjopi 29692 | Orthocomplement projection... |
| pjpoi 29693 | Projection in terms of ort... |
| pjoc1i 29694 | Projection of a vector in ... |
| pjchi 29695 | Projection of a vector in ... |
| pjoccl 29696 | The part of a vector that ... |
| pjoc1 29697 | Projection of a vector in ... |
| pjomli 29698 | Subspace form of orthomodu... |
| pjoml 29699 | Subspace form of orthomodu... |
| pjococi 29700 | Proof of orthocomplement t... |
| pjoc2i 29701 | Projection of a vector in ... |
| pjoc2 29702 | Projection of a vector in ... |
| sh0le 29703 | The zero subspace is the s... |
| ch0le 29704 | The zero subspace is the s... |
| shle0 29705 | No subspace is smaller tha... |
| chle0 29706 | No Hilbert lattice element... |
| chnlen0 29707 | A Hilbert lattice element ... |
| ch0pss 29708 | The zero subspace is a pro... |
| orthin 29709 | The intersection of orthog... |
| ssjo 29710 | The lattice join of a subs... |
| shne0i 29711 | A nonzero subspace has a n... |
| shs0i 29712 | Hilbert subspace sum with ... |
| shs00i 29713 | Two subspaces are zero iff... |
| ch0lei 29714 | The closed subspace zero i... |
| chle0i 29715 | No Hilbert closed subspace... |
| chne0i 29716 | A nonzero closed subspace ... |
| chocini 29717 | Intersection of a closed s... |
| chj0i 29718 | Join with lattice zero in ... |
| chm1i 29719 | Meet with lattice one in `... |
| chjcli 29720 | Closure of ` CH ` join. (... |
| chsleji 29721 | Subspace sum is smaller th... |
| chseli 29722 | Membership in subspace sum... |
| chincli 29723 | Closure of Hilbert lattice... |
| chsscon3i 29724 | Hilbert lattice contraposi... |
| chsscon1i 29725 | Hilbert lattice contraposi... |
| chsscon2i 29726 | Hilbert lattice contraposi... |
| chcon2i 29727 | Hilbert lattice contraposi... |
| chcon1i 29728 | Hilbert lattice contraposi... |
| chcon3i 29729 | Hilbert lattice contraposi... |
| chunssji 29730 | Union is smaller than ` CH... |
| chjcomi 29731 | Commutative law for join i... |
| chub1i 29732 | ` CH ` join is an upper bo... |
| chub2i 29733 | ` CH ` join is an upper bo... |
| chlubi 29734 | Hilbert lattice join is th... |
| chlubii 29735 | Hilbert lattice join is th... |
| chlej1i 29736 | Add join to both sides of ... |
| chlej2i 29737 | Add join to both sides of ... |
| chlej12i 29738 | Add join to both sides of ... |
| chlejb1i 29739 | Hilbert lattice ordering i... |
| chdmm1i 29740 | De Morgan's law for meet i... |
| chdmm2i 29741 | De Morgan's law for meet i... |
| chdmm3i 29742 | De Morgan's law for meet i... |
| chdmm4i 29743 | De Morgan's law for meet i... |
| chdmj1i 29744 | De Morgan's law for join i... |
| chdmj2i 29745 | De Morgan's law for join i... |
| chdmj3i 29746 | De Morgan's law for join i... |
| chdmj4i 29747 | De Morgan's law for join i... |
| chnlei 29748 | Equivalent expressions for... |
| chjassi 29749 | Associative law for Hilber... |
| chj00i 29750 | Two Hilbert lattice elemen... |
| chjoi 29751 | The join of a closed subsp... |
| chj1i 29752 | Join with Hilbert lattice ... |
| chm0i 29753 | Meet with Hilbert lattice ... |
| chm0 29754 | Meet with Hilbert lattice ... |
| shjshsi 29755 | Hilbert lattice join equal... |
| shjshseli 29756 | A closed subspace sum equa... |
| chne0 29757 | A nonzero closed subspace ... |
| chocin 29758 | Intersection of a closed s... |
| chssoc 29759 | A closed subspace less tha... |
| chj0 29760 | Join with Hilbert lattice ... |
| chslej 29761 | Subspace sum is smaller th... |
| chincl 29762 | Closure of Hilbert lattice... |
| chsscon3 29763 | Hilbert lattice contraposi... |
| chsscon1 29764 | Hilbert lattice contraposi... |
| chsscon2 29765 | Hilbert lattice contraposi... |
| chpsscon3 29766 | Hilbert lattice contraposi... |
| chpsscon1 29767 | Hilbert lattice contraposi... |
| chpsscon2 29768 | Hilbert lattice contraposi... |
| chjcom 29769 | Commutative law for Hilber... |
| chub1 29770 | Hilbert lattice join is gr... |
| chub2 29771 | Hilbert lattice join is gr... |
| chlub 29772 | Hilbert lattice join is th... |
| chlej1 29773 | Add join to both sides of ... |
| chlej2 29774 | Add join to both sides of ... |
| chlejb1 29775 | Hilbert lattice ordering i... |
| chlejb2 29776 | Hilbert lattice ordering i... |
| chnle 29777 | Equivalent expressions for... |
| chjo 29778 | The join of a closed subsp... |
| chabs1 29779 | Hilbert lattice absorption... |
| chabs2 29780 | Hilbert lattice absorption... |
| chabs1i 29781 | Hilbert lattice absorption... |
| chabs2i 29782 | Hilbert lattice absorption... |
| chjidm 29783 | Idempotent law for Hilbert... |
| chjidmi 29784 | Idempotent law for Hilbert... |
| chj12i 29785 | A rearrangement of Hilbert... |
| chj4i 29786 | Rearrangement of the join ... |
| chjjdiri 29787 | Hilbert lattice join distr... |
| chdmm1 29788 | De Morgan's law for meet i... |
| chdmm2 29789 | De Morgan's law for meet i... |
| chdmm3 29790 | De Morgan's law for meet i... |
| chdmm4 29791 | De Morgan's law for meet i... |
| chdmj1 29792 | De Morgan's law for join i... |
| chdmj2 29793 | De Morgan's law for join i... |
| chdmj3 29794 | De Morgan's law for join i... |
| chdmj4 29795 | De Morgan's law for join i... |
| chjass 29796 | Associative law for Hilber... |
| chj12 29797 | A rearrangement of Hilbert... |
| chj4 29798 | Rearrangement of the join ... |
| ledii 29799 | An ortholattice is distrib... |
| lediri 29800 | An ortholattice is distrib... |
| lejdii 29801 | An ortholattice is distrib... |
| lejdiri 29802 | An ortholattice is distrib... |
| ledi 29803 | An ortholattice is distrib... |
| spansn0 29804 | The span of the singleton ... |
| span0 29805 | The span of the empty set ... |
| elspani 29806 | Membership in the span of ... |
| spanuni 29807 | The span of a union is the... |
| spanun 29808 | The span of a union is the... |
| sshhococi 29809 | The join of two Hilbert sp... |
| hne0 29810 | Hilbert space has a nonzer... |
| chsup0 29811 | The supremum of the empty ... |
| h1deoi 29812 | Membership in orthocomplem... |
| h1dei 29813 | Membership in 1-dimensiona... |
| h1did 29814 | A generating vector belong... |
| h1dn0 29815 | A nonzero vector generates... |
| h1de2i 29816 | Membership in 1-dimensiona... |
| h1de2bi 29817 | Membership in 1-dimensiona... |
| h1de2ctlem 29818 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 29819 | Membership in 1-dimensiona... |
| spansni 29820 | The span of a singleton in... |
| elspansni 29821 | Membership in the span of ... |
| spansn 29822 | The span of a singleton in... |
| spansnch 29823 | The span of a Hilbert spac... |
| spansnsh 29824 | The span of a Hilbert spac... |
| spansnchi 29825 | The span of a singleton in... |
| spansnid 29826 | A vector belongs to the sp... |
| spansnmul 29827 | A scalar product with a ve... |
| elspansncl 29828 | A member of a span of a si... |
| elspansn 29829 | Membership in the span of ... |
| elspansn2 29830 | Membership in the span of ... |
| spansncol 29831 | The singletons of collinea... |
| spansneleqi 29832 | Membership relation implie... |
| spansneleq 29833 | Membership relation that i... |
| spansnss 29834 | The span of the singleton ... |
| elspansn3 29835 | A member of the span of th... |
| elspansn4 29836 | A span membership conditio... |
| elspansn5 29837 | A vector belonging to both... |
| spansnss2 29838 | The span of the singleton ... |
| normcan 29839 | Cancellation-type law that... |
| pjspansn 29840 | A projection on the span o... |
| spansnpji 29841 | A subset of Hilbert space ... |
| spanunsni 29842 | The span of the union of a... |
| spanpr 29843 | The span of a pair of vect... |
| h1datomi 29844 | A 1-dimensional subspace i... |
| h1datom 29845 | A 1-dimensional subspace i... |
| cmbr 29847 | Binary relation expressing... |
| pjoml2i 29848 | Variation of orthomodular ... |
| pjoml3i 29849 | Variation of orthomodular ... |
| pjoml4i 29850 | Variation of orthomodular ... |
| pjoml5i 29851 | The orthomodular law. Rem... |
| pjoml6i 29852 | An equivalent of the ortho... |
| cmbri 29853 | Binary relation expressing... |
| cmcmlem 29854 | Commutation is symmetric. ... |
| cmcmi 29855 | Commutation is symmetric. ... |
| cmcm2i 29856 | Commutation with orthocomp... |
| cmcm3i 29857 | Commutation with orthocomp... |
| cmcm4i 29858 | Commutation with orthocomp... |
| cmbr2i 29859 | Alternate definition of th... |
| cmcmii 29860 | Commutation is symmetric. ... |
| cmcm2ii 29861 | Commutation with orthocomp... |
| cmcm3ii 29862 | Commutation with orthocomp... |
| cmbr3i 29863 | Alternate definition for t... |
| cmbr4i 29864 | Alternate definition for t... |
| lecmi 29865 | Comparable Hilbert lattice... |
| lecmii 29866 | Comparable Hilbert lattice... |
| cmj1i 29867 | A Hilbert lattice element ... |
| cmj2i 29868 | A Hilbert lattice element ... |
| cmm1i 29869 | A Hilbert lattice element ... |
| cmm2i 29870 | A Hilbert lattice element ... |
| cmbr3 29871 | Alternate definition for t... |
| cm0 29872 | The zero Hilbert lattice e... |
| cmidi 29873 | The commutes relation is r... |
| pjoml2 29874 | Variation of orthomodular ... |
| pjoml3 29875 | Variation of orthomodular ... |
| pjoml5 29876 | The orthomodular law. Rem... |
| cmcm 29877 | Commutation is symmetric. ... |
| cmcm3 29878 | Commutation with orthocomp... |
| cmcm2 29879 | Commutation with orthocomp... |
| lecm 29880 | Comparable Hilbert lattice... |
| fh1 29881 | Foulis-Holland Theorem. I... |
| fh2 29882 | Foulis-Holland Theorem. I... |
| cm2j 29883 | A lattice element that com... |
| fh1i 29884 | Foulis-Holland Theorem. I... |
| fh2i 29885 | Foulis-Holland Theorem. I... |
| fh3i 29886 | Variation of the Foulis-Ho... |
| fh4i 29887 | Variation of the Foulis-Ho... |
| cm2ji 29888 | A lattice element that com... |
| cm2mi 29889 | A lattice element that com... |
| qlax1i 29890 | One of the equations showi... |
| qlax2i 29891 | One of the equations showi... |
| qlax3i 29892 | One of the equations showi... |
| qlax4i 29893 | One of the equations showi... |
| qlax5i 29894 | One of the equations showi... |
| qlaxr1i 29895 | One of the conditions show... |
| qlaxr2i 29896 | One of the conditions show... |
| qlaxr4i 29897 | One of the conditions show... |
| qlaxr5i 29898 | One of the conditions show... |
| qlaxr3i 29899 | A variation of the orthomo... |
| chscllem1 29900 | Lemma for ~ chscl . (Cont... |
| chscllem2 29901 | Lemma for ~ chscl . (Cont... |
| chscllem3 29902 | Lemma for ~ chscl . (Cont... |
| chscllem4 29903 | Lemma for ~ chscl . (Cont... |
| chscl 29904 | The subspace sum of two cl... |
| osumi 29905 | If two closed subspaces of... |
| osumcori 29906 | Corollary of ~ osumi . (C... |
| osumcor2i 29907 | Corollary of ~ osumi , sho... |
| osum 29908 | If two closed subspaces of... |
| spansnji 29909 | The subspace sum of a clos... |
| spansnj 29910 | The subspace sum of a clos... |
| spansnscl 29911 | The subspace sum of a clos... |
| sumspansn 29912 | The sum of two vectors bel... |
| spansnm0i 29913 | The meet of different one-... |
| nonbooli 29914 | A Hilbert lattice with two... |
| spansncvi 29915 | Hilbert space has the cove... |
| spansncv 29916 | Hilbert space has the cove... |
| 5oalem1 29917 | Lemma for orthoarguesian l... |
| 5oalem2 29918 | Lemma for orthoarguesian l... |
| 5oalem3 29919 | Lemma for orthoarguesian l... |
| 5oalem4 29920 | Lemma for orthoarguesian l... |
| 5oalem5 29921 | Lemma for orthoarguesian l... |
| 5oalem6 29922 | Lemma for orthoarguesian l... |
| 5oalem7 29923 | Lemma for orthoarguesian l... |
| 5oai 29924 | Orthoarguesian law 5OA. Th... |
| 3oalem1 29925 | Lemma for 3OA (weak) ortho... |
| 3oalem2 29926 | Lemma for 3OA (weak) ortho... |
| 3oalem3 29927 | Lemma for 3OA (weak) ortho... |
| 3oalem4 29928 | Lemma for 3OA (weak) ortho... |
| 3oalem5 29929 | Lemma for 3OA (weak) ortho... |
| 3oalem6 29930 | Lemma for 3OA (weak) ortho... |
| 3oai 29931 | 3OA (weak) orthoarguesian ... |
| pjorthi 29932 | Projection components on o... |
| pjch1 29933 | Property of identity proje... |
| pjo 29934 | The orthogonal projection.... |
| pjcompi 29935 | Component of a projection.... |
| pjidmi 29936 | A projection is idempotent... |
| pjadjii 29937 | A projection is self-adjoi... |
| pjaddii 29938 | Projection of vector sum i... |
| pjinormii 29939 | The inner product of a pro... |
| pjmulii 29940 | Projection of (scalar) pro... |
| pjsubii 29941 | Projection of vector diffe... |
| pjsslem 29942 | Lemma for subset relations... |
| pjss2i 29943 | Subset relationship for pr... |
| pjssmii 29944 | Projection meet property. ... |
| pjssge0ii 29945 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 29946 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 29947 | The projection on a subspa... |
| pjadji 29948 | A projection is self-adjoi... |
| pjaddi 29949 | Projection of vector sum i... |
| pjinormi 29950 | The inner product of a pro... |
| pjsubi 29951 | Projection of vector diffe... |
| pjmuli 29952 | Projection of scalar produ... |
| pjige0i 29953 | The inner product of a pro... |
| pjige0 29954 | The inner product of a pro... |
| pjcjt2 29955 | The projection on a subspa... |
| pj0i 29956 | The projection of the zero... |
| pjch 29957 | Projection of a vector in ... |
| pjid 29958 | The projection of a vector... |
| pjvec 29959 | The set of vectors belongi... |
| pjocvec 29960 | The set of vectors belongi... |
| pjocini 29961 | Membership of projection i... |
| pjini 29962 | Membership of projection i... |
| pjjsi 29963 | A sufficient condition for... |
| pjfni 29964 | Functionality of a project... |
| pjrni 29965 | The range of a projection.... |
| pjfoi 29966 | A projection maps onto its... |
| pjfi 29967 | The mapping of a projectio... |
| pjvi 29968 | The value of a projection ... |
| pjhfo 29969 | A projection maps onto its... |
| pjrn 29970 | The range of a projection.... |
| pjhf 29971 | The mapping of a projectio... |
| pjfn 29972 | Functionality of a project... |
| pjsumi 29973 | The projection on a subspa... |
| pj11i 29974 | One-to-one correspondence ... |
| pjdsi 29975 | Vector decomposition into ... |
| pjds3i 29976 | Vector decomposition into ... |
| pj11 29977 | One-to-one correspondence ... |
| pjmfn 29978 | Functionality of the proje... |
| pjmf1 29979 | The projector function map... |
| pjoi0 29980 | The inner product of proje... |
| pjoi0i 29981 | The inner product of proje... |
| pjopythi 29982 | Pythagorean theorem for pr... |
| pjopyth 29983 | Pythagorean theorem for pr... |
| pjnormi 29984 | The norm of the projection... |
| pjpythi 29985 | Pythagorean theorem for pr... |
| pjneli 29986 | If a vector does not belon... |
| pjnorm 29987 | The norm of the projection... |
| pjpyth 29988 | Pythagorean theorem for pr... |
| pjnel 29989 | If a vector does not belon... |
| pjnorm2 29990 | A vector belongs to the su... |
| mayete3i 29991 | Mayet's equation E_3. Par... |
| mayetes3i 29992 | Mayet's equation E^*_3, de... |
| hosmval 29998 | Value of the sum of two Hi... |
| hommval 29999 | Value of the scalar produc... |
| hodmval 30000 | Value of the difference of... |
| hfsmval 30001 | Value of the sum of two Hi... |
| hfmmval 30002 | Value of the scalar produc... |
| hosval 30003 | Value of the sum of two Hi... |
| homval 30004 | Value of the scalar produc... |
| hodval 30005 | Value of the difference of... |
| hfsval 30006 | Value of the sum of two Hi... |
| hfmval 30007 | Value of the scalar produc... |
| hoscl 30008 | Closure of the sum of two ... |
| homcl 30009 | Closure of the scalar prod... |
| hodcl 30010 | Closure of the difference ... |
| ho0val 30013 | Value of the zero Hilbert ... |
| ho0f 30014 | Functionality of the zero ... |
| df0op2 30015 | Alternate definition of Hi... |
| dfiop2 30016 | Alternate definition of Hi... |
| hoif 30017 | Functionality of the Hilbe... |
| hoival 30018 | The value of the Hilbert s... |
| hoico1 30019 | Composition with the Hilbe... |
| hoico2 30020 | Composition with the Hilbe... |
| hoaddcl 30021 | The sum of Hilbert space o... |
| homulcl 30022 | The scalar product of a Hi... |
| hoeq 30023 | Equality of Hilbert space ... |
| hoeqi 30024 | Equality of Hilbert space ... |
| hoscli 30025 | Closure of Hilbert space o... |
| hodcli 30026 | Closure of Hilbert space o... |
| hocoi 30027 | Composition of Hilbert spa... |
| hococli 30028 | Closure of composition of ... |
| hocofi 30029 | Mapping of composition of ... |
| hocofni 30030 | Functionality of compositi... |
| hoaddcli 30031 | Mapping of sum of Hilbert ... |
| hosubcli 30032 | Mapping of difference of H... |
| hoaddfni 30033 | Functionality of sum of Hi... |
| hosubfni 30034 | Functionality of differenc... |
| hoaddcomi 30035 | Commutativity of sum of Hi... |
| hosubcl 30036 | Mapping of difference of H... |
| hoaddcom 30037 | Commutativity of sum of Hi... |
| hodsi 30038 | Relationship between Hilbe... |
| hoaddassi 30039 | Associativity of sum of Hi... |
| hoadd12i 30040 | Commutative/associative la... |
| hoadd32i 30041 | Commutative/associative la... |
| hocadddiri 30042 | Distributive law for Hilbe... |
| hocsubdiri 30043 | Distributive law for Hilbe... |
| ho2coi 30044 | Double composition of Hilb... |
| hoaddass 30045 | Associativity of sum of Hi... |
| hoadd32 30046 | Commutative/associative la... |
| hoadd4 30047 | Rearrangement of 4 terms i... |
| hocsubdir 30048 | Distributive law for Hilbe... |
| hoaddid1i 30049 | Sum of a Hilbert space ope... |
| hodidi 30050 | Difference of a Hilbert sp... |
| ho0coi 30051 | Composition of the zero op... |
| hoid1i 30052 | Composition of Hilbert spa... |
| hoid1ri 30053 | Composition of Hilbert spa... |
| hoaddid1 30054 | Sum of a Hilbert space ope... |
| hodid 30055 | Difference of a Hilbert sp... |
| hon0 30056 | A Hilbert space operator i... |
| hodseqi 30057 | Subtraction and addition o... |
| ho0subi 30058 | Subtraction of Hilbert spa... |
| honegsubi 30059 | Relationship between Hilbe... |
| ho0sub 30060 | Subtraction of Hilbert spa... |
| hosubid1 30061 | The zero operator subtract... |
| honegsub 30062 | Relationship between Hilbe... |
| homulid2 30063 | An operator equals its sca... |
| homco1 30064 | Associative law for scalar... |
| homulass 30065 | Scalar product associative... |
| hoadddi 30066 | Scalar product distributiv... |
| hoadddir 30067 | Scalar product reverse dis... |
| homul12 30068 | Swap first and second fact... |
| honegneg 30069 | Double negative of a Hilbe... |
| hosubneg 30070 | Relationship between opera... |
| hosubdi 30071 | Scalar product distributiv... |
| honegdi 30072 | Distribution of negative o... |
| honegsubdi 30073 | Distribution of negative o... |
| honegsubdi2 30074 | Distribution of negative o... |
| hosubsub2 30075 | Law for double subtraction... |
| hosub4 30076 | Rearrangement of 4 terms i... |
| hosubadd4 30077 | Rearrangement of 4 terms i... |
| hoaddsubass 30078 | Associative-type law for a... |
| hoaddsub 30079 | Law for operator addition ... |
| hosubsub 30080 | Law for double subtraction... |
| hosubsub4 30081 | Law for double subtraction... |
| ho2times 30082 | Two times a Hilbert space ... |
| hoaddsubassi 30083 | Associativity of sum and d... |
| hoaddsubi 30084 | Law for sum and difference... |
| hosd1i 30085 | Hilbert space operator sum... |
| hosd2i 30086 | Hilbert space operator sum... |
| hopncani 30087 | Hilbert space operator can... |
| honpcani 30088 | Hilbert space operator can... |
| hosubeq0i 30089 | If the difference between ... |
| honpncani 30090 | Hilbert space operator can... |
| ho01i 30091 | A condition implying that ... |
| ho02i 30092 | A condition implying that ... |
| hoeq1 30093 | A condition implying that ... |
| hoeq2 30094 | A condition implying that ... |
| adjmo 30095 | Every Hilbert space operat... |
| adjsym 30096 | Symmetry property of an ad... |
| eigrei 30097 | A necessary and sufficient... |
| eigre 30098 | A necessary and sufficient... |
| eigposi 30099 | A sufficient condition (fi... |
| eigorthi 30100 | A necessary and sufficient... |
| eigorth 30101 | A necessary and sufficient... |
| nmopval 30119 | Value of the norm of a Hil... |
| elcnop 30120 | Property defining a contin... |
| ellnop 30121 | Property defining a linear... |
| lnopf 30122 | A linear Hilbert space ope... |
| elbdop 30123 | Property defining a bounde... |
| bdopln 30124 | A bounded linear Hilbert s... |
| bdopf 30125 | A bounded linear Hilbert s... |
| nmopsetretALT 30126 | The set in the supremum of... |
| nmopsetretHIL 30127 | The set in the supremum of... |
| nmopsetn0 30128 | The set in the supremum of... |
| nmopxr 30129 | The norm of a Hilbert spac... |
| nmoprepnf 30130 | The norm of a Hilbert spac... |
| nmopgtmnf 30131 | The norm of a Hilbert spac... |
| nmopreltpnf 30132 | The norm of a Hilbert spac... |
| nmopre 30133 | The norm of a bounded oper... |
| elbdop2 30134 | Property defining a bounde... |
| elunop 30135 | Property defining a unitar... |
| elhmop 30136 | Property defining a Hermit... |
| hmopf 30137 | A Hermitian operator is a ... |
| hmopex 30138 | The class of Hermitian ope... |
| nmfnval 30139 | Value of the norm of a Hil... |
| nmfnsetre 30140 | The set in the supremum of... |
| nmfnsetn0 30141 | The set in the supremum of... |
| nmfnxr 30142 | The norm of any Hilbert sp... |
| nmfnrepnf 30143 | The norm of a Hilbert spac... |
| nlfnval 30144 | Value of the null space of... |
| elcnfn 30145 | Property defining a contin... |
| ellnfn 30146 | Property defining a linear... |
| lnfnf 30147 | A linear Hilbert space fun... |
| dfadj2 30148 | Alternate definition of th... |
| funadj 30149 | Functionality of the adjoi... |
| dmadjss 30150 | The domain of the adjoint ... |
| dmadjop 30151 | A member of the domain of ... |
| adjeu 30152 | Elementhood in the domain ... |
| adjval 30153 | Value of the adjoint funct... |
| adjval2 30154 | Value of the adjoint funct... |
| cnvadj 30155 | The adjoint function equal... |
| funcnvadj 30156 | The converse of the adjoin... |
| adj1o 30157 | The adjoint function maps ... |
| dmadjrn 30158 | The adjoint of an operator... |
| eigvecval 30159 | The set of eigenvectors of... |
| eigvalfval 30160 | The eigenvalues of eigenve... |
| specval 30161 | The value of the spectrum ... |
| speccl 30162 | The spectrum of an operato... |
| hhlnoi 30163 | The linear operators of Hi... |
| hhnmoi 30164 | The norm of an operator in... |
| hhbloi 30165 | A bounded linear operator ... |
| hh0oi 30166 | The zero operator in Hilbe... |
| hhcno 30167 | The continuous operators o... |
| hhcnf 30168 | The continuous functionals... |
| dmadjrnb 30169 | The adjoint of an operator... |
| nmoplb 30170 | A lower bound for an opera... |
| nmopub 30171 | An upper bound for an oper... |
| nmopub2tALT 30172 | An upper bound for an oper... |
| nmopub2tHIL 30173 | An upper bound for an oper... |
| nmopge0 30174 | The norm of any Hilbert sp... |
| nmopgt0 30175 | A linear Hilbert space ope... |
| cnopc 30176 | Basic continuity property ... |
| lnopl 30177 | Basic linearity property o... |
| unop 30178 | Basic inner product proper... |
| unopf1o 30179 | A unitary operator in Hilb... |
| unopnorm 30180 | A unitary operator is idem... |
| cnvunop 30181 | The inverse (converse) of ... |
| unopadj 30182 | The inverse (converse) of ... |
| unoplin 30183 | A unitary operator is line... |
| counop 30184 | The composition of two uni... |
| hmop 30185 | Basic inner product proper... |
| hmopre 30186 | The inner product of the v... |
| nmfnlb 30187 | A lower bound for a functi... |
| nmfnleub 30188 | An upper bound for the nor... |
| nmfnleub2 30189 | An upper bound for the nor... |
| nmfnge0 30190 | The norm of any Hilbert sp... |
| elnlfn 30191 | Membership in the null spa... |
| elnlfn2 30192 | Membership in the null spa... |
| cnfnc 30193 | Basic continuity property ... |
| lnfnl 30194 | Basic linearity property o... |
| adjcl 30195 | Closure of the adjoint of ... |
| adj1 30196 | Property of an adjoint Hil... |
| adj2 30197 | Property of an adjoint Hil... |
| adjeq 30198 | A property that determines... |
| adjadj 30199 | Double adjoint. Theorem 3... |
| adjvalval 30200 | Value of the value of the ... |
| unopadj2 30201 | The adjoint of a unitary o... |
| hmopadj 30202 | A Hermitian operator is se... |
| hmdmadj 30203 | Every Hermitian operator h... |
| hmopadj2 30204 | An operator is Hermitian i... |
| hmoplin 30205 | A Hermitian operator is li... |
| brafval 30206 | The bra of a vector, expre... |
| braval 30207 | A bra-ket juxtaposition, e... |
| braadd 30208 | Linearity property of bra ... |
| bramul 30209 | Linearity property of bra ... |
| brafn 30210 | The bra function is a func... |
| bralnfn 30211 | The Dirac bra function is ... |
| bracl 30212 | Closure of the bra functio... |
| bra0 30213 | The Dirac bra of the zero ... |
| brafnmul 30214 | Anti-linearity property of... |
| kbfval 30215 | The outer product of two v... |
| kbop 30216 | The outer product of two v... |
| kbval 30217 | The value of the operator ... |
| kbmul 30218 | Multiplication property of... |
| kbpj 30219 | If a vector ` A ` has norm... |
| eleigvec 30220 | Membership in the set of e... |
| eleigvec2 30221 | Membership in the set of e... |
| eleigveccl 30222 | Closure of an eigenvector ... |
| eigvalval 30223 | The eigenvalue of an eigen... |
| eigvalcl 30224 | An eigenvalue is a complex... |
| eigvec1 30225 | Property of an eigenvector... |
| eighmre 30226 | The eigenvalues of a Hermi... |
| eighmorth 30227 | Eigenvectors of a Hermitia... |
| nmopnegi 30228 | Value of the norm of the n... |
| lnop0 30229 | The value of a linear Hilb... |
| lnopmul 30230 | Multiplicative property of... |
| lnopli 30231 | Basic scalar product prope... |
| lnopfi 30232 | A linear Hilbert space ope... |
| lnop0i 30233 | The value of a linear Hilb... |
| lnopaddi 30234 | Additive property of a lin... |
| lnopmuli 30235 | Multiplicative property of... |
| lnopaddmuli 30236 | Sum/product property of a ... |
| lnopsubi 30237 | Subtraction property for a... |
| lnopsubmuli 30238 | Subtraction/product proper... |
| lnopmulsubi 30239 | Product/subtraction proper... |
| homco2 30240 | Move a scalar product out ... |
| idunop 30241 | The identity function (res... |
| 0cnop 30242 | The identically zero funct... |
| 0cnfn 30243 | The identically zero funct... |
| idcnop 30244 | The identity function (res... |
| idhmop 30245 | The Hilbert space identity... |
| 0hmop 30246 | The identically zero funct... |
| 0lnop 30247 | The identically zero funct... |
| 0lnfn 30248 | The identically zero funct... |
| nmop0 30249 | The norm of the zero opera... |
| nmfn0 30250 | The norm of the identicall... |
| hmopbdoptHIL 30251 | A Hermitian operator is a ... |
| hoddii 30252 | Distributive law for Hilbe... |
| hoddi 30253 | Distributive law for Hilbe... |
| nmop0h 30254 | The norm of any operator o... |
| idlnop 30255 | The identity function (res... |
| 0bdop 30256 | The identically zero opera... |
| adj0 30257 | Adjoint of the zero operat... |
| nmlnop0iALT 30258 | A linear operator with a z... |
| nmlnop0iHIL 30259 | A linear operator with a z... |
| nmlnopgt0i 30260 | A linear Hilbert space ope... |
| nmlnop0 30261 | A linear operator with a z... |
| nmlnopne0 30262 | A linear operator with a n... |
| lnopmi 30263 | The scalar product of a li... |
| lnophsi 30264 | The sum of two linear oper... |
| lnophdi 30265 | The difference of two line... |
| lnopcoi 30266 | The composition of two lin... |
| lnopco0i 30267 | The composition of a linea... |
| lnopeq0lem1 30268 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 30269 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 30270 | A condition implying that ... |
| lnopeqi 30271 | Two linear Hilbert space o... |
| lnopeq 30272 | Two linear Hilbert space o... |
| lnopunilem1 30273 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 30274 | Lemma for ~ lnopunii . (C... |
| lnopunii 30275 | If a linear operator (whos... |
| elunop2 30276 | An operator is unitary iff... |
| nmopun 30277 | Norm of a unitary Hilbert ... |
| unopbd 30278 | A unitary operator is a bo... |
| lnophmlem1 30279 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 30280 | Lemma for ~ lnophmi . (Co... |
| lnophmi 30281 | A linear operator is Hermi... |
| lnophm 30282 | A linear operator is Hermi... |
| hmops 30283 | The sum of two Hermitian o... |
| hmopm 30284 | The scalar product of a He... |
| hmopd 30285 | The difference of two Herm... |
| hmopco 30286 | The composition of two com... |
| nmbdoplbi 30287 | A lower bound for the norm... |
| nmbdoplb 30288 | A lower bound for the norm... |
| nmcexi 30289 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 30290 | The norm of a continuous l... |
| nmcoplbi 30291 | A lower bound for the norm... |
| nmcopex 30292 | The norm of a continuous l... |
| nmcoplb 30293 | A lower bound for the norm... |
| nmophmi 30294 | The norm of the scalar pro... |
| bdophmi 30295 | The scalar product of a bo... |
| lnconi 30296 | Lemma for ~ lnopconi and ~... |
| lnopconi 30297 | A condition equivalent to ... |
| lnopcon 30298 | A condition equivalent to ... |
| lnopcnbd 30299 | A linear operator is conti... |
| lncnopbd 30300 | A continuous linear operat... |
| lncnbd 30301 | A continuous linear operat... |
| lnopcnre 30302 | A linear operator is conti... |
| lnfnli 30303 | Basic property of a linear... |
| lnfnfi 30304 | A linear Hilbert space fun... |
| lnfn0i 30305 | The value of a linear Hilb... |
| lnfnaddi 30306 | Additive property of a lin... |
| lnfnmuli 30307 | Multiplicative property of... |
| lnfnaddmuli 30308 | Sum/product property of a ... |
| lnfnsubi 30309 | Subtraction property for a... |
| lnfn0 30310 | The value of a linear Hilb... |
| lnfnmul 30311 | Multiplicative property of... |
| nmbdfnlbi 30312 | A lower bound for the norm... |
| nmbdfnlb 30313 | A lower bound for the norm... |
| nmcfnexi 30314 | The norm of a continuous l... |
| nmcfnlbi 30315 | A lower bound for the norm... |
| nmcfnex 30316 | The norm of a continuous l... |
| nmcfnlb 30317 | A lower bound of the norm ... |
| lnfnconi 30318 | A condition equivalent to ... |
| lnfncon 30319 | A condition equivalent to ... |
| lnfncnbd 30320 | A linear functional is con... |
| imaelshi 30321 | The image of a subspace un... |
| rnelshi 30322 | The range of a linear oper... |
| nlelshi 30323 | The null space of a linear... |
| nlelchi 30324 | The null space of a contin... |
| riesz3i 30325 | A continuous linear functi... |
| riesz4i 30326 | A continuous linear functi... |
| riesz4 30327 | A continuous linear functi... |
| riesz1 30328 | Part 1 of the Riesz repres... |
| riesz2 30329 | Part 2 of the Riesz repres... |
| cnlnadjlem1 30330 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 30331 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 30332 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 30333 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 30334 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 30335 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 30336 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 30337 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 30338 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 30339 | Every continuous linear op... |
| cnlnadjeui 30340 | Every continuous linear op... |
| cnlnadjeu 30341 | Every continuous linear op... |
| cnlnadj 30342 | Every continuous linear op... |
| cnlnssadj 30343 | Every continuous linear Hi... |
| bdopssadj 30344 | Every bounded linear Hilbe... |
| bdopadj 30345 | Every bounded linear Hilbe... |
| adjbdln 30346 | The adjoint of a bounded l... |
| adjbdlnb 30347 | An operator is bounded and... |
| adjbd1o 30348 | The mapping of adjoints of... |
| adjlnop 30349 | The adjoint of an operator... |
| adjsslnop 30350 | Every operator with an adj... |
| nmopadjlei 30351 | Property of the norm of an... |
| nmopadjlem 30352 | Lemma for ~ nmopadji . (C... |
| nmopadji 30353 | Property of the norm of an... |
| adjeq0 30354 | An operator is zero iff it... |
| adjmul 30355 | The adjoint of the scalar ... |
| adjadd 30356 | The adjoint of the sum of ... |
| nmoptrii 30357 | Triangle inequality for th... |
| nmopcoi 30358 | Upper bound for the norm o... |
| bdophsi 30359 | The sum of two bounded lin... |
| bdophdi 30360 | The difference between two... |
| bdopcoi 30361 | The composition of two bou... |
| nmoptri2i 30362 | Triangle-type inequality f... |
| adjcoi 30363 | The adjoint of a compositi... |
| nmopcoadji 30364 | The norm of an operator co... |
| nmopcoadj2i 30365 | The norm of an operator co... |
| nmopcoadj0i 30366 | An operator composed with ... |
| unierri 30367 | If we approximate a chain ... |
| branmfn 30368 | The norm of the bra functi... |
| brabn 30369 | The bra of a vector is a b... |
| rnbra 30370 | The set of bras equals the... |
| bra11 30371 | The bra function maps vect... |
| bracnln 30372 | A bra is a continuous line... |
| cnvbraval 30373 | Value of the converse of t... |
| cnvbracl 30374 | Closure of the converse of... |
| cnvbrabra 30375 | The converse bra of the br... |
| bracnvbra 30376 | The bra of the converse br... |
| bracnlnval 30377 | The vector that a continuo... |
| cnvbramul 30378 | Multiplication property of... |
| kbass1 30379 | Dirac bra-ket associative ... |
| kbass2 30380 | Dirac bra-ket associative ... |
| kbass3 30381 | Dirac bra-ket associative ... |
| kbass4 30382 | Dirac bra-ket associative ... |
| kbass5 30383 | Dirac bra-ket associative ... |
| kbass6 30384 | Dirac bra-ket associative ... |
| leopg 30385 | Ordering relation for posi... |
| leop 30386 | Ordering relation for oper... |
| leop2 30387 | Ordering relation for oper... |
| leop3 30388 | Operator ordering in terms... |
| leoppos 30389 | Binary relation defining a... |
| leoprf2 30390 | The ordering relation for ... |
| leoprf 30391 | The ordering relation for ... |
| leopsq 30392 | The square of a Hermitian ... |
| 0leop 30393 | The zero operator is a pos... |
| idleop 30394 | The identity operator is a... |
| leopadd 30395 | The sum of two positive op... |
| leopmuli 30396 | The scalar product of a no... |
| leopmul 30397 | The scalar product of a po... |
| leopmul2i 30398 | Scalar product applied to ... |
| leoptri 30399 | The positive operator orde... |
| leoptr 30400 | The positive operator orde... |
| leopnmid 30401 | A bounded Hermitian operat... |
| nmopleid 30402 | A nonzero, bounded Hermiti... |
| opsqrlem1 30403 | Lemma for opsqri . (Contr... |
| opsqrlem2 30404 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 30405 | Lemma for opsqri . (Contr... |
| opsqrlem4 30406 | Lemma for opsqri . (Contr... |
| opsqrlem5 30407 | Lemma for opsqri . (Contr... |
| opsqrlem6 30408 | Lemma for opsqri . (Contr... |
| pjhmopi 30409 | A projector is a Hermitian... |
| pjlnopi 30410 | A projector is a linear op... |
| pjnmopi 30411 | The operator norm of a pro... |
| pjbdlni 30412 | A projector is a bounded l... |
| pjhmop 30413 | A projection is a Hermitia... |
| hmopidmchi 30414 | An idempotent Hermitian op... |
| hmopidmpji 30415 | An idempotent Hermitian op... |
| hmopidmch 30416 | An idempotent Hermitian op... |
| hmopidmpj 30417 | An idempotent Hermitian op... |
| pjsdii 30418 | Distributive law for Hilbe... |
| pjddii 30419 | Distributive law for Hilbe... |
| pjsdi2i 30420 | Chained distributive law f... |
| pjcoi 30421 | Composition of projections... |
| pjcocli 30422 | Closure of composition of ... |
| pjcohcli 30423 | Closure of composition of ... |
| pjadjcoi 30424 | Adjoint of composition of ... |
| pjcofni 30425 | Functionality of compositi... |
| pjss1coi 30426 | Subset relationship for pr... |
| pjss2coi 30427 | Subset relationship for pr... |
| pjssmi 30428 | Projection meet property. ... |
| pjssge0i 30429 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 30430 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 30431 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 30432 | Composition of projections... |
| pjscji 30433 | The projection of orthogon... |
| pjssumi 30434 | The projection on a subspa... |
| pjssposi 30435 | Projector ordering can be ... |
| pjordi 30436 | The definition of projecto... |
| pjssdif2i 30437 | The projection subspace of... |
| pjssdif1i 30438 | A necessary and sufficient... |
| pjimai 30439 | The image of a projection.... |
| pjidmcoi 30440 | A projection is idempotent... |
| pjoccoi 30441 | Composition of projections... |
| pjtoi 30442 | Subspace sum of projection... |
| pjoci 30443 | Projection of orthocomplem... |
| pjidmco 30444 | A projection operator is i... |
| dfpjop 30445 | Definition of projection o... |
| pjhmopidm 30446 | Two ways to express the se... |
| elpjidm 30447 | A projection operator is i... |
| elpjhmop 30448 | A projection operator is H... |
| 0leopj 30449 | A projector is a positive ... |
| pjadj2 30450 | A projector is self-adjoin... |
| pjadj3 30451 | A projector is self-adjoin... |
| elpjch 30452 | Reconstruction of the subs... |
| elpjrn 30453 | Reconstruction of the subs... |
| pjinvari 30454 | A closed subspace ` H ` wi... |
| pjin1i 30455 | Lemma for Theorem 1.22 of ... |
| pjin2i 30456 | Lemma for Theorem 1.22 of ... |
| pjin3i 30457 | Lemma for Theorem 1.22 of ... |
| pjclem1 30458 | Lemma for projection commu... |
| pjclem2 30459 | Lemma for projection commu... |
| pjclem3 30460 | Lemma for projection commu... |
| pjclem4a 30461 | Lemma for projection commu... |
| pjclem4 30462 | Lemma for projection commu... |
| pjci 30463 | Two subspaces commute iff ... |
| pjcmul1i 30464 | A necessary and sufficient... |
| pjcmul2i 30465 | The projection subspace of... |
| pjcohocli 30466 | Closure of composition of ... |
| pjadj2coi 30467 | Adjoint of double composit... |
| pj2cocli 30468 | Closure of double composit... |
| pj3lem1 30469 | Lemma for projection tripl... |
| pj3si 30470 | Stronger projection triple... |
| pj3i 30471 | Projection triplet theorem... |
| pj3cor1i 30472 | Projection triplet corolla... |
| pjs14i 30473 | Theorem S-14 of Watanabe, ... |
| isst 30476 | Property of a state. (Con... |
| ishst 30477 | Property of a complex Hilb... |
| sticl 30478 | ` [ 0 , 1 ] ` closure of t... |
| stcl 30479 | Real closure of the value ... |
| hstcl 30480 | Closure of the value of a ... |
| hst1a 30481 | Unit value of a Hilbert-sp... |
| hstel2 30482 | Properties of a Hilbert-sp... |
| hstorth 30483 | Orthogonality property of ... |
| hstosum 30484 | Orthogonal sum property of... |
| hstoc 30485 | Sum of a Hilbert-space-val... |
| hstnmoc 30486 | Sum of norms of a Hilbert-... |
| stge0 30487 | The value of a state is no... |
| stle1 30488 | The value of a state is le... |
| hstle1 30489 | The norm of the value of a... |
| hst1h 30490 | The norm of a Hilbert-spac... |
| hst0h 30491 | The norm of a Hilbert-spac... |
| hstpyth 30492 | Pythagorean property of a ... |
| hstle 30493 | Ordering property of a Hil... |
| hstles 30494 | Ordering property of a Hil... |
| hstoh 30495 | A Hilbert-space-valued sta... |
| hst0 30496 | A Hilbert-space-valued sta... |
| sthil 30497 | The value of a state at th... |
| stj 30498 | The value of a state on a ... |
| sto1i 30499 | The state of a subspace pl... |
| sto2i 30500 | The state of the orthocomp... |
| stge1i 30501 | If a state is greater than... |
| stle0i 30502 | If a state is less than or... |
| stlei 30503 | Ordering law for states. ... |
| stlesi 30504 | Ordering law for states. ... |
| stji1i 30505 | Join of components of Sasa... |
| stm1i 30506 | State of component of unit... |
| stm1ri 30507 | State of component of unit... |
| stm1addi 30508 | Sum of states whose meet i... |
| staddi 30509 | If the sum of 2 states is ... |
| stm1add3i 30510 | Sum of states whose meet i... |
| stadd3i 30511 | If the sum of 3 states is ... |
| st0 30512 | The state of the zero subs... |
| strlem1 30513 | Lemma for strong state the... |
| strlem2 30514 | Lemma for strong state the... |
| strlem3a 30515 | Lemma for strong state the... |
| strlem3 30516 | Lemma for strong state the... |
| strlem4 30517 | Lemma for strong state the... |
| strlem5 30518 | Lemma for strong state the... |
| strlem6 30519 | Lemma for strong state the... |
| stri 30520 | Strong state theorem. The... |
| strb 30521 | Strong state theorem (bidi... |
| hstrlem2 30522 | Lemma for strong set of CH... |
| hstrlem3a 30523 | Lemma for strong set of CH... |
| hstrlem3 30524 | Lemma for strong set of CH... |
| hstrlem4 30525 | Lemma for strong set of CH... |
| hstrlem5 30526 | Lemma for strong set of CH... |
| hstrlem6 30527 | Lemma for strong set of CH... |
| hstri 30528 | Hilbert space admits a str... |
| hstrbi 30529 | Strong CH-state theorem (b... |
| largei 30530 | A Hilbert lattice admits a... |
| jplem1 30531 | Lemma for Jauch-Piron theo... |
| jplem2 30532 | Lemma for Jauch-Piron theo... |
| jpi 30533 | The function ` S ` , that ... |
| golem1 30534 | Lemma for Godowski's equat... |
| golem2 30535 | Lemma for Godowski's equat... |
| goeqi 30536 | Godowski's equation, shown... |
| stcltr1i 30537 | Property of a strong class... |
| stcltr2i 30538 | Property of a strong class... |
| stcltrlem1 30539 | Lemma for strong classical... |
| stcltrlem2 30540 | Lemma for strong classical... |
| stcltrthi 30541 | Theorem for classically st... |
| cvbr 30545 | Binary relation expressing... |
| cvbr2 30546 | Binary relation expressing... |
| cvcon3 30547 | Contraposition law for the... |
| cvpss 30548 | The covers relation implie... |
| cvnbtwn 30549 | The covers relation implie... |
| cvnbtwn2 30550 | The covers relation implie... |
| cvnbtwn3 30551 | The covers relation implie... |
| cvnbtwn4 30552 | The covers relation implie... |
| cvnsym 30553 | The covers relation is not... |
| cvnref 30554 | The covers relation is not... |
| cvntr 30555 | The covers relation is not... |
| spansncv2 30556 | Hilbert space has the cove... |
| mdbr 30557 | Binary relation expressing... |
| mdi 30558 | Consequence of the modular... |
| mdbr2 30559 | Binary relation expressing... |
| mdbr3 30560 | Binary relation expressing... |
| mdbr4 30561 | Binary relation expressing... |
| dmdbr 30562 | Binary relation expressing... |
| dmdmd 30563 | The dual modular pair prop... |
| mddmd 30564 | The modular pair property ... |
| dmdi 30565 | Consequence of the dual mo... |
| dmdbr2 30566 | Binary relation expressing... |
| dmdi2 30567 | Consequence of the dual mo... |
| dmdbr3 30568 | Binary relation expressing... |
| dmdbr4 30569 | Binary relation expressing... |
| dmdi4 30570 | Consequence of the dual mo... |
| dmdbr5 30571 | Binary relation expressing... |
| mddmd2 30572 | Relationship between modul... |
| mdsl0 30573 | A sublattice condition tha... |
| ssmd1 30574 | Ordering implies the modul... |
| ssmd2 30575 | Ordering implies the modul... |
| ssdmd1 30576 | Ordering implies the dual ... |
| ssdmd2 30577 | Ordering implies the dual ... |
| dmdsl3 30578 | Sublattice mapping for a d... |
| mdsl3 30579 | Sublattice mapping for a m... |
| mdslle1i 30580 | Order preservation of the ... |
| mdslle2i 30581 | Order preservation of the ... |
| mdslj1i 30582 | Join preservation of the o... |
| mdslj2i 30583 | Meet preservation of the r... |
| mdsl1i 30584 | If the modular pair proper... |
| mdsl2i 30585 | If the modular pair proper... |
| mdsl2bi 30586 | If the modular pair proper... |
| cvmdi 30587 | The covering property impl... |
| mdslmd1lem1 30588 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 30589 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 30590 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 30591 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 30592 | Preservation of the modula... |
| mdslmd2i 30593 | Preservation of the modula... |
| mdsldmd1i 30594 | Preservation of the dual m... |
| mdslmd3i 30595 | Modular pair conditions th... |
| mdslmd4i 30596 | Modular pair condition tha... |
| csmdsymi 30597 | Cross-symmetry implies M-s... |
| mdexchi 30598 | An exchange lemma for modu... |
| cvmd 30599 | The covering property impl... |
| cvdmd 30600 | The covering property impl... |
| ela 30602 | Atoms in a Hilbert lattice... |
| elat2 30603 | Expanded membership relati... |
| elatcv0 30604 | A Hilbert lattice element ... |
| atcv0 30605 | An atom covers the zero su... |
| atssch 30606 | Atoms are a subset of the ... |
| atelch 30607 | An atom is a Hilbert latti... |
| atne0 30608 | An atom is not the Hilbert... |
| atss 30609 | A lattice element smaller ... |
| atsseq 30610 | Two atoms in a subset rela... |
| atcveq0 30611 | A Hilbert lattice element ... |
| h1da 30612 | A 1-dimensional subspace i... |
| spansna 30613 | The span of the singleton ... |
| sh1dle 30614 | A 1-dimensional subspace i... |
| ch1dle 30615 | A 1-dimensional subspace i... |
| atom1d 30616 | The 1-dimensional subspace... |
| superpos 30617 | Superposition Principle. ... |
| chcv1 30618 | The Hilbert lattice has th... |
| chcv2 30619 | The Hilbert lattice has th... |
| chjatom 30620 | The join of a closed subsp... |
| shatomici 30621 | The lattice of Hilbert sub... |
| hatomici 30622 | The Hilbert lattice is ato... |
| hatomic 30623 | A Hilbert lattice is atomi... |
| shatomistici 30624 | The lattice of Hilbert sub... |
| hatomistici 30625 | ` CH ` is atomistic, i.e. ... |
| chpssati 30626 | Two Hilbert lattice elemen... |
| chrelati 30627 | The Hilbert lattice is rel... |
| chrelat2i 30628 | A consequence of relative ... |
| cvati 30629 | If a Hilbert lattice eleme... |
| cvbr4i 30630 | An alternate way to expres... |
| cvexchlem 30631 | Lemma for ~ cvexchi . (Co... |
| cvexchi 30632 | The Hilbert lattice satisf... |
| chrelat2 30633 | A consequence of relative ... |
| chrelat3 30634 | A consequence of relative ... |
| chrelat3i 30635 | A consequence of the relat... |
| chrelat4i 30636 | A consequence of relative ... |
| cvexch 30637 | The Hilbert lattice satisf... |
| cvp 30638 | The Hilbert lattice satisf... |
| atnssm0 30639 | The meet of a Hilbert latt... |
| atnemeq0 30640 | The meet of distinct atoms... |
| atssma 30641 | The meet with an atom's su... |
| atcv0eq 30642 | Two atoms covering the zer... |
| atcv1 30643 | Two atoms covering the zer... |
| atexch 30644 | The Hilbert lattice satisf... |
| atomli 30645 | An assertion holding in at... |
| atoml2i 30646 | An assertion holding in at... |
| atordi 30647 | An ordering law for a Hilb... |
| atcvatlem 30648 | Lemma for ~ atcvati . (Co... |
| atcvati 30649 | A nonzero Hilbert lattice ... |
| atcvat2i 30650 | A Hilbert lattice element ... |
| atord 30651 | An ordering law for a Hilb... |
| atcvat2 30652 | A Hilbert lattice element ... |
| chirredlem1 30653 | Lemma for ~ chirredi . (C... |
| chirredlem2 30654 | Lemma for ~ chirredi . (C... |
| chirredlem3 30655 | Lemma for ~ chirredi . (C... |
| chirredlem4 30656 | Lemma for ~ chirredi . (C... |
| chirredi 30657 | The Hilbert lattice is irr... |
| chirred 30658 | The Hilbert lattice is irr... |
| atcvat3i 30659 | A condition implying that ... |
| atcvat4i 30660 | A condition implying exist... |
| atdmd 30661 | Two Hilbert lattice elemen... |
| atmd 30662 | Two Hilbert lattice elemen... |
| atmd2 30663 | Two Hilbert lattice elemen... |
| atabsi 30664 | Absorption of an incompara... |
| atabs2i 30665 | Absorption of an incompara... |
| mdsymlem1 30666 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 30667 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 30668 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 30669 | Lemma for ~ mdsymi . This... |
| mdsymlem5 30670 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 30671 | Lemma for ~ mdsymi . This... |
| mdsymlem7 30672 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 30673 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 30674 | M-symmetry of the Hilbert ... |
| mdsym 30675 | M-symmetry of the Hilbert ... |
| dmdsym 30676 | Dual M-symmetry of the Hil... |
| atdmd2 30677 | Two Hilbert lattice elemen... |
| sumdmdii 30678 | If the subspace sum of two... |
| cmmdi 30679 | Commuting subspaces form a... |
| cmdmdi 30680 | Commuting subspaces form a... |
| sumdmdlem 30681 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 30682 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 30683 | The subspace sum of two Hi... |
| dmdbr4ati 30684 | Dual modular pair property... |
| dmdbr5ati 30685 | Dual modular pair property... |
| dmdbr6ati 30686 | Dual modular pair property... |
| dmdbr7ati 30687 | Dual modular pair property... |
| mdoc1i 30688 | Orthocomplements form a mo... |
| mdoc2i 30689 | Orthocomplements form a mo... |
| dmdoc1i 30690 | Orthocomplements form a du... |
| dmdoc2i 30691 | Orthocomplements form a du... |
| mdcompli 30692 | A condition equivalent to ... |
| dmdcompli 30693 | A condition equivalent to ... |
| mddmdin0i 30694 | If dual modular implies mo... |
| cdjreui 30695 | A member of the sum of dis... |
| cdj1i 30696 | Two ways to express " ` A ... |
| cdj3lem1 30697 | A property of " ` A ` and ... |
| cdj3lem2 30698 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 30699 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 30700 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 30701 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 30702 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 30703 | Lemma for ~ cdj3i . The s... |
| cdj3i 30704 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 30705 | (_This theorem is a dummy ... |
| sa-abvi 30706 | A theorem about the univer... |
| xfree 30707 | A partial converse to ~ 19... |
| xfree2 30708 | A partial converse to ~ 19... |
| addltmulALT 30709 | A proof readability experi... |
| bian1d 30710 | Adding a superfluous conju... |
| or3di 30711 | Distributive law for disju... |
| or3dir 30712 | Distributive law for disju... |
| 3o1cs 30713 | Deduction eliminating disj... |
| 3o2cs 30714 | Deduction eliminating disj... |
| 3o3cs 30715 | Deduction eliminating disj... |
| sbc2iedf 30716 | Conversion of implicit sub... |
| rspc2daf 30717 | Double restricted speciali... |
| nelbOLDOLD 30718 | Obsolete version of ~ nelb... |
| ralcom4f 30719 | Commutation of restricted ... |
| rexcom4f 30720 | Commutation of restricted ... |
| 19.9d2rf 30721 | A deduction version of one... |
| 19.9d2r 30722 | A deduction version of one... |
| r19.29ffa 30723 | A commonly used pattern ba... |
| eqtrb 30724 | A transposition of equalit... |
| opsbc2ie 30725 | Conversion of implicit sub... |
| opreu2reuALT 30726 | Correspondence between uni... |
| 2reucom 30729 | Double restricted existent... |
| 2reu2rex1 30730 | Double restricted existent... |
| 2reureurex 30731 | Double restricted existent... |
| 2reu2reu2 30732 | Double restricted existent... |
| opreu2reu1 30733 | Equivalent definition of t... |
| sq2reunnltb 30734 | There exists a unique deco... |
| addsqnot2reu 30735 | For each complex number ` ... |
| sbceqbidf 30736 | Equality theorem for class... |
| sbcies 30737 | A special version of class... |
| mo5f 30738 | Alternate definition of "a... |
| nmo 30739 | Negation of "at most one".... |
| reuxfrdf 30740 | Transfer existential uniqu... |
| rexunirn 30741 | Restricted existential qua... |
| rmoxfrd 30742 | Transfer "at most one" res... |
| rmoun 30743 | "At most one" restricted e... |
| rmounid 30744 | A case where an "at most o... |
| dmrab 30745 | Domain of a restricted cla... |
| difrab2 30746 | Difference of two restrict... |
| rabexgfGS 30747 | Separation Scheme in terms... |
| rabsnel 30748 | Truth implied by equality ... |
| rabeqsnd 30749 | Conditions for a restricte... |
| eqrrabd 30750 | Deduce equality with a res... |
| foresf1o 30751 | From a surjective function... |
| rabfodom 30752 | Domination relation for re... |
| abrexdomjm 30753 | An indexed set is dominate... |
| abrexdom2jm 30754 | An indexed set is dominate... |
| abrexexd 30755 | Existence of a class abstr... |
| elabreximd 30756 | Class substitution in an i... |
| elabreximdv 30757 | Class substitution in an i... |
| abrexss 30758 | A necessary condition for ... |
| elunsn 30759 | Elementhood to a union wit... |
| nelun 30760 | Negated membership for a u... |
| snsssng 30761 | If a singleton is a subset... |
| rabss3d 30762 | Subclass law for restricte... |
| inin 30763 | Intersection with an inter... |
| inindif 30764 | See ~ inundif . (Contribu... |
| difininv 30765 | Condition for the intersec... |
| difeq 30766 | Rewriting an equation with... |
| eqdif 30767 | If both set differences of... |
| undif5 30768 | An equality involving clas... |
| indifbi 30769 | Two ways to express equali... |
| diffib 30770 | Case where ~ diffi is a bi... |
| difxp1ss 30771 | Difference law for Cartesi... |
| difxp2ss 30772 | Difference law for Cartesi... |
| undifr 30773 | Union of complementary par... |
| indifundif 30774 | A remarkable equation with... |
| elpwincl1 30775 | Closure of intersection wi... |
| elpwdifcl 30776 | Closure of class differenc... |
| elpwiuncl 30777 | Closure of indexed union w... |
| eqsnd 30778 | Deduce that a set is a sin... |
| elpreq 30779 | Equality wihin a pair. (C... |
| nelpr 30780 | A set ` A ` not in a pair ... |
| inpr0 30781 | Rewrite an empty intersect... |
| neldifpr1 30782 | The first element of a pai... |
| neldifpr2 30783 | The second element of a pa... |
| unidifsnel 30784 | The other element of a pai... |
| unidifsnne 30785 | The other element of a pai... |
| ifeqeqx 30786 | An equality theorem tailor... |
| elimifd 30787 | Elimination of a condition... |
| elim2if 30788 | Elimination of two conditi... |
| elim2ifim 30789 | Elimination of two conditi... |
| ifeq3da 30790 | Given an expression ` C ` ... |
| uniinn0 30791 | Sufficient and necessary c... |
| uniin1 30792 | Union of intersection. Ge... |
| uniin2 30793 | Union of intersection. Ge... |
| difuncomp 30794 | Express a class difference... |
| elpwunicl 30795 | Closure of a set union wit... |
| cbviunf 30796 | Rule used to change the bo... |
| iuneq12daf 30797 | Equality deduction for ind... |
| iunin1f 30798 | Indexed union of intersect... |
| ssiun3 30799 | Subset equivalence for an ... |
| ssiun2sf 30800 | Subset relationship for an... |
| iuninc 30801 | The union of an increasing... |
| iundifdifd 30802 | The intersection of a set ... |
| iundifdif 30803 | The intersection of a set ... |
| iunrdx 30804 | Re-index an indexed union.... |
| iunpreima 30805 | Preimage of an indexed uni... |
| iunrnmptss 30806 | A subset relation for an i... |
| iunxunsn 30807 | Appending a set to an inde... |
| iunxunpr 30808 | Appending two sets to an i... |
| iinabrex 30809 | Rewriting an indexed inter... |
| disjnf 30810 | In case ` x ` is not free ... |
| cbvdisjf 30811 | Change bound variables in ... |
| disjss1f 30812 | A subset of a disjoint col... |
| disjeq1f 30813 | Equality theorem for disjo... |
| disjxun0 30814 | Simplify a disjoint union.... |
| disjdifprg 30815 | A trivial partition into a... |
| disjdifprg2 30816 | A trivial partition of a s... |
| disji2f 30817 | Property of a disjoint col... |
| disjif 30818 | Property of a disjoint col... |
| disjorf 30819 | Two ways to say that a col... |
| disjorsf 30820 | Two ways to say that a col... |
| disjif2 30821 | Property of a disjoint col... |
| disjabrex 30822 | Rewriting a disjoint colle... |
| disjabrexf 30823 | Rewriting a disjoint colle... |
| disjpreima 30824 | A preimage of a disjoint s... |
| disjrnmpt 30825 | Rewriting a disjoint colle... |
| disjin 30826 | If a collection is disjoin... |
| disjin2 30827 | If a collection is disjoin... |
| disjxpin 30828 | Derive a disjunction over ... |
| iundisjf 30829 | Rewrite a countable union ... |
| iundisj2f 30830 | A disjoint union is disjoi... |
| disjrdx 30831 | Re-index a disjunct collec... |
| disjex 30832 | Two ways to say that two c... |
| disjexc 30833 | A variant of ~ disjex , ap... |
| disjunsn 30834 | Append an element to a dis... |
| disjun0 30835 | Adding the empty element p... |
| disjiunel 30836 | A set of elements B of a d... |
| disjuniel 30837 | A set of elements B of a d... |
| xpdisjres 30838 | Restriction of a constant ... |
| opeldifid 30839 | Ordered pair elementhood o... |
| difres 30840 | Case when class difference... |
| imadifxp 30841 | Image of the difference wi... |
| relfi 30842 | A relation (set) is finite... |
| reldisjun 30843 | Split a relation into two ... |
| 0res 30844 | Restriction of the empty f... |
| funresdm1 30845 | Restriction of a disjoint ... |
| fnunres1 30846 | Restriction of a disjoint ... |
| fcoinver 30847 | Build an equivalence relat... |
| fcoinvbr 30848 | Binary relation for the eq... |
| brabgaf 30849 | The law of concretion for ... |
| brelg 30850 | Two things in a binary rel... |
| br8d 30851 | Substitution for an eight-... |
| opabdm 30852 | Domain of an ordered-pair ... |
| opabrn 30853 | Range of an ordered-pair c... |
| opabssi 30854 | Sufficient condition for a... |
| opabid2ss 30855 | One direction of ~ opabid2... |
| ssrelf 30856 | A subclass relationship de... |
| eqrelrd2 30857 | A version of ~ eqrelrdv2 w... |
| erbr3b 30858 | Biconditional for equivale... |
| iunsnima 30859 | Image of a singleton by an... |
| iunsnima2 30860 | Version of ~ iunsnima with... |
| ac6sf2 30861 | Alternate version of ~ ac6... |
| fnresin 30862 | Restriction of a function ... |
| f1o3d 30863 | Describe an implicit one-t... |
| eldmne0 30864 | A function of nonempty dom... |
| f1rnen 30865 | Equinumerosity of the rang... |
| rinvf1o 30866 | Sufficient conditions for ... |
| fresf1o 30867 | Conditions for a restricti... |
| nfpconfp 30868 | The set of fixed points of... |
| fmptco1f1o 30869 | The action of composing (t... |
| cofmpt2 30870 | Express composition of a m... |
| f1mptrn 30871 | Express injection for a ma... |
| dfimafnf 30872 | Alternate definition of th... |
| funimass4f 30873 | Membership relation for th... |
| elimampt 30874 | Membership in the image of... |
| suppss2f 30875 | Show that the support of a... |
| fovcld 30876 | Closure law for an operati... |
| ofrn 30877 | The range of the function ... |
| ofrn2 30878 | The range of the function ... |
| off2 30879 | The function operation pro... |
| ofresid 30880 | Applying an operation rest... |
| fimarab 30881 | Expressing the image of a ... |
| unipreima 30882 | Preimage of a class union.... |
| opfv 30883 | Value of a function produc... |
| xppreima 30884 | The preimage of a Cartesia... |
| 2ndimaxp 30885 | Image of a cartesian produ... |
| djussxp2 30886 | Stronger version of ~ djus... |
| 2ndresdju 30887 | The ` 2nd ` function restr... |
| 2ndresdjuf1o 30888 | The ` 2nd ` function restr... |
| xppreima2 30889 | The preimage of a Cartesia... |
| elunirn2 30890 | Condition for the membersh... |
| abfmpunirn 30891 | Membership in a union of a... |
| rabfmpunirn 30892 | Membership in a union of a... |
| abfmpeld 30893 | Membership in an element o... |
| abfmpel 30894 | Membership in an element o... |
| fmptdF 30895 | Domain and codomain of the... |
| fmptcof2 30896 | Composition of two functio... |
| fcomptf 30897 | Express composition of two... |
| acunirnmpt 30898 | Axiom of choice for the un... |
| acunirnmpt2 30899 | Axiom of choice for the un... |
| acunirnmpt2f 30900 | Axiom of choice for the un... |
| aciunf1lem 30901 | Choice in an index union. ... |
| aciunf1 30902 | Choice in an index union. ... |
| ofoprabco 30903 | Function operation as a co... |
| ofpreima 30904 | Express the preimage of a ... |
| ofpreima2 30905 | Express the preimage of a ... |
| funcnvmpt 30906 | Condition for a function i... |
| funcnv5mpt 30907 | Two ways to say that a fun... |
| funcnv4mpt 30908 | Two ways to say that a fun... |
| preimane 30909 | Different elements have di... |
| fnpreimac 30910 | Choose a set ` x ` contain... |
| fgreu 30911 | Exactly one point of a fun... |
| fcnvgreu 30912 | If the converse of a relat... |
| rnmposs 30913 | The range of an operation ... |
| mptssALT 30914 | Deduce subset relation of ... |
| dfcnv2 30915 | Alternative definition of ... |
| fnimatp 30916 | The image of an unordered ... |
| fnunres2 30917 | Restriction of a disjoint ... |
| mpomptxf 30918 | Express a two-argument fun... |
| suppovss 30919 | A bound for the support of... |
| fvdifsupp 30920 | Function value is zero out... |
| fmptssfisupp 30921 | The restriction of a mappi... |
| suppiniseg 30922 | Relation between the suppo... |
| fsuppinisegfi 30923 | The initial segment ` ( ``... |
| fressupp 30924 | The restriction of a funct... |
| fdifsuppconst 30925 | A function is a zero const... |
| ressupprn 30926 | The range of a function re... |
| supppreima 30927 | Express the support of a f... |
| fsupprnfi 30928 | Finite support implies fin... |
| cosnopne 30929 | Composition of two ordered... |
| cosnop 30930 | Composition of two ordered... |
| cnvprop 30931 | Converse of a pair of orde... |
| brprop 30932 | Binary relation for a pair... |
| mptprop 30933 | Rewrite pairs of ordered p... |
| coprprop 30934 | Composition of two pairs o... |
| gtiso 30935 | Two ways to write a strict... |
| isoun 30936 | Infer an isomorphism from ... |
| disjdsct 30937 | A disjoint collection is d... |
| df1stres 30938 | Definition for a restricti... |
| df2ndres 30939 | Definition for a restricti... |
| 1stpreimas 30940 | The preimage of a singleto... |
| 1stpreima 30941 | The preimage by ` 1st ` is... |
| 2ndpreima 30942 | The preimage by ` 2nd ` is... |
| curry2ima 30943 | The image of a curried fun... |
| preiman0 30944 | The preimage of a nonempty... |
| intimafv 30945 | The intersection of an ima... |
| supssd 30946 | Inequality deduction for s... |
| infssd 30947 | Inequality deduction for i... |
| imafi2 30948 | The image by a finite set ... |
| unifi3 30949 | If a union is finite, then... |
| snct 30950 | A singleton is countable. ... |
| prct 30951 | An unordered pair is count... |
| mpocti 30952 | An operation is countable ... |
| abrexct 30953 | An image set of a countabl... |
| mptctf 30954 | A countable mapping set is... |
| abrexctf 30955 | An image set of a countabl... |
| padct 30956 | Index a countable set with... |
| cnvoprabOLD 30957 | The converse of a class ab... |
| f1od2 30958 | Sufficient condition for a... |
| fcobij 30959 | Composing functions with a... |
| fcobijfs 30960 | Composing finitely support... |
| suppss3 30961 | Deduce a function's suppor... |
| fsuppcurry1 30962 | Finite support of a currie... |
| fsuppcurry2 30963 | Finite support of a currie... |
| offinsupp1 30964 | Finite support for a funct... |
| ffs2 30965 | Rewrite a function's suppo... |
| ffsrn 30966 | The range of a finitely su... |
| resf1o 30967 | Restriction of functions t... |
| maprnin 30968 | Restricting the range of t... |
| fpwrelmapffslem 30969 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 30970 | Define a canonical mapping... |
| fpwrelmapffs 30971 | Define a canonical mapping... |
| creq0 30972 | The real representation of... |
| 1nei 30973 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 30974 | An integer unit times itse... |
| nnmulge 30975 | Multiplying by a positive ... |
| lt2addrd 30976 | If the right-hand side of ... |
| xrlelttric 30977 | Trichotomy law for extende... |
| xaddeq0 30978 | Two extended reals which a... |
| xrinfm 30979 | The extended real numbers ... |
| le2halvesd 30980 | A sum is less than the who... |
| xraddge02 30981 | A number is less than or e... |
| xrge0addge 30982 | A number is less than or e... |
| xlt2addrd 30983 | If the right-hand side of ... |
| xrsupssd 30984 | Inequality deduction for s... |
| xrge0infss 30985 | Any subset of nonnegative ... |
| xrge0infssd 30986 | Inequality deduction for i... |
| xrge0addcld 30987 | Nonnegative extended reals... |
| xrge0subcld 30988 | Condition for closure of n... |
| infxrge0lb 30989 | A member of a set of nonne... |
| infxrge0glb 30990 | The infimum of a set of no... |
| infxrge0gelb 30991 | The infimum of a set of no... |
| xrofsup 30992 | The supremum is preserved ... |
| supxrnemnf 30993 | The supremum of a nonempty... |
| xnn0gt0 30994 | Nonzero extended nonnegati... |
| xnn01gt 30995 | An extended nonnegative in... |
| nn0xmulclb 30996 | Finite multiplication in t... |
| joiniooico 30997 | Disjoint joining an open i... |
| ubico 30998 | A right-open interval does... |
| xeqlelt 30999 | Equality in terms of 'less... |
| eliccelico 31000 | Relate elementhood to a cl... |
| elicoelioo 31001 | Relate elementhood to a cl... |
| iocinioc2 31002 | Intersection between two o... |
| xrdifh 31003 | Class difference of a half... |
| iocinif 31004 | Relate intersection of two... |
| difioo 31005 | The difference between two... |
| difico 31006 | The difference between two... |
| uzssico 31007 | Upper integer sets are a s... |
| fz2ssnn0 31008 | A finite set of sequential... |
| nndiffz1 31009 | Upper set of the positive ... |
| ssnnssfz 31010 | For any finite subset of `... |
| fzne1 31011 | Elementhood in a finite se... |
| fzm1ne1 31012 | Elementhood of an integer ... |
| fzspl 31013 | Split the last element of ... |
| fzdif2 31014 | Split the last element of ... |
| fzodif2 31015 | Split the last element of ... |
| fzodif1 31016 | Set difference of two half... |
| fzsplit3 31017 | Split a finite interval of... |
| bcm1n 31018 | The proportion of one bino... |
| iundisjfi 31019 | Rewrite a countable union ... |
| iundisj2fi 31020 | A disjoint union is disjoi... |
| iundisjcnt 31021 | Rewrite a countable union ... |
| iundisj2cnt 31022 | A countable disjoint union... |
| fzone1 31023 | Elementhood in a half-open... |
| fzom1ne1 31024 | Elementhood in a half-open... |
| f1ocnt 31025 | Given a countable set ` A ... |
| fz1nnct 31026 | NN and integer ranges star... |
| fz1nntr 31027 | NN and integer ranges star... |
| hashunif 31028 | The cardinality of a disjo... |
| hashxpe 31029 | The size of the Cartesian ... |
| hashgt1 31030 | Restate "set contains at l... |
| dvdszzq 31031 | Divisibility for an intege... |
| prmdvdsbc 31032 | Condition for a prime numb... |
| numdenneg 31033 | Numerator and denominator ... |
| divnumden2 31034 | Calculate the reduced form... |
| nnindf 31035 | Principle of Mathematical ... |
| nn0min 31036 | Extracting the minimum pos... |
| subne0nn 31037 | A nonnegative difference i... |
| ltesubnnd 31038 | Subtracting an integer num... |
| fprodeq02 31039 | If one of the factors is z... |
| pr01ssre 31040 | The range of the indicator... |
| fprodex01 31041 | A product of factors equal... |
| prodpr 31042 | A product over a pair is t... |
| prodtp 31043 | A product over a triple is... |
| fsumub 31044 | An upper bound for a term ... |
| fsumiunle 31045 | Upper bound for a sum of n... |
| dfdec100 31046 | Split the hundreds from a ... |
| dp2eq1 31049 | Equality theorem for the d... |
| dp2eq2 31050 | Equality theorem for the d... |
| dp2eq1i 31051 | Equality theorem for the d... |
| dp2eq2i 31052 | Equality theorem for the d... |
| dp2eq12i 31053 | Equality theorem for the d... |
| dp20u 31054 | Add a zero in the tenths (... |
| dp20h 31055 | Add a zero in the unit pla... |
| dp2cl 31056 | Closure for the decimal fr... |
| dp2clq 31057 | Closure for a decimal frac... |
| rpdp2cl 31058 | Closure for a decimal frac... |
| rpdp2cl2 31059 | Closure for a decimal frac... |
| dp2lt10 31060 | Decimal fraction builds re... |
| dp2lt 31061 | Comparing two decimal frac... |
| dp2ltsuc 31062 | Comparing a decimal fracti... |
| dp2ltc 31063 | Comparing two decimal expa... |
| dpval 31066 | Define the value of the de... |
| dpcl 31067 | Prove that the closure of ... |
| dpfrac1 31068 | Prove a simple equivalence... |
| dpval2 31069 | Value of the decimal point... |
| dpval3 31070 | Value of the decimal point... |
| dpmul10 31071 | Multiply by 10 a decimal e... |
| decdiv10 31072 | Divide a decimal number by... |
| dpmul100 31073 | Multiply by 100 a decimal ... |
| dp3mul10 31074 | Multiply by 10 a decimal e... |
| dpmul1000 31075 | Multiply by 1000 a decimal... |
| dpval3rp 31076 | Value of the decimal point... |
| dp0u 31077 | Add a zero in the tenths p... |
| dp0h 31078 | Remove a zero in the units... |
| rpdpcl 31079 | Closure of the decimal poi... |
| dplt 31080 | Comparing two decimal expa... |
| dplti 31081 | Comparing a decimal expans... |
| dpgti 31082 | Comparing a decimal expans... |
| dpltc 31083 | Comparing two decimal inte... |
| dpexpp1 31084 | Add one zero to the mantis... |
| 0dp2dp 31085 | Multiply by 10 a decimal e... |
| dpadd2 31086 | Addition with one decimal,... |
| dpadd 31087 | Addition with one decimal.... |
| dpadd3 31088 | Addition with two decimals... |
| dpmul 31089 | Multiplication with one de... |
| dpmul4 31090 | An upper bound to multipli... |
| threehalves 31091 | Example theorem demonstrat... |
| 1mhdrd 31092 | Example theorem demonstrat... |
| xdivval 31095 | Value of division: the (un... |
| xrecex 31096 | Existence of reciprocal of... |
| xmulcand 31097 | Cancellation law for exten... |
| xreceu 31098 | Existential uniqueness of ... |
| xdivcld 31099 | Closure law for the extend... |
| xdivcl 31100 | Closure law for the extend... |
| xdivmul 31101 | Relationship between divis... |
| rexdiv 31102 | The extended real division... |
| xdivrec 31103 | Relationship between divis... |
| xdivid 31104 | A number divided by itself... |
| xdiv0 31105 | Division into zero is zero... |
| xdiv0rp 31106 | Division into zero is zero... |
| eliccioo 31107 | Membership in a closed int... |
| elxrge02 31108 | Elementhood in the set of ... |
| xdivpnfrp 31109 | Plus infinity divided by a... |
| rpxdivcld 31110 | Closure law for extended d... |
| xrpxdivcld 31111 | Closure law for extended d... |
| wrdfd 31112 | A word is a zero-based seq... |
| wrdres 31113 | Condition for the restrict... |
| wrdsplex 31114 | Existence of a split of a ... |
| pfx1s2 31115 | The prefix of length 1 of ... |
| pfxrn2 31116 | The range of a prefix of a... |
| pfxrn3 31117 | Express the range of a pre... |
| pfxf1 31118 | Condition for a prefix to ... |
| s1f1 31119 | Conditions for a length 1 ... |
| s2rn 31120 | Range of a length 2 string... |
| s2f1 31121 | Conditions for a length 2 ... |
| s3rn 31122 | Range of a length 3 string... |
| s3f1 31123 | Conditions for a length 3 ... |
| s3clhash 31124 | Closure of the words of le... |
| ccatf1 31125 | Conditions for a concatena... |
| pfxlsw2ccat 31126 | Reconstruct a word from it... |
| wrdt2ind 31127 | Perform an induction over ... |
| swrdrn2 31128 | The range of a subword is ... |
| swrdrn3 31129 | Express the range of a sub... |
| swrdf1 31130 | Condition for a subword to... |
| swrdrndisj 31131 | Condition for the range of... |
| splfv3 31132 | Symbols to the right of a ... |
| 1cshid 31133 | Cyclically shifting a sing... |
| cshw1s2 31134 | Cyclically shifting a leng... |
| cshwrnid 31135 | Cyclically shifting a word... |
| cshf1o 31136 | Condition for the cyclic s... |
| ressplusf 31137 | The group operation functi... |
| ressnm 31138 | The norm in a restricted s... |
| abvpropd2 31139 | Weaker version of ~ abvpro... |
| oppgle 31140 | less-than relation of an o... |
| oppgleOLD 31141 | Obsolete version of ~ oppg... |
| oppglt 31142 | less-than relation of an o... |
| ressprs 31143 | The restriction of a prose... |
| oduprs 31144 | Being a proset is a self-d... |
| posrasymb 31145 | A poset ordering is asymet... |
| resspos 31146 | The restriction of a Poset... |
| resstos 31147 | The restriction of a Toset... |
| odutos 31148 | Being a toset is a self-du... |
| tlt2 31149 | In a Toset, two elements m... |
| tlt3 31150 | In a Toset, two elements m... |
| trleile 31151 | In a Toset, two elements m... |
| toslublem 31152 | Lemma for ~ toslub and ~ x... |
| toslub 31153 | In a toset, the lowest upp... |
| tosglblem 31154 | Lemma for ~ tosglb and ~ x... |
| tosglb 31155 | Same theorem as ~ toslub ,... |
| clatp0cl 31156 | The poset zero of a comple... |
| clatp1cl 31157 | The poset one of a complet... |
| mntoval 31162 | Operation value of the mon... |
| ismnt 31163 | Express the statement " ` ... |
| ismntd 31164 | Property of being a monoto... |
| mntf 31165 | A monotone function is a f... |
| mgcoval 31166 | Operation value of the mon... |
| mgcval 31167 | Monotone Galois connection... |
| mgcf1 31168 | The lower adjoint ` F ` of... |
| mgcf2 31169 | The upper adjoint ` G ` of... |
| mgccole1 31170 | An inequality for the kern... |
| mgccole2 31171 | Inequality for the closure... |
| mgcmnt1 31172 | The lower adjoint ` F ` of... |
| mgcmnt2 31173 | The upper adjoint ` G ` of... |
| mgcmntco 31174 | A Galois connection like s... |
| dfmgc2lem 31175 | Lemma for dfmgc2, backward... |
| dfmgc2 31176 | Alternate definition of th... |
| mgcmnt1d 31177 | Galois connection implies ... |
| mgcmnt2d 31178 | Galois connection implies ... |
| mgccnv 31179 | The inverse Galois connect... |
| pwrssmgc 31180 | Given a function ` F ` , e... |
| mgcf1olem1 31181 | Property of a Galois conne... |
| mgcf1olem2 31182 | Property of a Galois conne... |
| mgcf1o 31183 | Given a Galois connection,... |
| xrs0 31186 | The zero of the extended r... |
| xrslt 31187 | The "strictly less than" r... |
| xrsinvgval 31188 | The inversion operation in... |
| xrsmulgzz 31189 | The "multiple" function in... |
| xrstos 31190 | The extended real numbers ... |
| xrsclat 31191 | The extended real numbers ... |
| xrsp0 31192 | The poset 0 of the extende... |
| xrsp1 31193 | The poset 1 of the extende... |
| ressmulgnn 31194 | Values for the group multi... |
| ressmulgnn0 31195 | Values for the group multi... |
| xrge0base 31196 | The base of the extended n... |
| xrge00 31197 | The zero of the extended n... |
| xrge0plusg 31198 | The additive law of the ex... |
| xrge0le 31199 | The "less than or equal to... |
| xrge0mulgnn0 31200 | The group multiple functio... |
| xrge0addass 31201 | Associativity of extended ... |
| xrge0addgt0 31202 | The sum of nonnegative and... |
| xrge0adddir 31203 | Right-distributivity of ex... |
| xrge0adddi 31204 | Left-distributivity of ext... |
| xrge0npcan 31205 | Extended nonnegative real ... |
| fsumrp0cl 31206 | Closure of a finite sum of... |
| abliso 31207 | The image of an Abelian gr... |
| gsumsubg 31208 | The group sum in a subgrou... |
| gsumsra 31209 | The group sum in a subring... |
| gsummpt2co 31210 | Split a finite sum into a ... |
| gsummpt2d 31211 | Express a finite sum over ... |
| lmodvslmhm 31212 | Scalar multiplication in a... |
| gsumvsmul1 31213 | Pull a scalar multiplicati... |
| gsummptres 31214 | Extend a finite group sum ... |
| gsummptres2 31215 | Extend a finite group sum ... |
| gsumzresunsn 31216 | Append an element to a fin... |
| gsumpart 31217 | Express a group sum as a d... |
| gsumhashmul 31218 | Express a group sum by gro... |
| xrge0tsmsd 31219 | Any finite or infinite sum... |
| xrge0tsmsbi 31220 | Any limit of a finite or i... |
| xrge0tsmseq 31221 | Any limit of a finite or i... |
| cntzun 31222 | The centralizer of a union... |
| cntzsnid 31223 | The centralizer of the ide... |
| cntrcrng 31224 | The center of a ring is a ... |
| isomnd 31229 | A (left) ordered monoid is... |
| isogrp 31230 | A (left-)ordered group is ... |
| ogrpgrp 31231 | A left-ordered group is a ... |
| omndmnd 31232 | A left-ordered monoid is a... |
| omndtos 31233 | A left-ordered monoid is a... |
| omndadd 31234 | In an ordered monoid, the ... |
| omndaddr 31235 | In a right ordered monoid,... |
| omndadd2d 31236 | In a commutative left orde... |
| omndadd2rd 31237 | In a left- and right- orde... |
| submomnd 31238 | A submonoid of an ordered ... |
| xrge0omnd 31239 | The nonnegative extended r... |
| omndmul2 31240 | In an ordered monoid, the ... |
| omndmul3 31241 | In an ordered monoid, the ... |
| omndmul 31242 | In a commutative ordered m... |
| ogrpinv0le 31243 | In an ordered group, the o... |
| ogrpsub 31244 | In an ordered group, the o... |
| ogrpaddlt 31245 | In an ordered group, stric... |
| ogrpaddltbi 31246 | In a right ordered group, ... |
| ogrpaddltrd 31247 | In a right ordered group, ... |
| ogrpaddltrbid 31248 | In a right ordered group, ... |
| ogrpsublt 31249 | In an ordered group, stric... |
| ogrpinv0lt 31250 | In an ordered group, the o... |
| ogrpinvlt 31251 | In an ordered group, the o... |
| gsumle 31252 | A finite sum in an ordered... |
| symgfcoeu 31253 | Uniqueness property of per... |
| symgcom 31254 | Two permutations ` X ` and... |
| symgcom2 31255 | Two permutations ` X ` and... |
| symgcntz 31256 | All elements of a (finite)... |
| odpmco 31257 | The composition of two odd... |
| symgsubg 31258 | The value of the group sub... |
| pmtrprfv2 31259 | In a transposition of two ... |
| pmtrcnel 31260 | Composing a permutation ` ... |
| pmtrcnel2 31261 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 31262 | Composing a permutation ` ... |
| pmtridf1o 31263 | Transpositions of ` X ` an... |
| pmtridfv1 31264 | Value at X of the transpos... |
| pmtridfv2 31265 | Value at Y of the transpos... |
| psgnid 31266 | Permutation sign of the id... |
| psgndmfi 31267 | For a finite base set, the... |
| pmtrto1cl 31268 | Useful lemma for the follo... |
| psgnfzto1stlem 31269 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 31270 | Value of our permutation `... |
| fzto1st1 31271 | Special case where the per... |
| fzto1st 31272 | The function moving one el... |
| fzto1stinvn 31273 | Value of the inverse of ou... |
| psgnfzto1st 31274 | The permutation sign for m... |
| tocycval 31277 | Value of the cycle builder... |
| tocycfv 31278 | Function value of a permut... |
| tocycfvres1 31279 | A cyclic permutation is a ... |
| tocycfvres2 31280 | A cyclic permutation is th... |
| cycpmfvlem 31281 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 31282 | Value of a cycle function ... |
| cycpmfv2 31283 | Value of a cycle function ... |
| cycpmfv3 31284 | Values outside of the orbi... |
| cycpmcl 31285 | Cyclic permutations are pe... |
| tocycf 31286 | The permutation cycle buil... |
| tocyc01 31287 | Permutation cycles built f... |
| cycpm2tr 31288 | A cyclic permutation of 2 ... |
| cycpm2cl 31289 | Closure for the 2-cycles. ... |
| cyc2fv1 31290 | Function value of a 2-cycl... |
| cyc2fv2 31291 | Function value of a 2-cycl... |
| trsp2cyc 31292 | Exhibit the word a transpo... |
| cycpmco2f1 31293 | The word U used in ~ cycpm... |
| cycpmco2rn 31294 | The orbit of the compositi... |
| cycpmco2lem1 31295 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 31296 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 31297 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 31298 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 31299 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 31300 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 31301 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 31302 | The composition of a cycli... |
| cyc2fvx 31303 | Function value of a 2-cycl... |
| cycpm3cl 31304 | Closure of the 3-cycles in... |
| cycpm3cl2 31305 | Closure of the 3-cycles in... |
| cyc3fv1 31306 | Function value of a 3-cycl... |
| cyc3fv2 31307 | Function value of a 3-cycl... |
| cyc3fv3 31308 | Function value of a 3-cycl... |
| cyc3co2 31309 | Represent a 3-cycle as a c... |
| cycpmconjvlem 31310 | Lemma for ~ cycpmconjv . ... |
| cycpmconjv 31311 | A formula for computing co... |
| cycpmrn 31312 | The range of the word used... |
| tocyccntz 31313 | All elements of a (finite)... |
| evpmval 31314 | Value of the set of even p... |
| cnmsgn0g 31315 | The neutral element of the... |
| evpmsubg 31316 | The alternating group is a... |
| evpmid 31317 | The identity is an even pe... |
| altgnsg 31318 | The alternating group ` ( ... |
| cyc3evpm 31319 | 3-Cycles are even permutat... |
| cyc3genpmlem 31320 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 31321 | The alternating group ` A ... |
| cycpmgcl 31322 | Cyclic permutations are pe... |
| cycpmconjslem1 31323 | Lemma for ~ cycpmconjs . ... |
| cycpmconjslem2 31324 | Lemma for ~ cycpmconjs . ... |
| cycpmconjs 31325 | All cycles of the same len... |
| cyc3conja 31326 | All 3-cycles are conjugate... |
| sgnsv 31329 | The sign mapping. (Contri... |
| sgnsval 31330 | The sign value. (Contribu... |
| sgnsf 31331 | The sign function. (Contr... |
| inftmrel 31336 | The infinitesimal relation... |
| isinftm 31337 | Express ` x ` is infinites... |
| isarchi 31338 | Express the predicate " ` ... |
| pnfinf 31339 | Plus infinity is an infini... |
| xrnarchi 31340 | The completed real line is... |
| isarchi2 31341 | Alternative way to express... |
| submarchi 31342 | A submonoid is archimedean... |
| isarchi3 31343 | This is the usual definiti... |
| archirng 31344 | Property of Archimedean or... |
| archirngz 31345 | Property of Archimedean le... |
| archiexdiv 31346 | In an Archimedean group, g... |
| archiabllem1a 31347 | Lemma for ~ archiabl : In... |
| archiabllem1b 31348 | Lemma for ~ archiabl . (C... |
| archiabllem1 31349 | Archimedean ordered groups... |
| archiabllem2a 31350 | Lemma for ~ archiabl , whi... |
| archiabllem2c 31351 | Lemma for ~ archiabl . (C... |
| archiabllem2b 31352 | Lemma for ~ archiabl . (C... |
| archiabllem2 31353 | Archimedean ordered groups... |
| archiabl 31354 | Archimedean left- and righ... |
| isslmd 31357 | The predicate "is a semimo... |
| slmdlema 31358 | Lemma for properties of a ... |
| lmodslmd 31359 | Left semimodules generaliz... |
| slmdcmn 31360 | A semimodule is a commutat... |
| slmdmnd 31361 | A semimodule is a monoid. ... |
| slmdsrg 31362 | The scalar component of a ... |
| slmdbn0 31363 | The base set of a semimodu... |
| slmdacl 31364 | Closure of ring addition f... |
| slmdmcl 31365 | Closure of ring multiplica... |
| slmdsn0 31366 | The set of scalars in a se... |
| slmdvacl 31367 | Closure of vector addition... |
| slmdass 31368 | Semiring left module vecto... |
| slmdvscl 31369 | Closure of scalar product ... |
| slmdvsdi 31370 | Distributive law for scala... |
| slmdvsdir 31371 | Distributive law for scala... |
| slmdvsass 31372 | Associative law for scalar... |
| slmd0cl 31373 | The ring zero in a semimod... |
| slmd1cl 31374 | The ring unit in a semirin... |
| slmdvs1 31375 | Scalar product with ring u... |
| slmd0vcl 31376 | The zero vector is a vecto... |
| slmd0vlid 31377 | Left identity law for the ... |
| slmd0vrid 31378 | Right identity law for the... |
| slmd0vs 31379 | Zero times a vector is the... |
| slmdvs0 31380 | Anything times the zero ve... |
| gsumvsca1 31381 | Scalar product of a finite... |
| gsumvsca2 31382 | Scalar product of a finite... |
| prmsimpcyc 31383 | A group of prime order is ... |
| rngurd 31384 | Deduce the unit of a ring ... |
| dvdschrmulg 31385 | In a ring, any multiple of... |
| freshmansdream 31386 | For a prime number ` P ` ,... |
| frobrhm 31387 | In a commutative ring with... |
| ress1r 31388 | ` 1r ` is unaffected by re... |
| dvrdir 31389 | Distributive law for the d... |
| rdivmuldivd 31390 | Multiplication of two rati... |
| ringinvval 31391 | The ring inverse expressed... |
| dvrcan5 31392 | Cancellation law for commo... |
| subrgchr 31393 | If ` A ` is a subring of `... |
| rmfsupp2 31394 | A mapping of a multiplicat... |
| primefldchr 31395 | The characteristic of a pr... |
| isorng 31400 | An ordered ring is a ring ... |
| orngring 31401 | An ordered ring is a ring.... |
| orngogrp 31402 | An ordered ring is an orde... |
| isofld 31403 | An ordered field is a fiel... |
| orngmul 31404 | In an ordered ring, the or... |
| orngsqr 31405 | In an ordered ring, all sq... |
| ornglmulle 31406 | In an ordered ring, multip... |
| orngrmulle 31407 | In an ordered ring, multip... |
| ornglmullt 31408 | In an ordered ring, multip... |
| orngrmullt 31409 | In an ordered ring, multip... |
| orngmullt 31410 | In an ordered ring, the st... |
| ofldfld 31411 | An ordered field is a fiel... |
| ofldtos 31412 | An ordered field is a tota... |
| orng0le1 31413 | In an ordered ring, the ri... |
| ofldlt1 31414 | In an ordered field, the r... |
| ofldchr 31415 | The characteristic of an o... |
| suborng 31416 | Every subring of an ordere... |
| subofld 31417 | Every subfield of an order... |
| isarchiofld 31418 | Axiom of Archimedes : a ch... |
| rhmdvdsr 31419 | A ring homomorphism preser... |
| rhmopp 31420 | A ring homomorphism is als... |
| elrhmunit 31421 | Ring homomorphisms preserv... |
| rhmdvd 31422 | A ring homomorphism preser... |
| rhmunitinv 31423 | Ring homomorphisms preserv... |
| kerunit 31424 | If a unit element lies in ... |
| reldmresv 31427 | The scalar restriction is ... |
| resvval 31428 | Value of structure restric... |
| resvid2 31429 | General behavior of trivia... |
| resvval2 31430 | Value of nontrivial struct... |
| resvsca 31431 | Base set of a structure re... |
| resvlem 31432 | Other elements of a scalar... |
| resvlemOLD 31433 | Obsolete version of ~ resv... |
| resvbas 31434 | ` Base ` is unaffected by ... |
| resvbasOLD 31435 | Obsolete proof of ~ resvba... |
| resvplusg 31436 | ` +g ` is unaffected by sc... |
| resvplusgOLD 31437 | Obsolete proof of ~ resvpl... |
| resvvsca 31438 | ` .s ` is unaffected by sc... |
| resvvscaOLD 31439 | Obsolete proof of ~ resvvs... |
| resvmulr 31440 | ` .r ` is unaffected by sc... |
| resvmulrOLD 31441 | Obsolete proof of ~ resvmu... |
| resv0g 31442 | ` 0g ` is unaffected by sc... |
| resv1r 31443 | ` 1r ` is unaffected by sc... |
| resvcmn 31444 | Scalar restriction preserv... |
| gzcrng 31445 | The gaussian integers form... |
| reofld 31446 | The real numbers form an o... |
| nn0omnd 31447 | The nonnegative integers f... |
| rearchi 31448 | The field of the real numb... |
| nn0archi 31449 | The monoid of the nonnegat... |
| xrge0slmod 31450 | The extended nonnegative r... |
| qusker 31451 | The kernel of a quotient m... |
| eqgvscpbl 31452 | The left coset equivalence... |
| qusvscpbl 31453 | The quotient map distribut... |
| qusscaval 31454 | Value of the scalar multip... |
| imaslmod 31455 | The image structure of a l... |
| quslmod 31456 | If ` G ` is a submodule in... |
| quslmhm 31457 | If ` G ` is a submodule of... |
| ecxpid 31458 | The equivalence class of a... |
| eqg0el 31459 | Equivalence class of a quo... |
| qsxpid 31460 | The quotient set of a cart... |
| qusxpid 31461 | The Group quotient equival... |
| qustriv 31462 | The quotient of a group ` ... |
| qustrivr 31463 | Converse of ~ qustriv . (... |
| znfermltl 31464 | Fermat's little theorem in... |
| islinds5 31465 | A set is linearly independ... |
| ellspds 31466 | Variation on ~ ellspd . (... |
| 0ellsp 31467 | Zero is in all spans. (Co... |
| 0nellinds 31468 | The group identity cannot ... |
| rspsnel 31469 | Membership in a principal ... |
| rspsnid 31470 | A principal ideal contains... |
| elrsp 31471 | Write the elements of a ri... |
| rspidlid 31472 | The ideal span of an ideal... |
| pidlnz 31473 | A principal ideal generate... |
| lbslsp 31474 | Any element of a left modu... |
| lindssn 31475 | Any singleton of a nonzero... |
| lindflbs 31476 | Conditions for an independ... |
| linds2eq 31477 | Deduce equality of element... |
| lindfpropd 31478 | Property deduction for lin... |
| lindspropd 31479 | Property deduction for lin... |
| elgrplsmsn 31480 | Membership in a sumset wit... |
| lsmsnorb 31481 | The sumset of a group with... |
| lsmsnorb2 31482 | The sumset of a single ele... |
| elringlsm 31483 | Membership in a product of... |
| elringlsmd 31484 | Membership in a product of... |
| ringlsmss 31485 | Closure of the product of ... |
| ringlsmss1 31486 | The product of an ideal ` ... |
| ringlsmss2 31487 | The product with an ideal ... |
| lsmsnpridl 31488 | The product of the ring wi... |
| lsmsnidl 31489 | The product of the ring wi... |
| lsmidllsp 31490 | The sum of two ideals is t... |
| lsmidl 31491 | The sum of two ideals is a... |
| lsmssass 31492 | Group sum is associative, ... |
| grplsm0l 31493 | Sumset with the identity s... |
| grplsmid 31494 | The direct sum of an eleme... |
| quslsm 31495 | Express the image by the q... |
| qusima 31496 | The image of a subgroup by... |
| nsgqus0 31497 | A normal subgroup ` N ` is... |
| nsgmgclem 31498 | Lemma for ~ nsgmgc . (Con... |
| nsgmgc 31499 | There is a monotone Galois... |
| nsgqusf1olem1 31500 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem2 31501 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1olem3 31502 | Lemma for ~ nsgqusf1o . (... |
| nsgqusf1o 31503 | The canonical projection h... |
| intlidl 31504 | The intersection of a none... |
| rhmpreimaidl 31505 | The preimage of an ideal b... |
| kerlidl 31506 | The kernel of a ring homom... |
| 0ringidl 31507 | The zero ideal is the only... |
| elrspunidl 31508 | Elementhood to the span of... |
| lidlincl 31509 | Ideals are closed under in... |
| idlinsubrg 31510 | The intersection between a... |
| rhmimaidl 31511 | The image of an ideal ` I ... |
| prmidlval 31514 | The class of prime ideals ... |
| isprmidl 31515 | The predicate "is a prime ... |
| prmidlnr 31516 | A prime ideal is a proper ... |
| prmidl 31517 | The main property of a pri... |
| prmidl2 31518 | A condition that shows an ... |
| idlmulssprm 31519 | Let ` P ` be a prime ideal... |
| pridln1 31520 | A proper ideal cannot cont... |
| prmidlidl 31521 | A prime ideal is an ideal.... |
| prmidlssidl 31522 | Prime ideals as a subset o... |
| lidlnsg 31523 | An ideal is a normal subgr... |
| cringm4 31524 | Commutative/associative la... |
| isprmidlc 31525 | The predicate "is prime id... |
| prmidlc 31526 | Property of a prime ideal ... |
| 0ringprmidl 31527 | The trivial ring does not ... |
| prmidl0 31528 | The zero ideal of a commut... |
| rhmpreimaprmidl 31529 | The preimage of a prime id... |
| qsidomlem1 31530 | If the quotient ring of a ... |
| qsidomlem2 31531 | A quotient by a prime idea... |
| qsidom 31532 | An ideal ` I ` in the comm... |
| mxidlval 31535 | The set of maximal ideals ... |
| ismxidl 31536 | The predicate "is a maxima... |
| mxidlidl 31537 | A maximal ideal is an idea... |
| mxidlnr 31538 | A maximal ideal is proper.... |
| mxidlmax 31539 | A maximal ideal is a maxim... |
| mxidln1 31540 | One is not contained in an... |
| mxidlnzr 31541 | A ring with a maximal idea... |
| mxidlprm 31542 | Every maximal ideal is pri... |
| ssmxidllem 31543 | The set ` P ` used in the ... |
| ssmxidl 31544 | Let ` R ` be a ring, and l... |
| krull 31545 | Krull's theorem: Any nonz... |
| mxidlnzrb 31546 | A ring is nonzero if and o... |
| idlsrgstr 31549 | A constructed semiring of ... |
| idlsrgval 31550 | Lemma for ~ idlsrgbas thro... |
| idlsrgbas 31551 | Baae of the ideals of a ri... |
| idlsrgplusg 31552 | Additive operation of the ... |
| idlsrg0g 31553 | The zero ideal is the addi... |
| idlsrgmulr 31554 | Multiplicative operation o... |
| idlsrgtset 31555 | Topology component of the ... |
| idlsrgmulrval 31556 | Value of the ring multipli... |
| idlsrgmulrcl 31557 | Ideals of a ring ` R ` are... |
| idlsrgmulrss1 31558 | In a commutative ring, the... |
| idlsrgmulrss2 31559 | The product of two ideals ... |
| idlsrgmulrssin 31560 | In a commutative ring, the... |
| idlsrgmnd 31561 | The ideals of a ring form ... |
| idlsrgcmnd 31562 | The ideals of a ring form ... |
| isufd 31565 | The property of being a Un... |
| rprmval 31566 | The prime elements of a ri... |
| isrprm 31567 | Property for ` P ` to be a... |
| asclmulg 31568 | Apply group multiplication... |
| fply1 31569 | Conditions for a function ... |
| ply1scleq 31570 | Equality of a constant pol... |
| ply1chr 31571 | The characteristic of a po... |
| ply1fermltl 31572 | Fermat's little theorem fo... |
| sra1r 31573 | The multiplicative neutral... |
| sraring 31574 | Condition for a subring al... |
| sradrng 31575 | Condition for a subring al... |
| srasubrg 31576 | A subring of the original ... |
| sralvec 31577 | Given a sub division ring ... |
| srafldlvec 31578 | Given a subfield ` F ` of ... |
| drgext0g 31579 | The additive neutral eleme... |
| drgextvsca 31580 | The scalar multiplication ... |
| drgext0gsca 31581 | The additive neutral eleme... |
| drgextsubrg 31582 | The scalar field is a subr... |
| drgextlsp 31583 | The scalar field is a subs... |
| drgextgsum 31584 | Group sum in a division ri... |
| lvecdimfi 31585 | Finite version of ~ lvecdi... |
| dimval 31588 | The dimension of a vector ... |
| dimvalfi 31589 | The dimension of a vector ... |
| dimcl 31590 | Closure of the vector spac... |
| lvecdim0i 31591 | A vector space of dimensio... |
| lvecdim0 31592 | A vector space of dimensio... |
| lssdimle 31593 | The dimension of a linear ... |
| dimpropd 31594 | If two structures have the... |
| rgmoddim 31595 | The left vector space indu... |
| frlmdim 31596 | Dimension of a free left m... |
| tnglvec 31597 | Augmenting a structure wit... |
| tngdim 31598 | Dimension of a left vector... |
| rrxdim 31599 | Dimension of the generaliz... |
| matdim 31600 | Dimension of the space of ... |
| lbslsat 31601 | A nonzero vector ` X ` is ... |
| lsatdim 31602 | A line, spanned by a nonze... |
| drngdimgt0 31603 | The dimension of a vector ... |
| lmhmlvec2 31604 | A homomorphism of left vec... |
| kerlmhm 31605 | The kernel of a vector spa... |
| imlmhm 31606 | The image of a vector spac... |
| lindsunlem 31607 | Lemma for ~ lindsun . (Co... |
| lindsun 31608 | Condition for the union of... |
| lbsdiflsp0 31609 | The linear spans of two di... |
| dimkerim 31610 | Given a linear map ` F ` b... |
| qusdimsum 31611 | Let ` W ` be a vector spac... |
| fedgmullem1 31612 | Lemma for ~ fedgmul . (Co... |
| fedgmullem2 31613 | Lemma for ~ fedgmul . (Co... |
| fedgmul 31614 | The multiplicativity formu... |
| relfldext 31623 | The field extension is a r... |
| brfldext 31624 | The field extension relati... |
| ccfldextrr 31625 | The field of the complex n... |
| fldextfld1 31626 | A field extension is only ... |
| fldextfld2 31627 | A field extension is only ... |
| fldextsubrg 31628 | Field extension implies a ... |
| fldextress 31629 | Field extension implies a ... |
| brfinext 31630 | The finite field extension... |
| extdgval 31631 | Value of the field extensi... |
| fldextsralvec 31632 | The subring algebra associ... |
| extdgcl 31633 | Closure of the field exten... |
| extdggt0 31634 | Degrees of field extension... |
| fldexttr 31635 | Field extension is a trans... |
| fldextid 31636 | The field extension relati... |
| extdgid 31637 | A trivial field extension ... |
| extdgmul 31638 | The multiplicativity formu... |
| finexttrb 31639 | The extension ` E ` of ` K... |
| extdg1id 31640 | If the degree of the exten... |
| extdg1b 31641 | The degree of the extensio... |
| fldextchr 31642 | The characteristic of a su... |
| ccfldsrarelvec 31643 | The subring algebra of the... |
| ccfldextdgrr 31644 | The degree of the field ex... |
| smatfval 31647 | Value of the submatrix. (... |
| smatrcl 31648 | Closure of the rectangular... |
| smatlem 31649 | Lemma for the next theorem... |
| smattl 31650 | Entries of a submatrix, to... |
| smattr 31651 | Entries of a submatrix, to... |
| smatbl 31652 | Entries of a submatrix, bo... |
| smatbr 31653 | Entries of a submatrix, bo... |
| smatcl 31654 | Closure of the square subm... |
| matmpo 31655 | Write a square matrix as a... |
| 1smat1 31656 | The submatrix of the ident... |
| submat1n 31657 | One case where the submatr... |
| submatres 31658 | Special case where the sub... |
| submateqlem1 31659 | Lemma for ~ submateq . (C... |
| submateqlem2 31660 | Lemma for ~ submateq . (C... |
| submateq 31661 | Sufficient condition for t... |
| submatminr1 31662 | If we take a submatrix by ... |
| lmatval 31665 | Value of the literal matri... |
| lmatfval 31666 | Entries of a literal matri... |
| lmatfvlem 31667 | Useful lemma to extract li... |
| lmatcl 31668 | Closure of the literal mat... |
| lmat22lem 31669 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 31670 | Entry of a 2x2 literal mat... |
| lmat22e12 31671 | Entry of a 2x2 literal mat... |
| lmat22e21 31672 | Entry of a 2x2 literal mat... |
| lmat22e22 31673 | Entry of a 2x2 literal mat... |
| lmat22det 31674 | The determinant of a liter... |
| mdetpmtr1 31675 | The determinant of a matri... |
| mdetpmtr2 31676 | The determinant of a matri... |
| mdetpmtr12 31677 | The determinant of a matri... |
| mdetlap1 31678 | A Laplace expansion of the... |
| madjusmdetlem1 31679 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 31680 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 31681 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 31682 | Lemma for ~ madjusmdet . ... |
| madjusmdet 31683 | Express the cofactor of th... |
| mdetlap 31684 | Laplace expansion of the d... |
| ist0cld 31685 | The predicate "is a T_0 sp... |
| txomap 31686 | Given two open maps ` F ` ... |
| qtopt1 31687 | If every equivalence class... |
| qtophaus 31688 | If an open map's graph in ... |
| circtopn 31689 | The topology of the unit c... |
| circcn 31690 | The function gluing the re... |
| reff 31691 | For any cover refinement, ... |
| locfinreflem 31692 | A locally finite refinemen... |
| locfinref 31693 | A locally finite refinemen... |
| iscref 31696 | The property that every op... |
| crefeq 31697 | Equality theorem for the "... |
| creftop 31698 | A space where every open c... |
| crefi 31699 | The property that every op... |
| crefdf 31700 | A formulation of ~ crefi e... |
| crefss 31701 | The "every open cover has ... |
| cmpcref 31702 | Equivalent definition of c... |
| cmpfiref 31703 | Every open cover of a Comp... |
| ldlfcntref 31706 | Every open cover of a Lind... |
| ispcmp 31709 | The predicate "is a paraco... |
| cmppcmp 31710 | Every compact space is par... |
| dispcmp 31711 | Every discrete space is pa... |
| pcmplfin 31712 | Given a paracompact topolo... |
| pcmplfinf 31713 | Given a paracompact topolo... |
| rspecval 31716 | Value of the spectrum of t... |
| rspecbas 31717 | The prime ideals form the ... |
| rspectset 31718 | Topology component of the ... |
| rspectopn 31719 | The topology component of ... |
| zarcls0 31720 | The closure of the identit... |
| zarcls1 31721 | The unit ideal ` B ` is th... |
| zarclsun 31722 | The union of two closed se... |
| zarclsiin 31723 | In a Zariski topology, the... |
| zarclsint 31724 | The intersection of a fami... |
| zarclssn 31725 | The closed points of Zaris... |
| zarcls 31726 | The open sets of the Zaris... |
| zartopn 31727 | The Zariski topology is a ... |
| zartop 31728 | The Zariski topology is a ... |
| zartopon 31729 | The points of the Zariski ... |
| zar0ring 31730 | The Zariski Topology of th... |
| zart0 31731 | The Zariski topology is T_... |
| zarmxt1 31732 | The Zariski topology restr... |
| zarcmplem 31733 | Lemma for ~ zarcmp . (Con... |
| zarcmp 31734 | The Zariski topology is co... |
| rspectps 31735 | The spectrum of a ring ` R... |
| rhmpreimacnlem 31736 | Lemma for ~ rhmpreimacn . ... |
| rhmpreimacn 31737 | The function mapping a pri... |
| metidval 31742 | Value of the metric identi... |
| metidss 31743 | As a relation, the metric ... |
| metidv 31744 | ` A ` and ` B ` identify b... |
| metideq 31745 | Basic property of the metr... |
| metider 31746 | The metric identification ... |
| pstmval 31747 | Value of the metric induce... |
| pstmfval 31748 | Function value of the metr... |
| pstmxmet 31749 | The metric induced by a ps... |
| hauseqcn 31750 | In a Hausdorff topology, t... |
| elunitge0 31751 | An element of the closed u... |
| unitssxrge0 31752 | The closed unit interval i... |
| unitdivcld 31753 | Necessary conditions for a... |
| iistmd 31754 | The closed unit interval f... |
| unicls 31755 | The union of the closed se... |
| tpr2tp 31756 | The usual topology on ` ( ... |
| tpr2uni 31757 | The usual topology on ` ( ... |
| xpinpreima 31758 | Rewrite the cartesian prod... |
| xpinpreima2 31759 | Rewrite the cartesian prod... |
| sqsscirc1 31760 | The complex square of side... |
| sqsscirc2 31761 | The complex square of side... |
| cnre2csqlem 31762 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 31763 | Image of a centered square... |
| tpr2rico 31764 | For any point of an open s... |
| cnvordtrestixx 31765 | The restriction of the 'gr... |
| prsdm 31766 | Domain of the relation of ... |
| prsrn 31767 | Range of the relation of a... |
| prsss 31768 | Relation of a subproset. ... |
| prsssdm 31769 | Domain of a subproset rela... |
| ordtprsval 31770 | Value of the order topolog... |
| ordtprsuni 31771 | Value of the order topolog... |
| ordtcnvNEW 31772 | The order dual generates t... |
| ordtrestNEW 31773 | The subspace topology of a... |
| ordtrest2NEWlem 31774 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 31775 | An interval-closed set ` A... |
| ordtconnlem1 31776 | Connectedness in the order... |
| ordtconn 31777 | Connectedness in the order... |
| mndpluscn 31778 | A mapping that is both a h... |
| mhmhmeotmd 31779 | Deduce a Topological Monoi... |
| rmulccn 31780 | Multiplication by a real c... |
| raddcn 31781 | Addition in the real numbe... |
| xrmulc1cn 31782 | The operation multiplying ... |
| fmcncfil 31783 | The image of a Cauchy filt... |
| xrge0hmph 31784 | The extended nonnegative r... |
| xrge0iifcnv 31785 | Define a bijection from ` ... |
| xrge0iifcv 31786 | The defined function's val... |
| xrge0iifiso 31787 | The defined bijection from... |
| xrge0iifhmeo 31788 | Expose a homeomorphism fro... |
| xrge0iifhom 31789 | The defined function from ... |
| xrge0iif1 31790 | Condition for the defined ... |
| xrge0iifmhm 31791 | The defined function from ... |
| xrge0pluscn 31792 | The addition operation of ... |
| xrge0mulc1cn 31793 | The operation multiplying ... |
| xrge0tps 31794 | The extended nonnegative r... |
| xrge0topn 31795 | The topology of the extend... |
| xrge0haus 31796 | The topology of the extend... |
| xrge0tmd 31797 | The extended nonnegative r... |
| xrge0tmdALT 31798 | Alternate proof of ~ xrge0... |
| lmlim 31799 | Relate a limit in a given ... |
| lmlimxrge0 31800 | Relate a limit in the nonn... |
| rge0scvg 31801 | Implication of convergence... |
| fsumcvg4 31802 | A serie with finite suppor... |
| pnfneige0 31803 | A neighborhood of ` +oo ` ... |
| lmxrge0 31804 | Express "sequence ` F ` co... |
| lmdvg 31805 | If a monotonic sequence of... |
| lmdvglim 31806 | If a monotonic real number... |
| pl1cn 31807 | A univariate polynomial is... |
| zringnm 31810 | The norm (function) for a ... |
| zzsnm 31811 | The norm of the ring of th... |
| zlm0 31812 | Zero of a ` ZZ ` -module. ... |
| zlm1 31813 | Unit of a ` ZZ ` -module (... |
| zlmds 31814 | Distance in a ` ZZ ` -modu... |
| zlmtset 31815 | Topology in a ` ZZ ` -modu... |
| zlmnm 31816 | Norm of a ` ZZ ` -module (... |
| zhmnrg 31817 | The ` ZZ ` -module built f... |
| nmmulg 31818 | The norm of a group produc... |
| zrhnm 31819 | The norm of the image by `... |
| cnzh 31820 | The ` ZZ ` -module of ` CC... |
| rezh 31821 | The ` ZZ ` -module of ` RR... |
| qqhval 31824 | Value of the canonical hom... |
| zrhf1ker 31825 | The kernel of the homomorp... |
| zrhchr 31826 | The kernel of the homomorp... |
| zrhker 31827 | The kernel of the homomorp... |
| zrhunitpreima 31828 | The preimage by ` ZRHom ` ... |
| elzrhunit 31829 | Condition for the image by... |
| elzdif0 31830 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 31831 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 31832 | Value of the canonical hom... |
| qqhvval 31833 | Value of the canonical hom... |
| qqh0 31834 | The image of ` 0 ` by the ... |
| qqh1 31835 | The image of ` 1 ` by the ... |
| qqhf 31836 | ` QQHom ` as a function. ... |
| qqhvq 31837 | The image of a quotient by... |
| qqhghm 31838 | The ` QQHom ` homomorphism... |
| qqhrhm 31839 | The ` QQHom ` homomorphism... |
| qqhnm 31840 | The norm of the image by `... |
| qqhcn 31841 | The ` QQHom ` homomorphism... |
| qqhucn 31842 | The ` QQHom ` homomorphism... |
| rrhval 31846 | Value of the canonical hom... |
| rrhcn 31847 | If the topology of ` R ` i... |
| rrhf 31848 | If the topology of ` R ` i... |
| isrrext 31850 | Express the property " ` R... |
| rrextnrg 31851 | An extension of ` RR ` is ... |
| rrextdrg 31852 | An extension of ` RR ` is ... |
| rrextnlm 31853 | The norm of an extension o... |
| rrextchr 31854 | The ring characteristic of... |
| rrextcusp 31855 | An extension of ` RR ` is ... |
| rrexttps 31856 | An extension of ` RR ` is ... |
| rrexthaus 31857 | The topology of an extensi... |
| rrextust 31858 | The uniformity of an exten... |
| rerrext 31859 | The field of the real numb... |
| cnrrext 31860 | The field of the complex n... |
| qqtopn 31861 | The topology of the field ... |
| rrhfe 31862 | If ` R ` is an extension o... |
| rrhcne 31863 | If ` R ` is an extension o... |
| rrhqima 31864 | The ` RRHom ` homomorphism... |
| rrh0 31865 | The image of ` 0 ` by the ... |
| xrhval 31868 | The value of the embedding... |
| zrhre 31869 | The ` ZRHom ` homomorphism... |
| qqhre 31870 | The ` QQHom ` homomorphism... |
| rrhre 31871 | The ` RRHom ` homomorphism... |
| relmntop 31874 | Manifold is a relation. (... |
| ismntoplly 31875 | Property of being a manifo... |
| ismntop 31876 | Property of being a manifo... |
| nexple 31877 | A lower bound for an expon... |
| indv 31880 | Value of the indicator fun... |
| indval 31881 | Value of the indicator fun... |
| indval2 31882 | Alternate value of the ind... |
| indf 31883 | An indicator function as a... |
| indfval 31884 | Value of the indicator fun... |
| ind1 31885 | Value of the indicator fun... |
| ind0 31886 | Value of the indicator fun... |
| ind1a 31887 | Value of the indicator fun... |
| indpi1 31888 | Preimage of the singleton ... |
| indsum 31889 | Finite sum of a product wi... |
| indsumin 31890 | Finite sum of a product wi... |
| prodindf 31891 | The product of indicators ... |
| indf1o 31892 | The bijection between a po... |
| indpreima 31893 | A function with range ` { ... |
| indf1ofs 31894 | The bijection between fini... |
| esumex 31897 | An extended sum is a set b... |
| esumcl 31898 | Closure for extended sum i... |
| esumeq12dvaf 31899 | Equality deduction for ext... |
| esumeq12dva 31900 | Equality deduction for ext... |
| esumeq12d 31901 | Equality deduction for ext... |
| esumeq1 31902 | Equality theorem for an ex... |
| esumeq1d 31903 | Equality theorem for an ex... |
| esumeq2 31904 | Equality theorem for exten... |
| esumeq2d 31905 | Equality deduction for ext... |
| esumeq2dv 31906 | Equality deduction for ext... |
| esumeq2sdv 31907 | Equality deduction for ext... |
| nfesum1 31908 | Bound-variable hypothesis ... |
| nfesum2 31909 | Bound-variable hypothesis ... |
| cbvesum 31910 | Change bound variable in a... |
| cbvesumv 31911 | Change bound variable in a... |
| esumid 31912 | Identify the extended sum ... |
| esumgsum 31913 | A finite extended sum is t... |
| esumval 31914 | Develop the value of the e... |
| esumel 31915 | The extended sum is a limi... |
| esumnul 31916 | Extended sum over the empt... |
| esum0 31917 | Extended sum of zero. (Co... |
| esumf1o 31918 | Re-index an extended sum u... |
| esumc 31919 | Convert from the collectio... |
| esumrnmpt 31920 | Rewrite an extended sum in... |
| esumsplit 31921 | Split an extended sum into... |
| esummono 31922 | Extended sum is monotonic.... |
| esumpad 31923 | Extend an extended sum by ... |
| esumpad2 31924 | Remove zeroes from an exte... |
| esumadd 31925 | Addition of infinite sums.... |
| esumle 31926 | If all of the terms of an ... |
| gsumesum 31927 | Relate a group sum on ` ( ... |
| esumlub 31928 | The extended sum is the lo... |
| esumaddf 31929 | Addition of infinite sums.... |
| esumlef 31930 | If all of the terms of an ... |
| esumcst 31931 | The extended sum of a cons... |
| esumsnf 31932 | The extended sum of a sing... |
| esumsn 31933 | The extended sum of a sing... |
| esumpr 31934 | Extended sum over a pair. ... |
| esumpr2 31935 | Extended sum over a pair, ... |
| esumrnmpt2 31936 | Rewrite an extended sum in... |
| esumfzf 31937 | Formulating a partial exte... |
| esumfsup 31938 | Formulating an extended su... |
| esumfsupre 31939 | Formulating an extended su... |
| esumss 31940 | Change the index set to a ... |
| esumpinfval 31941 | The value of the extended ... |
| esumpfinvallem 31942 | Lemma for ~ esumpfinval . ... |
| esumpfinval 31943 | The value of the extended ... |
| esumpfinvalf 31944 | Same as ~ esumpfinval , mi... |
| esumpinfsum 31945 | The value of the extended ... |
| esumpcvgval 31946 | The value of the extended ... |
| esumpmono 31947 | The partial sums in an ext... |
| esumcocn 31948 | Lemma for ~ esummulc2 and ... |
| esummulc1 31949 | An extended sum multiplied... |
| esummulc2 31950 | An extended sum multiplied... |
| esumdivc 31951 | An extended sum divided by... |
| hashf2 31952 | Lemma for ~ hasheuni . (C... |
| hasheuni 31953 | The cardinality of a disjo... |
| esumcvg 31954 | The sequence of partial su... |
| esumcvg2 31955 | Simpler version of ~ esumc... |
| esumcvgsum 31956 | The value of the extended ... |
| esumsup 31957 | Express an extended sum as... |
| esumgect 31958 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 31959 | All terms of a converging ... |
| esum2dlem 31960 | Lemma for ~ esum2d (finite... |
| esum2d 31961 | Write a double extended su... |
| esumiun 31962 | Sum over a nonnecessarily ... |
| ofceq 31965 | Equality theorem for funct... |
| ofcfval 31966 | Value of an operation appl... |
| ofcval 31967 | Evaluate a function/consta... |
| ofcfn 31968 | The function operation pro... |
| ofcfeqd2 31969 | Equality theorem for funct... |
| ofcfval3 31970 | General value of ` ( F oFC... |
| ofcf 31971 | The function/constant oper... |
| ofcfval2 31972 | The function operation exp... |
| ofcfval4 31973 | The function/constant oper... |
| ofcc 31974 | Left operation by a consta... |
| ofcof 31975 | Relate function operation ... |
| sigaex 31978 | Lemma for ~ issiga and ~ i... |
| sigaval 31979 | The set of sigma-algebra w... |
| issiga 31980 | An alternative definition ... |
| isrnsiga 31981 | The property of being a si... |
| 0elsiga 31982 | A sigma-algebra contains t... |
| baselsiga 31983 | A sigma-algebra contains i... |
| sigasspw 31984 | A sigma-algebra is a set o... |
| sigaclcu 31985 | A sigma-algebra is closed ... |
| sigaclcuni 31986 | A sigma-algebra is closed ... |
| sigaclfu 31987 | A sigma-algebra is closed ... |
| sigaclcu2 31988 | A sigma-algebra is closed ... |
| sigaclfu2 31989 | A sigma-algebra is closed ... |
| sigaclcu3 31990 | A sigma-algebra is closed ... |
| issgon 31991 | Property of being a sigma-... |
| sgon 31992 | A sigma-algebra is a sigma... |
| elsigass 31993 | An element of a sigma-alge... |
| elrnsiga 31994 | Dropping the base informat... |
| isrnsigau 31995 | The property of being a si... |
| unielsiga 31996 | A sigma-algebra contains i... |
| dmvlsiga 31997 | Lebesgue-measurable subset... |
| pwsiga 31998 | Any power set forms a sigm... |
| prsiga 31999 | The smallest possible sigm... |
| sigaclci 32000 | A sigma-algebra is closed ... |
| difelsiga 32001 | A sigma-algebra is closed ... |
| unelsiga 32002 | A sigma-algebra is closed ... |
| inelsiga 32003 | A sigma-algebra is closed ... |
| sigainb 32004 | Building a sigma-algebra f... |
| insiga 32005 | The intersection of a coll... |
| sigagenval 32008 | Value of the generated sig... |
| sigagensiga 32009 | A generated sigma-algebra ... |
| sgsiga 32010 | A generated sigma-algebra ... |
| unisg 32011 | The sigma-algebra generate... |
| dmsigagen 32012 | A sigma-algebra can be gen... |
| sssigagen 32013 | A set is a subset of the s... |
| sssigagen2 32014 | A subset of the generating... |
| elsigagen 32015 | Any element of a set is al... |
| elsigagen2 32016 | Any countable union of ele... |
| sigagenss 32017 | The generated sigma-algebr... |
| sigagenss2 32018 | Sufficient condition for i... |
| sigagenid 32019 | The sigma-algebra generate... |
| ispisys 32020 | The property of being a pi... |
| ispisys2 32021 | The property of being a pi... |
| inelpisys 32022 | Pi-systems are closed unde... |
| sigapisys 32023 | All sigma-algebras are pi-... |
| isldsys 32024 | The property of being a la... |
| pwldsys 32025 | The power set of the unive... |
| unelldsys 32026 | Lambda-systems are closed ... |
| sigaldsys 32027 | All sigma-algebras are lam... |
| ldsysgenld 32028 | The intersection of all la... |
| sigapildsyslem 32029 | Lemma for ~ sigapildsys . ... |
| sigapildsys 32030 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 32031 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 32032 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 32033 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 32034 | The lambda system ` E ` ge... |
| dynkin 32035 | Dynkin's lambda-pi theorem... |
| isros 32036 | The property of being a ri... |
| rossspw 32037 | A ring of sets is a collec... |
| 0elros 32038 | A ring of sets contains th... |
| unelros 32039 | A ring of sets is closed u... |
| difelros 32040 | A ring of sets is closed u... |
| inelros 32041 | A ring of sets is closed u... |
| fiunelros 32042 | A ring of sets is closed u... |
| issros 32043 | The property of being a se... |
| srossspw 32044 | A semiring of sets is a co... |
| 0elsros 32045 | A semiring of sets contain... |
| inelsros 32046 | A semiring of sets is clos... |
| diffiunisros 32047 | In semiring of sets, compl... |
| rossros 32048 | Rings of sets are semiring... |
| brsiga 32051 | The Borel Algebra on real ... |
| brsigarn 32052 | The Borel Algebra is a sig... |
| brsigasspwrn 32053 | The Borel Algebra is a set... |
| unibrsiga 32054 | The union of the Borel Alg... |
| cldssbrsiga 32055 | A Borel Algebra contains a... |
| sxval 32058 | Value of the product sigma... |
| sxsiga 32059 | A product sigma-algebra is... |
| sxsigon 32060 | A product sigma-algebra is... |
| sxuni 32061 | The base set of a product ... |
| elsx 32062 | The cartesian product of t... |
| measbase 32065 | The base set of a measure ... |
| measval 32066 | The value of the ` measure... |
| ismeas 32067 | The property of being a me... |
| isrnmeas 32068 | The property of being a me... |
| dmmeas 32069 | The domain of a measure is... |
| measbasedom 32070 | The base set of a measure ... |
| measfrge0 32071 | A measure is a function ov... |
| measfn 32072 | A measure is a function on... |
| measvxrge0 32073 | The values of a measure ar... |
| measvnul 32074 | The measure of the empty s... |
| measge0 32075 | A measure is nonnegative. ... |
| measle0 32076 | If the measure of a given ... |
| measvun 32077 | The measure of a countable... |
| measxun2 32078 | The measure the union of t... |
| measun 32079 | The measure the union of t... |
| measvunilem 32080 | Lemma for ~ measvuni . (C... |
| measvunilem0 32081 | Lemma for ~ measvuni . (C... |
| measvuni 32082 | The measure of a countable... |
| measssd 32083 | A measure is monotone with... |
| measunl 32084 | A measure is sub-additive ... |
| measiuns 32085 | The measure of the union o... |
| measiun 32086 | A measure is sub-additive.... |
| meascnbl 32087 | A measure is continuous fr... |
| measinblem 32088 | Lemma for ~ measinb . (Co... |
| measinb 32089 | Building a measure restric... |
| measres 32090 | Building a measure restric... |
| measinb2 32091 | Building a measure restric... |
| measdivcst 32092 | Division of a measure by a... |
| measdivcstALTV 32093 | Alternate version of ~ mea... |
| cntmeas 32094 | The Counting measure is a ... |
| pwcntmeas 32095 | The counting measure is a ... |
| cntnevol 32096 | Counting and Lebesgue meas... |
| voliune 32097 | The Lebesgue measure funct... |
| volfiniune 32098 | The Lebesgue measure funct... |
| volmeas 32099 | The Lebesgue measure is a ... |
| ddeval1 32102 | Value of the delta measure... |
| ddeval0 32103 | Value of the delta measure... |
| ddemeas 32104 | The Dirac delta measure is... |
| relae 32108 | 'almost everywhere' is a r... |
| brae 32109 | 'almost everywhere' relati... |
| braew 32110 | 'almost everywhere' relati... |
| truae 32111 | A truth holds almost every... |
| aean 32112 | A conjunction holds almost... |
| faeval 32114 | Value of the 'almost every... |
| relfae 32115 | The 'almost everywhere' bu... |
| brfae 32116 | 'almost everywhere' relati... |
| ismbfm 32119 | The predicate " ` F ` is a... |
| elunirnmbfm 32120 | The property of being a me... |
| mbfmfun 32121 | A measurable function is a... |
| mbfmf 32122 | A measurable function as a... |
| isanmbfm 32123 | The predicate to be a meas... |
| mbfmcnvima 32124 | The preimage by a measurab... |
| mbfmbfm 32125 | A measurable function to a... |
| mbfmcst 32126 | A constant function is mea... |
| 1stmbfm 32127 | The first projection map i... |
| 2ndmbfm 32128 | The second projection map ... |
| imambfm 32129 | If the sigma-algebra in th... |
| cnmbfm 32130 | A continuous function is m... |
| mbfmco 32131 | The composition of two mea... |
| mbfmco2 32132 | The pair building of two m... |
| mbfmvolf 32133 | Measurable functions with ... |
| elmbfmvol2 32134 | Measurable functions with ... |
| mbfmcnt 32135 | All functions are measurab... |
| br2base 32136 | The base set for the gener... |
| dya2ub 32137 | An upper bound for a dyadi... |
| sxbrsigalem0 32138 | The closed half-spaces of ... |
| sxbrsigalem3 32139 | The sigma-algebra generate... |
| dya2iocival 32140 | The function ` I ` returns... |
| dya2iocress 32141 | Dyadic intervals are subse... |
| dya2iocbrsiga 32142 | Dyadic intervals are Borel... |
| dya2icobrsiga 32143 | Dyadic intervals are Borel... |
| dya2icoseg 32144 | For any point and any clos... |
| dya2icoseg2 32145 | For any point and any open... |
| dya2iocrfn 32146 | The function returning dya... |
| dya2iocct 32147 | The dyadic rectangle set i... |
| dya2iocnrect 32148 | For any point of an open r... |
| dya2iocnei 32149 | For any point of an open s... |
| dya2iocuni 32150 | Every open set of ` ( RR X... |
| dya2iocucvr 32151 | The dyadic rectangular set... |
| sxbrsigalem1 32152 | The Borel algebra on ` ( R... |
| sxbrsigalem2 32153 | The sigma-algebra generate... |
| sxbrsigalem4 32154 | The Borel algebra on ` ( R... |
| sxbrsigalem5 32155 | First direction for ~ sxbr... |
| sxbrsigalem6 32156 | First direction for ~ sxbr... |
| sxbrsiga 32157 | The product sigma-algebra ... |
| omsval 32160 | Value of the function mapp... |
| omsfval 32161 | Value of the outer measure... |
| omscl 32162 | A closure lemma for the co... |
| omsf 32163 | A constructed outer measur... |
| oms0 32164 | A constructed outer measur... |
| omsmon 32165 | A constructed outer measur... |
| omssubaddlem 32166 | For any small margin ` E `... |
| omssubadd 32167 | A constructed outer measur... |
| carsgval 32170 | Value of the Caratheodory ... |
| carsgcl 32171 | Closure of the Caratheodor... |
| elcarsg 32172 | Property of being a Carath... |
| baselcarsg 32173 | The universe set, ` O ` , ... |
| 0elcarsg 32174 | The empty set is Caratheod... |
| carsguni 32175 | The union of all Caratheod... |
| elcarsgss 32176 | Caratheodory measurable se... |
| difelcarsg 32177 | The Caratheodory measurabl... |
| inelcarsg 32178 | The Caratheodory measurabl... |
| unelcarsg 32179 | The Caratheodory-measurabl... |
| difelcarsg2 32180 | The Caratheodory-measurabl... |
| carsgmon 32181 | Utility lemma: Apply mono... |
| carsgsigalem 32182 | Lemma for the following th... |
| fiunelcarsg 32183 | The Caratheodory measurabl... |
| carsgclctunlem1 32184 | Lemma for ~ carsgclctun . ... |
| carsggect 32185 | The outer measure is count... |
| carsgclctunlem2 32186 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 32187 | Lemma for ~ carsgclctun . ... |
| carsgclctun 32188 | The Caratheodory measurabl... |
| carsgsiga 32189 | The Caratheodory measurabl... |
| omsmeas 32190 | The restriction of a const... |
| pmeasmono 32191 | This theorem's hypotheses ... |
| pmeasadd 32192 | A premeasure on a ring of ... |
| itgeq12dv 32193 | Equality theorem for an in... |
| sitgval 32199 | Value of the simple functi... |
| issibf 32200 | The predicate " ` F ` is a... |
| sibf0 32201 | The constant zero function... |
| sibfmbl 32202 | A simple function is measu... |
| sibff 32203 | A simple function is a fun... |
| sibfrn 32204 | A simple function has fini... |
| sibfima 32205 | Any preimage of a singleto... |
| sibfinima 32206 | The measure of the interse... |
| sibfof 32207 | Applying function operatio... |
| sitgfval 32208 | Value of the Bochner integ... |
| sitgclg 32209 | Closure of the Bochner int... |
| sitgclbn 32210 | Closure of the Bochner int... |
| sitgclcn 32211 | Closure of the Bochner int... |
| sitgclre 32212 | Closure of the Bochner int... |
| sitg0 32213 | The integral of the consta... |
| sitgf 32214 | The integral for simple fu... |
| sitgaddlemb 32215 | Lemma for * sitgadd . (Co... |
| sitmval 32216 | Value of the simple functi... |
| sitmfval 32217 | Value of the integral dist... |
| sitmcl 32218 | Closure of the integral di... |
| sitmf 32219 | The integral metric as a f... |
| oddpwdc 32221 | Lemma for ~ eulerpart . T... |
| oddpwdcv 32222 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 32223 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 32224 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 32225 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 32226 | Lemma for ~ eulerpart . (... |
| eulerpartlems 32227 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 32228 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 32229 | Lemma for ~ eulerpart . (... |
| eulerpartleme 32230 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 32231 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 32232 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 32233 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 32234 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 32235 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 32236 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 32237 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 32238 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 32239 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 32240 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 32241 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 32242 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 32243 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 32244 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 32245 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 32246 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 32247 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 32248 | Lemma for ~ eulerpart . (... |
| eulerpart 32249 | Euler's theorem on partiti... |
| subiwrd 32252 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 32253 | Length of a subword of an ... |
| iwrdsplit 32254 | Lemma for ~ sseqp1 . (Con... |
| sseqval 32255 | Value of the strong sequen... |
| sseqfv1 32256 | Value of the strong sequen... |
| sseqfn 32257 | A strong recursive sequenc... |
| sseqmw 32258 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 32259 | A strong recursive sequenc... |
| sseqfres 32260 | The first elements in the ... |
| sseqfv2 32261 | Value of the strong sequen... |
| sseqp1 32262 | Value of the strong sequen... |
| fiblem 32265 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 32266 | Value of the Fibonacci seq... |
| fib1 32267 | Value of the Fibonacci seq... |
| fibp1 32268 | Value of the Fibonacci seq... |
| fib2 32269 | Value of the Fibonacci seq... |
| fib3 32270 | Value of the Fibonacci seq... |
| fib4 32271 | Value of the Fibonacci seq... |
| fib5 32272 | Value of the Fibonacci seq... |
| fib6 32273 | Value of the Fibonacci seq... |
| elprob 32276 | The property of being a pr... |
| domprobmeas 32277 | A probability measure is a... |
| domprobsiga 32278 | The domain of a probabilit... |
| probtot 32279 | The probability of the uni... |
| prob01 32280 | A probability is an elemen... |
| probnul 32281 | The probability of the emp... |
| unveldomd 32282 | The universe is an element... |
| unveldom 32283 | The universe is an element... |
| nuleldmp 32284 | The empty set is an elemen... |
| probcun 32285 | The probability of the uni... |
| probun 32286 | The probability of the uni... |
| probdif 32287 | The probability of the dif... |
| probinc 32288 | A probability law is incre... |
| probdsb 32289 | The probability of the com... |
| probmeasd 32290 | A probability measure is a... |
| probvalrnd 32291 | The value of a probability... |
| probtotrnd 32292 | The probability of the uni... |
| totprobd 32293 | Law of total probability, ... |
| totprob 32294 | Law of total probability. ... |
| probfinmeasb 32295 | Build a probability measur... |
| probfinmeasbALTV 32296 | Alternate version of ~ pro... |
| probmeasb 32297 | Build a probability from a... |
| cndprobval 32300 | The value of the condition... |
| cndprobin 32301 | An identity linking condit... |
| cndprob01 32302 | The conditional probabilit... |
| cndprobtot 32303 | The conditional probabilit... |
| cndprobnul 32304 | The conditional probabilit... |
| cndprobprob 32305 | The conditional probabilit... |
| bayesth 32306 | Bayes Theorem. (Contribut... |
| rrvmbfm 32309 | A real-valued random varia... |
| isrrvv 32310 | Elementhood to the set of ... |
| rrvvf 32311 | A real-valued random varia... |
| rrvfn 32312 | A real-valued random varia... |
| rrvdm 32313 | The domain of a random var... |
| rrvrnss 32314 | The range of a random vari... |
| rrvf2 32315 | A real-valued random varia... |
| rrvdmss 32316 | The domain of a random var... |
| rrvfinvima 32317 | For a real-value random va... |
| 0rrv 32318 | The constant function equa... |
| rrvadd 32319 | The sum of two random vari... |
| rrvmulc 32320 | A random variable multipli... |
| rrvsum 32321 | An indexed sum of random v... |
| orvcval 32324 | Value of the preimage mapp... |
| orvcval2 32325 | Another way to express the... |
| elorvc 32326 | Elementhood of a preimage.... |
| orvcval4 32327 | The value of the preimage ... |
| orvcoel 32328 | If the relation produces o... |
| orvccel 32329 | If the relation produces c... |
| elorrvc 32330 | Elementhood of a preimage ... |
| orrvcval4 32331 | The value of the preimage ... |
| orrvcoel 32332 | If the relation produces o... |
| orrvccel 32333 | If the relation produces c... |
| orvcgteel 32334 | Preimage maps produced by ... |
| orvcelval 32335 | Preimage maps produced by ... |
| orvcelel 32336 | Preimage maps produced by ... |
| dstrvval 32337 | The value of the distribut... |
| dstrvprob 32338 | The distribution of a rand... |
| orvclteel 32339 | Preimage maps produced by ... |
| dstfrvel 32340 | Elementhood of preimage ma... |
| dstfrvunirn 32341 | The limit of all preimage ... |
| orvclteinc 32342 | Preimage maps produced by ... |
| dstfrvinc 32343 | A cumulative distribution ... |
| dstfrvclim1 32344 | The limit of the cumulativ... |
| coinfliplem 32345 | Division in the extended r... |
| coinflipprob 32346 | The ` P ` we defined for c... |
| coinflipspace 32347 | The space of our coin-flip... |
| coinflipuniv 32348 | The universe of our coin-f... |
| coinfliprv 32349 | The ` X ` we defined for c... |
| coinflippv 32350 | The probability of heads i... |
| coinflippvt 32351 | The probability of tails i... |
| ballotlemoex 32352 | ` O ` is a set. (Contribu... |
| ballotlem1 32353 | The size of the universe i... |
| ballotlemelo 32354 | Elementhood in ` O ` . (C... |
| ballotlem2 32355 | The probability that the f... |
| ballotlemfval 32356 | The value of ` F ` . (Con... |
| ballotlemfelz 32357 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 32358 | If the ` J ` th ballot is ... |
| ballotlemfc0 32359 | ` F ` takes value 0 betwee... |
| ballotlemfcc 32360 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 32361 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 32362 | ` ( F `` C ) ` always star... |
| ballotleme 32363 | Elements of ` E ` . (Cont... |
| ballotlemodife 32364 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 32365 | If the first pick is a vot... |
| ballotlem5 32366 | If A is not ahead througho... |
| ballotlemi 32367 | Value of ` I ` for a given... |
| ballotlemiex 32368 | Properties of ` ( I `` C )... |
| ballotlemi1 32369 | The first tie cannot be re... |
| ballotlemii 32370 | The first tie cannot be re... |
| ballotlemsup 32371 | The set of zeroes of ` F `... |
| ballotlemimin 32372 | ` ( I `` C ) ` is the firs... |
| ballotlemic 32373 | If the first vote is for B... |
| ballotlem1c 32374 | If the first vote is for A... |
| ballotlemsval 32375 | Value of ` S ` . (Contrib... |
| ballotlemsv 32376 | Value of ` S ` evaluated a... |
| ballotlemsgt1 32377 | ` S ` maps values less tha... |
| ballotlemsdom 32378 | Domain of ` S ` for a give... |
| ballotlemsel1i 32379 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 32380 | The defined ` S ` is a bij... |
| ballotlemsi 32381 | The image by ` S ` of the ... |
| ballotlemsima 32382 | The image by ` S ` of an i... |
| ballotlemieq 32383 | If two countings share the... |
| ballotlemrval 32384 | Value of ` R ` . (Contrib... |
| ballotlemscr 32385 | The image of ` ( R `` C ) ... |
| ballotlemrv 32386 | Value of ` R ` evaluated a... |
| ballotlemrv1 32387 | Value of ` R ` before the ... |
| ballotlemrv2 32388 | Value of ` R ` after the t... |
| ballotlemro 32389 | Range of ` R ` is included... |
| ballotlemgval 32390 | Expand the value of ` .^ `... |
| ballotlemgun 32391 | A property of the defined ... |
| ballotlemfg 32392 | Express the value of ` ( F... |
| ballotlemfrc 32393 | Express the value of ` ( F... |
| ballotlemfrci 32394 | Reverse counting preserves... |
| ballotlemfrceq 32395 | Value of ` F ` for a rever... |
| ballotlemfrcn0 32396 | Value of ` F ` for a rever... |
| ballotlemrc 32397 | Range of ` R ` . (Contrib... |
| ballotlemirc 32398 | Applying ` R ` does not ch... |
| ballotlemrinv0 32399 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 32400 | ` R ` is its own inverse :... |
| ballotlem1ri 32401 | When the vote on the first... |
| ballotlem7 32402 | ` R ` is a bijection betwe... |
| ballotlem8 32403 | There are as many counting... |
| ballotth 32404 | Bertrand's ballot problem ... |
| sgncl 32405 | Closure of the signum. (C... |
| sgnclre 32406 | Closure of the signum. (C... |
| sgnneg 32407 | Negation of the signum. (... |
| sgn3da 32408 | A conditional containing a... |
| sgnmul 32409 | Signum of a product. (Con... |
| sgnmulrp2 32410 | Multiplication by a positi... |
| sgnsub 32411 | Subtraction of a number of... |
| sgnnbi 32412 | Negative signum. (Contrib... |
| sgnpbi 32413 | Positive signum. (Contrib... |
| sgn0bi 32414 | Zero signum. (Contributed... |
| sgnsgn 32415 | Signum is idempotent. (Co... |
| sgnmulsgn 32416 | If two real numbers are of... |
| sgnmulsgp 32417 | If two real numbers are of... |
| fzssfzo 32418 | Condition for an integer i... |
| gsumncl 32419 | Closure of a group sum in ... |
| gsumnunsn 32420 | Closure of a group sum in ... |
| ccatmulgnn0dir 32421 | Concatenation of words fol... |
| ofcccat 32422 | Letterwise operations on w... |
| ofcs1 32423 | Letterwise operations on a... |
| ofcs2 32424 | Letterwise operations on a... |
| plymul02 32425 | Product of a polynomial wi... |
| plymulx0 32426 | Coefficients of a polynomi... |
| plymulx 32427 | Coefficients of a polynomi... |
| plyrecld 32428 | Closure of a polynomial wi... |
| signsplypnf 32429 | The quotient of a polynomi... |
| signsply0 32430 | Lemma for the rule of sign... |
| signspval 32431 | The value of the skipping ... |
| signsw0glem 32432 | Neutral element property o... |
| signswbase 32433 | The base of ` W ` is the u... |
| signswplusg 32434 | The operation of ` W ` . ... |
| signsw0g 32435 | The neutral element of ` W... |
| signswmnd 32436 | ` W ` is a monoid structur... |
| signswrid 32437 | The zero-skipping operatio... |
| signswlid 32438 | The zero-skipping operatio... |
| signswn0 32439 | The zero-skipping operatio... |
| signswch 32440 | The zero-skipping operatio... |
| signslema 32441 | Computational part of ~~? ... |
| signstfv 32442 | Value of the zero-skipping... |
| signstfval 32443 | Value of the zero-skipping... |
| signstcl 32444 | Closure of the zero skippi... |
| signstf 32445 | The zero skipping sign wor... |
| signstlen 32446 | Length of the zero skippin... |
| signstf0 32447 | Sign of a single letter wo... |
| signstfvn 32448 | Zero-skipping sign in a wo... |
| signsvtn0 32449 | If the last letter is nonz... |
| signstfvp 32450 | Zero-skipping sign in a wo... |
| signstfvneq0 32451 | In case the first letter i... |
| signstfvcl 32452 | Closure of the zero skippi... |
| signstfvc 32453 | Zero-skipping sign in a wo... |
| signstres 32454 | Restriction of a zero skip... |
| signstfveq0a 32455 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 32456 | In case the last letter is... |
| signsvvfval 32457 | The value of ` V ` , which... |
| signsvvf 32458 | ` V ` is a function. (Con... |
| signsvf0 32459 | There is no change of sign... |
| signsvf1 32460 | In a single-letter word, w... |
| signsvfn 32461 | Number of changes in a wor... |
| signsvtp 32462 | Adding a letter of the sam... |
| signsvtn 32463 | Adding a letter of a diffe... |
| signsvfpn 32464 | Adding a letter of the sam... |
| signsvfnn 32465 | Adding a letter of a diffe... |
| signlem0 32466 | Adding a zero as the highe... |
| signshf 32467 | ` H ` , corresponding to t... |
| signshwrd 32468 | ` H ` , corresponding to t... |
| signshlen 32469 | Length of ` H ` , correspo... |
| signshnz 32470 | ` H ` is not the empty wor... |
| efcld 32471 | Closure law for the expone... |
| iblidicc 32472 | The identity function is i... |
| rpsqrtcn 32473 | Continuity of the real pos... |
| divsqrtid 32474 | A real number divided by i... |
| cxpcncf1 32475 | The power function on comp... |
| efmul2picn 32476 | Multiplying by ` ( _i x. (... |
| fct2relem 32477 | Lemma for ~ ftc2re . (Con... |
| ftc2re 32478 | The Fundamental Theorem of... |
| fdvposlt 32479 | Functions with a positive ... |
| fdvneggt 32480 | Functions with a negative ... |
| fdvposle 32481 | Functions with a nonnegati... |
| fdvnegge 32482 | Functions with a nonpositi... |
| prodfzo03 32483 | A product of three factors... |
| actfunsnf1o 32484 | The action ` F ` of extend... |
| actfunsnrndisj 32485 | The action ` F ` of extend... |
| itgexpif 32486 | The basis for the circle m... |
| fsum2dsub 32487 | Lemma for ~ breprexp - Re-... |
| reprval 32490 | Value of the representatio... |
| repr0 32491 | There is exactly one repre... |
| reprf 32492 | Members of the representat... |
| reprsum 32493 | Sums of values of the memb... |
| reprle 32494 | Upper bound to the terms i... |
| reprsuc 32495 | Express the representation... |
| reprfi 32496 | Bounded representations ar... |
| reprss 32497 | Representations with terms... |
| reprinrn 32498 | Representations with term ... |
| reprlt 32499 | There are no representatio... |
| hashreprin 32500 | Express a sum of represent... |
| reprgt 32501 | There are no representatio... |
| reprinfz1 32502 | For the representation of ... |
| reprfi2 32503 | Corollary of ~ reprinfz1 .... |
| reprfz1 32504 | Corollary of ~ reprinfz1 .... |
| hashrepr 32505 | Develop the number of repr... |
| reprpmtf1o 32506 | Transposing ` 0 ` and ` X ... |
| reprdifc 32507 | Express the representation... |
| chpvalz 32508 | Value of the second Chebys... |
| chtvalz 32509 | Value of the Chebyshev fun... |
| breprexplema 32510 | Lemma for ~ breprexp (indu... |
| breprexplemb 32511 | Lemma for ~ breprexp (clos... |
| breprexplemc 32512 | Lemma for ~ breprexp (indu... |
| breprexp 32513 | Express the ` S ` th power... |
| breprexpnat 32514 | Express the ` S ` th power... |
| vtsval 32517 | Value of the Vinogradov tr... |
| vtscl 32518 | Closure of the Vinogradov ... |
| vtsprod 32519 | Express the Vinogradov tri... |
| circlemeth 32520 | The Hardy, Littlewood and ... |
| circlemethnat 32521 | The Hardy, Littlewood and ... |
| circlevma 32522 | The Circle Method, where t... |
| circlemethhgt 32523 | The circle method, where t... |
| hgt750lemc 32527 | An upper bound to the summ... |
| hgt750lemd 32528 | An upper bound to the summ... |
| hgt749d 32529 | A deduction version of ~ a... |
| logdivsqrle 32530 | Conditions for ` ( ( log `... |
| hgt750lem 32531 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 32532 | Decimal multiplication gal... |
| hgt750lemf 32533 | Lemma for the statement 7.... |
| hgt750lemg 32534 | Lemma for the statement 7.... |
| oddprm2 32535 | Two ways to write the set ... |
| hgt750lemb 32536 | An upper bound on the cont... |
| hgt750lema 32537 | An upper bound on the cont... |
| hgt750leme 32538 | An upper bound on the cont... |
| tgoldbachgnn 32539 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 32540 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 32541 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 32542 | Odd integers greater than ... |
| tgoldbachgt 32543 | Odd integers greater than ... |
| istrkg2d 32546 | Property of fulfilling dim... |
| axtglowdim2ALTV 32547 | Alternate version of ~ axt... |
| axtgupdim2ALTV 32548 | Alternate version of ~ axt... |
| afsval 32551 | Value of the AFS relation ... |
| brafs 32552 | Binary relation form of th... |
| tg5segofs 32553 | Rephrase ~ axtg5seg using ... |
| lpadval 32556 | Value of the ` leftpad ` f... |
| lpadlem1 32557 | Lemma for the ` leftpad ` ... |
| lpadlem3 32558 | Lemma for ~ lpadlen1 . (C... |
| lpadlen1 32559 | Length of a left-padded wo... |
| lpadlem2 32560 | Lemma for the ` leftpad ` ... |
| lpadlen2 32561 | Length of a left-padded wo... |
| lpadmax 32562 | Length of a left-padded wo... |
| lpadleft 32563 | The contents of prefix of ... |
| lpadright 32564 | The suffix of a left-padde... |
| bnj170 32577 | ` /\ ` -manipulation. (Co... |
| bnj240 32578 | ` /\ ` -manipulation. (Co... |
| bnj248 32579 | ` /\ ` -manipulation. (Co... |
| bnj250 32580 | ` /\ ` -manipulation. (Co... |
| bnj251 32581 | ` /\ ` -manipulation. (Co... |
| bnj252 32582 | ` /\ ` -manipulation. (Co... |
| bnj253 32583 | ` /\ ` -manipulation. (Co... |
| bnj255 32584 | ` /\ ` -manipulation. (Co... |
| bnj256 32585 | ` /\ ` -manipulation. (Co... |
| bnj257 32586 | ` /\ ` -manipulation. (Co... |
| bnj258 32587 | ` /\ ` -manipulation. (Co... |
| bnj268 32588 | ` /\ ` -manipulation. (Co... |
| bnj290 32589 | ` /\ ` -manipulation. (Co... |
| bnj291 32590 | ` /\ ` -manipulation. (Co... |
| bnj312 32591 | ` /\ ` -manipulation. (Co... |
| bnj334 32592 | ` /\ ` -manipulation. (Co... |
| bnj345 32593 | ` /\ ` -manipulation. (Co... |
| bnj422 32594 | ` /\ ` -manipulation. (Co... |
| bnj432 32595 | ` /\ ` -manipulation. (Co... |
| bnj446 32596 | ` /\ ` -manipulation. (Co... |
| bnj23 32597 | First-order logic and set ... |
| bnj31 32598 | First-order logic and set ... |
| bnj62 32599 | First-order logic and set ... |
| bnj89 32600 | First-order logic and set ... |
| bnj90 32601 | First-order logic and set ... |
| bnj101 32602 | First-order logic and set ... |
| bnj105 32603 | First-order logic and set ... |
| bnj115 32604 | First-order logic and set ... |
| bnj132 32605 | First-order logic and set ... |
| bnj133 32606 | First-order logic and set ... |
| bnj156 32607 | First-order logic and set ... |
| bnj158 32608 | First-order logic and set ... |
| bnj168 32609 | First-order logic and set ... |
| bnj206 32610 | First-order logic and set ... |
| bnj216 32611 | First-order logic and set ... |
| bnj219 32612 | First-order logic and set ... |
| bnj226 32613 | First-order logic and set ... |
| bnj228 32614 | First-order logic and set ... |
| bnj519 32615 | First-order logic and set ... |
| bnj521 32616 | First-order logic and set ... |
| bnj524 32617 | First-order logic and set ... |
| bnj525 32618 | First-order logic and set ... |
| bnj534 32619 | First-order logic and set ... |
| bnj538 32620 | First-order logic and set ... |
| bnj529 32621 | First-order logic and set ... |
| bnj551 32622 | First-order logic and set ... |
| bnj563 32623 | First-order logic and set ... |
| bnj564 32624 | First-order logic and set ... |
| bnj593 32625 | First-order logic and set ... |
| bnj596 32626 | First-order logic and set ... |
| bnj610 32627 | Pass from equality ( ` x =... |
| bnj642 32628 | ` /\ ` -manipulation. (Co... |
| bnj643 32629 | ` /\ ` -manipulation. (Co... |
| bnj645 32630 | ` /\ ` -manipulation. (Co... |
| bnj658 32631 | ` /\ ` -manipulation. (Co... |
| bnj667 32632 | ` /\ ` -manipulation. (Co... |
| bnj705 32633 | ` /\ ` -manipulation. (Co... |
| bnj706 32634 | ` /\ ` -manipulation. (Co... |
| bnj707 32635 | ` /\ ` -manipulation. (Co... |
| bnj708 32636 | ` /\ ` -manipulation. (Co... |
| bnj721 32637 | ` /\ ` -manipulation. (Co... |
| bnj832 32638 | ` /\ ` -manipulation. (Co... |
| bnj835 32639 | ` /\ ` -manipulation. (Co... |
| bnj836 32640 | ` /\ ` -manipulation. (Co... |
| bnj837 32641 | ` /\ ` -manipulation. (Co... |
| bnj769 32642 | ` /\ ` -manipulation. (Co... |
| bnj770 32643 | ` /\ ` -manipulation. (Co... |
| bnj771 32644 | ` /\ ` -manipulation. (Co... |
| bnj887 32645 | ` /\ ` -manipulation. (Co... |
| bnj918 32646 | First-order logic and set ... |
| bnj919 32647 | First-order logic and set ... |
| bnj923 32648 | First-order logic and set ... |
| bnj927 32649 | First-order logic and set ... |
| bnj931 32650 | First-order logic and set ... |
| bnj937 32651 | First-order logic and set ... |
| bnj941 32652 | First-order logic and set ... |
| bnj945 32653 | Technical lemma for ~ bnj6... |
| bnj946 32654 | First-order logic and set ... |
| bnj951 32655 | ` /\ ` -manipulation. (Co... |
| bnj956 32656 | First-order logic and set ... |
| bnj976 32657 | First-order logic and set ... |
| bnj982 32658 | First-order logic and set ... |
| bnj1019 32659 | First-order logic and set ... |
| bnj1023 32660 | First-order logic and set ... |
| bnj1095 32661 | First-order logic and set ... |
| bnj1096 32662 | First-order logic and set ... |
| bnj1098 32663 | First-order logic and set ... |
| bnj1101 32664 | First-order logic and set ... |
| bnj1113 32665 | First-order logic and set ... |
| bnj1109 32666 | First-order logic and set ... |
| bnj1131 32667 | First-order logic and set ... |
| bnj1138 32668 | First-order logic and set ... |
| bnj1142 32669 | First-order logic and set ... |
| bnj1143 32670 | First-order logic and set ... |
| bnj1146 32671 | First-order logic and set ... |
| bnj1149 32672 | First-order logic and set ... |
| bnj1185 32673 | First-order logic and set ... |
| bnj1196 32674 | First-order logic and set ... |
| bnj1198 32675 | First-order logic and set ... |
| bnj1209 32676 | First-order logic and set ... |
| bnj1211 32677 | First-order logic and set ... |
| bnj1213 32678 | First-order logic and set ... |
| bnj1212 32679 | First-order logic and set ... |
| bnj1219 32680 | First-order logic and set ... |
| bnj1224 32681 | First-order logic and set ... |
| bnj1230 32682 | First-order logic and set ... |
| bnj1232 32683 | First-order logic and set ... |
| bnj1235 32684 | First-order logic and set ... |
| bnj1239 32685 | First-order logic and set ... |
| bnj1238 32686 | First-order logic and set ... |
| bnj1241 32687 | First-order logic and set ... |
| bnj1247 32688 | First-order logic and set ... |
| bnj1254 32689 | First-order logic and set ... |
| bnj1262 32690 | First-order logic and set ... |
| bnj1266 32691 | First-order logic and set ... |
| bnj1265 32692 | First-order logic and set ... |
| bnj1275 32693 | First-order logic and set ... |
| bnj1276 32694 | First-order logic and set ... |
| bnj1292 32695 | First-order logic and set ... |
| bnj1293 32696 | First-order logic and set ... |
| bnj1294 32697 | First-order logic and set ... |
| bnj1299 32698 | First-order logic and set ... |
| bnj1304 32699 | First-order logic and set ... |
| bnj1316 32700 | First-order logic and set ... |
| bnj1317 32701 | First-order logic and set ... |
| bnj1322 32702 | First-order logic and set ... |
| bnj1340 32703 | First-order logic and set ... |
| bnj1345 32704 | First-order logic and set ... |
| bnj1350 32705 | First-order logic and set ... |
| bnj1351 32706 | First-order logic and set ... |
| bnj1352 32707 | First-order logic and set ... |
| bnj1361 32708 | First-order logic and set ... |
| bnj1366 32709 | First-order logic and set ... |
| bnj1379 32710 | First-order logic and set ... |
| bnj1383 32711 | First-order logic and set ... |
| bnj1385 32712 | First-order logic and set ... |
| bnj1386 32713 | First-order logic and set ... |
| bnj1397 32714 | First-order logic and set ... |
| bnj1400 32715 | First-order logic and set ... |
| bnj1405 32716 | First-order logic and set ... |
| bnj1422 32717 | First-order logic and set ... |
| bnj1424 32718 | First-order logic and set ... |
| bnj1436 32719 | First-order logic and set ... |
| bnj1441 32720 | First-order logic and set ... |
| bnj1441g 32721 | First-order logic and set ... |
| bnj1454 32722 | First-order logic and set ... |
| bnj1459 32723 | First-order logic and set ... |
| bnj1464 32724 | Conversion of implicit sub... |
| bnj1465 32725 | First-order logic and set ... |
| bnj1468 32726 | Conversion of implicit sub... |
| bnj1476 32727 | First-order logic and set ... |
| bnj1502 32728 | First-order logic and set ... |
| bnj1503 32729 | First-order logic and set ... |
| bnj1517 32730 | First-order logic and set ... |
| bnj1521 32731 | First-order logic and set ... |
| bnj1533 32732 | First-order logic and set ... |
| bnj1534 32733 | First-order logic and set ... |
| bnj1536 32734 | First-order logic and set ... |
| bnj1538 32735 | First-order logic and set ... |
| bnj1541 32736 | First-order logic and set ... |
| bnj1542 32737 | First-order logic and set ... |
| bnj110 32738 | Well-founded induction res... |
| bnj157 32739 | Well-founded induction res... |
| bnj66 32740 | Technical lemma for ~ bnj6... |
| bnj91 32741 | First-order logic and set ... |
| bnj92 32742 | First-order logic and set ... |
| bnj93 32743 | Technical lemma for ~ bnj9... |
| bnj95 32744 | Technical lemma for ~ bnj1... |
| bnj96 32745 | Technical lemma for ~ bnj1... |
| bnj97 32746 | Technical lemma for ~ bnj1... |
| bnj98 32747 | Technical lemma for ~ bnj1... |
| bnj106 32748 | First-order logic and set ... |
| bnj118 32749 | First-order logic and set ... |
| bnj121 32750 | First-order logic and set ... |
| bnj124 32751 | Technical lemma for ~ bnj1... |
| bnj125 32752 | Technical lemma for ~ bnj1... |
| bnj126 32753 | Technical lemma for ~ bnj1... |
| bnj130 32754 | Technical lemma for ~ bnj1... |
| bnj149 32755 | Technical lemma for ~ bnj1... |
| bnj150 32756 | Technical lemma for ~ bnj1... |
| bnj151 32757 | Technical lemma for ~ bnj1... |
| bnj154 32758 | Technical lemma for ~ bnj1... |
| bnj155 32759 | Technical lemma for ~ bnj1... |
| bnj153 32760 | Technical lemma for ~ bnj8... |
| bnj207 32761 | Technical lemma for ~ bnj8... |
| bnj213 32762 | First-order logic and set ... |
| bnj222 32763 | Technical lemma for ~ bnj2... |
| bnj229 32764 | Technical lemma for ~ bnj5... |
| bnj517 32765 | Technical lemma for ~ bnj5... |
| bnj518 32766 | Technical lemma for ~ bnj8... |
| bnj523 32767 | Technical lemma for ~ bnj8... |
| bnj526 32768 | Technical lemma for ~ bnj8... |
| bnj528 32769 | Technical lemma for ~ bnj8... |
| bnj535 32770 | Technical lemma for ~ bnj8... |
| bnj539 32771 | Technical lemma for ~ bnj8... |
| bnj540 32772 | Technical lemma for ~ bnj8... |
| bnj543 32773 | Technical lemma for ~ bnj8... |
| bnj544 32774 | Technical lemma for ~ bnj8... |
| bnj545 32775 | Technical lemma for ~ bnj8... |
| bnj546 32776 | Technical lemma for ~ bnj8... |
| bnj548 32777 | Technical lemma for ~ bnj8... |
| bnj553 32778 | Technical lemma for ~ bnj8... |
| bnj554 32779 | Technical lemma for ~ bnj8... |
| bnj556 32780 | Technical lemma for ~ bnj8... |
| bnj557 32781 | Technical lemma for ~ bnj8... |
| bnj558 32782 | Technical lemma for ~ bnj8... |
| bnj561 32783 | Technical lemma for ~ bnj8... |
| bnj562 32784 | Technical lemma for ~ bnj8... |
| bnj570 32785 | Technical lemma for ~ bnj8... |
| bnj571 32786 | Technical lemma for ~ bnj8... |
| bnj605 32787 | Technical lemma. This lem... |
| bnj581 32788 | Technical lemma for ~ bnj5... |
| bnj589 32789 | Technical lemma for ~ bnj8... |
| bnj590 32790 | Technical lemma for ~ bnj8... |
| bnj591 32791 | Technical lemma for ~ bnj8... |
| bnj594 32792 | Technical lemma for ~ bnj8... |
| bnj580 32793 | Technical lemma for ~ bnj5... |
| bnj579 32794 | Technical lemma for ~ bnj8... |
| bnj602 32795 | Equality theorem for the `... |
| bnj607 32796 | Technical lemma for ~ bnj8... |
| bnj609 32797 | Technical lemma for ~ bnj8... |
| bnj611 32798 | Technical lemma for ~ bnj8... |
| bnj600 32799 | Technical lemma for ~ bnj8... |
| bnj601 32800 | Technical lemma for ~ bnj8... |
| bnj852 32801 | Technical lemma for ~ bnj6... |
| bnj864 32802 | Technical lemma for ~ bnj6... |
| bnj865 32803 | Technical lemma for ~ bnj6... |
| bnj873 32804 | Technical lemma for ~ bnj6... |
| bnj849 32805 | Technical lemma for ~ bnj6... |
| bnj882 32806 | Definition (using hypothes... |
| bnj18eq1 32807 | Equality theorem for trans... |
| bnj893 32808 | Property of ` _trCl ` . U... |
| bnj900 32809 | Technical lemma for ~ bnj6... |
| bnj906 32810 | Property of ` _trCl ` . (... |
| bnj908 32811 | Technical lemma for ~ bnj6... |
| bnj911 32812 | Technical lemma for ~ bnj6... |
| bnj916 32813 | Technical lemma for ~ bnj6... |
| bnj917 32814 | Technical lemma for ~ bnj6... |
| bnj934 32815 | Technical lemma for ~ bnj6... |
| bnj929 32816 | Technical lemma for ~ bnj6... |
| bnj938 32817 | Technical lemma for ~ bnj6... |
| bnj944 32818 | Technical lemma for ~ bnj6... |
| bnj953 32819 | Technical lemma for ~ bnj6... |
| bnj958 32820 | Technical lemma for ~ bnj6... |
| bnj1000 32821 | Technical lemma for ~ bnj8... |
| bnj965 32822 | Technical lemma for ~ bnj8... |
| bnj964 32823 | Technical lemma for ~ bnj6... |
| bnj966 32824 | Technical lemma for ~ bnj6... |
| bnj967 32825 | Technical lemma for ~ bnj6... |
| bnj969 32826 | Technical lemma for ~ bnj6... |
| bnj970 32827 | Technical lemma for ~ bnj6... |
| bnj910 32828 | Technical lemma for ~ bnj6... |
| bnj978 32829 | Technical lemma for ~ bnj6... |
| bnj981 32830 | Technical lemma for ~ bnj6... |
| bnj983 32831 | Technical lemma for ~ bnj6... |
| bnj984 32832 | Technical lemma for ~ bnj6... |
| bnj985v 32833 | Version of ~ bnj985 with a... |
| bnj985 32834 | Technical lemma for ~ bnj6... |
| bnj986 32835 | Technical lemma for ~ bnj6... |
| bnj996 32836 | Technical lemma for ~ bnj6... |
| bnj998 32837 | Technical lemma for ~ bnj6... |
| bnj999 32838 | Technical lemma for ~ bnj6... |
| bnj1001 32839 | Technical lemma for ~ bnj6... |
| bnj1006 32840 | Technical lemma for ~ bnj6... |
| bnj1014 32841 | Technical lemma for ~ bnj6... |
| bnj1015 32842 | Technical lemma for ~ bnj6... |
| bnj1018g 32843 | Version of ~ bnj1018 with ... |
| bnj1018 32844 | Technical lemma for ~ bnj6... |
| bnj1020 32845 | Technical lemma for ~ bnj6... |
| bnj1021 32846 | Technical lemma for ~ bnj6... |
| bnj907 32847 | Technical lemma for ~ bnj6... |
| bnj1029 32848 | Property of ` _trCl ` . (... |
| bnj1033 32849 | Technical lemma for ~ bnj6... |
| bnj1034 32850 | Technical lemma for ~ bnj6... |
| bnj1039 32851 | Technical lemma for ~ bnj6... |
| bnj1040 32852 | Technical lemma for ~ bnj6... |
| bnj1047 32853 | Technical lemma for ~ bnj6... |
| bnj1049 32854 | Technical lemma for ~ bnj6... |
| bnj1052 32855 | Technical lemma for ~ bnj6... |
| bnj1053 32856 | Technical lemma for ~ bnj6... |
| bnj1071 32857 | Technical lemma for ~ bnj6... |
| bnj1083 32858 | Technical lemma for ~ bnj6... |
| bnj1090 32859 | Technical lemma for ~ bnj6... |
| bnj1093 32860 | Technical lemma for ~ bnj6... |
| bnj1097 32861 | Technical lemma for ~ bnj6... |
| bnj1110 32862 | Technical lemma for ~ bnj6... |
| bnj1112 32863 | Technical lemma for ~ bnj6... |
| bnj1118 32864 | Technical lemma for ~ bnj6... |
| bnj1121 32865 | Technical lemma for ~ bnj6... |
| bnj1123 32866 | Technical lemma for ~ bnj6... |
| bnj1030 32867 | Technical lemma for ~ bnj6... |
| bnj1124 32868 | Property of ` _trCl ` . (... |
| bnj1133 32869 | Technical lemma for ~ bnj6... |
| bnj1128 32870 | Technical lemma for ~ bnj6... |
| bnj1127 32871 | Property of ` _trCl ` . (... |
| bnj1125 32872 | Property of ` _trCl ` . (... |
| bnj1145 32873 | Technical lemma for ~ bnj6... |
| bnj1147 32874 | Property of ` _trCl ` . (... |
| bnj1137 32875 | Property of ` _trCl ` . (... |
| bnj1148 32876 | Property of ` _pred ` . (... |
| bnj1136 32877 | Technical lemma for ~ bnj6... |
| bnj1152 32878 | Technical lemma for ~ bnj6... |
| bnj1154 32879 | Property of ` Fr ` . (Con... |
| bnj1171 32880 | Technical lemma for ~ bnj6... |
| bnj1172 32881 | Technical lemma for ~ bnj6... |
| bnj1173 32882 | Technical lemma for ~ bnj6... |
| bnj1174 32883 | Technical lemma for ~ bnj6... |
| bnj1175 32884 | Technical lemma for ~ bnj6... |
| bnj1176 32885 | Technical lemma for ~ bnj6... |
| bnj1177 32886 | Technical lemma for ~ bnj6... |
| bnj1186 32887 | Technical lemma for ~ bnj6... |
| bnj1190 32888 | Technical lemma for ~ bnj6... |
| bnj1189 32889 | Technical lemma for ~ bnj6... |
| bnj69 32890 | Existence of a minimal ele... |
| bnj1228 32891 | Existence of a minimal ele... |
| bnj1204 32892 | Well-founded induction. T... |
| bnj1234 32893 | Technical lemma for ~ bnj6... |
| bnj1245 32894 | Technical lemma for ~ bnj6... |
| bnj1256 32895 | Technical lemma for ~ bnj6... |
| bnj1259 32896 | Technical lemma for ~ bnj6... |
| bnj1253 32897 | Technical lemma for ~ bnj6... |
| bnj1279 32898 | Technical lemma for ~ bnj6... |
| bnj1286 32899 | Technical lemma for ~ bnj6... |
| bnj1280 32900 | Technical lemma for ~ bnj6... |
| bnj1296 32901 | Technical lemma for ~ bnj6... |
| bnj1309 32902 | Technical lemma for ~ bnj6... |
| bnj1307 32903 | Technical lemma for ~ bnj6... |
| bnj1311 32904 | Technical lemma for ~ bnj6... |
| bnj1318 32905 | Technical lemma for ~ bnj6... |
| bnj1326 32906 | Technical lemma for ~ bnj6... |
| bnj1321 32907 | Technical lemma for ~ bnj6... |
| bnj1364 32908 | Property of ` _FrSe ` . (... |
| bnj1371 32909 | Technical lemma for ~ bnj6... |
| bnj1373 32910 | Technical lemma for ~ bnj6... |
| bnj1374 32911 | Technical lemma for ~ bnj6... |
| bnj1384 32912 | Technical lemma for ~ bnj6... |
| bnj1388 32913 | Technical lemma for ~ bnj6... |
| bnj1398 32914 | Technical lemma for ~ bnj6... |
| bnj1413 32915 | Property of ` _trCl ` . (... |
| bnj1408 32916 | Technical lemma for ~ bnj1... |
| bnj1414 32917 | Property of ` _trCl ` . (... |
| bnj1415 32918 | Technical lemma for ~ bnj6... |
| bnj1416 32919 | Technical lemma for ~ bnj6... |
| bnj1418 32920 | Property of ` _pred ` . (... |
| bnj1417 32921 | Technical lemma for ~ bnj6... |
| bnj1421 32922 | Technical lemma for ~ bnj6... |
| bnj1444 32923 | Technical lemma for ~ bnj6... |
| bnj1445 32924 | Technical lemma for ~ bnj6... |
| bnj1446 32925 | Technical lemma for ~ bnj6... |
| bnj1447 32926 | Technical lemma for ~ bnj6... |
| bnj1448 32927 | Technical lemma for ~ bnj6... |
| bnj1449 32928 | Technical lemma for ~ bnj6... |
| bnj1442 32929 | Technical lemma for ~ bnj6... |
| bnj1450 32930 | Technical lemma for ~ bnj6... |
| bnj1423 32931 | Technical lemma for ~ bnj6... |
| bnj1452 32932 | Technical lemma for ~ bnj6... |
| bnj1466 32933 | Technical lemma for ~ bnj6... |
| bnj1467 32934 | Technical lemma for ~ bnj6... |
| bnj1463 32935 | Technical lemma for ~ bnj6... |
| bnj1489 32936 | Technical lemma for ~ bnj6... |
| bnj1491 32937 | Technical lemma for ~ bnj6... |
| bnj1312 32938 | Technical lemma for ~ bnj6... |
| bnj1493 32939 | Technical lemma for ~ bnj6... |
| bnj1497 32940 | Technical lemma for ~ bnj6... |
| bnj1498 32941 | Technical lemma for ~ bnj6... |
| bnj60 32942 | Well-founded recursion, pa... |
| bnj1514 32943 | Technical lemma for ~ bnj1... |
| bnj1518 32944 | Technical lemma for ~ bnj1... |
| bnj1519 32945 | Technical lemma for ~ bnj1... |
| bnj1520 32946 | Technical lemma for ~ bnj1... |
| bnj1501 32947 | Technical lemma for ~ bnj1... |
| bnj1500 32948 | Well-founded recursion, pa... |
| bnj1525 32949 | Technical lemma for ~ bnj1... |
| bnj1529 32950 | Technical lemma for ~ bnj1... |
| bnj1523 32951 | Technical lemma for ~ bnj1... |
| bnj1522 32952 | Well-founded recursion, pa... |
| exdifsn 32953 | There exists an element in... |
| srcmpltd 32954 | If a statement is true for... |
| prsrcmpltd 32955 | If a statement is true for... |
| dff15 32956 | A one-to-one function in t... |
| f1resveqaeq 32957 | If a function restricted t... |
| f1resrcmplf1dlem 32958 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 32959 | If a function's restrictio... |
| funen1cnv 32960 | If a function is equinumer... |
| fnrelpredd 32961 | A function that preserves ... |
| cardpred 32962 | The cardinality function p... |
| nummin 32963 | Every nonempty class of nu... |
| fineqvrep 32964 | If the Axiom of Infinity i... |
| fineqvpow 32965 | If the Axiom of Infinity i... |
| fineqvac 32966 | If the Axiom of Infinity i... |
| fineqvacALT 32967 | Shorter proof of ~ fineqva... |
| zltp1ne 32968 | Integer ordering relation.... |
| nnltp1ne 32969 | Positive integer ordering ... |
| nn0ltp1ne 32970 | Nonnegative integer orderi... |
| 0nn0m1nnn0 32971 | A number is zero if and on... |
| f1resfz0f1d 32972 | If a function with a seque... |
| fisshasheq 32973 | A finite set is equal to i... |
| hashfundm 32974 | The size of a set function... |
| hashf1dmrn 32975 | The size of the domain of ... |
| hashf1dmcdm 32976 | The size of the domain of ... |
| revpfxsfxrev 32977 | The reverse of a prefix of... |
| swrdrevpfx 32978 | A subword expressed in ter... |
| lfuhgr 32979 | A hypergraph is loop-free ... |
| lfuhgr2 32980 | A hypergraph is loop-free ... |
| lfuhgr3 32981 | A hypergraph is loop-free ... |
| cplgredgex 32982 | Any two (distinct) vertice... |
| cusgredgex 32983 | Any two (distinct) vertice... |
| cusgredgex2 32984 | Any two distinct vertices ... |
| pfxwlk 32985 | A prefix of a walk is a wa... |
| revwlk 32986 | The reverse of a walk is a... |
| revwlkb 32987 | Two words represent a walk... |
| swrdwlk 32988 | Two matching subwords of a... |
| pthhashvtx 32989 | A graph containing a path ... |
| pthisspthorcycl 32990 | A path is either a simple ... |
| spthcycl 32991 | A walk is a trivial path i... |
| usgrgt2cycl 32992 | A non-trivial cycle in a s... |
| usgrcyclgt2v 32993 | A simple graph with a non-... |
| subgrwlk 32994 | If a walk exists in a subg... |
| subgrtrl 32995 | If a trail exists in a sub... |
| subgrpth 32996 | If a path exists in a subg... |
| subgrcycl 32997 | If a cycle exists in a sub... |
| cusgr3cyclex 32998 | Every complete simple grap... |
| loop1cycl 32999 | A hypergraph has a cycle o... |
| 2cycld 33000 | Construction of a 2-cycle ... |
| 2cycl2d 33001 | Construction of a 2-cycle ... |
| umgr2cycllem 33002 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 33003 | A multigraph with two dist... |
| dfacycgr1 33006 | An alternate definition of... |
| isacycgr 33007 | The property of being an a... |
| isacycgr1 33008 | The property of being an a... |
| acycgrcycl 33009 | Any cycle in an acyclic gr... |
| acycgr0v 33010 | A null graph (with no vert... |
| acycgr1v 33011 | A multigraph with one vert... |
| acycgr2v 33012 | A simple graph with two ve... |
| prclisacycgr 33013 | A proper class (representi... |
| acycgrislfgr 33014 | An acyclic hypergraph is a... |
| upgracycumgr 33015 | An acyclic pseudograph is ... |
| umgracycusgr 33016 | An acyclic multigraph is a... |
| upgracycusgr 33017 | An acyclic pseudograph is ... |
| cusgracyclt3v 33018 | A complete simple graph is... |
| pthacycspth 33019 | A path in an acyclic graph... |
| acycgrsubgr 33020 | The subgraph of an acyclic... |
| quartfull 33027 | The quartic equation, writ... |
| deranglem 33028 | Lemma for derangements. (... |
| derangval 33029 | Define the derangement fun... |
| derangf 33030 | The derangement number is ... |
| derang0 33031 | The derangement number of ... |
| derangsn 33032 | The derangement number of ... |
| derangenlem 33033 | One half of ~ derangen . ... |
| derangen 33034 | The derangement number is ... |
| subfacval 33035 | The subfactorial is define... |
| derangen2 33036 | Write the derangement numb... |
| subfacf 33037 | The subfactorial is a func... |
| subfaclefac 33038 | The subfactorial is less t... |
| subfac0 33039 | The subfactorial at zero. ... |
| subfac1 33040 | The subfactorial at one. ... |
| subfacp1lem1 33041 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 33042 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 33043 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 33044 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 33045 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 33046 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 33047 | Lemma for ~ subfacp1 . By... |
| subfacp1 33048 | A two-term recurrence for ... |
| subfacval2 33049 | A closed-form expression f... |
| subfaclim 33050 | The subfactorial converges... |
| subfacval3 33051 | Another closed form expres... |
| derangfmla 33052 | The derangements formula, ... |
| erdszelem1 33053 | Lemma for ~ erdsze . (Con... |
| erdszelem2 33054 | Lemma for ~ erdsze . (Con... |
| erdszelem3 33055 | Lemma for ~ erdsze . (Con... |
| erdszelem4 33056 | Lemma for ~ erdsze . (Con... |
| erdszelem5 33057 | Lemma for ~ erdsze . (Con... |
| erdszelem6 33058 | Lemma for ~ erdsze . (Con... |
| erdszelem7 33059 | Lemma for ~ erdsze . (Con... |
| erdszelem8 33060 | Lemma for ~ erdsze . (Con... |
| erdszelem9 33061 | Lemma for ~ erdsze . (Con... |
| erdszelem10 33062 | Lemma for ~ erdsze . (Con... |
| erdszelem11 33063 | Lemma for ~ erdsze . (Con... |
| erdsze 33064 | The Erdős-Szekeres th... |
| erdsze2lem1 33065 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 33066 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 33067 | Generalize the statement o... |
| kur14lem1 33068 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 33069 | Lemma for ~ kur14 . Write... |
| kur14lem3 33070 | Lemma for ~ kur14 . A clo... |
| kur14lem4 33071 | Lemma for ~ kur14 . Compl... |
| kur14lem5 33072 | Lemma for ~ kur14 . Closu... |
| kur14lem6 33073 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 33074 | Lemma for ~ kur14 : main p... |
| kur14lem8 33075 | Lemma for ~ kur14 . Show ... |
| kur14lem9 33076 | Lemma for ~ kur14 . Since... |
| kur14lem10 33077 | Lemma for ~ kur14 . Disch... |
| kur14 33078 | Kuratowski's closure-compl... |
| ispconn 33085 | The property of being a pa... |
| pconncn 33086 | The property of being a pa... |
| pconntop 33087 | A simply connected space i... |
| issconn 33088 | The property of being a si... |
| sconnpconn 33089 | A simply connected space i... |
| sconntop 33090 | A simply connected space i... |
| sconnpht 33091 | A closed path in a simply ... |
| cnpconn 33092 | An image of a path-connect... |
| pconnconn 33093 | A path-connected space is ... |
| txpconn 33094 | The topological product of... |
| ptpconn 33095 | The topological product of... |
| indispconn 33096 | The indiscrete topology (o... |
| connpconn 33097 | A connected and locally pa... |
| qtoppconn 33098 | A quotient of a path-conne... |
| pconnpi1 33099 | All fundamental groups in ... |
| sconnpht2 33100 | Any two paths in a simply ... |
| sconnpi1 33101 | A path-connected topologic... |
| txsconnlem 33102 | Lemma for ~ txsconn . (Co... |
| txsconn 33103 | The topological product of... |
| cvxpconn 33104 | A convex subset of the com... |
| cvxsconn 33105 | A convex subset of the com... |
| blsconn 33106 | An open ball in the comple... |
| cnllysconn 33107 | The topology of the comple... |
| resconn 33108 | A subset of ` RR ` is simp... |
| ioosconn 33109 | An open interval is simply... |
| iccsconn 33110 | A closed interval is simpl... |
| retopsconn 33111 | The real numbers are simpl... |
| iccllysconn 33112 | A closed interval is local... |
| rellysconn 33113 | The real numbers are local... |
| iisconn 33114 | The unit interval is simpl... |
| iillysconn 33115 | The unit interval is local... |
| iinllyconn 33116 | The unit interval is local... |
| fncvm 33119 | Lemma for covering maps. ... |
| cvmscbv 33120 | Change bound variables in ... |
| iscvm 33121 | The property of being a co... |
| cvmtop1 33122 | Reverse closure for a cove... |
| cvmtop2 33123 | Reverse closure for a cove... |
| cvmcn 33124 | A covering map is a contin... |
| cvmcov 33125 | Property of a covering map... |
| cvmsrcl 33126 | Reverse closure for an eve... |
| cvmsi 33127 | One direction of ~ cvmsval... |
| cvmsval 33128 | Elementhood in the set ` S... |
| cvmsss 33129 | An even covering is a subs... |
| cvmsn0 33130 | An even covering is nonemp... |
| cvmsuni 33131 | An even covering of ` U ` ... |
| cvmsdisj 33132 | An even covering of ` U ` ... |
| cvmshmeo 33133 | Every element of an even c... |
| cvmsf1o 33134 | ` F ` , localized to an el... |
| cvmscld 33135 | The sets of an even coveri... |
| cvmsss2 33136 | An open subset of an evenl... |
| cvmcov2 33137 | The covering map property ... |
| cvmseu 33138 | Every element in ` U. T ` ... |
| cvmsiota 33139 | Identify the unique elemen... |
| cvmopnlem 33140 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 33141 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 33142 | A covering map is an open ... |
| cvmliftmolem1 33143 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 33144 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 33145 | A lift of a continuous fun... |
| cvmliftmo 33146 | A lift of a continuous fun... |
| cvmliftlem1 33147 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 33148 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 33149 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 33150 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 33151 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 33152 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 33153 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 33154 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 33155 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 33156 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 33157 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 33158 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 33159 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 33160 | Lemma for ~ cvmlift . Dis... |
| cvmlift 33161 | One of the important prope... |
| cvmfo 33162 | A covering map is an onto ... |
| cvmliftiota 33163 | Write out a function ` H `... |
| cvmlift2lem1 33164 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 33165 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 33166 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 33167 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 33168 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 33169 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 33170 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 33171 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 33172 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 33173 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 33174 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 33175 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 33176 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 33177 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 33178 | A two-dimensional version ... |
| cvmliftphtlem 33179 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 33180 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 33181 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 33182 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 33183 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 33184 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 33185 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 33186 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 33187 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 33188 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 33189 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 33190 | A general version of ~ cvm... |
| snmlff 33191 | The function ` F ` from ~ ... |
| snmlfval 33192 | The function ` F ` from ~ ... |
| snmlval 33193 | The property " ` A ` is si... |
| snmlflim 33194 | If ` A ` is simply normal,... |
| goel 33209 | A "Godel-set of membership... |
| goelel3xp 33210 | A "Godel-set of membership... |
| goeleq12bg 33211 | Two "Godel-set of membersh... |
| gonafv 33212 | The "Godel-set for the She... |
| goaleq12d 33213 | Equality of the "Godel-set... |
| gonanegoal 33214 | The Godel-set for the Shef... |
| satf 33215 | The satisfaction predicate... |
| satfsucom 33216 | The satisfaction predicate... |
| satfn 33217 | The satisfaction predicate... |
| satom 33218 | The satisfaction predicate... |
| satfvsucom 33219 | The satisfaction predicate... |
| satfv0 33220 | The value of the satisfact... |
| satfvsuclem1 33221 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 33222 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 33223 | The value of the satisfact... |
| satfv1lem 33224 | Lemma for ~ satfv1 . (Con... |
| satfv1 33225 | The value of the satisfact... |
| satfsschain 33226 | The binary relation of a s... |
| satfvsucsuc 33227 | The satisfaction predicate... |
| satfbrsuc 33228 | The binary relation of a s... |
| satfrel 33229 | The value of the satisfact... |
| satfdmlem 33230 | Lemma for ~ satfdm . (Con... |
| satfdm 33231 | The domain of the satisfac... |
| satfrnmapom 33232 | The range of the satisfact... |
| satfv0fun 33233 | The value of the satisfact... |
| satf0 33234 | The satisfaction predicate... |
| satf0sucom 33235 | The satisfaction predicate... |
| satf00 33236 | The value of the satisfact... |
| satf0suclem 33237 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 33238 | The value of the satisfact... |
| satf0op 33239 | An element of a value of t... |
| satf0n0 33240 | The value of the satisfact... |
| sat1el2xp 33241 | The first component of an ... |
| fmlafv 33242 | The valid Godel formulas o... |
| fmla 33243 | The set of all valid Godel... |
| fmla0 33244 | The valid Godel formulas o... |
| fmla0xp 33245 | The valid Godel formulas o... |
| fmlasuc0 33246 | The valid Godel formulas o... |
| fmlafvel 33247 | A class is a valid Godel f... |
| fmlasuc 33248 | The valid Godel formulas o... |
| fmla1 33249 | The valid Godel formulas o... |
| isfmlasuc 33250 | The characterization of a ... |
| fmlasssuc 33251 | The Godel formulas of heig... |
| fmlaomn0 33252 | The empty set is not a God... |
| fmlan0 33253 | The empty set is not a God... |
| gonan0 33254 | The "Godel-set of NAND" is... |
| goaln0 33255 | The "Godel-set of universa... |
| gonarlem 33256 | Lemma for ~ gonar (inducti... |
| gonar 33257 | If the "Godel-set of NAND"... |
| goalrlem 33258 | Lemma for ~ goalr (inducti... |
| goalr 33259 | If the "Godel-set of unive... |
| fmla0disjsuc 33260 | The set of valid Godel for... |
| fmlasucdisj 33261 | The valid Godel formulas o... |
| satfdmfmla 33262 | The domain of the satisfac... |
| satffunlem 33263 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 33264 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 33265 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 33266 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 33267 | The domain of an ordered p... |
| satffunlem2lem2 33268 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 33269 | Lemma 1 for ~ satffun : in... |
| satffunlem2 33270 | Lemma 2 for ~ satffun : in... |
| satffun 33271 | The value of the satisfact... |
| satff 33272 | The satisfaction predicate... |
| satfun 33273 | The satisfaction predicate... |
| satfvel 33274 | An element of the value of... |
| satfv0fvfmla0 33275 | The value of the satisfact... |
| satefv 33276 | The simplified satisfactio... |
| sate0 33277 | The simplified satisfactio... |
| satef 33278 | The simplified satisfactio... |
| sate0fv0 33279 | A simplified satisfaction ... |
| satefvfmla0 33280 | The simplified satisfactio... |
| sategoelfvb 33281 | Characterization of a valu... |
| sategoelfv 33282 | Condition of a valuation `... |
| ex-sategoelel 33283 | Example of a valuation of ... |
| ex-sategoel 33284 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 33285 | The value of the satisfact... |
| 2goelgoanfmla1 33286 | Two Godel-sets of membersh... |
| satefvfmla1 33287 | The simplified satisfactio... |
| ex-sategoelelomsuc 33288 | Example of a valuation of ... |
| ex-sategoelel12 33289 | Example of a valuation of ... |
| prv 33290 | The "proves" relation on a... |
| elnanelprv 33291 | The wff ` ( A e. B -/\ B e... |
| prv0 33292 | Every wff encoded as ` U `... |
| prv1n 33293 | No wff encoded as a Godel-... |
| mvtval 33362 | The set of variable typeco... |
| mrexval 33363 | The set of "raw expression... |
| mexval 33364 | The set of expressions, wh... |
| mexval2 33365 | The set of expressions, wh... |
| mdvval 33366 | The set of disjoint variab... |
| mvrsval 33367 | The set of variables in an... |
| mvrsfpw 33368 | The set of variables in an... |
| mrsubffval 33369 | The substitution of some v... |
| mrsubfval 33370 | The substitution of some v... |
| mrsubval 33371 | The substitution of some v... |
| mrsubcv 33372 | The value of a substituted... |
| mrsubvr 33373 | The value of a substituted... |
| mrsubff 33374 | A substitution is a functi... |
| mrsubrn 33375 | Although it is defined for... |
| mrsubff1 33376 | When restricted to complet... |
| mrsubff1o 33377 | When restricted to complet... |
| mrsub0 33378 | The value of the substitut... |
| mrsubf 33379 | A substitution is a functi... |
| mrsubccat 33380 | Substitution distributes o... |
| mrsubcn 33381 | A substitution does not ch... |
| elmrsubrn 33382 | Characterization of the su... |
| mrsubco 33383 | The composition of two sub... |
| mrsubvrs 33384 | The set of variables in a ... |
| msubffval 33385 | A substitution applied to ... |
| msubfval 33386 | A substitution applied to ... |
| msubval 33387 | A substitution applied to ... |
| msubrsub 33388 | A substitution applied to ... |
| msubty 33389 | The type of a substituted ... |
| elmsubrn 33390 | Characterization of substi... |
| msubrn 33391 | Although it is defined for... |
| msubff 33392 | A substitution is a functi... |
| msubco 33393 | The composition of two sub... |
| msubf 33394 | A substitution is a functi... |
| mvhfval 33395 | Value of the function mapp... |
| mvhval 33396 | Value of the function mapp... |
| mpstval 33397 | A pre-statement is an orde... |
| elmpst 33398 | Property of being a pre-st... |
| msrfval 33399 | Value of the reduct of a p... |
| msrval 33400 | Value of the reduct of a p... |
| mpstssv 33401 | A pre-statement is an orde... |
| mpst123 33402 | Decompose a pre-statement ... |
| mpstrcl 33403 | The elements of a pre-stat... |
| msrf 33404 | The reduct of a pre-statem... |
| msrrcl 33405 | If ` X ` and ` Y ` have th... |
| mstaval 33406 | Value of the set of statem... |
| msrid 33407 | The reduct of a statement ... |
| msrfo 33408 | The reduct of a pre-statem... |
| mstapst 33409 | A statement is a pre-state... |
| elmsta 33410 | Property of being a statem... |
| ismfs 33411 | A formal system is a tuple... |
| mfsdisj 33412 | The constants and variable... |
| mtyf2 33413 | The type function maps var... |
| mtyf 33414 | The type function maps var... |
| mvtss 33415 | The set of variable typeco... |
| maxsta 33416 | An axiom is a statement. ... |
| mvtinf 33417 | Each variable typecode has... |
| msubff1 33418 | When restricted to complet... |
| msubff1o 33419 | When restricted to complet... |
| mvhf 33420 | The function mapping varia... |
| mvhf1 33421 | The function mapping varia... |
| msubvrs 33422 | The set of variables in a ... |
| mclsrcl 33423 | Reverse closure for the cl... |
| mclsssvlem 33424 | Lemma for ~ mclsssv . (Co... |
| mclsval 33425 | The function mapping varia... |
| mclsssv 33426 | The closure of a set of ex... |
| ssmclslem 33427 | Lemma for ~ ssmcls . (Con... |
| vhmcls 33428 | All variable hypotheses ar... |
| ssmcls 33429 | The original expressions a... |
| ss2mcls 33430 | The closure is monotonic u... |
| mclsax 33431 | The closure is closed unde... |
| mclsind 33432 | Induction theorem for clos... |
| mppspstlem 33433 | Lemma for ~ mppspst . (Co... |
| mppsval 33434 | Definition of a provable p... |
| elmpps 33435 | Definition of a provable p... |
| mppspst 33436 | A provable pre-statement i... |
| mthmval 33437 | A theorem is a pre-stateme... |
| elmthm 33438 | A theorem is a pre-stateme... |
| mthmi 33439 | A statement whose reduct i... |
| mthmsta 33440 | A theorem is a pre-stateme... |
| mppsthm 33441 | A provable pre-statement i... |
| mthmblem 33442 | Lemma for ~ mthmb . (Cont... |
| mthmb 33443 | If two statements have the... |
| mthmpps 33444 | Given a theorem, there is ... |
| mclsppslem 33445 | The closure is closed unde... |
| mclspps 33446 | The closure is closed unde... |
| problem1 33523 | Practice problem 1. Clues... |
| problem2 33524 | Practice problem 2. Clues... |
| problem3 33525 | Practice problem 3. Clues... |
| problem4 33526 | Practice problem 4. Clues... |
| problem5 33527 | Practice problem 5. Clues... |
| quad3 33528 | Variant of quadratic equat... |
| climuzcnv 33529 | Utility lemma to convert b... |
| sinccvglem 33530 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 33531 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 33532 | The circumference of a cir... |
| elfzm12 33533 | Membership in a curtailed ... |
| nn0seqcvg 33534 | A strictly-decreasing nonn... |
| lediv2aALT 33535 | Division of both sides of ... |
| abs2sqlei 33536 | The absolute values of two... |
| abs2sqlti 33537 | The absolute values of two... |
| abs2sqle 33538 | The absolute values of two... |
| abs2sqlt 33539 | The absolute values of two... |
| abs2difi 33540 | Difference of absolute val... |
| abs2difabsi 33541 | Absolute value of differen... |
| axextprim 33542 | ~ ax-ext without distinct ... |
| axrepprim 33543 | ~ ax-rep without distinct ... |
| axunprim 33544 | ~ ax-un without distinct v... |
| axpowprim 33545 | ~ ax-pow without distinct ... |
| axregprim 33546 | ~ ax-reg without distinct ... |
| axinfprim 33547 | ~ ax-inf without distinct ... |
| axacprim 33548 | ~ ax-ac without distinct v... |
| untelirr 33549 | We call a class "untanged"... |
| untuni 33550 | The union of a class is un... |
| untsucf 33551 | If a class is untangled, t... |
| unt0 33552 | The null set is untangled.... |
| untint 33553 | If there is an untangled e... |
| efrunt 33554 | If ` A ` is well-founded b... |
| untangtr 33555 | A transitive class is unta... |
| 3orel2 33556 | Partial elimination of a t... |
| 3orel3 33557 | Partial elimination of a t... |
| 3pm3.2ni 33558 | Triple negated disjunction... |
| 3jaodd 33559 | Double deduction form of ~... |
| 3orit 33560 | Closed form of ~ 3ori . (... |
| biimpexp 33561 | A biconditional in the ant... |
| 3orel13 33562 | Elimination of two disjunc... |
| onelssex 33563 | Ordinal less than is equiv... |
| nepss 33564 | Two classes are unequal if... |
| 3ccased 33565 | Triple disjunction form of... |
| dfso3 33566 | Expansion of the definitio... |
| brtpid1 33567 | A binary relation involvin... |
| brtpid2 33568 | A binary relation involvin... |
| brtpid3 33569 | A binary relation involvin... |
| ceqsrexv2 33570 | Alternate elimitation of a... |
| iota5f 33571 | A method for computing iot... |
| ceqsralv2 33572 | Alternate elimination of a... |
| dford5 33573 | A class is ordinal iff it ... |
| jath 33574 | Closed form of ~ ja . Pro... |
| riotarab 33575 | Restricted iota of a restr... |
| reurab 33576 | Restricted existential uni... |
| snres0 33577 | Condition for restriction ... |
| fnssintima 33578 | Condition for subset of an... |
| xpab 33579 | Cross product of two class... |
| dfse3 33580 | Alternate definition of se... |
| ralxpes 33581 | A version of ~ ralxp with ... |
| ot2elxp 33582 | Ordered triple membership ... |
| ot21std 33583 | Extract the first member o... |
| ot22ndd 33584 | Extract the second member ... |
| otthne 33585 | Contrapositive of the orde... |
| elxpxp 33586 | Membership in a triple cro... |
| elxpxpss 33587 | Version of ~ elrel for tri... |
| ralxp3f 33588 | Restricted for all over a ... |
| ralxp3 33589 | Restricted for-all over a ... |
| sbcoteq1a 33590 | Equality theorem for subst... |
| ralxp3es 33591 | Restricted for-all over a ... |
| onunel 33592 | The union of two ordinals ... |
| imaeqsexv 33593 | Substitute a function valu... |
| imaeqsalv 33594 | Substitute a function valu... |
| nnuni 33595 | The union of a finite ordi... |
| nnasmo 33596 | Finite ordinal subtraction... |
| eldifsucnn 33597 | Condition for membership i... |
| rdg0n 33598 | If ` A ` is a proper class... |
| sqdivzi 33599 | Distribution of square ove... |
| supfz 33600 | The supremum of a finite s... |
| inffz 33601 | The infimum of a finite se... |
| fz0n 33602 | The sequence ` ( 0 ... ( N... |
| shftvalg 33603 | Value of a sequence shifte... |
| divcnvlin 33604 | Limit of the ratio of two ... |
| climlec3 33605 | Comparison of a constant t... |
| logi 33606 | Calculate the logarithm of... |
| iexpire 33607 | ` _i ` raised to itself is... |
| bcneg1 33608 | The binomial coefficent ov... |
| bcm1nt 33609 | The proportion of one bion... |
| bcprod 33610 | A product identity for bin... |
| bccolsum 33611 | A column-sum rule for bino... |
| iprodefisumlem 33612 | Lemma for ~ iprodefisum . ... |
| iprodefisum 33613 | Applying the exponential f... |
| iprodgam 33614 | An infinite product versio... |
| faclimlem1 33615 | Lemma for ~ faclim . Clos... |
| faclimlem2 33616 | Lemma for ~ faclim . Show... |
| faclimlem3 33617 | Lemma for ~ faclim . Alge... |
| faclim 33618 | An infinite product expres... |
| iprodfac 33619 | An infinite product expres... |
| faclim2 33620 | Another factorial limit du... |
| gcd32 33621 | Swap the second and third ... |
| gcdabsorb 33622 | Absorption law for gcd. (... |
| brtp 33623 | A condition for a binary r... |
| dftr6 33624 | A potential definition of ... |
| coep 33625 | Composition with the membe... |
| coepr 33626 | Composition with the conve... |
| dffr5 33627 | A quantifier-free definiti... |
| dfso2 33628 | Quantifier-free definition... |
| br8 33629 | Substitution for an eight-... |
| br6 33630 | Substitution for a six-pla... |
| br4 33631 | Substitution for a four-pl... |
| cnvco1 33632 | Another distributive law o... |
| cnvco2 33633 | Another distributive law o... |
| eldm3 33634 | Quantifier-free definition... |
| elrn3 33635 | Quantifier-free definition... |
| pocnv 33636 | The converse of a partial ... |
| socnv 33637 | The converse of a strict o... |
| sotrd 33638 | Transitivity law for stric... |
| sotr3 33639 | Transitivity law for stric... |
| sotrine 33640 | Trichotomy law for strict ... |
| eqfunresadj 33641 | Law for adjoining an eleme... |
| eqfunressuc 33642 | Law for equality of restri... |
| funeldmb 33643 | If ` (/) ` is not part of ... |
| elintfv 33644 | Membership in an intersect... |
| funpsstri 33645 | A condition for subset tri... |
| fundmpss 33646 | If a class ` F ` is a prop... |
| fvresval 33647 | The value of a function at... |
| funsseq 33648 | Given two functions with e... |
| fununiq 33649 | The uniqueness condition o... |
| funbreq 33650 | An equality condition for ... |
| br1steq 33651 | Uniqueness condition for t... |
| br2ndeq 33652 | Uniqueness condition for t... |
| dfdm5 33653 | Definition of domain in te... |
| dfrn5 33654 | Definition of range in ter... |
| opelco3 33655 | Alternate way of saying th... |
| elima4 33656 | Quantifier-free expression... |
| fv1stcnv 33657 | The value of the converse ... |
| fv2ndcnv 33658 | The value of the converse ... |
| imaindm 33659 | The image is unaffected by... |
| setinds 33660 | Principle of set induction... |
| setinds2f 33661 | ` _E ` induction schema, u... |
| setinds2 33662 | ` _E ` induction schema, u... |
| elpotr 33663 | A class of transitive sets... |
| dford5reg 33664 | Given ~ ax-reg , an ordina... |
| dfon2lem1 33665 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 33666 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 33667 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 33668 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 33669 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 33670 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 33671 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 33672 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 33673 | Lemma for ~ dfon2 . A cla... |
| dfon2 33674 | ` On ` consists of all set... |
| rdgprc0 33675 | The value of the recursive... |
| rdgprc 33676 | The value of the recursive... |
| dfrdg2 33677 | Alternate definition of th... |
| dfrdg3 33678 | Generalization of ~ dfrdg2... |
| axextdfeq 33679 | A version of ~ ax-ext for ... |
| ax8dfeq 33680 | A version of ~ ax-8 for us... |
| axextdist 33681 | ~ ax-ext with distinctors ... |
| axextbdist 33682 | ~ axextb with distinctors ... |
| 19.12b 33683 | Version of ~ 19.12vv with ... |
| exnel 33684 | There is always a set not ... |
| distel 33685 | Distinctors in terms of me... |
| axextndbi 33686 | ~ axextnd as a bicondition... |
| hbntg 33687 | A more general form of ~ h... |
| hbimtg 33688 | A more general and closed ... |
| hbaltg 33689 | A more general and closed ... |
| hbng 33690 | A more general form of ~ h... |
| hbimg 33691 | A more general form of ~ h... |
| tfisg 33692 | A closed form of ~ tfis . ... |
| ttrcleq 33695 | Equality theorem for trans... |
| nfttrcld 33696 | Bound variable hypothesis ... |
| nfttrcl 33697 | Bound variable hypothesis ... |
| relttrcl 33698 | The transitive closure of ... |
| brttrcl 33699 | Characterization of elemen... |
| brttrcl2 33700 | Characterization of elemen... |
| ssttrcl 33701 | If ` R ` is a relation, th... |
| ttrcltr 33702 | The transitive closure of ... |
| ttrclresv 33703 | The transitive closure of ... |
| ttrclco 33704 | Composition law for the tr... |
| cottrcl 33705 | Composition law for the tr... |
| ttrclss 33706 | If ` R ` is a subclass of ... |
| dmttrcl 33707 | The domain of a transitive... |
| rnttrcl 33708 | The range of a transitive ... |
| ttrclexg 33709 | If ` R ` is a set, then so... |
| dfttrcl2 33710 | When ` R ` is a set and a ... |
| ttrclselem1 33711 | Lemma for ~ ttrclse . Sho... |
| ttrclselem2 33712 | Lemma for ~ ttrclse . Sho... |
| ttrclse 33713 | If ` R ` is set-like over ... |
| frpoins3xpg 33714 | Special case of founded pa... |
| frpoins3xp3g 33715 | Special case of founded pa... |
| xpord2lem 33716 | Lemma for cross product or... |
| poxp2 33717 | Another way of partially o... |
| frxp2 33718 | Another way of giving a fo... |
| xpord2pred 33719 | Calculate the predecessor ... |
| sexp2 33720 | Condition for the relation... |
| xpord2ind 33721 | Induction over the cross p... |
| xpord3lem 33722 | Lemma for triple ordering.... |
| poxp3 33723 | Triple cross product parti... |
| frxp3 33724 | Give foundedness over a tr... |
| xpord3pred 33725 | Calculate the predecsessor... |
| sexp3 33726 | Show that the triple order... |
| xpord3ind 33727 | Induction over the triple ... |
| orderseqlem 33728 | Lemma for ~ poseq and ~ so... |
| poseq 33729 | A partial ordering of sequ... |
| soseq 33730 | A linear ordering of seque... |
| wsuceq123 33735 | Equality theorem for well-... |
| wsuceq1 33736 | Equality theorem for well-... |
| wsuceq2 33737 | Equality theorem for well-... |
| wsuceq3 33738 | Equality theorem for well-... |
| nfwsuc 33739 | Bound-variable hypothesis ... |
| wlimeq12 33740 | Equality theorem for the l... |
| wlimeq1 33741 | Equality theorem for the l... |
| wlimeq2 33742 | Equality theorem for the l... |
| nfwlim 33743 | Bound-variable hypothesis ... |
| elwlim 33744 | Membership in the limit cl... |
| wzel 33745 | The zero of a well-founded... |
| wsuclem 33746 | Lemma for the supremum pro... |
| wsucex 33747 | Existence theorem for well... |
| wsuccl 33748 | If ` X ` is a set with an ... |
| wsuclb 33749 | A well-founded successor i... |
| wlimss 33750 | The class of limit points ... |
| on2recsfn 33753 | Show that double recursion... |
| on2recsov 33754 | Calculate the value of the... |
| on2ind 33755 | Double induction over ordi... |
| on3ind 33756 | Triple induction over ordi... |
| naddfn 33757 | Natural addition is a func... |
| naddcllem 33758 | Lemma for ordinal addition... |
| naddcl 33759 | Closure law for natural ad... |
| naddov 33760 | The value of natural addit... |
| naddov2 33761 | Alternate expression for n... |
| naddcom 33762 | Natural addition commutes.... |
| naddid1 33763 | Ordinal zero is the additi... |
| naddssim 33764 | Ordinal less-than-or-equal... |
| naddelim 33765 | Ordinal less-than is prese... |
| naddel1 33766 | Ordinal less-than is not a... |
| naddel2 33767 | Ordinal less-than is not a... |
| naddss1 33768 | Ordinal less-than-or-equal... |
| naddss2 33769 | Ordinal less-than-or-equal... |
| elno 33776 | Membership in the surreals... |
| sltval 33777 | The value of the surreal l... |
| bdayval 33778 | The value of the birthday ... |
| nofun 33779 | A surreal is a function. ... |
| nodmon 33780 | The domain of a surreal is... |
| norn 33781 | The range of a surreal is ... |
| nofnbday 33782 | A surreal is a function ov... |
| nodmord 33783 | The domain of a surreal ha... |
| elno2 33784 | An alternative condition f... |
| elno3 33785 | Another condition for memb... |
| sltval2 33786 | Alternate expression for s... |
| nofv 33787 | The function value of a su... |
| nosgnn0 33788 | ` (/) ` is not a surreal s... |
| nosgnn0i 33789 | If ` X ` is a surreal sign... |
| noreson 33790 | The restriction of a surre... |
| sltintdifex 33791 |
If ` A |
| sltres 33792 | If the restrictions of two... |
| noxp1o 33793 | The Cartesian product of a... |
| noseponlem 33794 | Lemma for ~ nosepon . Con... |
| nosepon 33795 | Given two unequal surreals... |
| noextend 33796 | Extending a surreal by one... |
| noextendseq 33797 | Extend a surreal by a sequ... |
| noextenddif 33798 | Calculate the place where ... |
| noextendlt 33799 | Extending a surreal with a... |
| noextendgt 33800 | Extending a surreal with a... |
| nolesgn2o 33801 | Given ` A ` less than or e... |
| nolesgn2ores 33802 | Given ` A ` less than or e... |
| nogesgn1o 33803 | Given ` A ` greater than o... |
| nogesgn1ores 33804 | Given ` A ` greater than o... |
| sltsolem1 33805 | Lemma for ~ sltso . The s... |
| sltso 33806 | Surreal less than totally ... |
| bdayfo 33807 | The birthday function maps... |
| fvnobday 33808 | The value of a surreal at ... |
| nosepnelem 33809 | Lemma for ~ nosepne . (Co... |
| nosepne 33810 | The value of two non-equal... |
| nosep1o 33811 | If the value of a surreal ... |
| nosep2o 33812 | If the value of a surreal ... |
| nosepdmlem 33813 | Lemma for ~ nosepdm . (Co... |
| nosepdm 33814 | The first place two surrea... |
| nosepeq 33815 | The values of two surreals... |
| nosepssdm 33816 | Given two non-equal surrea... |
| nodenselem4 33817 | Lemma for ~ nodense . Sho... |
| nodenselem5 33818 | Lemma for ~ nodense . If ... |
| nodenselem6 33819 | The restriction of a surre... |
| nodenselem7 33820 | Lemma for ~ nodense . ` A ... |
| nodenselem8 33821 | Lemma for ~ nodense . Giv... |
| nodense 33822 | Given two distinct surreal... |
| bdayimaon 33823 | Lemma for full-eta propert... |
| nolt02olem 33824 | Lemma for ~ nolt02o . If ... |
| nolt02o 33825 | Given ` A ` less than ` B ... |
| nogt01o 33826 | Given ` A ` greater than `... |
| noresle 33827 | Restriction law for surrea... |
| nomaxmo 33828 | A class of surreals has at... |
| nominmo 33829 | A class of surreals has at... |
| nosupprefixmo 33830 | In any class of surreals, ... |
| noinfprefixmo 33831 | In any class of surreals, ... |
| nosupcbv 33832 | Lemma to change bound vari... |
| nosupno 33833 | The next several theorems ... |
| nosupdm 33834 | The domain of the surreal ... |
| nosupbday 33835 | Birthday bounding law for ... |
| nosupfv 33836 | The value of surreal supre... |
| nosupres 33837 | A restriction law for surr... |
| nosupbnd1lem1 33838 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 33839 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 33840 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 33841 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 33842 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 33843 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 33844 | Bounding law from below fo... |
| nosupbnd2lem1 33845 | Bounding law from above wh... |
| nosupbnd2 33846 | Bounding law from above fo... |
| noinfcbv 33847 | Change bound variables for... |
| noinfno 33848 | The next several theorems ... |
| noinfdm 33849 | Next, we calculate the dom... |
| noinfbday 33850 | Birthday bounding law for ... |
| noinffv 33851 | The value of surreal infim... |
| noinfres 33852 | The restriction of surreal... |
| noinfbnd1lem1 33853 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1lem2 33854 | Lemma for ~ noinfbnd1 . W... |
| noinfbnd1lem3 33855 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem4 33856 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem5 33857 | Lemma for ~ noinfbnd1 . I... |
| noinfbnd1lem6 33858 | Lemma for ~ noinfbnd1 . E... |
| noinfbnd1 33859 | Bounding law from above fo... |
| noinfbnd2lem1 33860 | Bounding law from below wh... |
| noinfbnd2 33861 | Bounding law from below fo... |
| nosupinfsep 33862 | Given two sets of surreals... |
| noetasuplem1 33863 | Lemma for ~ noeta . Estab... |
| noetasuplem2 33864 | Lemma for ~ noeta . The r... |
| noetasuplem3 33865 | Lemma for ~ noeta . ` Z ` ... |
| noetasuplem4 33866 | Lemma for ~ noeta . When ... |
| noetainflem1 33867 | Lemma for ~ noeta . Estab... |
| noetainflem2 33868 | Lemma for ~ noeta . The r... |
| noetainflem3 33869 | Lemma for ~ noeta . ` W ` ... |
| noetainflem4 33870 | Lemma for ~ noeta . If ` ... |
| noetalem1 33871 | Lemma for ~ noeta . Eithe... |
| noetalem2 33872 | Lemma for ~ noeta . The f... |
| noeta 33873 | The full-eta axiom for the... |
| sltirr 33876 | Surreal less than is irref... |
| slttr 33877 | Surreal less than is trans... |
| sltasym 33878 | Surreal less than is asymm... |
| sltlin 33879 | Surreal less than obeys tr... |
| slttrieq2 33880 | Trichotomy law for surreal... |
| slttrine 33881 | Trichotomy law for surreal... |
| slenlt 33882 | Surreal less than or equal... |
| sltnle 33883 | Surreal less than in terms... |
| sleloe 33884 | Surreal less than or equal... |
| sletri3 33885 | Trichotomy law for surreal... |
| sltletr 33886 | Surreal transitive law. (... |
| slelttr 33887 | Surreal transitive law. (... |
| sletr 33888 | Surreal transitive law. (... |
| slttrd 33889 | Surreal less than is trans... |
| sltletrd 33890 | Surreal less than is trans... |
| slelttrd 33891 | Surreal less than is trans... |
| sletrd 33892 | Surreal less than or equal... |
| slerflex 33893 | Surreal less than or equal... |
| bdayfun 33894 | The birthday function is a... |
| bdayfn 33895 | The birthday function is a... |
| bdaydm 33896 | The birthday function's do... |
| bdayrn 33897 | The birthday function's ra... |
| bdayelon 33898 | The value of the birthday ... |
| nocvxminlem 33899 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 33900 | Given a nonempty convex cl... |
| noprc 33901 | The surreal numbers are a ... |
| noeta2 33906 | A version of ~ noeta with ... |
| brsslt 33907 | Binary relation form of th... |
| ssltex1 33908 | The first argument of surr... |
| ssltex2 33909 | The second argument of sur... |
| ssltss1 33910 | The first argument of surr... |
| ssltss2 33911 | The second argument of sur... |
| ssltsep 33912 | The separation property of... |
| ssltd 33913 | Deduce surreal set less th... |
| ssltsepc 33914 | Two elements of separated ... |
| ssltsepcd 33915 | Two elements of separated ... |
| sssslt1 33916 | Relationship between surre... |
| sssslt2 33917 | Relationship between surre... |
| nulsslt 33918 | The empty set is less than... |
| nulssgt 33919 | The empty set is greater t... |
| conway 33920 | Conway's Simplicity Theore... |
| scutval 33921 | The value of the surreal c... |
| scutcut 33922 | Cut properties of the surr... |
| scutcl 33923 | Closure law for surreal cu... |
| scutcld 33924 | Closure law for surreal cu... |
| scutbday 33925 | The birthday of the surrea... |
| eqscut 33926 | Condition for equality to ... |
| eqscut2 33927 | Condition for equality to ... |
| sslttr 33928 | Transitive law for surreal... |
| ssltun1 33929 | Union law for surreal set ... |
| ssltun2 33930 | Union law for surreal set ... |
| scutun12 33931 | Union law for surreal cuts... |
| dmscut 33932 | The domain of the surreal ... |
| scutf 33933 | Functionality statement fo... |
| etasslt 33934 | A restatement of ~ noeta u... |
| etasslt2 33935 | A version of ~ etasslt wit... |
| scutbdaybnd 33936 | An upper bound on the birt... |
| scutbdaybnd2 33937 | An upper bound on the birt... |
| scutbdaybnd2lim 33938 | An upper bound on the birt... |
| scutbdaylt 33939 | If a surreal lies in a gap... |
| slerec 33940 | A comparison law for surre... |
| sltrec 33941 | A comparison law for surre... |
| ssltdisj 33942 | If ` A ` preceeds ` B ` , ... |
| 0sno 33947 | Surreal zero is a surreal.... |
| 1sno 33948 | Surreal one is a surreal. ... |
| bday0s 33949 | Calculate the birthday of ... |
| 0slt1s 33950 | Surreal zero is less than ... |
| bday0b 33951 | The only surreal with birt... |
| bday1s 33952 | The birthday of surreal on... |
| madeval 33963 | The value of the made by f... |
| madeval2 33964 | Alternative characterizati... |
| oldval 33965 | The value of the old optio... |
| newval 33966 | The value of the new optio... |
| madef 33967 | The made function is a fun... |
| oldf 33968 | The older function is a fu... |
| newf 33969 | The new function is a func... |
| old0 33970 | No surreal is older than `... |
| madessno 33971 | Made sets are surreals. (... |
| oldssno 33972 | Old sets are surreals. (C... |
| newssno 33973 | New sets are surreals. (C... |
| leftval 33974 | The value of the left opti... |
| rightval 33975 | The value of the right opt... |
| leftf 33976 | The functionality of the l... |
| rightf 33977 | The functionality of the r... |
| elmade 33978 | Membership in the made fun... |
| elmade2 33979 | Membership in the made fun... |
| elold 33980 | Membership in an old set. ... |
| ssltleft 33981 | A surreal is greater than ... |
| ssltright 33982 | A surreal is less than its... |
| lltropt 33983 | The left options of a surr... |
| made0 33984 | The only surreal made on d... |
| new0 33985 | The only surreal new on da... |
| madess 33986 | If ` A ` is less than or e... |
| oldssmade 33987 | The older-than set is a su... |
| leftssold 33988 | The left options are a sub... |
| rightssold 33989 | The right options are a su... |
| leftssno 33990 | The left set of a surreal ... |
| rightssno 33991 | The right set of a surreal... |
| madecut 33992 | Given a section that is a ... |
| madeun 33993 | The made set is the union ... |
| madeoldsuc 33994 | The made set is the old se... |
| oldsuc 33995 | The value of the old set a... |
| oldlim 33996 | The value of the old set a... |
| madebdayim 33997 | If a surreal is a member o... |
| oldbdayim 33998 | If ` X ` is in the old set... |
| oldirr 33999 | No surreal is a member of ... |
| leftirr 34000 | No surreal is a member of ... |
| rightirr 34001 | No surreal is a member of ... |
| left0s 34002 | The left set of ` 0s ` is ... |
| right0s 34003 | The right set of ` 0s ` is... |
| lrold 34004 | The union of the left and ... |
| madebdaylemold 34005 | Lemma for ~ madebday . If... |
| madebdaylemlrcut 34006 | Lemma for ~ madebday . If... |
| madebday 34007 | A surreal is part of the s... |
| oldbday 34008 | A surreal is part of the s... |
| newbday 34009 | A surreal is an element of... |
| lrcut 34010 | A surreal is equal to the ... |
| scutfo 34011 | The surreal cut function i... |
| sltn0 34012 | If ` X ` is less than ` Y ... |
| lruneq 34013 | If two surreals share a bi... |
| sltlpss 34014 | If two surreals share a bi... |
| cofsslt 34015 | If every element of ` A ` ... |
| coinitsslt 34016 | If ` B ` is coinitial with... |
| cofcut1 34017 | If ` C ` is cofinal with `... |
| cofcut2 34018 | If ` A ` and ` C ` are mut... |
| cofcutr 34019 | If ` X ` is the cut of ` A... |
| cofcutrtime 34020 | If ` X ` is the cut of ` A... |
| lrrecval 34023 | The next step in the devel... |
| lrrecval2 34024 | Next, we establish an alte... |
| lrrecpo 34025 | Now, we establish that ` R... |
| lrrecse 34026 | Next, we show that ` R ` i... |
| lrrecfr 34027 | Now we show that ` R ` is ... |
| lrrecpred 34028 | Finally, we calculate the ... |
| noinds 34029 | Induction principle for a ... |
| norecfn 34030 | Surreal recursion over one... |
| norecov 34031 | Calculate the value of the... |
| noxpordpo 34034 | To get through most of the... |
| noxpordfr 34035 | Next we establish the foun... |
| noxpordse 34036 | Next we establish the set-... |
| noxpordpred 34037 | Next we calculate the pred... |
| no2indslem 34038 | Double induction on surrea... |
| no2inds 34039 | Double induction on surrea... |
| norec2fn 34040 | The double-recursion opera... |
| norec2ov 34041 | The value of the double-re... |
| no3inds 34042 | Triple induction over surr... |
| negsfn 34049 | Surreal negation is a func... |
| negsval 34050 | The value of the surreal n... |
| negs0s 34051 | Negative surreal zero is s... |
| addsfn 34052 | Surreal addition is a func... |
| addsval 34053 | The value of surreal addit... |
| addsid1 34054 | Surreal addition to zero i... |
| addsid1d 34055 | Surreal addition to zero i... |
| addscom 34056 | Surreal addition commutes.... |
| addscomd 34057 | Surreal addition commutes.... |
| addscllem1 34058 | Lemma for addscl (future) ... |
| txpss3v 34107 | A tail Cartesian product i... |
| txprel 34108 | A tail Cartesian product i... |
| brtxp 34109 | Characterize a ternary rel... |
| brtxp2 34110 | The binary relation over a... |
| dfpprod2 34111 | Expanded definition of par... |
| pprodcnveq 34112 | A converse law for paralle... |
| pprodss4v 34113 | The parallel product is a ... |
| brpprod 34114 | Characterize a quaternary ... |
| brpprod3a 34115 | Condition for parallel pro... |
| brpprod3b 34116 | Condition for parallel pro... |
| relsset 34117 | The subset class is a bina... |
| brsset 34118 | For sets, the ` SSet ` bin... |
| idsset 34119 | ` _I ` is equal to the int... |
| eltrans 34120 | Membership in the class of... |
| dfon3 34121 | A quantifier-free definiti... |
| dfon4 34122 | Another quantifier-free de... |
| brtxpsd 34123 | Expansion of a common form... |
| brtxpsd2 34124 | Another common abbreviatio... |
| brtxpsd3 34125 | A third common abbreviatio... |
| relbigcup 34126 | The ` Bigcup ` relationshi... |
| brbigcup 34127 | Binary relation over ` Big... |
| dfbigcup2 34128 | ` Bigcup ` using maps-to n... |
| fobigcup 34129 | ` Bigcup ` maps the univer... |
| fnbigcup 34130 | ` Bigcup ` is a function o... |
| fvbigcup 34131 | For sets, ` Bigcup ` yield... |
| elfix 34132 | Membership in the fixpoint... |
| elfix2 34133 | Alternative membership in ... |
| dffix2 34134 | The fixpoints of a class i... |
| fixssdm 34135 | The fixpoints of a class a... |
| fixssrn 34136 | The fixpoints of a class a... |
| fixcnv 34137 | The fixpoints of a class a... |
| fixun 34138 | The fixpoint operator dist... |
| ellimits 34139 | Membership in the class of... |
| limitssson 34140 | The class of all limit ord... |
| dfom5b 34141 | A quantifier-free definiti... |
| sscoid 34142 | A condition for subset and... |
| dffun10 34143 | Another potential definiti... |
| elfuns 34144 | Membership in the class of... |
| elfunsg 34145 | Closed form of ~ elfuns . ... |
| brsingle 34146 | The binary relation form o... |
| elsingles 34147 | Membership in the class of... |
| fnsingle 34148 | The singleton relationship... |
| fvsingle 34149 | The value of the singleton... |
| dfsingles2 34150 | Alternate definition of th... |
| snelsingles 34151 | A singleton is a member of... |
| dfiota3 34152 | A definition of iota using... |
| dffv5 34153 | Another quantifier-free de... |
| unisnif 34154 | Express union of singleton... |
| brimage 34155 | Binary relation form of th... |
| brimageg 34156 | Closed form of ~ brimage .... |
| funimage 34157 | ` Image A ` is a function.... |
| fnimage 34158 | ` Image R ` is a function ... |
| imageval 34159 | The image functor in maps-... |
| fvimage 34160 | Value of the image functor... |
| brcart 34161 | Binary relation form of th... |
| brdomain 34162 | Binary relation form of th... |
| brrange 34163 | Binary relation form of th... |
| brdomaing 34164 | Closed form of ~ brdomain ... |
| brrangeg 34165 | Closed form of ~ brrange .... |
| brimg 34166 | Binary relation form of th... |
| brapply 34167 | Binary relation form of th... |
| brcup 34168 | Binary relation form of th... |
| brcap 34169 | Binary relation form of th... |
| brsuccf 34170 | Binary relation form of th... |
| funpartlem 34171 | Lemma for ~ funpartfun . ... |
| funpartfun 34172 | The functional part of ` F... |
| funpartss 34173 | The functional part of ` F... |
| funpartfv 34174 | The function value of the ... |
| fullfunfnv 34175 | The full functional part o... |
| fullfunfv 34176 | The function value of the ... |
| brfullfun 34177 | A binary relation form con... |
| brrestrict 34178 | Binary relation form of th... |
| dfrecs2 34179 | A quantifier-free definiti... |
| dfrdg4 34180 | A quantifier-free definiti... |
| dfint3 34181 | Quantifier-free definition... |
| imagesset 34182 | The Image functor applied ... |
| brub 34183 | Binary relation form of th... |
| brlb 34184 | Binary relation form of th... |
| altopex 34189 | Alternative ordered pairs ... |
| altopthsn 34190 | Two alternate ordered pair... |
| altopeq12 34191 | Equality for alternate ord... |
| altopeq1 34192 | Equality for alternate ord... |
| altopeq2 34193 | Equality for alternate ord... |
| altopth1 34194 | Equality of the first memb... |
| altopth2 34195 | Equality of the second mem... |
| altopthg 34196 | Alternate ordered pair the... |
| altopthbg 34197 | Alternate ordered pair the... |
| altopth 34198 | The alternate ordered pair... |
| altopthb 34199 | Alternate ordered pair the... |
| altopthc 34200 | Alternate ordered pair the... |
| altopthd 34201 | Alternate ordered pair the... |
| altxpeq1 34202 | Equality for alternate Car... |
| altxpeq2 34203 | Equality for alternate Car... |
| elaltxp 34204 | Membership in alternate Ca... |
| altopelaltxp 34205 | Alternate ordered pair mem... |
| altxpsspw 34206 | An inclusion rule for alte... |
| altxpexg 34207 | The alternate Cartesian pr... |
| rankaltopb 34208 | Compute the rank of an alt... |
| nfaltop 34209 | Bound-variable hypothesis ... |
| sbcaltop 34210 | Distribution of class subs... |
| cgrrflx2d 34213 | Deduction form of ~ axcgrr... |
| cgrtr4d 34214 | Deduction form of ~ axcgrt... |
| cgrtr4and 34215 | Deduction form of ~ axcgrt... |
| cgrrflx 34216 | Reflexivity law for congru... |
| cgrrflxd 34217 | Deduction form of ~ cgrrfl... |
| cgrcomim 34218 | Congruence commutes on the... |
| cgrcom 34219 | Congruence commutes betwee... |
| cgrcomand 34220 | Deduction form of ~ cgrcom... |
| cgrtr 34221 | Transitivity law for congr... |
| cgrtrand 34222 | Deduction form of ~ cgrtr ... |
| cgrtr3 34223 | Transitivity law for congr... |
| cgrtr3and 34224 | Deduction form of ~ cgrtr3... |
| cgrcoml 34225 | Congruence commutes on the... |
| cgrcomr 34226 | Congruence commutes on the... |
| cgrcomlr 34227 | Congruence commutes on bot... |
| cgrcomland 34228 | Deduction form of ~ cgrcom... |
| cgrcomrand 34229 | Deduction form of ~ cgrcom... |
| cgrcomlrand 34230 | Deduction form of ~ cgrcom... |
| cgrtriv 34231 | Degenerate segments are co... |
| cgrid2 34232 | Identity law for congruenc... |
| cgrdegen 34233 | Two congruent segments are... |
| brofs 34234 | Binary relation form of th... |
| 5segofs 34235 | Rephrase ~ ax5seg using th... |
| ofscom 34236 | The outer five segment pre... |
| cgrextend 34237 | Link congruence over a pai... |
| cgrextendand 34238 | Deduction form of ~ cgrext... |
| segconeq 34239 | Two points that satisfy th... |
| segconeu 34240 | Existential uniqueness ver... |
| btwntriv2 34241 | Betweenness always holds f... |
| btwncomim 34242 | Betweenness commutes. Imp... |
| btwncom 34243 | Betweenness commutes. (Co... |
| btwncomand 34244 | Deduction form of ~ btwnco... |
| btwntriv1 34245 | Betweenness always holds f... |
| btwnswapid 34246 | If you can swap the first ... |
| btwnswapid2 34247 | If you can swap arguments ... |
| btwnintr 34248 | Inner transitivity law for... |
| btwnexch3 34249 | Exchange the first endpoin... |
| btwnexch3and 34250 | Deduction form of ~ btwnex... |
| btwnouttr2 34251 | Outer transitivity law for... |
| btwnexch2 34252 | Exchange the outer point o... |
| btwnouttr 34253 | Outer transitivity law for... |
| btwnexch 34254 | Outer transitivity law for... |
| btwnexchand 34255 | Deduction form of ~ btwnex... |
| btwndiff 34256 | There is always a ` c ` di... |
| trisegint 34257 | A line segment between two... |
| funtransport 34260 | The ` TransportTo ` relati... |
| fvtransport 34261 | Calculate the value of the... |
| transportcl 34262 | Closure law for segment tr... |
| transportprops 34263 | Calculate the defining pro... |
| brifs 34272 | Binary relation form of th... |
| ifscgr 34273 | Inner five segment congrue... |
| cgrsub 34274 | Removing identical parts f... |
| brcgr3 34275 | Binary relation form of th... |
| cgr3permute3 34276 | Permutation law for three-... |
| cgr3permute1 34277 | Permutation law for three-... |
| cgr3permute2 34278 | Permutation law for three-... |
| cgr3permute4 34279 | Permutation law for three-... |
| cgr3permute5 34280 | Permutation law for three-... |
| cgr3tr4 34281 | Transitivity law for three... |
| cgr3com 34282 | Commutativity law for thre... |
| cgr3rflx 34283 | Identity law for three-pla... |
| cgrxfr 34284 | A line segment can be divi... |
| btwnxfr 34285 | A condition for extending ... |
| colinrel 34286 | Colinearity is a relations... |
| brcolinear2 34287 | Alternate colinearity bina... |
| brcolinear 34288 | The binary relation form o... |
| colinearex 34289 | The colinear predicate exi... |
| colineardim1 34290 | If ` A ` is colinear with ... |
| colinearperm1 34291 | Permutation law for coline... |
| colinearperm3 34292 | Permutation law for coline... |
| colinearperm2 34293 | Permutation law for coline... |
| colinearperm4 34294 | Permutation law for coline... |
| colinearperm5 34295 | Permutation law for coline... |
| colineartriv1 34296 | Trivial case of colinearit... |
| colineartriv2 34297 | Trivial case of colinearit... |
| btwncolinear1 34298 | Betweenness implies coline... |
| btwncolinear2 34299 | Betweenness implies coline... |
| btwncolinear3 34300 | Betweenness implies coline... |
| btwncolinear4 34301 | Betweenness implies coline... |
| btwncolinear5 34302 | Betweenness implies coline... |
| btwncolinear6 34303 | Betweenness implies coline... |
| colinearxfr 34304 | Transfer law for colineari... |
| lineext 34305 | Extend a line with a missi... |
| brofs2 34306 | Change some conditions for... |
| brifs2 34307 | Change some conditions for... |
| brfs 34308 | Binary relation form of th... |
| fscgr 34309 | Congruence law for the gen... |
| linecgr 34310 | Congruence rule for lines.... |
| linecgrand 34311 | Deduction form of ~ linecg... |
| lineid 34312 | Identity law for points on... |
| idinside 34313 | Law for finding a point in... |
| endofsegid 34314 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 34315 | Deduction form of ~ endofs... |
| btwnconn1lem1 34316 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 34317 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 34318 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 34319 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 34320 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 34321 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 34322 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 34323 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 34324 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 34325 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 34326 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 34327 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 34328 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 34329 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 34330 | Connectitivy law for betwe... |
| btwnconn2 34331 | Another connectivity law f... |
| btwnconn3 34332 | Inner connectivity law for... |
| midofsegid 34333 | If two points fall in the ... |
| segcon2 34334 | Generalization of ~ axsegc... |
| brsegle 34337 | Binary relation form of th... |
| brsegle2 34338 | Alternate characterization... |
| seglecgr12im 34339 | Substitution law for segme... |
| seglecgr12 34340 | Substitution law for segme... |
| seglerflx 34341 | Segment comparison is refl... |
| seglemin 34342 | Any segment is at least as... |
| segletr 34343 | Segment less than is trans... |
| segleantisym 34344 | Antisymmetry law for segme... |
| seglelin 34345 | Linearity law for segment ... |
| btwnsegle 34346 | If ` B ` falls between ` A... |
| colinbtwnle 34347 | Given three colinear point... |
| broutsideof 34350 | Binary relation form of ` ... |
| broutsideof2 34351 | Alternate form of ` Outsid... |
| outsidene1 34352 | Outsideness implies inequa... |
| outsidene2 34353 | Outsideness implies inequa... |
| btwnoutside 34354 | A principle linking outsid... |
| broutsideof3 34355 | Characterization of outsid... |
| outsideofrflx 34356 | Reflexivity of outsideness... |
| outsideofcom 34357 | Commutativity law for outs... |
| outsideoftr 34358 | Transitivity law for outsi... |
| outsideofeq 34359 | Uniqueness law for ` Outsi... |
| outsideofeu 34360 | Given a nondegenerate ray,... |
| outsidele 34361 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 34362 | Outside of implies colinea... |
| funray 34369 | Show that the ` Ray ` rela... |
| fvray 34370 | Calculate the value of the... |
| funline 34371 | Show that the ` Line ` rel... |
| linedegen 34372 | When ` Line ` is applied w... |
| fvline 34373 | Calculate the value of the... |
| liness 34374 | A line is a subset of the ... |
| fvline2 34375 | Alternate definition of a ... |
| lineunray 34376 | A line is composed of a po... |
| lineelsb2 34377 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 34378 | Reflexivity law for line m... |
| linecom 34379 | Commutativity law for line... |
| linerflx2 34380 | Reflexivity law for line m... |
| ellines 34381 | Membership in the set of a... |
| linethru 34382 | If ` A ` is a line contain... |
| hilbert1.1 34383 | There is a line through an... |
| hilbert1.2 34384 | There is at most one line ... |
| linethrueu 34385 | There is a unique line goi... |
| lineintmo 34386 | Two distinct lines interse... |
| fwddifval 34391 | Calculate the value of the... |
| fwddifnval 34392 | The value of the forward d... |
| fwddifn0 34393 | The value of the n-iterate... |
| fwddifnp1 34394 | The value of the n-iterate... |
| rankung 34395 | The rank of the union of t... |
| ranksng 34396 | The rank of a singleton. ... |
| rankelg 34397 | The membership relation is... |
| rankpwg 34398 | The rank of a power set. ... |
| rank0 34399 | The rank of the empty set ... |
| rankeq1o 34400 | The only set with rank ` 1... |
| elhf 34403 | Membership in the heredita... |
| elhf2 34404 | Alternate form of membersh... |
| elhf2g 34405 | Hereditarily finiteness vi... |
| 0hf 34406 | The empty set is a heredit... |
| hfun 34407 | The union of two HF sets i... |
| hfsn 34408 | The singleton of an HF set... |
| hfadj 34409 | Adjoining one HF element t... |
| hfelhf 34410 | Any member of an HF set is... |
| hftr 34411 | The class of all hereditar... |
| hfext 34412 | Extensionality for HF sets... |
| hfuni 34413 | The union of an HF set is ... |
| hfpw 34414 | The power class of an HF s... |
| hfninf 34415 | ` _om ` is not hereditaril... |
| a1i14 34416 | Add two antecedents to a w... |
| a1i24 34417 | Add two antecedents to a w... |
| exp5d 34418 | An exportation inference. ... |
| exp5g 34419 | An exportation inference. ... |
| exp5k 34420 | An exportation inference. ... |
| exp56 34421 | An exportation inference. ... |
| exp58 34422 | An exportation inference. ... |
| exp510 34423 | An exportation inference. ... |
| exp511 34424 | An exportation inference. ... |
| exp512 34425 | An exportation inference. ... |
| 3com12d 34426 | Commutation in consequent.... |
| imp5p 34427 | A triple importation infer... |
| imp5q 34428 | A triple importation infer... |
| ecase13d 34429 | Deduction for elimination ... |
| subtr 34430 | Transitivity of implicit s... |
| subtr2 34431 | Transitivity of implicit s... |
| trer 34432 | A relation intersected wit... |
| elicc3 34433 | An equivalent membership c... |
| finminlem 34434 | A useful lemma about finit... |
| gtinf 34435 | Any number greater than an... |
| opnrebl 34436 | A set is open in the stand... |
| opnrebl2 34437 | A set is open in the stand... |
| nn0prpwlem 34438 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 34439 | Two nonnegative integers a... |
| topbnd 34440 | Two equivalent expressions... |
| opnbnd 34441 | A set is open iff it is di... |
| cldbnd 34442 | A set is closed iff it con... |
| ntruni 34443 | A union of interiors is a ... |
| clsun 34444 | A pairwise union of closur... |
| clsint2 34445 | The closure of an intersec... |
| opnregcld 34446 | A set is regularly closed ... |
| cldregopn 34447 | A set if regularly open if... |
| neiin 34448 | Two neighborhoods intersec... |
| hmeoclda 34449 | Homeomorphisms preserve cl... |
| hmeocldb 34450 | Homeomorphisms preserve cl... |
| ivthALT 34451 | An alternate proof of the ... |
| fnerel 34454 | Fineness is a relation. (... |
| isfne 34455 | The predicate " ` B ` is f... |
| isfne4 34456 | The predicate " ` B ` is f... |
| isfne4b 34457 | A condition for a topology... |
| isfne2 34458 | The predicate " ` B ` is f... |
| isfne3 34459 | The predicate " ` B ` is f... |
| fnebas 34460 | A finer cover covers the s... |
| fnetg 34461 | A finer cover generates a ... |
| fnessex 34462 | If ` B ` is finer than ` A... |
| fneuni 34463 | If ` B ` is finer than ` A... |
| fneint 34464 | If a cover is finer than a... |
| fness 34465 | A cover is finer than its ... |
| fneref 34466 | Reflexivity of the finenes... |
| fnetr 34467 | Transitivity of the finene... |
| fneval 34468 | Two covers are finer than ... |
| fneer 34469 | Fineness intersected with ... |
| topfne 34470 | Fineness for covers corres... |
| topfneec 34471 | A cover is equivalent to a... |
| topfneec2 34472 | A topology is precisely id... |
| fnessref 34473 | A cover is finer iff it ha... |
| refssfne 34474 | A cover is a refinement if... |
| neibastop1 34475 | A collection of neighborho... |
| neibastop2lem 34476 | Lemma for ~ neibastop2 . ... |
| neibastop2 34477 | In the topology generated ... |
| neibastop3 34478 | The topology generated by ... |
| topmtcl 34479 | The meet of a collection o... |
| topmeet 34480 | Two equivalent formulation... |
| topjoin 34481 | Two equivalent formulation... |
| fnemeet1 34482 | The meet of a collection o... |
| fnemeet2 34483 | The meet of equivalence cl... |
| fnejoin1 34484 | Join of equivalence classe... |
| fnejoin2 34485 | Join of equivalence classe... |
| fgmin 34486 | Minimality property of a g... |
| neifg 34487 | The neighborhood filter of... |
| tailfval 34488 | The tail function for a di... |
| tailval 34489 | The tail of an element in ... |
| eltail 34490 | An element of a tail. (Co... |
| tailf 34491 | The tail function of a dir... |
| tailini 34492 | A tail contains its initia... |
| tailfb 34493 | The collection of tails of... |
| filnetlem1 34494 | Lemma for ~ filnet . Chan... |
| filnetlem2 34495 | Lemma for ~ filnet . The ... |
| filnetlem3 34496 | Lemma for ~ filnet . (Con... |
| filnetlem4 34497 | Lemma for ~ filnet . (Con... |
| filnet 34498 | A filter has the same conv... |
| tb-ax1 34499 | The first of three axioms ... |
| tb-ax2 34500 | The second of three axioms... |
| tb-ax3 34501 | The third of three axioms ... |
| tbsyl 34502 | The weak syllogism from Ta... |
| re1ax2lem 34503 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 34504 | ~ ax-2 rederived from the ... |
| naim1 34505 | Constructor theorem for ` ... |
| naim2 34506 | Constructor theorem for ` ... |
| naim1i 34507 | Constructor rule for ` -/\... |
| naim2i 34508 | Constructor rule for ` -/\... |
| naim12i 34509 | Constructor rule for ` -/\... |
| nabi1i 34510 | Constructor rule for ` -/\... |
| nabi2i 34511 | Constructor rule for ` -/\... |
| nabi12i 34512 | Constructor rule for ` -/\... |
| df3nandALT1 34515 | The double nand expressed ... |
| df3nandALT2 34516 | The double nand expressed ... |
| andnand1 34517 | Double and in terms of dou... |
| imnand2 34518 | An ` -> ` nand relation. ... |
| nalfal 34519 | Not all sets hold ` F. ` a... |
| nexntru 34520 | There does not exist a set... |
| nexfal 34521 | There does not exist a set... |
| neufal 34522 | There does not exist exact... |
| neutru 34523 | There does not exist exact... |
| nmotru 34524 | There does not exist at mo... |
| mofal 34525 | There exist at most one se... |
| nrmo 34526 | "At most one" restricted e... |
| meran1 34527 | A single axiom for proposi... |
| meran2 34528 | A single axiom for proposi... |
| meran3 34529 | A single axiom for proposi... |
| waj-ax 34530 | A single axiom for proposi... |
| lukshef-ax2 34531 | A single axiom for proposi... |
| arg-ax 34532 | A single axiom for proposi... |
| negsym1 34533 | In the paper "On Variable ... |
| imsym1 34534 | A symmetry with ` -> ` . ... |
| bisym1 34535 | A symmetry with ` <-> ` . ... |
| consym1 34536 | A symmetry with ` /\ ` . ... |
| dissym1 34537 | A symmetry with ` \/ ` . ... |
| nandsym1 34538 | A symmetry with ` -/\ ` . ... |
| unisym1 34539 | A symmetry with ` A. ` . ... |
| exisym1 34540 | A symmetry with ` E. ` . ... |
| unqsym1 34541 | A symmetry with ` E! ` . ... |
| amosym1 34542 | A symmetry with ` E* ` . ... |
| subsym1 34543 | A symmetry with ` [ x / y ... |
| ontopbas 34544 | An ordinal number is a top... |
| onsstopbas 34545 | The class of ordinal numbe... |
| onpsstopbas 34546 | The class of ordinal numbe... |
| ontgval 34547 | The topology generated fro... |
| ontgsucval 34548 | The topology generated fro... |
| onsuctop 34549 | A successor ordinal number... |
| onsuctopon 34550 | One of the topologies on a... |
| ordtoplem 34551 | Membership of the class of... |
| ordtop 34552 | An ordinal is a topology i... |
| onsucconni 34553 | A successor ordinal number... |
| onsucconn 34554 | A successor ordinal number... |
| ordtopconn 34555 | An ordinal topology is con... |
| onintopssconn 34556 | An ordinal topology is con... |
| onsuct0 34557 | A successor ordinal number... |
| ordtopt0 34558 | An ordinal topology is T_0... |
| onsucsuccmpi 34559 | The successor of a success... |
| onsucsuccmp 34560 | The successor of a success... |
| limsucncmpi 34561 | The successor of a limit o... |
| limsucncmp 34562 | The successor of a limit o... |
| ordcmp 34563 | An ordinal topology is com... |
| ssoninhaus 34564 | The ordinal topologies ` 1... |
| onint1 34565 | The ordinal T_1 spaces are... |
| oninhaus 34566 | The ordinal Hausdorff spac... |
| fveleq 34567 | Please add description her... |
| findfvcl 34568 | Please add description her... |
| findreccl 34569 | Please add description her... |
| findabrcl 34570 | Please add description her... |
| nnssi2 34571 | Convert a theorem for real... |
| nnssi3 34572 | Convert a theorem for real... |
| nndivsub 34573 | Please add description her... |
| nndivlub 34574 | A factor of a positive int... |
| ee7.2aOLD 34577 | Lemma for Euclid's Element... |
| dnival 34578 | Value of the "distance to ... |
| dnicld1 34579 | Closure theorem for the "d... |
| dnicld2 34580 | Closure theorem for the "d... |
| dnif 34581 | The "distance to nearest i... |
| dnizeq0 34582 | The distance to nearest in... |
| dnizphlfeqhlf 34583 | The distance to nearest in... |
| rddif2 34584 | Variant of ~ rddif . (Con... |
| dnibndlem1 34585 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 34586 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 34587 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 34588 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 34589 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 34590 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 34591 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 34592 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 34593 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 34594 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 34595 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 34596 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 34597 | Lemma for ~ dnibnd . (Con... |
| dnibnd 34598 | The "distance to nearest i... |
| dnicn 34599 | The "distance to nearest i... |
| knoppcnlem1 34600 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 34601 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 34602 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 34603 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 34604 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 34605 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 34606 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 34607 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 34608 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 34609 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 34610 | Lemma for ~ knoppcn . (Co... |
| knoppcn 34611 | The continuous nowhere dif... |
| knoppcld 34612 | Closure theorem for Knopp'... |
| unblimceq0lem 34613 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 34614 | If ` F ` is unbounded near... |
| unbdqndv1 34615 | If the difference quotient... |
| unbdqndv2lem1 34616 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 34617 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 34618 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 34619 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 34620 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 34621 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 34622 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 34623 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 34624 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 34625 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 34626 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 34627 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 34628 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 34629 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 34630 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 34631 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 34632 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 34633 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 34634 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 34635 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 34636 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 34637 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 34638 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 34639 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 34640 | Lemma for ~ knoppndv . (C... |
| knoppndv 34641 | The continuous nowhere dif... |
| knoppf 34642 | Knopp's function is a func... |
| knoppcn2 34643 | Variant of ~ knoppcn with ... |
| cnndvlem1 34644 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 34645 | Lemma for ~ cnndv . (Cont... |
| cnndv 34646 | There exists a continuous ... |
| bj-mp2c 34647 | A double modus ponens infe... |
| bj-mp2d 34648 | A double modus ponens infe... |
| bj-0 34649 | A syntactic theorem. See ... |
| bj-1 34650 | In this proof, the use of ... |
| bj-a1k 34651 | Weakening of ~ ax-1 . As ... |
| bj-poni 34652 | Inference associated with ... |
| bj-nnclav 34653 | When ` F. ` is substituted... |
| bj-nnclavi 34654 | Inference associated with ... |
| bj-nnclavc 34655 | Commuted form of ~ bj-nncl... |
| bj-nnclavci 34656 | Inference associated with ... |
| bj-jarrii 34657 | Inference associated with ... |
| bj-imim21 34658 | The propositional function... |
| bj-imim21i 34659 | Inference associated with ... |
| bj-peircestab 34660 | Over minimal implicational... |
| bj-stabpeirce 34661 | This minimal implicational... |
| bj-syl66ib 34662 | A mixed syllogism inferenc... |
| bj-orim2 34663 | Proof of ~ orim2 from the ... |
| bj-currypeirce 34664 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 34665 | Peirce's axiom ~ peirce im... |
| bj-animbi 34666 | Conjunction in terms of im... |
| bj-currypara 34667 | Curry's paradox. Note tha... |
| bj-con2com 34668 | A commuted form of the con... |
| bj-con2comi 34669 | Inference associated with ... |
| bj-pm2.01i 34670 | Inference associated with ... |
| bj-nimn 34671 | If a formula is true, then... |
| bj-nimni 34672 | Inference associated with ... |
| bj-peircei 34673 | Inference associated with ... |
| bj-looinvi 34674 | Inference associated with ... |
| bj-looinvii 34675 | Inference associated with ... |
| bj-mt2bi 34676 | Version of ~ mt2 where the... |
| bj-ntrufal 34677 | The negation of a theorem ... |
| bj-fal 34678 | Shortening of ~ fal using ... |
| bj-jaoi1 34679 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 34680 | Shortens ~ consensus (110>... |
| bj-dfbi4 34681 | Alternate definition of th... |
| bj-dfbi5 34682 | Alternate definition of th... |
| bj-dfbi6 34683 | Alternate definition of th... |
| bj-bijust0ALT 34684 | Alternate proof of ~ bijus... |
| bj-bijust00 34685 | A self-implication does no... |
| bj-consensus 34686 | Version of ~ consensus exp... |
| bj-consensusALT 34687 | Alternate proof of ~ bj-co... |
| bj-df-ifc 34688 | Candidate definition for t... |
| bj-dfif 34689 | Alternate definition of th... |
| bj-ififc 34690 | A biconditional connecting... |
| bj-imbi12 34691 | Uncurried (imported) form ... |
| bj-biorfi 34692 | This should be labeled "bi... |
| bj-falor 34693 | Dual of ~ truan (which has... |
| bj-falor2 34694 | Dual of ~ truan . (Contri... |
| bj-bibibi 34695 | A property of the bicondit... |
| bj-imn3ani 34696 | Duplication of ~ bnj1224 .... |
| bj-andnotim 34697 | Two ways of expressing a c... |
| bj-bi3ant 34698 | This used to be in the mai... |
| bj-bisym 34699 | This used to be in the mai... |
| bj-bixor 34700 | Equivalence of two ternary... |
| bj-axdd2 34701 | This implication, proved u... |
| bj-axd2d 34702 | This implication, proved u... |
| bj-axtd 34703 | This implication, proved f... |
| bj-gl4 34704 | In a normal modal logic, t... |
| bj-axc4 34705 | Over minimal calculus, the... |
| prvlem1 34710 | An elementary property of ... |
| prvlem2 34711 | An elementary property of ... |
| bj-babygodel 34712 | See the section header com... |
| bj-babylob 34713 | See the section header com... |
| bj-godellob 34714 | Proof of Gödel's theo... |
| bj-genr 34715 | Generalization rule on the... |
| bj-genl 34716 | Generalization rule on the... |
| bj-genan 34717 | Generalization rule on a c... |
| bj-mpgs 34718 | From a closed form theorem... |
| bj-2alim 34719 | Closed form of ~ 2alimi . ... |
| bj-2exim 34720 | Closed form of ~ 2eximi . ... |
| bj-alanim 34721 | Closed form of ~ alanimi .... |
| bj-2albi 34722 | Closed form of ~ 2albii . ... |
| bj-notalbii 34723 | Equivalence of universal q... |
| bj-2exbi 34724 | Closed form of ~ 2exbii . ... |
| bj-3exbi 34725 | Closed form of ~ 3exbii . ... |
| bj-sylgt2 34726 | Uncurried (imported) form ... |
| bj-alrimg 34727 | The general form of the *a... |
| bj-alrimd 34728 | A slightly more general ~ ... |
| bj-sylget 34729 | Dual statement of ~ sylgt ... |
| bj-sylget2 34730 | Uncurried (imported) form ... |
| bj-exlimg 34731 | The general form of the *e... |
| bj-sylge 34732 | Dual statement of ~ sylg (... |
| bj-exlimd 34733 | A slightly more general ~ ... |
| bj-nfimexal 34734 | A weak from of nonfreeness... |
| bj-alexim 34735 | Closed form of ~ aleximi .... |
| bj-nexdh 34736 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 34737 | Uncurried (imported) form ... |
| bj-hbxfrbi 34738 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 34739 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 34740 | Distribute quantifiers ove... |
| bj-exalimi 34741 | An inference for distribut... |
| bj-exalims 34742 | Distributing quantifiers o... |
| bj-exalimsi 34743 | An inference for distribut... |
| bj-ax12ig 34744 | A lemma used to prove a we... |
| bj-ax12i 34745 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 34746 | Closed form of ~ nfim and ... |
| bj-cbvalimt 34747 | A lemma in closed form use... |
| bj-cbveximt 34748 | A lemma in closed form use... |
| bj-eximALT 34749 | Alternate proof of ~ exim ... |
| bj-aleximiALT 34750 | Alternate proof of ~ alexi... |
| bj-eximcom 34751 | A commuted form of ~ exim ... |
| bj-ax12wlem 34752 | A lemma used to prove a we... |
| bj-cbvalim 34753 | A lemma used to prove ~ bj... |
| bj-cbvexim 34754 | A lemma used to prove ~ bj... |
| bj-cbvalimi 34755 | An equality-free general i... |
| bj-cbveximi 34756 | An equality-free general i... |
| bj-cbval 34757 | Changing a bound variable ... |
| bj-cbvex 34758 | Changing a bound variable ... |
| bj-ssbeq 34761 | Substitution in an equalit... |
| bj-ssblem1 34762 | A lemma for the definiens ... |
| bj-ssblem2 34763 | An instance of ~ ax-11 pro... |
| bj-ax12v 34764 | A weaker form of ~ ax-12 a... |
| bj-ax12 34765 | Remove a DV condition from... |
| bj-ax12ssb 34766 | Axiom ~ bj-ax12 expressed ... |
| bj-19.41al 34767 | Special case of ~ 19.41 pr... |
| bj-equsexval 34768 | Special case of ~ equsexv ... |
| bj-subst 34769 | Proof of ~ sbalex from cor... |
| bj-ssbid2 34770 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 34771 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 34772 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 34773 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 34774 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 34775 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 34776 | Proof of ~ ax6e (hence ~ a... |
| bj-spimvwt 34777 | Closed form of ~ spimvw . ... |
| bj-spnfw 34778 | Theorem close to a closed ... |
| bj-cbvexiw 34779 | Change bound variable. Th... |
| bj-cbvexivw 34780 | Change bound variable. Th... |
| bj-modald 34781 | A short form of the axiom ... |
| bj-denot 34782 | A weakening of ~ ax-6 and ... |
| bj-eqs 34783 | A lemma for substitutions,... |
| bj-cbvexw 34784 | Change bound variable. Th... |
| bj-ax12w 34785 | The general statement that... |
| bj-ax89 34786 | A theorem which could be u... |
| bj-elequ12 34787 | An identity law for the no... |
| bj-cleljusti 34788 | One direction of ~ cleljus... |
| bj-alcomexcom 34789 | Commutation of universal q... |
| bj-hbalt 34790 | Closed form of ~ hbal . W... |
| axc11n11 34791 | Proof of ~ axc11n from { ~... |
| axc11n11r 34792 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 34793 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 34794 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 34795 | Alternate proof of ~ bj-ax... |
| bj-sb 34796 | A weak variant of ~ sbid2 ... |
| bj-modalbe 34797 | The predicate-calculus ver... |
| bj-spst 34798 | Closed form of ~ sps . On... |
| bj-19.21bit 34799 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 34800 | Closed form of ~ 19.23bi .... |
| bj-nexrt 34801 | Closed form of ~ nexr . C... |
| bj-alrim 34802 | Closed form of ~ alrimi . ... |
| bj-alrim2 34803 | Uncurried (imported) form ... |
| bj-nfdt0 34804 | A theorem close to a close... |
| bj-nfdt 34805 | Closed form of ~ nf5d and ... |
| bj-nexdt 34806 | Closed form of ~ nexd . (... |
| bj-nexdvt 34807 | Closed form of ~ nexdv . ... |
| bj-alexbiex 34808 | Adding a second quantifier... |
| bj-exexbiex 34809 | Adding a second quantifier... |
| bj-alalbial 34810 | Adding a second quantifier... |
| bj-exalbial 34811 | Adding a second quantifier... |
| bj-19.9htbi 34812 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 34813 | Strengthening ~ hbnt by re... |
| bj-biexal1 34814 | A general FOL biconditiona... |
| bj-biexal2 34815 | When ` ph ` is substituted... |
| bj-biexal3 34816 | When ` ph ` is substituted... |
| bj-bialal 34817 | When ` ph ` is substituted... |
| bj-biexex 34818 | When ` ph ` is substituted... |
| bj-hbext 34819 | Closed form of ~ hbex . (... |
| bj-nfalt 34820 | Closed form of ~ nfal . (... |
| bj-nfext 34821 | Closed form of ~ nfex . (... |
| bj-eeanvw 34822 | Version of ~ exdistrv with... |
| bj-modal4 34823 | First-order logic form of ... |
| bj-modal4e 34824 | First-order logic form of ... |
| bj-modalb 34825 | A short form of the axiom ... |
| bj-wnf1 34826 | When ` ph ` is substituted... |
| bj-wnf2 34827 | When ` ph ` is substituted... |
| bj-wnfanf 34828 | When ` ph ` is substituted... |
| bj-wnfenf 34829 | When ` ph ` is substituted... |
| bj-substax12 34830 | Equivalent form of the axi... |
| bj-substw 34831 | Weak form of the LHS of ~ ... |
| bj-nnfbi 34834 | If two formulas are equiva... |
| bj-nnfbd 34835 | If two formulas are equiva... |
| bj-nnfbii 34836 | If two formulas are equiva... |
| bj-nnfa 34837 | Nonfreeness implies the eq... |
| bj-nnfad 34838 | Nonfreeness implies the eq... |
| bj-nnfai 34839 | Nonfreeness implies the eq... |
| bj-nnfe 34840 | Nonfreeness implies the eq... |
| bj-nnfed 34841 | Nonfreeness implies the eq... |
| bj-nnfei 34842 | Nonfreeness implies the eq... |
| bj-nnfea 34843 | Nonfreeness implies the eq... |
| bj-nnfead 34844 | Nonfreeness implies the eq... |
| bj-nnfeai 34845 | Nonfreeness implies the eq... |
| bj-dfnnf2 34846 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 34847 | New nonfreeness implies ol... |
| bj-wnfnf 34848 | When ` ph ` is substituted... |
| bj-nnfnt 34849 | A variable is nonfree in a... |
| bj-nnftht 34850 | A variable is nonfree in a... |
| bj-nnfth 34851 | A variable is nonfree in a... |
| bj-nnfnth 34852 | A variable is nonfree in t... |
| bj-nnfim1 34853 | A consequence of nonfreene... |
| bj-nnfim2 34854 | A consequence of nonfreene... |
| bj-nnfim 34855 | Nonfreeness in the anteced... |
| bj-nnfimd 34856 | Nonfreeness in the anteced... |
| bj-nnfan 34857 | Nonfreeness in both conjun... |
| bj-nnfand 34858 | Nonfreeness in both conjun... |
| bj-nnfor 34859 | Nonfreeness in both disjun... |
| bj-nnford 34860 | Nonfreeness in both disjun... |
| bj-nnfbit 34861 | Nonfreeness in both sides ... |
| bj-nnfbid 34862 | Nonfreeness in both sides ... |
| bj-nnfv 34863 | A non-occurring variable i... |
| bj-nnf-alrim 34864 | Proof of the closed form o... |
| bj-nnf-exlim 34865 | Proof of the closed form o... |
| bj-dfnnf3 34866 | Alternate definition of no... |
| bj-nfnnfTEMP 34867 | New nonfreeness is equival... |
| bj-nnfa1 34868 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 34869 | See ~ nfe1 . (Contributed... |
| bj-19.12 34870 | See ~ 19.12 . Could be la... |
| bj-nnflemaa 34871 | One of four lemmas for non... |
| bj-nnflemee 34872 | One of four lemmas for non... |
| bj-nnflemae 34873 | One of four lemmas for non... |
| bj-nnflemea 34874 | One of four lemmas for non... |
| bj-nnfalt 34875 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 34876 | See ~ nfex and ~ bj-nfext ... |
| bj-stdpc5t 34877 | Alias of ~ bj-nnf-alrim fo... |
| bj-19.21t 34878 | Statement ~ 19.21t proved ... |
| bj-19.23t 34879 | Statement ~ 19.23t proved ... |
| bj-19.36im 34880 | One direction of ~ 19.36 f... |
| bj-19.37im 34881 | One direction of ~ 19.37 f... |
| bj-19.42t 34882 | Closed form of ~ 19.42 fro... |
| bj-19.41t 34883 | Closed form of ~ 19.41 fro... |
| bj-sbft 34884 | Version of ~ sbft using ` ... |
| bj-pm11.53vw 34885 | Version of ~ pm11.53v with... |
| bj-pm11.53v 34886 | Version of ~ pm11.53v with... |
| bj-pm11.53a 34887 | A variant of ~ pm11.53v . ... |
| bj-equsvt 34888 | A variant of ~ equsv . (C... |
| bj-equsalvwd 34889 | Variant of ~ equsalvw . (... |
| bj-equsexvwd 34890 | Variant of ~ equsexvw . (... |
| bj-sbievwd 34891 | Variant of ~ sbievw . (Co... |
| bj-axc10 34892 | Alternate proof of ~ axc10... |
| bj-alequex 34893 | A fol lemma. See ~ aleque... |
| bj-spimt2 34894 | A step in the proof of ~ s... |
| bj-cbv3ta 34895 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 34896 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 34897 | A theorem close to a close... |
| bj-hbsb3 34898 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 34899 | A theorem close to a close... |
| bj-nfs1t2 34900 | A theorem close to a close... |
| bj-nfs1 34901 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 34902 | Version of ~ axc10 with a ... |
| bj-spimtv 34903 | Version of ~ spimt with a ... |
| bj-cbv3hv2 34904 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 34905 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 34906 | Version of ~ cbv2h with a ... |
| bj-cbv2v 34907 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 34908 | Version of ~ cbvald with a... |
| bj-cbvexdv 34909 | Version of ~ cbvexd with a... |
| bj-cbval2vv 34910 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 34911 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 34912 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 34913 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 34914 | Version of ~ cbvex4v with ... |
| bj-equsalhv 34915 | Version of ~ equsalh with ... |
| bj-axc11nv 34916 | Version of ~ axc11n with a... |
| bj-aecomsv 34917 | Version of ~ aecoms with a... |
| bj-axc11v 34918 | Version of ~ axc11 with a ... |
| bj-drnf2v 34919 | Version of ~ drnf2 with a ... |
| bj-equs45fv 34920 | Version of ~ equs45f with ... |
| bj-hbs1 34921 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 34922 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 34923 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 34924 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 34925 | Remove dependency on ~ ax-... |
| bj-dtru 34926 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 34927 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 34928 | Biconditional version of a... |
| bj-hbaeb 34929 | Biconditional version of ~... |
| bj-hbnaeb 34930 | Biconditional version of ~... |
| bj-dvv 34931 | A special instance of ~ bj... |
| bj-equsal1t 34932 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 34933 | Inference associated with ... |
| bj-equsal1 34934 | One direction of ~ equsal ... |
| bj-equsal2 34935 | One direction of ~ equsal ... |
| bj-equsal 34936 | Shorter proof of ~ equsal ... |
| stdpc5t 34937 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 34938 | More direct proof of ~ std... |
| 2stdpc5 34939 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 34940 | Proof of ~ 19.21t from ~ s... |
| exlimii 34941 | Inference associated with ... |
| ax11-pm 34942 | Proof of ~ ax-11 similar t... |
| ax6er 34943 | Commuted form of ~ ax6e . ... |
| exlimiieq1 34944 | Inferring a theorem when i... |
| exlimiieq2 34945 | Inferring a theorem when i... |
| ax11-pm2 34946 | Proof of ~ ax-11 from the ... |
| bj-sbsb 34947 | Biconditional showing two ... |
| bj-dfsb2 34948 | Alternate (dual) definitio... |
| bj-sbf3 34949 | Substitution has no effect... |
| bj-sbf4 34950 | Substitution has no effect... |
| bj-sbnf 34951 | Move nonfree predicate in ... |
| bj-eu3f 34952 | Version of ~ eu3v where th... |
| bj-sblem1 34953 | Lemma for substitution. (... |
| bj-sblem2 34954 | Lemma for substitution. (... |
| bj-sblem 34955 | Lemma for substitution. (... |
| bj-sbievw1 34956 | Lemma for substitution. (... |
| bj-sbievw2 34957 | Lemma for substitution. (... |
| bj-sbievw 34958 | Lemma for substitution. C... |
| bj-sbievv 34959 | Version of ~ sbie with a s... |
| bj-moeub 34960 | Uniqueness is equivalent t... |
| bj-sbidmOLD 34961 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 34962 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 34963 | Curried (exported) form of... |
| bj-dvelimv 34964 | A version of ~ dvelim usin... |
| bj-nfeel2 34965 | Nonfreeness in a membershi... |
| bj-axc14nf 34966 | Proof of a version of ~ ax... |
| bj-axc14 34967 | Alternate proof of ~ axc14... |
| mobidvALT 34968 | Alternate proof of ~ mobid... |
| sbn1ALT 34969 | Alternate proof of ~ sbn1 ... |
| eliminable1 34970 | A theorem used to prove th... |
| eliminable2a 34971 | A theorem used to prove th... |
| eliminable2b 34972 | A theorem used to prove th... |
| eliminable2c 34973 | A theorem used to prove th... |
| eliminable3a 34974 | A theorem used to prove th... |
| eliminable3b 34975 | A theorem used to prove th... |
| eliminable-velab 34976 | A theorem used to prove th... |
| eliminable-veqab 34977 | A theorem used to prove th... |
| eliminable-abeqv 34978 | A theorem used to prove th... |
| eliminable-abeqab 34979 | A theorem used to prove th... |
| eliminable-abelv 34980 | A theorem used to prove th... |
| eliminable-abelab 34981 | A theorem used to prove th... |
| bj-denoteslem 34982 | Lemma for ~ bj-denotes . ... |
| bj-denotes 34983 | This would be the justific... |
| bj-issettru 34984 | Weak version of ~ isset wi... |
| bj-elabtru 34985 | This is as close as we can... |
| bj-issetwt 34986 | Closed form of ~ bj-issetw... |
| bj-issetw 34987 | The closest one can get to... |
| bj-elissetALT 34988 | Alternate proof of ~ eliss... |
| bj-issetiv 34989 | Version of ~ bj-isseti wit... |
| bj-isseti 34990 | Version of ~ isseti with a... |
| bj-ralvw 34991 | A weak version of ~ ralv n... |
| bj-rexvw 34992 | A weak version of ~ rexv n... |
| bj-rababw 34993 | A weak version of ~ rabab ... |
| bj-rexcom4bv 34994 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 34995 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 34996 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 34997 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 34998 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 34999 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 35000 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 35001 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 35002 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 35003 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 35004 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 35005 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 35006 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 35007 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 35008 | Remove from ~ spcimdv depe... |
| elelb 35009 | Equivalence between two co... |
| bj-pwvrelb 35010 | Characterization of the el... |
| bj-nfcsym 35011 | The nonfreeness quantifier... |
| bj-sbeqALT 35012 | Substitution in an equalit... |
| bj-sbeq 35013 | Distribute proper substitu... |
| bj-sbceqgALT 35014 | Distribute proper substitu... |
| bj-csbsnlem 35015 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 35016 | Substitution in a singleto... |
| bj-sbel1 35017 | Version of ~ sbcel1g when ... |
| bj-abv 35018 | The class of sets verifyin... |
| bj-abvALT 35019 | Alternate version of ~ bj-... |
| bj-ab0 35020 | The class of sets verifyin... |
| bj-abf 35021 | Shorter proof of ~ abf (wh... |
| bj-csbprc 35022 | More direct proof of ~ csb... |
| bj-exlimvmpi 35023 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 35024 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 35025 | Lemma for theorems of the ... |
| bj-exlimmpbir 35026 | Lemma for theorems of the ... |
| bj-vtoclf 35027 | Remove dependency on ~ ax-... |
| bj-vtocl 35028 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 35029 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 35030 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 35031 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 35032 | A version of ~ vtoclg with... |
| bj-rabbida2 35033 | Version of ~ rabbidva2 wit... |
| bj-rabeqd 35034 | Deduction form of ~ rabeq ... |
| bj-rabeqbid 35035 | Version of ~ rabeqbidv wit... |
| bj-rabeqbida 35036 | Version of ~ rabeqbidva wi... |
| bj-seex 35037 | Version of ~ seex with a d... |
| bj-nfcf 35038 | Version of ~ df-nfc with a... |
| bj-zfauscl 35039 | General version of ~ zfaus... |
| bj-elabd2ALT 35040 | Alternate proof of ~ elabd... |
| bj-unrab 35041 | Generalization of ~ unrab ... |
| bj-inrab 35042 | Generalization of ~ inrab ... |
| bj-inrab2 35043 | Shorter proof of ~ inrab .... |
| bj-inrab3 35044 | Generalization of ~ dfrab3... |
| bj-rabtr 35045 | Restricted class abstracti... |
| bj-rabtrALT 35046 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 35047 | Proof of ~ bj-rabtr found ... |
| bj-gabss 35050 | Inclusion of generalized c... |
| bj-gabssd 35051 | Inclusion of generalized c... |
| bj-gabeqd 35052 | Equality of generalized cl... |
| bj-gabeqis 35053 | Equality of generalized cl... |
| bj-elgab 35054 | Elements of a generalized ... |
| bj-gabima 35055 | Generalized class abstract... |
| bj-ru0 35058 | The FOL part of Russell's ... |
| bj-ru1 35059 | A version of Russell's par... |
| bj-ru 35060 | Remove dependency on ~ ax-... |
| currysetlem 35061 | Lemma for ~ currysetlem , ... |
| curryset 35062 | Curry's paradox in set the... |
| currysetlem1 35063 | Lemma for ~ currysetALT . ... |
| currysetlem2 35064 | Lemma for ~ currysetALT . ... |
| currysetlem3 35065 | Lemma for ~ currysetALT . ... |
| currysetALT 35066 | Alternate proof of ~ curry... |
| bj-n0i 35067 | Inference associated with ... |
| bj-disjcsn 35068 | A class is disjoint from i... |
| bj-disjsn01 35069 | Disjointness of the single... |
| bj-0nel1 35070 | The empty set does not bel... |
| bj-1nel0 35071 | ` 1o ` does not belong to ... |
| bj-xpimasn 35072 | The image of a singleton, ... |
| bj-xpima1sn 35073 | The image of a singleton b... |
| bj-xpima1snALT 35074 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 35075 | The image of a singleton b... |
| bj-xpnzex 35076 | If the first factor of a p... |
| bj-xpexg2 35077 | Curried (exported) form of... |
| bj-xpnzexb 35078 | If the first factor of a p... |
| bj-cleq 35079 | Substitution property for ... |
| bj-snsetex 35080 | The class of sets "whose s... |
| bj-clex 35081 | Sethood of certain classes... |
| bj-sngleq 35084 | Substitution property for ... |
| bj-elsngl 35085 | Characterization of the el... |
| bj-snglc 35086 | Characterization of the el... |
| bj-snglss 35087 | The singletonization of a ... |
| bj-0nelsngl 35088 | The empty set is not a mem... |
| bj-snglinv 35089 | Inverse of singletonizatio... |
| bj-snglex 35090 | A class is a set if and on... |
| bj-tageq 35093 | Substitution property for ... |
| bj-eltag 35094 | Characterization of the el... |
| bj-0eltag 35095 | The empty set belongs to t... |
| bj-tagn0 35096 | The tagging of a class is ... |
| bj-tagss 35097 | The tagging of a class is ... |
| bj-snglsstag 35098 | The singletonization is in... |
| bj-sngltagi 35099 | The singletonization is in... |
| bj-sngltag 35100 | The singletonization and t... |
| bj-tagci 35101 | Characterization of the el... |
| bj-tagcg 35102 | Characterization of the el... |
| bj-taginv 35103 | Inverse of tagging. (Cont... |
| bj-tagex 35104 | A class is a set if and on... |
| bj-xtageq 35105 | The products of a given cl... |
| bj-xtagex 35106 | The product of a set and t... |
| bj-projeq 35109 | Substitution property for ... |
| bj-projeq2 35110 | Substitution property for ... |
| bj-projun 35111 | The class projection on a ... |
| bj-projex 35112 | Sethood of the class proje... |
| bj-projval 35113 | Value of the class project... |
| bj-1upleq 35116 | Substitution property for ... |
| bj-pr1eq 35119 | Substitution property for ... |
| bj-pr1un 35120 | The first projection prese... |
| bj-pr1val 35121 | Value of the first project... |
| bj-pr11val 35122 | Value of the first project... |
| bj-pr1ex 35123 | Sethood of the first proje... |
| bj-1uplth 35124 | The characteristic propert... |
| bj-1uplex 35125 | A monuple is a set if and ... |
| bj-1upln0 35126 | A monuple is nonempty. (C... |
| bj-2upleq 35129 | Substitution property for ... |
| bj-pr21val 35130 | Value of the first project... |
| bj-pr2eq 35133 | Substitution property for ... |
| bj-pr2un 35134 | The second projection pres... |
| bj-pr2val 35135 | Value of the second projec... |
| bj-pr22val 35136 | Value of the second projec... |
| bj-pr2ex 35137 | Sethood of the second proj... |
| bj-2uplth 35138 | The characteristic propert... |
| bj-2uplex 35139 | A couple is a set if and o... |
| bj-2upln0 35140 | A couple is nonempty. (Co... |
| bj-2upln1upl 35141 | A couple is never equal to... |
| bj-rcleqf 35142 | Relative version of ~ cleq... |
| bj-rcleq 35143 | Relative version of ~ dfcl... |
| bj-reabeq 35144 | Relative form of ~ abeq2 .... |
| bj-disj2r 35145 | Relative version of ~ ssdi... |
| bj-sscon 35146 | Contraposition law for rel... |
| eleq2w2ALT 35147 | Alternate proof of ~ eleq2... |
| bj-clel3gALT 35148 | Alternate proof of ~ clel3... |
| bj-pw0ALT 35149 | Alternate proof of ~ pw0 .... |
| bj-sselpwuni 35150 | Quantitative version of ~ ... |
| bj-unirel 35151 | Quantitative version of ~ ... |
| bj-elpwg 35152 | If the intersection of two... |
| bj-vjust 35153 | Justification theorem for ... |
| bj-nul 35154 | Two formulations of the ax... |
| bj-nuliota 35155 | Definition of the empty se... |
| bj-nuliotaALT 35156 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 35157 | Alternate proof of ~ vtocl... |
| bj-elsn12g 35158 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 35159 | Biconditional version of ~... |
| bj-pwcfsdom 35160 | Remove hypothesis from ~ p... |
| bj-grur1 35161 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 35162 | The extension of a predica... |
| bj-dfid2ALT 35163 | Alternate version of ~ dfi... |
| bj-0nelopab 35164 | The empty set is never an ... |
| bj-brrelex12ALT 35165 | Two classes related by a b... |
| bj-epelg 35166 | The membership relation an... |
| bj-epelb 35167 | Two classes are related by... |
| bj-nsnid 35168 | A set does not contain the... |
| bj-rdg0gALT 35169 | Alternate proof of ~ rdg0g... |
| bj-evaleq 35170 | Equality theorem for the `... |
| bj-evalfun 35171 | The evaluation at a class ... |
| bj-evalfn 35172 | The evaluation at a class ... |
| bj-evalval 35173 | Value of the evaluation at... |
| bj-evalid 35174 | The evaluation at a set of... |
| bj-ndxarg 35175 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 35176 | Closed general form of ~ s... |
| bj-rest00 35179 | An elementwise intersectio... |
| bj-restsn 35180 | An elementwise intersectio... |
| bj-restsnss 35181 | Special case of ~ bj-rests... |
| bj-restsnss2 35182 | Special case of ~ bj-rests... |
| bj-restsn0 35183 | An elementwise intersectio... |
| bj-restsn10 35184 | Special case of ~ bj-rests... |
| bj-restsnid 35185 | The elementwise intersecti... |
| bj-rest10 35186 | An elementwise intersectio... |
| bj-rest10b 35187 | Alternate version of ~ bj-... |
| bj-restn0 35188 | An elementwise intersectio... |
| bj-restn0b 35189 | Alternate version of ~ bj-... |
| bj-restpw 35190 | The elementwise intersecti... |
| bj-rest0 35191 | An elementwise intersectio... |
| bj-restb 35192 | An elementwise intersectio... |
| bj-restv 35193 | An elementwise intersectio... |
| bj-resta 35194 | An elementwise intersectio... |
| bj-restuni 35195 | The union of an elementwis... |
| bj-restuni2 35196 | The union of an elementwis... |
| bj-restreg 35197 | A reformulation of the axi... |
| bj-raldifsn 35198 | All elements in a set sati... |
| bj-0int 35199 | If ` A ` is a collection o... |
| bj-mooreset 35200 | A Moore collection is a se... |
| bj-ismoore 35203 | Characterization of Moore ... |
| bj-ismoored0 35204 | Necessary condition to be ... |
| bj-ismoored 35205 | Necessary condition to be ... |
| bj-ismoored2 35206 | Necessary condition to be ... |
| bj-ismooredr 35207 | Sufficient condition to be... |
| bj-ismooredr2 35208 | Sufficient condition to be... |
| bj-discrmoore 35209 | The powerclass ` ~P A ` is... |
| bj-0nmoore 35210 | The empty set is not a Moo... |
| bj-snmoore 35211 | A singleton is a Moore col... |
| bj-snmooreb 35212 | A singleton is a Moore col... |
| bj-prmoore 35213 | A pair formed of two neste... |
| bj-0nelmpt 35214 | The empty set is not an el... |
| bj-mptval 35215 | Value of a function given ... |
| bj-dfmpoa 35216 | An equivalent definition o... |
| bj-mpomptALT 35217 | Alternate proof of ~ mpomp... |
| setsstrset 35234 | Relation between ~ df-sets... |
| bj-nfald 35235 | Variant of ~ nfald . (Con... |
| bj-nfexd 35236 | Variant of ~ nfexd . (Con... |
| copsex2d 35237 | Implicit substitution dedu... |
| copsex2b 35238 | Biconditional form of ~ co... |
| opelopabd 35239 | Membership of an ordere pa... |
| opelopabb 35240 | Membership of an ordered p... |
| opelopabbv 35241 | Membership of an ordered p... |
| bj-opelrelex 35242 | The coordinates of an orde... |
| bj-opelresdm 35243 | If an ordered pair is in a... |
| bj-brresdm 35244 | If two classes are related... |
| brabd0 35245 | Expressing that two sets a... |
| brabd 35246 | Expressing that two sets a... |
| bj-brab2a1 35247 | "Unbounded" version of ~ b... |
| bj-opabssvv 35248 | A variant of ~ relopabiv (... |
| bj-funidres 35249 | The restricted identity re... |
| bj-opelidb 35250 | Characterization of the or... |
| bj-opelidb1 35251 | Characterization of the or... |
| bj-inexeqex 35252 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 35253 | If the intersection of two... |
| bj-opelid 35254 | Characterization of the or... |
| bj-ideqg 35255 | Characterization of the cl... |
| bj-ideqgALT 35256 | Alternate proof of ~ bj-id... |
| bj-ideqb 35257 | Characterization of classe... |
| bj-idres 35258 | Alternate expression for t... |
| bj-opelidres 35259 | Characterization of the or... |
| bj-idreseq 35260 | Sufficient condition for t... |
| bj-idreseqb 35261 | Characterization for two c... |
| bj-ideqg1 35262 | For sets, the identity rel... |
| bj-ideqg1ALT 35263 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 35264 | Characterization of the co... |
| bj-elid3 35265 | Characterization of the co... |
| bj-elid4 35266 | Characterization of the el... |
| bj-elid5 35267 | Characterization of the el... |
| bj-elid6 35268 | Characterization of the el... |
| bj-elid7 35269 | Characterization of the el... |
| bj-diagval 35272 | Value of the functionalize... |
| bj-diagval2 35273 | Value of the functionalize... |
| bj-eldiag 35274 | Characterization of the el... |
| bj-eldiag2 35275 | Characterization of the el... |
| bj-imdirvallem 35278 | Lemma for ~ bj-imdirval an... |
| bj-imdirval 35279 | Value of the functionalize... |
| bj-imdirval2lem 35280 | Lemma for ~ bj-imdirval2 a... |
| bj-imdirval2 35281 | Value of the functionalize... |
| bj-imdirval3 35282 | Value of the functionalize... |
| bj-imdiridlem 35283 | Lemma for ~ bj-imdirid and... |
| bj-imdirid 35284 | Functorial property of the... |
| bj-opelopabid 35285 | Membership in an ordered-p... |
| bj-opabco 35286 | Composition of ordered-pai... |
| bj-xpcossxp 35287 | The composition of two Car... |
| bj-imdirco 35288 | Functorial property of the... |
| bj-iminvval 35291 | Value of the functionalize... |
| bj-iminvval2 35292 | Value of the functionalize... |
| bj-iminvid 35293 | Functorial property of the... |
| bj-inftyexpitaufo 35300 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 35303 | An element of the circle a... |
| bj-inftyexpiinv 35306 | Utility theorem for the in... |
| bj-inftyexpiinj 35307 | Injectivity of the paramet... |
| bj-inftyexpidisj 35308 | An element of the circle a... |
| bj-ccinftydisj 35311 | The circle at infinity is ... |
| bj-elccinfty 35312 | A lemma for infinite exten... |
| bj-ccssccbar 35315 | Complex numbers are extend... |
| bj-ccinftyssccbar 35316 | Infinite extended complex ... |
| bj-pinftyccb 35319 | The class ` pinfty ` is an... |
| bj-pinftynrr 35320 | The extended complex numbe... |
| bj-minftyccb 35323 | The class ` minfty ` is an... |
| bj-minftynrr 35324 | The extended complex numbe... |
| bj-pinftynminfty 35325 | The extended complex numbe... |
| bj-rrhatsscchat 35334 | The real projective line i... |
| bj-imafv 35349 | If the direct image of a s... |
| bj-funun 35350 | Value of a function expres... |
| bj-fununsn1 35351 | Value of a function expres... |
| bj-fununsn2 35352 | Value of a function expres... |
| bj-fvsnun1 35353 | The value of a function wi... |
| bj-fvsnun2 35354 | The value of a function wi... |
| bj-fvmptunsn1 35355 | Value of a function expres... |
| bj-fvmptunsn2 35356 | Value of a function expres... |
| bj-iomnnom 35357 | The canonical bijection fr... |
| bj-smgrpssmgm 35366 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 35367 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 35368 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 35369 | Monoids are semigroups (el... |
| bj-cmnssmnd 35370 | Commutative monoids are mo... |
| bj-cmnssmndel 35371 | Commutative monoids are mo... |
| bj-grpssmnd 35372 | Groups are monoids. (Cont... |
| bj-grpssmndel 35373 | Groups are monoids (elemen... |
| bj-ablssgrp 35374 | Abelian groups are groups.... |
| bj-ablssgrpel 35375 | Abelian groups are groups ... |
| bj-ablsscmn 35376 | Abelian groups are commuta... |
| bj-ablsscmnel 35377 | Abelian groups are commuta... |
| bj-modssabl 35378 | (The additive groups of) m... |
| bj-vecssmod 35379 | Vector spaces are modules.... |
| bj-vecssmodel 35380 | Vector spaces are modules ... |
| bj-finsumval0 35383 | Value of a finite sum. (C... |
| bj-fvimacnv0 35384 | Variant of ~ fvimacnv wher... |
| bj-isvec 35385 | The predicate "is a vector... |
| bj-fldssdrng 35386 | Fields are division rings.... |
| bj-flddrng 35387 | Fields are division rings ... |
| bj-rrdrg 35388 | The field of real numbers ... |
| bj-isclm 35389 | The predicate "is a subcom... |
| bj-isrvec 35392 | The predicate "is a real v... |
| bj-rvecmod 35393 | Real vector spaces are mod... |
| bj-rvecssmod 35394 | Real vector spaces are mod... |
| bj-rvecrr 35395 | The field of scalars of a ... |
| bj-isrvecd 35396 | The predicate "is a real v... |
| bj-rvecvec 35397 | Real vector spaces are vec... |
| bj-isrvec2 35398 | The predicate "is a real v... |
| bj-rvecssvec 35399 | Real vector spaces are vec... |
| bj-rveccmod 35400 | Real vector spaces are sub... |
| bj-rvecsscmod 35401 | Real vector spaces are sub... |
| bj-rvecsscvec 35402 | Real vector spaces are sub... |
| bj-rveccvec 35403 | Real vector spaces are sub... |
| bj-rvecssabl 35404 | (The additive groups of) r... |
| bj-rvecabl 35405 | (The additive groups of) r... |
| bj-subcom 35406 | A consequence of commutati... |
| bj-lineqi 35407 | Solution of a (scalar) lin... |
| bj-bary1lem 35408 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 35409 | Lemma for bj-bary1: comput... |
| bj-bary1 35410 | Barycentric coordinates in... |
| bj-endval 35413 | Value of the monoid of end... |
| bj-endbase 35414 | Base set of the monoid of ... |
| bj-endcomp 35415 | Composition law of the mon... |
| bj-endmnd 35416 | The monoid of endomorphism... |
| taupilem3 35417 | Lemma for tau-related theo... |
| taupilemrplb 35418 | A set of positive reals ha... |
| taupilem1 35419 | Lemma for ~ taupi . A pos... |
| taupilem2 35420 | Lemma for ~ taupi . The s... |
| taupi 35421 | Relationship between ` _ta... |
| dfgcd3 35422 | Alternate definition of th... |
| irrdifflemf 35423 | Lemma for ~ irrdiff . The... |
| irrdiff 35424 | The irrationals are exactl... |
| iccioo01 35425 | The closed unit interval i... |
| csbrecsg 35426 | Move class substitution in... |
| csbrdgg 35427 | Move class substitution in... |
| csboprabg 35428 | Move class substitution in... |
| csbmpo123 35429 | Move class substitution in... |
| con1bii2 35430 | A contraposition inference... |
| con2bii2 35431 | A contraposition inference... |
| vtoclefex 35432 | Implicit substitution of a... |
| rnmptsn 35433 | The range of a function ma... |
| f1omptsnlem 35434 | This is the core of the pr... |
| f1omptsn 35435 | A function mapping to sing... |
| mptsnunlem 35436 | This is the core of the pr... |
| mptsnun 35437 | A class ` B ` is equal to ... |
| dissneqlem 35438 | This is the core of the pr... |
| dissneq 35439 | Any topology that contains... |
| exlimim 35440 | Closed form of ~ exlimimd ... |
| exlimimd 35441 | Existential elimination ru... |
| exellim 35442 | Closed form of ~ exellimdd... |
| exellimddv 35443 | Eliminate an antecedent wh... |
| topdifinfindis 35444 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 35445 | This is the core of the pr... |
| topdifinffin 35446 | Part of Exercise 3 of [Mun... |
| topdifinf 35447 | Part of Exercise 3 of [Mun... |
| topdifinfeq 35448 | Two different ways of defi... |
| icorempo 35449 | Closed-below, open-above i... |
| icoreresf 35450 | Closed-below, open-above i... |
| icoreval 35451 | Value of the closed-below,... |
| icoreelrnab 35452 | Elementhood in the set of ... |
| isbasisrelowllem1 35453 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 35454 | Lemma for ~ isbasisrelowl ... |
| icoreclin 35455 | The set of closed-below, o... |
| isbasisrelowl 35456 | The set of all closed-belo... |
| icoreunrn 35457 | The union of all closed-be... |
| istoprelowl 35458 | The set of all closed-belo... |
| icoreelrn 35459 | A class abstraction which ... |
| iooelexlt 35460 | An element of an open inte... |
| relowlssretop 35461 | The lower limit topology o... |
| relowlpssretop 35462 | The lower limit topology o... |
| sucneqond 35463 | Inequality of an ordinal s... |
| sucneqoni 35464 | Inequality of an ordinal s... |
| onsucuni3 35465 | If an ordinal number has a... |
| 1oequni2o 35466 | The ordinal number ` 1o ` ... |
| rdgsucuni 35467 | If an ordinal number has a... |
| rdgeqoa 35468 | If a recursive function wi... |
| elxp8 35469 | Membership in a Cartesian ... |
| cbveud 35470 | Deduction used to change b... |
| cbvreud 35471 | Deduction used to change b... |
| difunieq 35472 | The difference of unions i... |
| inunissunidif 35473 | Theorem about subsets of t... |
| rdgellim 35474 | Elementhood in a recursive... |
| rdglimss 35475 | A recursive definition at ... |
| rdgssun 35476 | In a recursive definition ... |
| exrecfnlem 35477 | Lemma for ~ exrecfn . (Co... |
| exrecfn 35478 | Theorem about the existenc... |
| exrecfnpw 35479 | For any base set, a set wh... |
| finorwe 35480 | If the Axiom of Infinity i... |
| dffinxpf 35483 | This theorem is the same a... |
| finxpeq1 35484 | Equality theorem for Carte... |
| finxpeq2 35485 | Equality theorem for Carte... |
| csbfinxpg 35486 | Distribute proper substitu... |
| finxpreclem1 35487 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 35488 | Lemma for ` ^^ ` recursion... |
| finxp0 35489 | The value of Cartesian exp... |
| finxp1o 35490 | The value of Cartesian exp... |
| finxpreclem3 35491 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 35492 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 35493 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 35494 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 35495 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 35496 | The value of Cartesian exp... |
| finxp2o 35497 | The value of Cartesian exp... |
| finxp3o 35498 | The value of Cartesian exp... |
| finxpnom 35499 | Cartesian exponentiation w... |
| finxp00 35500 | Cartesian exponentiation o... |
| iunctb2 35501 | Using the axiom of countab... |
| domalom 35502 | A class which dominates ev... |
| isinf2 35503 | The converse of ~ isinf . ... |
| ctbssinf 35504 | Using the axiom of choice,... |
| ralssiun 35505 | The index set of an indexe... |
| nlpineqsn 35506 | For every point ` p ` of a... |
| nlpfvineqsn 35507 | Given a subset ` A ` of ` ... |
| fvineqsnf1 35508 | A theorem about functions ... |
| fvineqsneu 35509 | A theorem about functions ... |
| fvineqsneq 35510 | A theorem about functions ... |
| pibp16 35511 | Property P000016 of pi-bas... |
| pibp19 35512 | Property P000019 of pi-bas... |
| pibp21 35513 | Property P000021 of pi-bas... |
| pibt1 35514 | Theorem T000001 of pi-base... |
| pibt2 35515 | Theorem T000002 of pi-base... |
| wl-section-prop 35516 | Intuitionistic logic is no... |
| wl-section-boot 35520 | In this section, I provide... |
| wl-luk-imim1i 35521 | Inference adding common co... |
| wl-luk-syl 35522 | An inference version of th... |
| wl-luk-imtrid 35523 | A syllogism rule of infere... |
| wl-luk-pm2.18d 35524 | Deduction based on reducti... |
| wl-luk-con4i 35525 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 35526 | Inference rule. Copy of ~... |
| wl-luk-a1i 35527 | Inference rule. Copy of ~... |
| wl-luk-mpi 35528 | A nested modus ponens infe... |
| wl-luk-imim2i 35529 | Inference adding common an... |
| wl-luk-imtrdi 35530 | A syllogism rule of infere... |
| wl-luk-ax3 35531 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 35532 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 35533 | This theorem, called "Asse... |
| wl-luk-com12 35534 | Inference that swaps (comm... |
| wl-luk-pm2.21 35535 | From a wff and its negatio... |
| wl-luk-con1i 35536 | A contraposition inference... |
| wl-luk-ja 35537 | Inference joining the ante... |
| wl-luk-imim2 35538 | A closed form of syllogism... |
| wl-luk-a1d 35539 | Deduction introducing an e... |
| wl-luk-ax2 35540 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 35541 | Principle of identity. Th... |
| wl-luk-notnotr 35542 | Converse of double negatio... |
| wl-luk-pm2.04 35543 | Swap antecedents. Theorem... |
| wl-section-impchain 35544 | An implication like ` ( ps... |
| wl-impchain-mp-x 35545 | This series of theorems pr... |
| wl-impchain-mp-0 35546 | This theorem is the start ... |
| wl-impchain-mp-1 35547 | This theorem is in fact a ... |
| wl-impchain-mp-2 35548 | This theorem is in fact a ... |
| wl-impchain-com-1.x 35549 | It is often convenient to ... |
| wl-impchain-com-1.1 35550 | A degenerate form of antec... |
| wl-impchain-com-1.2 35551 | This theorem is in fact a ... |
| wl-impchain-com-1.3 35552 | This theorem is in fact a ... |
| wl-impchain-com-1.4 35553 | This theorem is in fact a ... |
| wl-impchain-com-n.m 35554 | This series of theorems al... |
| wl-impchain-com-2.3 35555 | This theorem is in fact a ... |
| wl-impchain-com-2.4 35556 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 35557 | This theorem is in fact a ... |
| wl-impchain-a1-x 35558 | If an implication chain is... |
| wl-impchain-a1-1 35559 | Inference rule, a copy of ... |
| wl-impchain-a1-2 35560 | Inference rule, a copy of ... |
| wl-impchain-a1-3 35561 | Inference rule, a copy of ... |
| wl-ifp-ncond1 35562 | If one case of an ` if- ` ... |
| wl-ifp-ncond2 35563 | If one case of an ` if- ` ... |
| wl-ifpimpr 35564 | If one case of an ` if- ` ... |
| wl-ifp4impr 35565 | If one case of an ` if- ` ... |
| wl-df-3xor 35566 | Alternative definition of ... |
| wl-df3xor2 35567 | Alternative definition of ... |
| wl-df3xor3 35568 | Alternative form of ~ wl-d... |
| wl-3xortru 35569 | If the first input is true... |
| wl-3xorfal 35570 | If the first input is fals... |
| wl-3xorbi 35571 | Triple xor can be replaced... |
| wl-3xorbi2 35572 | Alternative form of ~ wl-3... |
| wl-3xorbi123d 35573 | Equivalence theorem for tr... |
| wl-3xorbi123i 35574 | Equivalence theorem for tr... |
| wl-3xorrot 35575 | Rotation law for triple xo... |
| wl-3xorcoma 35576 | Commutative law for triple... |
| wl-3xorcomb 35577 | Commutative law for triple... |
| wl-3xornot1 35578 | Flipping the first input f... |
| wl-3xornot 35579 | Triple xor distributes ove... |
| wl-1xor 35580 | In the recursive scheme ... |
| wl-2xor 35581 | In the recursive scheme ... |
| wl-df-3mintru2 35582 | Alternative definition of ... |
| wl-df2-3mintru2 35583 | The adder carry in disjunc... |
| wl-df3-3mintru2 35584 | The adder carry in conjunc... |
| wl-df4-3mintru2 35585 | An alternative definition ... |
| wl-1mintru1 35586 | Using the recursion formul... |
| wl-1mintru2 35587 | Using the recursion formul... |
| wl-2mintru1 35588 | Using the recursion formul... |
| wl-2mintru2 35589 | Using the recursion formul... |
| wl-df3maxtru1 35590 | Assuming "(n+1)-maxtru1" `... |
| wl-ax13lem1 35592 | A version of ~ ax-wl-13v w... |
| wl-mps 35593 | Replacing a nested consequ... |
| wl-syls1 35594 | Replacing a nested consequ... |
| wl-syls2 35595 | Replacing a nested anteced... |
| wl-embant 35596 | A true wff can always be a... |
| wl-orel12 35597 | In a conjunctive normal fo... |
| wl-cases2-dnf 35598 | A particular instance of ~... |
| wl-cbvmotv 35599 | Change bound variable. Us... |
| wl-moteq 35600 | Change bound variable. Us... |
| wl-motae 35601 | Change bound variable. Us... |
| wl-moae 35602 | Two ways to express "at mo... |
| wl-euae 35603 | Two ways to express "exact... |
| wl-nax6im 35604 | The following series of th... |
| wl-hbae1 35605 | This specialization of ~ h... |
| wl-naevhba1v 35606 | An instance of ~ hbn1w app... |
| wl-spae 35607 | Prove an instance of ~ sp ... |
| wl-speqv 35608 | Under the assumption ` -. ... |
| wl-19.8eqv 35609 | Under the assumption ` -. ... |
| wl-19.2reqv 35610 | Under the assumption ` -. ... |
| wl-nfalv 35611 | If ` x ` is not present in... |
| wl-nfimf1 35612 | An antecedent is irrelevan... |
| wl-nfae1 35613 | Unlike ~ nfae , this speci... |
| wl-nfnae1 35614 | Unlike ~ nfnae , this spec... |
| wl-aetr 35615 | A transitive law for varia... |
| wl-axc11r 35616 | Same as ~ axc11r , but usi... |
| wl-dral1d 35617 | A version of ~ dral1 with ... |
| wl-cbvalnaed 35618 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 35619 | A more general version of ... |
| wl-exeq 35620 | The semantics of ` E. x y ... |
| wl-aleq 35621 | The semantics of ` A. x y ... |
| wl-nfeqfb 35622 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 35623 | If ` y ` is not free in ` ... |
| wl-equsalvw 35624 | Version of ~ equsalv with ... |
| wl-equsald 35625 | Deduction version of ~ equ... |
| wl-equsal 35626 | A useful equivalence relat... |
| wl-equsal1t 35627 | The expression ` x = y ` i... |
| wl-equsalcom 35628 | This simple equivalence ea... |
| wl-equsal1i 35629 | The antecedent ` x = y ` i... |
| wl-sb6rft 35630 | A specialization of ~ wl-e... |
| wl-cbvalsbi 35631 | Change bounded variables i... |
| wl-sbrimt 35632 | Substitution with a variab... |
| wl-sblimt 35633 | Substitution with a variab... |
| wl-sb8t 35634 | Substitution of variable i... |
| wl-sb8et 35635 | Substitution of variable i... |
| wl-sbhbt 35636 | Closed form of ~ sbhb . C... |
| wl-sbnf1 35637 | Two ways expressing that `... |
| wl-equsb3 35638 | ~ equsb3 with a distinctor... |
| wl-equsb4 35639 | Substitution applied to an... |
| wl-2sb6d 35640 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 35641 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 35642 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 35643 | Version of ~ sbcom2 with a... |
| wl-sbalnae 35644 | A theorem used in eliminat... |
| wl-sbal1 35645 | A theorem used in eliminat... |
| wl-sbal2 35646 | Move quantifier in and out... |
| wl-2spsbbi 35647 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 35648 | This theorem provides a ba... |
| wl-lem-nexmo 35649 | This theorem provides a ba... |
| wl-lem-moexsb 35650 | The antecedent ` A. x ( ph... |
| wl-alanbii 35651 | This theorem extends ~ ala... |
| wl-mo2df 35652 | Version of ~ mof with a co... |
| wl-mo2tf 35653 | Closed form of ~ mof with ... |
| wl-eudf 35654 | Version of ~ eu6 with a co... |
| wl-eutf 35655 | Closed form of ~ eu6 with ... |
| wl-euequf 35656 | ~ euequ proved with a dist... |
| wl-mo2t 35657 | Closed form of ~ mof . (C... |
| wl-mo3t 35658 | Closed form of ~ mo3 . (C... |
| wl-sb8eut 35659 | Substitution of variable i... |
| wl-sb8mot 35660 | Substitution of variable i... |
| wl-axc11rc11 35661 | Proving ~ axc11r from ~ ax... |
| wl-ax11-lem1 35663 | A transitive law for varia... |
| wl-ax11-lem2 35664 | Lemma. (Contributed by Wo... |
| wl-ax11-lem3 35665 | Lemma. (Contributed by Wo... |
| wl-ax11-lem4 35666 | Lemma. (Contributed by Wo... |
| wl-ax11-lem5 35667 | Lemma. (Contributed by Wo... |
| wl-ax11-lem6 35668 | Lemma. (Contributed by Wo... |
| wl-ax11-lem7 35669 | Lemma. (Contributed by Wo... |
| wl-ax11-lem8 35670 | Lemma. (Contributed by Wo... |
| wl-ax11-lem9 35671 | The easy part when ` x ` c... |
| wl-ax11-lem10 35672 | We now have prepared every... |
| wl-clabv 35673 | Variant of ~ df-clab , whe... |
| wl-dfclab 35674 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 35675 | Using class abstraction in... |
| wl-clabt 35676 | Using class abstraction in... |
| rabiun 35677 | Abstraction restricted to ... |
| iundif1 35678 | Indexed union of class dif... |
| imadifss 35679 | The difference of images i... |
| cureq 35680 | Equality theorem for curry... |
| unceq 35681 | Equality theorem for uncur... |
| curf 35682 | Functional property of cur... |
| uncf 35683 | Functional property of unc... |
| curfv 35684 | Value of currying. (Contr... |
| uncov 35685 | Value of uncurrying. (Con... |
| curunc 35686 | Currying of uncurrying. (... |
| unccur 35687 | Uncurrying of currying. (... |
| phpreu 35688 | Theorem related to pigeonh... |
| finixpnum 35689 | A finite Cartesian product... |
| fin2solem 35690 | Lemma for ~ fin2so . (Con... |
| fin2so 35691 | Any totally ordered Tarski... |
| ltflcei 35692 | Theorem to move the floor ... |
| leceifl 35693 | Theorem to move the floor ... |
| sin2h 35694 | Half-angle rule for sine. ... |
| cos2h 35695 | Half-angle rule for cosine... |
| tan2h 35696 | Half-angle rule for tangen... |
| lindsadd 35697 | In a vector space, the uni... |
| lindsdom 35698 | A linearly independent set... |
| lindsenlbs 35699 | A maximal linearly indepen... |
| matunitlindflem1 35700 | One direction of ~ matunit... |
| matunitlindflem2 35701 | One direction of ~ matunit... |
| matunitlindf 35702 | A matrix over a field is i... |
| ptrest 35703 | Expressing a restriction o... |
| ptrecube 35704 | Any point in an open set o... |
| poimirlem1 35705 | Lemma for ~ poimir - the v... |
| poimirlem2 35706 | Lemma for ~ poimir - conse... |
| poimirlem3 35707 | Lemma for ~ poimir to add ... |
| poimirlem4 35708 | Lemma for ~ poimir connect... |
| poimirlem5 35709 | Lemma for ~ poimir to esta... |
| poimirlem6 35710 | Lemma for ~ poimir establi... |
| poimirlem7 35711 | Lemma for ~ poimir , simil... |
| poimirlem8 35712 | Lemma for ~ poimir , estab... |
| poimirlem9 35713 | Lemma for ~ poimir , estab... |
| poimirlem10 35714 | Lemma for ~ poimir establi... |
| poimirlem11 35715 | Lemma for ~ poimir connect... |
| poimirlem12 35716 | Lemma for ~ poimir connect... |
| poimirlem13 35717 | Lemma for ~ poimir - for a... |
| poimirlem14 35718 | Lemma for ~ poimir - for a... |
| poimirlem15 35719 | Lemma for ~ poimir , that ... |
| poimirlem16 35720 | Lemma for ~ poimir establi... |
| poimirlem17 35721 | Lemma for ~ poimir establi... |
| poimirlem18 35722 | Lemma for ~ poimir stating... |
| poimirlem19 35723 | Lemma for ~ poimir establi... |
| poimirlem20 35724 | Lemma for ~ poimir establi... |
| poimirlem21 35725 | Lemma for ~ poimir stating... |
| poimirlem22 35726 | Lemma for ~ poimir , that ... |
| poimirlem23 35727 | Lemma for ~ poimir , two w... |
| poimirlem24 35728 | Lemma for ~ poimir , two w... |
| poimirlem25 35729 | Lemma for ~ poimir stating... |
| poimirlem26 35730 | Lemma for ~ poimir showing... |
| poimirlem27 35731 | Lemma for ~ poimir showing... |
| poimirlem28 35732 | Lemma for ~ poimir , a var... |
| poimirlem29 35733 | Lemma for ~ poimir connect... |
| poimirlem30 35734 | Lemma for ~ poimir combini... |
| poimirlem31 35735 | Lemma for ~ poimir , assig... |
| poimirlem32 35736 | Lemma for ~ poimir , combi... |
| poimir 35737 | Poincare-Miranda theorem. ... |
| broucube 35738 | Brouwer - or as Kulpa call... |
| heicant 35739 | Heine-Cantor theorem: a co... |
| opnmbllem0 35740 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 35741 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 35742 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 35743 | The difference between two... |
| mblfinlem4 35744 | Backward direction of ~ is... |
| ismblfin 35745 | Measurability in terms of ... |
| ovoliunnfl 35746 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 35747 | Demonstration of ~ ovoliun... |
| voliunnfl 35748 | ~ voliun is incompatible w... |
| volsupnfl 35749 | ~ volsup is incompatible w... |
| mbfresfi 35750 | Measurability of a piecewi... |
| mbfposadd 35751 | If the sum of two measurab... |
| cnambfre 35752 | A real-valued, a.e. contin... |
| dvtanlem 35753 | Lemma for ~ dvtan - the do... |
| dvtan 35754 | Derivative of tangent. (C... |
| itg2addnclem 35755 | An alternate expression fo... |
| itg2addnclem2 35756 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 35757 | Lemma incomprehensible in ... |
| itg2addnc 35758 | Alternate proof of ~ itg2a... |
| itg2gt0cn 35759 | ~ itg2gt0 holds on functio... |
| ibladdnclem 35760 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 35761 | Choice-free analogue of ~ ... |
| itgaddnclem1 35762 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 35763 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 35764 | Choice-free analogue of ~ ... |
| iblsubnc 35765 | Choice-free analogue of ~ ... |
| itgsubnc 35766 | Choice-free analogue of ~ ... |
| iblabsnclem 35767 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 35768 | Choice-free analogue of ~ ... |
| iblmulc2nc 35769 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 35770 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 35771 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 35772 | Choice-free analogue of ~ ... |
| itgabsnc 35773 | Choice-free analogue of ~ ... |
| itggt0cn 35774 | ~ itggt0 holds for continu... |
| ftc1cnnclem 35775 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 35776 | Choice-free proof of ~ ftc... |
| ftc1anclem1 35777 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 35778 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 35779 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 35780 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 35781 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 35782 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 35783 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 35784 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 35785 | ~ ftc1a holds for function... |
| ftc2nc 35786 | Choice-free proof of ~ ftc... |
| asindmre 35787 | Real part of domain of dif... |
| dvasin 35788 | Derivative of arcsine. (C... |
| dvacos 35789 | Derivative of arccosine. ... |
| dvreasin 35790 | Real derivative of arcsine... |
| dvreacos 35791 | Real derivative of arccosi... |
| areacirclem1 35792 | Antiderivative of cross-se... |
| areacirclem2 35793 | Endpoint-inclusive continu... |
| areacirclem3 35794 | Integrability of cross-sec... |
| areacirclem4 35795 | Endpoint-inclusive continu... |
| areacirclem5 35796 | Finding the cross-section ... |
| areacirc 35797 | The area of a circle of ra... |
| unirep 35798 | Define a quantity whose de... |
| cover2 35799 | Two ways of expressing the... |
| cover2g 35800 | Two ways of expressing the... |
| brabg2 35801 | Relation by a binary relat... |
| opelopab3 35802 | Ordered pair membership in... |
| cocanfo 35803 | Cancellation of a surjecti... |
| brresi2 35804 | Restriction of a binary re... |
| fnopabeqd 35805 | Equality deduction for fun... |
| fvopabf4g 35806 | Function value of an opera... |
| eqfnun 35807 | Two functions on ` A u. B ... |
| fnopabco 35808 | Composition of a function ... |
| opropabco 35809 | Composition of an operator... |
| cocnv 35810 | Composition with a functio... |
| f1ocan1fv 35811 | Cancel a composition by a ... |
| f1ocan2fv 35812 | Cancel a composition by th... |
| inixp 35813 | Intersection of Cartesian ... |
| upixp 35814 | Universal property of the ... |
| abrexdom 35815 | An indexed set is dominate... |
| abrexdom2 35816 | An indexed set is dominate... |
| ac6gf 35817 | Axiom of Choice. (Contrib... |
| indexa 35818 | If for every element of an... |
| indexdom 35819 | If for every element of an... |
| frinfm 35820 | A subset of a well-founded... |
| welb 35821 | A nonempty subset of a wel... |
| supex2g 35822 | Existence of supremum. (C... |
| supclt 35823 | Closure of supremum. (Con... |
| supubt 35824 | Upper bound property of su... |
| filbcmb 35825 | Combine a finite set of lo... |
| fzmul 35826 | Membership of a product in... |
| sdclem2 35827 | Lemma for ~ sdc . (Contri... |
| sdclem1 35828 | Lemma for ~ sdc . (Contri... |
| sdc 35829 | Strong dependent choice. ... |
| fdc 35830 | Finite version of dependen... |
| fdc1 35831 | Variant of ~ fdc with no s... |
| seqpo 35832 | Two ways to say that a seq... |
| incsequz 35833 | An increasing sequence of ... |
| incsequz2 35834 | An increasing sequence of ... |
| nnubfi 35835 | A bounded above set of pos... |
| nninfnub 35836 | An infinite set of positiv... |
| subspopn 35837 | An open set is open in the... |
| neificl 35838 | Neighborhoods are closed u... |
| lpss2 35839 | Limit points of a subset a... |
| metf1o 35840 | Use a bijection with a met... |
| blssp 35841 | A ball in the subspace met... |
| mettrifi 35842 | Generalized triangle inequ... |
| lmclim2 35843 | A sequence in a metric spa... |
| geomcau 35844 | If the distance between co... |
| caures 35845 | The restriction of a Cauch... |
| caushft 35846 | A shifted Cauchy sequence ... |
| constcncf 35847 | A constant function is a c... |
| cnres2 35848 | The restriction of a conti... |
| cnresima 35849 | A continuous function is c... |
| cncfres 35850 | A continuous function on c... |
| istotbnd 35854 | The predicate "is a totall... |
| istotbnd2 35855 | The predicate "is a totall... |
| istotbnd3 35856 | A metric space is totally ... |
| totbndmet 35857 | The predicate "totally bou... |
| 0totbnd 35858 | The metric (there is only ... |
| sstotbnd2 35859 | Condition for a subset of ... |
| sstotbnd 35860 | Condition for a subset of ... |
| sstotbnd3 35861 | Use a net that is not nece... |
| totbndss 35862 | A subset of a totally boun... |
| equivtotbnd 35863 | If the metric ` M ` is "st... |
| isbnd 35865 | The predicate "is a bounde... |
| bndmet 35866 | A bounded metric space is ... |
| isbndx 35867 | A "bounded extended metric... |
| isbnd2 35868 | The predicate "is a bounde... |
| isbnd3 35869 | A metric space is bounded ... |
| isbnd3b 35870 | A metric space is bounded ... |
| bndss 35871 | A subset of a bounded metr... |
| blbnd 35872 | A ball is bounded. (Contr... |
| ssbnd 35873 | A subset of a metric space... |
| totbndbnd 35874 | A totally bounded metric s... |
| equivbnd 35875 | If the metric ` M ` is "st... |
| bnd2lem 35876 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 35877 | If balls are totally bound... |
| prdsbnd 35878 | The product metric over fi... |
| prdstotbnd 35879 | The product metric over fi... |
| prdsbnd2 35880 | If balls are totally bound... |
| cntotbnd 35881 | A subset of the complex nu... |
| cnpwstotbnd 35882 | A subset of ` A ^ I ` , wh... |
| ismtyval 35885 | The set of isometries betw... |
| isismty 35886 | The condition "is an isome... |
| ismtycnv 35887 | The inverse of an isometry... |
| ismtyima 35888 | The image of a ball under ... |
| ismtyhmeolem 35889 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 35890 | An isometry is a homeomorp... |
| ismtybndlem 35891 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 35892 | Isometries preserve bounde... |
| ismtyres 35893 | A restriction of an isomet... |
| heibor1lem 35894 | Lemma for ~ heibor1 . A c... |
| heibor1 35895 | One half of ~ heibor , tha... |
| heiborlem1 35896 | Lemma for ~ heibor . We w... |
| heiborlem2 35897 | Lemma for ~ heibor . Subs... |
| heiborlem3 35898 | Lemma for ~ heibor . Usin... |
| heiborlem4 35899 | Lemma for ~ heibor . Usin... |
| heiborlem5 35900 | Lemma for ~ heibor . The ... |
| heiborlem6 35901 | Lemma for ~ heibor . Sinc... |
| heiborlem7 35902 | Lemma for ~ heibor . Sinc... |
| heiborlem8 35903 | Lemma for ~ heibor . The ... |
| heiborlem9 35904 | Lemma for ~ heibor . Disc... |
| heiborlem10 35905 | Lemma for ~ heibor . The ... |
| heibor 35906 | Generalized Heine-Borel Th... |
| bfplem1 35907 | Lemma for ~ bfp . The seq... |
| bfplem2 35908 | Lemma for ~ bfp . Using t... |
| bfp 35909 | Banach fixed point theorem... |
| rrnval 35912 | The n-dimensional Euclidea... |
| rrnmval 35913 | The value of the Euclidean... |
| rrnmet 35914 | Euclidean space is a metri... |
| rrndstprj1 35915 | The distance between two p... |
| rrndstprj2 35916 | Bound on the distance betw... |
| rrncmslem 35917 | Lemma for ~ rrncms . (Con... |
| rrncms 35918 | Euclidean space is complet... |
| repwsmet 35919 | The supremum metric on ` R... |
| rrnequiv 35920 | The supremum metric on ` R... |
| rrntotbnd 35921 | A set in Euclidean space i... |
| rrnheibor 35922 | Heine-Borel theorem for Eu... |
| ismrer1 35923 | An isometry between ` RR `... |
| reheibor 35924 | Heine-Borel theorem for re... |
| iccbnd 35925 | A closed interval in ` RR ... |
| icccmpALT 35926 | A closed interval in ` RR ... |
| isass 35931 | The predicate "is an assoc... |
| isexid 35932 | The predicate ` G ` has a ... |
| ismgmOLD 35935 | Obsolete version of ~ ismg... |
| clmgmOLD 35936 | Obsolete version of ~ mgmc... |
| opidonOLD 35937 | Obsolete version of ~ mndp... |
| rngopidOLD 35938 | Obsolete version of ~ mndp... |
| opidon2OLD 35939 | Obsolete version of ~ mndp... |
| isexid2 35940 | If ` G e. ( Magma i^i ExId... |
| exidu1 35941 | Uniqueness of the left and... |
| idrval 35942 | The value of the identity ... |
| iorlid 35943 | A magma right and left ide... |
| cmpidelt 35944 | A magma right and left ide... |
| smgrpismgmOLD 35947 | Obsolete version of ~ sgrp... |
| issmgrpOLD 35948 | Obsolete version of ~ issg... |
| smgrpmgm 35949 | A semigroup is a magma. (... |
| smgrpassOLD 35950 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 35953 | Obsolete version of ~ mnds... |
| mndoisexid 35954 | A monoid has an identity e... |
| mndoismgmOLD 35955 | Obsolete version of ~ mndm... |
| mndomgmid 35956 | A monoid is a magma with a... |
| ismndo 35957 | The predicate "is a monoid... |
| ismndo1 35958 | The predicate "is a monoid... |
| ismndo2 35959 | The predicate "is a monoid... |
| grpomndo 35960 | A group is a monoid. (Con... |
| exidcl 35961 | Closure of the binary oper... |
| exidreslem 35962 | Lemma for ~ exidres and ~ ... |
| exidres 35963 | The restriction of a binar... |
| exidresid 35964 | The restriction of a binar... |
| ablo4pnp 35965 | A commutative/associative ... |
| grpoeqdivid 35966 | Two group elements are equ... |
| grposnOLD 35967 | The group operation for th... |
| elghomlem1OLD 35970 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 35971 | Obsolete as of 15-Mar-2020... |
| elghomOLD 35972 | Obsolete version of ~ isgh... |
| ghomlinOLD 35973 | Obsolete version of ~ ghml... |
| ghomidOLD 35974 | Obsolete version of ~ ghmi... |
| ghomf 35975 | Mapping property of a grou... |
| ghomco 35976 | The composition of two gro... |
| ghomdiv 35977 | Group homomorphisms preser... |
| grpokerinj 35978 | A group homomorphism is in... |
| relrngo 35981 | The class of all unital ri... |
| isrngo 35982 | The predicate "is a (unita... |
| isrngod 35983 | Conditions that determine ... |
| rngoi 35984 | The properties of a unital... |
| rngosm 35985 | Functionality of the multi... |
| rngocl 35986 | Closure of the multiplicat... |
| rngoid 35987 | The multiplication operati... |
| rngoideu 35988 | The unit element of a ring... |
| rngodi 35989 | Distributive law for the m... |
| rngodir 35990 | Distributive law for the m... |
| rngoass 35991 | Associative law for the mu... |
| rngo2 35992 | A ring element plus itself... |
| rngoablo 35993 | A ring's addition operatio... |
| rngoablo2 35994 | In a unital ring the addit... |
| rngogrpo 35995 | A ring's addition operatio... |
| rngone0 35996 | The base set of a ring is ... |
| rngogcl 35997 | Closure law for the additi... |
| rngocom 35998 | The addition operation of ... |
| rngoaass 35999 | The addition operation of ... |
| rngoa32 36000 | The addition operation of ... |
| rngoa4 36001 | Rearrangement of 4 terms i... |
| rngorcan 36002 | Right cancellation law for... |
| rngolcan 36003 | Left cancellation law for ... |
| rngo0cl 36004 | A ring has an additive ide... |
| rngo0rid 36005 | The additive identity of a... |
| rngo0lid 36006 | The additive identity of a... |
| rngolz 36007 | The zero of a unital ring ... |
| rngorz 36008 | The zero of a unital ring ... |
| rngosn3 36009 | Obsolete as of 25-Jan-2020... |
| rngosn4 36010 | Obsolete as of 25-Jan-2020... |
| rngosn6 36011 | Obsolete as of 25-Jan-2020... |
| rngonegcl 36012 | A ring is closed under neg... |
| rngoaddneg1 36013 | Adding the negative in a r... |
| rngoaddneg2 36014 | Adding the negative in a r... |
| rngosub 36015 | Subtraction in a ring, in ... |
| rngmgmbs4 36016 | The range of an internal o... |
| rngodm1dm2 36017 | In a unital ring the domai... |
| rngorn1 36018 | In a unital ring the range... |
| rngorn1eq 36019 | In a unital ring the range... |
| rngomndo 36020 | In a unital ring the multi... |
| rngoidmlem 36021 | The unit of a ring is an i... |
| rngolidm 36022 | The unit of a ring is an i... |
| rngoridm 36023 | The unit of a ring is an i... |
| rngo1cl 36024 | The unit of a ring belongs... |
| rngoueqz 36025 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 36026 | Negation in a ring is the ... |
| rngonegmn1r 36027 | Negation in a ring is the ... |
| rngoneglmul 36028 | Negation of a product in a... |
| rngonegrmul 36029 | Negation of a product in a... |
| rngosubdi 36030 | Ring multiplication distri... |
| rngosubdir 36031 | Ring multiplication distri... |
| zerdivemp1x 36032 | In a unitary ring a left i... |
| isdivrngo 36035 | The predicate "is a divisi... |
| drngoi 36036 | The properties of a divisi... |
| gidsn 36037 | Obsolete as of 23-Jan-2020... |
| zrdivrng 36038 | The zero ring is not a div... |
| dvrunz 36039 | In a division ring the uni... |
| isgrpda 36040 | Properties that determine ... |
| isdrngo1 36041 | The predicate "is a divisi... |
| divrngcl 36042 | The product of two nonzero... |
| isdrngo2 36043 | A division ring is a ring ... |
| isdrngo3 36044 | A division ring is a ring ... |
| rngohomval 36049 | The set of ring homomorphi... |
| isrngohom 36050 | The predicate "is a ring h... |
| rngohomf 36051 | A ring homomorphism is a f... |
| rngohomcl 36052 | Closure law for a ring hom... |
| rngohom1 36053 | A ring homomorphism preser... |
| rngohomadd 36054 | Ring homomorphisms preserv... |
| rngohommul 36055 | Ring homomorphisms preserv... |
| rngogrphom 36056 | A ring homomorphism is a g... |
| rngohom0 36057 | A ring homomorphism preser... |
| rngohomsub 36058 | Ring homomorphisms preserv... |
| rngohomco 36059 | The composition of two rin... |
| rngokerinj 36060 | A ring homomorphism is inj... |
| rngoisoval 36062 | The set of ring isomorphis... |
| isrngoiso 36063 | The predicate "is a ring i... |
| rngoiso1o 36064 | A ring isomorphism is a bi... |
| rngoisohom 36065 | A ring isomorphism is a ri... |
| rngoisocnv 36066 | The inverse of a ring isom... |
| rngoisoco 36067 | The composition of two rin... |
| isriscg 36069 | The ring isomorphism relat... |
| isrisc 36070 | The ring isomorphism relat... |
| risc 36071 | The ring isomorphism relat... |
| risci 36072 | Determine that two rings a... |
| riscer 36073 | Ring isomorphism is an equ... |
| iscom2 36080 | A device to add commutativ... |
| iscrngo 36081 | The predicate "is a commut... |
| iscrngo2 36082 | The predicate "is a commut... |
| iscringd 36083 | Conditions that determine ... |
| flddivrng 36084 | A field is a division ring... |
| crngorngo 36085 | A commutative ring is a ri... |
| crngocom 36086 | The multiplication operati... |
| crngm23 36087 | Commutative/associative la... |
| crngm4 36088 | Commutative/associative la... |
| fldcrng 36089 | A field is a commutative r... |
| isfld2 36090 | The predicate "is a field"... |
| crngohomfo 36091 | The image of a homomorphis... |
| idlval 36098 | The class of ideals of a r... |
| isidl 36099 | The predicate "is an ideal... |
| isidlc 36100 | The predicate "is an ideal... |
| idlss 36101 | An ideal of ` R ` is a sub... |
| idlcl 36102 | An element of an ideal is ... |
| idl0cl 36103 | An ideal contains ` 0 ` . ... |
| idladdcl 36104 | An ideal is closed under a... |
| idllmulcl 36105 | An ideal is closed under m... |
| idlrmulcl 36106 | An ideal is closed under m... |
| idlnegcl 36107 | An ideal is closed under n... |
| idlsubcl 36108 | An ideal is closed under s... |
| rngoidl 36109 | A ring ` R ` is an ` R ` i... |
| 0idl 36110 | The set containing only ` ... |
| 1idl 36111 | Two ways of expressing the... |
| 0rngo 36112 | In a ring, ` 0 = 1 ` iff t... |
| divrngidl 36113 | The only ideals in a divis... |
| intidl 36114 | The intersection of a none... |
| inidl 36115 | The intersection of two id... |
| unichnidl 36116 | The union of a nonempty ch... |
| keridl 36117 | The kernel of a ring homom... |
| pridlval 36118 | The class of prime ideals ... |
| ispridl 36119 | The predicate "is a prime ... |
| pridlidl 36120 | A prime ideal is an ideal.... |
| pridlnr 36121 | A prime ideal is a proper ... |
| pridl 36122 | The main property of a pri... |
| ispridl2 36123 | A condition that shows an ... |
| maxidlval 36124 | The set of maximal ideals ... |
| ismaxidl 36125 | The predicate "is a maxima... |
| maxidlidl 36126 | A maximal ideal is an idea... |
| maxidlnr 36127 | A maximal ideal is proper.... |
| maxidlmax 36128 | A maximal ideal is a maxim... |
| maxidln1 36129 | One is not contained in an... |
| maxidln0 36130 | A ring with a maximal idea... |
| isprrngo 36135 | The predicate "is a prime ... |
| prrngorngo 36136 | A prime ring is a ring. (... |
| smprngopr 36137 | A simple ring (one whose o... |
| divrngpr 36138 | A division ring is a prime... |
| isdmn 36139 | The predicate "is a domain... |
| isdmn2 36140 | The predicate "is a domain... |
| dmncrng 36141 | A domain is a commutative ... |
| dmnrngo 36142 | A domain is a ring. (Cont... |
| flddmn 36143 | A field is a domain. (Con... |
| igenval 36146 | The ideal generated by a s... |
| igenss 36147 | A set is a subset of the i... |
| igenidl 36148 | The ideal generated by a s... |
| igenmin 36149 | The ideal generated by a s... |
| igenidl2 36150 | The ideal generated by an ... |
| igenval2 36151 | The ideal generated by a s... |
| prnc 36152 | A principal ideal (an idea... |
| isfldidl 36153 | Determine if a ring is a f... |
| isfldidl2 36154 | Determine if a ring is a f... |
| ispridlc 36155 | The predicate "is a prime ... |
| pridlc 36156 | Property of a prime ideal ... |
| pridlc2 36157 | Property of a prime ideal ... |
| pridlc3 36158 | Property of a prime ideal ... |
| isdmn3 36159 | The predicate "is a domain... |
| dmnnzd 36160 | A domain has no zero-divis... |
| dmncan1 36161 | Cancellation law for domai... |
| dmncan2 36162 | Cancellation law for domai... |
| efald2 36163 | A proof by contradiction. ... |
| notbinot1 36164 | Simplification rule of neg... |
| bicontr 36165 | Biconditional of its own n... |
| impor 36166 | An equivalent formula for ... |
| orfa 36167 | The falsum ` F. ` can be r... |
| notbinot2 36168 | Commutation rule between n... |
| biimpor 36169 | A rewriting rule for bicon... |
| orfa1 36170 | Add a contradicting disjun... |
| orfa2 36171 | Remove a contradicting dis... |
| bifald 36172 | Infer the equivalence to a... |
| orsild 36173 | A lemma for not-or-not eli... |
| orsird 36174 | A lemma for not-or-not eli... |
| cnf1dd 36175 | A lemma for Conjunctive No... |
| cnf2dd 36176 | A lemma for Conjunctive No... |
| cnfn1dd 36177 | A lemma for Conjunctive No... |
| cnfn2dd 36178 | A lemma for Conjunctive No... |
| or32dd 36179 | A rearrangement of disjunc... |
| notornotel1 36180 | A lemma for not-or-not eli... |
| notornotel2 36181 | A lemma for not-or-not eli... |
| contrd 36182 | A proof by contradiction, ... |
| an12i 36183 | An inference from commutin... |
| exmid2 36184 | An excluded middle law. (... |
| selconj 36185 | An inference for selecting... |
| truconj 36186 | Add true as a conjunct. (... |
| orel 36187 | An inference for disjuncti... |
| negel 36188 | An inference for negation ... |
| botel 36189 | An inference for bottom el... |
| tradd 36190 | Add top ad a conjunct. (C... |
| gm-sbtru 36191 | Substitution does not chan... |
| sbfal 36192 | Substitution does not chan... |
| sbcani 36193 | Distribution of class subs... |
| sbcori 36194 | Distribution of class subs... |
| sbcimi 36195 | Distribution of class subs... |
| sbcni 36196 | Move class substitution in... |
| sbali 36197 | Discard class substitution... |
| sbexi 36198 | Discard class substitution... |
| sbcalf 36199 | Move universal quantifier ... |
| sbcexf 36200 | Move existential quantifie... |
| sbcalfi 36201 | Move universal quantifier ... |
| sbcexfi 36202 | Move existential quantifie... |
| spsbcdi 36203 | A lemma for eliminating a ... |
| alrimii 36204 | A lemma for introducing a ... |
| spesbcdi 36205 | A lemma for introducing an... |
| exlimddvf 36206 | A lemma for eliminating an... |
| exlimddvfi 36207 | A lemma for eliminating an... |
| sbceq1ddi 36208 | A lemma for eliminating in... |
| sbccom2lem 36209 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 36210 | Commutative law for double... |
| sbccom2f 36211 | Commutative law for double... |
| sbccom2fi 36212 | Commutative law for double... |
| csbcom2fi 36213 | Commutative law for double... |
| fald 36214 | Refutation of falsity, in ... |
| tsim1 36215 | A Tseitin axiom for logica... |
| tsim2 36216 | A Tseitin axiom for logica... |
| tsim3 36217 | A Tseitin axiom for logica... |
| tsbi1 36218 | A Tseitin axiom for logica... |
| tsbi2 36219 | A Tseitin axiom for logica... |
| tsbi3 36220 | A Tseitin axiom for logica... |
| tsbi4 36221 | A Tseitin axiom for logica... |
| tsxo1 36222 | A Tseitin axiom for logica... |
| tsxo2 36223 | A Tseitin axiom for logica... |
| tsxo3 36224 | A Tseitin axiom for logica... |
| tsxo4 36225 | A Tseitin axiom for logica... |
| tsan1 36226 | A Tseitin axiom for logica... |
| tsan2 36227 | A Tseitin axiom for logica... |
| tsan3 36228 | A Tseitin axiom for logica... |
| tsna1 36229 | A Tseitin axiom for logica... |
| tsna2 36230 | A Tseitin axiom for logica... |
| tsna3 36231 | A Tseitin axiom for logica... |
| tsor1 36232 | A Tseitin axiom for logica... |
| tsor2 36233 | A Tseitin axiom for logica... |
| tsor3 36234 | A Tseitin axiom for logica... |
| ts3an1 36235 | A Tseitin axiom for triple... |
| ts3an2 36236 | A Tseitin axiom for triple... |
| ts3an3 36237 | A Tseitin axiom for triple... |
| ts3or1 36238 | A Tseitin axiom for triple... |
| ts3or2 36239 | A Tseitin axiom for triple... |
| ts3or3 36240 | A Tseitin axiom for triple... |
| iuneq2f 36241 | Equality deduction for ind... |
| rabeq12f 36242 | Equality deduction for res... |
| csbeq12 36243 | Equality deduction for sub... |
| sbeqi 36244 | Equality deduction for sub... |
| ralbi12f 36245 | Equality deduction for res... |
| oprabbi 36246 | Equality deduction for cla... |
| mpobi123f 36247 | Equality deduction for map... |
| iuneq12f 36248 | Equality deduction for ind... |
| iineq12f 36249 | Equality deduction for ind... |
| opabbi 36250 | Equality deduction for cla... |
| mptbi12f 36251 | Equality deduction for map... |
| orcomdd 36252 | Commutativity of logic dis... |
| scottexf 36253 | A version of ~ scottex wit... |
| scott0f 36254 | A version of ~ scott0 with... |
| scottn0f 36255 | A version of ~ scott0f wit... |
| ac6s3f 36256 | Generalization of the Axio... |
| ac6s6 36257 | Generalization of the Axio... |
| ac6s6f 36258 | Generalization of the Axio... |
| el2v1 36297 | New way ( ~ elv , and the ... |
| el3v 36298 | New way ( ~ elv , and the ... |
| el3v1 36299 | New way ( ~ elv , and the ... |
| el3v2 36300 | New way ( ~ elv , and the ... |
| el3v3 36301 | New way ( ~ elv , and the ... |
| el3v12 36302 | New way ( ~ elv , and the ... |
| el3v13 36303 | New way ( ~ elv , and the ... |
| el3v23 36304 | New way ( ~ elv , and the ... |
| an2anr 36305 | Double commutation in conj... |
| anan 36306 | Multiple commutations in c... |
| triantru3 36307 | A wff is equivalent to its... |
| eqeltr 36308 | Substitution of equal clas... |
| eqelb 36309 | Substitution of equal clas... |
| eqeqan2d 36310 | Implication of introducing... |
| inres2 36311 | Two ways of expressing the... |
| coideq 36312 | Equality theorem for compo... |
| nexmo1 36313 | If there is no case where ... |
| 3albii 36314 | Inference adding three uni... |
| 3ralbii 36315 | Inference adding three res... |
| ssrabi 36316 | Inference of restricted ab... |
| rabbieq 36317 | Equivalent wff's correspon... |
| rabimbieq 36318 | Restricted equivalent wff'... |
| abeqin 36319 | Intersection with class ab... |
| abeqinbi 36320 | Intersection with class ab... |
| rabeqel 36321 | Class element of a restric... |
| eqrelf 36322 | The equality connective be... |
| releleccnv 36323 | Elementhood in a converse ... |
| releccnveq 36324 | Equality of converse ` R `... |
| opelvvdif 36325 | Negated elementhood of ord... |
| vvdifopab 36326 | Ordered-pair class abstrac... |
| brvdif 36327 | Binary relation with unive... |
| brvdif2 36328 | Binary relation with unive... |
| brvvdif 36329 | Binary relation with the c... |
| brvbrvvdif 36330 | Binary relation with the c... |
| brcnvep 36331 | The converse of the binary... |
| elecALTV 36332 | Elementhood in the ` R ` -... |
| brcnvepres 36333 | Restricted converse epsilo... |
| brres2 36334 | Binary relation on a restr... |
| eldmres 36335 | Elementhood in the domain ... |
| eldm4 36336 | Elementhood in a domain. ... |
| eldmres2 36337 | Elementhood in the domain ... |
| eceq1i 36338 | Equality theorem for ` C `... |
| elecres 36339 | Elementhood in the restric... |
| ecres 36340 | Restricted coset of ` B ` ... |
| ecres2 36341 | The restricted coset of ` ... |
| eccnvepres 36342 | Restricted converse epsilo... |
| eleccnvep 36343 | Elementhood in the convers... |
| eccnvep 36344 | The converse epsilon coset... |
| extep 36345 | Property of epsilon relati... |
| eccnvepres2 36346 | The restricted converse ep... |
| eccnvepres3 36347 | Condition for a restricted... |
| eldmqsres 36348 | Elementhood in a restricte... |
| eldmqsres2 36349 | Elementhood in a restricte... |
| qsss1 36350 | Subclass theorem for quoti... |
| qseq1i 36351 | Equality theorem for quoti... |
| qseq1d 36352 | Equality theorem for quoti... |
| brinxprnres 36353 | Binary relation on a restr... |
| inxprnres 36354 | Restriction of a class as ... |
| dfres4 36355 | Alternate definition of th... |
| exan3 36356 | Equivalent expressions wit... |
| exanres 36357 | Equivalent expressions wit... |
| exanres3 36358 | Equivalent expressions wit... |
| exanres2 36359 | Equivalent expressions wit... |
| cnvepres 36360 | Restricted converse epsilo... |
| ssrel3 36361 | Subclass relation in anoth... |
| eqrel2 36362 | Equality of relations. (C... |
| rncnv 36363 | Range of converse is the d... |
| dfdm6 36364 | Alternate definition of do... |
| dfrn6 36365 | Alternate definition of ra... |
| rncnvepres 36366 | The range of the restricte... |
| dmecd 36367 | Equality of the coset of `... |
| dmec2d 36368 | Equality of the coset of `... |
| brid 36369 | Property of the identity b... |
| ideq2 36370 | For sets, the identity bin... |
| idresssidinxp 36371 | Condition for the identity... |
| idreseqidinxp 36372 | Condition for the identity... |
| extid 36373 | Property of identity relat... |
| inxpss 36374 | Two ways to say that an in... |
| idinxpss 36375 | Two ways to say that an in... |
| inxpss3 36376 | Two ways to say that an in... |
| inxpss2 36377 | Two ways to say that inter... |
| inxpssidinxp 36378 | Two ways to say that inter... |
| idinxpssinxp 36379 | Two ways to say that inter... |
| idinxpssinxp2 36380 | Identity intersection with... |
| idinxpssinxp3 36381 | Identity intersection with... |
| idinxpssinxp4 36382 | Identity intersection with... |
| relcnveq3 36383 | Two ways of saying a relat... |
| relcnveq 36384 | Two ways of saying a relat... |
| relcnveq2 36385 | Two ways of saying a relat... |
| relcnveq4 36386 | Two ways of saying a relat... |
| qsresid 36387 | Simplification of a specia... |
| n0elqs 36388 | Two ways of expressing tha... |
| n0elqs2 36389 | Two ways of expressing tha... |
| ecex2 36390 | Condition for a coset to b... |
| uniqsALTV 36391 | The union of a quotient se... |
| imaexALTV 36392 | Existence of an image of a... |
| ecexALTV 36393 | Existence of a coset, like... |
| rnresequniqs 36394 | The range of a restriction... |
| n0el2 36395 | Two ways of expressing tha... |
| cnvepresex 36396 | Sethood condition for the ... |
| eccnvepex 36397 | The converse epsilon coset... |
| cnvepimaex 36398 | The image of converse epsi... |
| cnvepima 36399 | The image of converse epsi... |
| inex3 36400 | Sufficient condition for t... |
| inxpex 36401 | Sufficient condition for a... |
| eqres 36402 | Converting a class constan... |
| brrabga 36403 | The law of concretion for ... |
| brcnvrabga 36404 | The law of concretion for ... |
| opideq 36405 | Equality conditions for or... |
| iss2 36406 | A subclass of the identity... |
| eldmcnv 36407 | Elementhood in a domain of... |
| dfrel5 36408 | Alternate definition of th... |
| dfrel6 36409 | Alternate definition of th... |
| cnvresrn 36410 | Converse restricted to ran... |
| ecin0 36411 | Two ways of saying that th... |
| ecinn0 36412 | Two ways of saying that th... |
| ineleq 36413 | Equivalence of restricted ... |
| inecmo 36414 | Equivalence of a double re... |
| inecmo2 36415 | Equivalence of a double re... |
| ineccnvmo 36416 | Equivalence of a double re... |
| alrmomorn 36417 | Equivalence of an "at most... |
| alrmomodm 36418 | Equivalence of an "at most... |
| ineccnvmo2 36419 | Equivalence of a double un... |
| inecmo3 36420 | Equivalence of a double un... |
| moantr 36421 | Sufficient condition for t... |
| brabidgaw 36422 | The law of concretion for ... |
| brabidga 36423 | The law of concretion for ... |
| inxp2 36424 | Intersection with a Cartes... |
| opabf 36425 | A class abstraction of a c... |
| ec0 36426 | The empty-coset of a class... |
| 0qs 36427 | Quotient set with the empt... |
| xrnss3v 36429 | A range Cartesian product ... |
| xrnrel 36430 | A range Cartesian product ... |
| brxrn 36431 | Characterize a ternary rel... |
| brxrn2 36432 | A characterization of the ... |
| dfxrn2 36433 | Alternate definition of th... |
| xrneq1 36434 | Equality theorem for the r... |
| xrneq1i 36435 | Equality theorem for the r... |
| xrneq1d 36436 | Equality theorem for the r... |
| xrneq2 36437 | Equality theorem for the r... |
| xrneq2i 36438 | Equality theorem for the r... |
| xrneq2d 36439 | Equality theorem for the r... |
| xrneq12 36440 | Equality theorem for the r... |
| xrneq12i 36441 | Equality theorem for the r... |
| xrneq12d 36442 | Equality theorem for the r... |
| elecxrn 36443 | Elementhood in the ` ( R |... |
| ecxrn 36444 | The ` ( R |X. S ) ` -coset... |
| xrninxp 36445 | Intersection of a range Ca... |
| xrninxp2 36446 | Intersection of a range Ca... |
| xrninxpex 36447 | Sufficient condition for t... |
| inxpxrn 36448 | Two ways to express the in... |
| br1cnvxrn2 36449 | The converse of a binary r... |
| elec1cnvxrn2 36450 | Elementhood in the convers... |
| rnxrn 36451 | Range of the range Cartesi... |
| rnxrnres 36452 | Range of a range Cartesian... |
| rnxrncnvepres 36453 | Range of a range Cartesian... |
| rnxrnidres 36454 | Range of a range Cartesian... |
| xrnres 36455 | Two ways to express restri... |
| xrnres2 36456 | Two ways to express restri... |
| xrnres3 36457 | Two ways to express restri... |
| xrnres4 36458 | Two ways to express restri... |
| xrnresex 36459 | Sufficient condition for a... |
| xrnidresex 36460 | Sufficient condition for a... |
| xrncnvepresex 36461 | Sufficient condition for a... |
| brin2 36462 | Binary relation on an inte... |
| brin3 36463 | Binary relation on an inte... |
| dfcoss2 36466 | Alternate definition of th... |
| dfcoss3 36467 | Alternate definition of th... |
| dfcoss4 36468 | Alternate definition of th... |
| cossex 36469 | If ` A ` is a set then the... |
| cosscnvex 36470 | If ` A ` is a set then the... |
| 1cosscnvepresex 36471 | Sufficient condition for a... |
| 1cossxrncnvepresex 36472 | Sufficient condition for a... |
| relcoss 36473 | Cosets by ` R ` is a relat... |
| relcoels 36474 | Coelements on ` A ` is a r... |
| cossss 36475 | Subclass theorem for the c... |
| cosseq 36476 | Equality theorem for the c... |
| cosseqi 36477 | Equality theorem for the c... |
| cosseqd 36478 | Equality theorem for the c... |
| 1cossres 36479 | The class of cosets by a r... |
| dfcoels 36480 | Alternate definition of th... |
| brcoss 36481 | ` A ` and ` B ` are cosets... |
| brcoss2 36482 | Alternate form of the ` A ... |
| brcoss3 36483 | Alternate form of the ` A ... |
| brcosscnvcoss 36484 | For sets, the ` A ` and ` ... |
| brcoels 36485 | ` B ` and ` C ` are coelem... |
| cocossss 36486 | Two ways of saying that co... |
| cnvcosseq 36487 | The converse of cosets by ... |
| br2coss 36488 | Cosets by ` ,~ R ` binary ... |
| br1cossres 36489 | ` B ` and ` C ` are cosets... |
| br1cossres2 36490 | ` B ` and ` C ` are cosets... |
| relbrcoss 36491 | ` A ` and ` B ` are cosets... |
| br1cossinres 36492 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 36493 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 36494 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 36495 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 36496 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 36497 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 36498 | The domain of cosets is th... |
| dmcoss2 36499 | The domain of cosets is th... |
| rncossdmcoss 36500 | The range of cosets is the... |
| dm1cosscnvepres 36501 | The domain of cosets of th... |
| dmcoels 36502 | The domain of coelements i... |
| eldmcoss 36503 | Elementhood in the domain ... |
| eldmcoss2 36504 | Elementhood in the domain ... |
| eldm1cossres 36505 | Elementhood in the domain ... |
| eldm1cossres2 36506 | Elementhood in the domain ... |
| refrelcosslem 36507 | Lemma for the left side of... |
| refrelcoss3 36508 | The class of cosets by ` R... |
| refrelcoss2 36509 | The class of cosets by ` R... |
| symrelcoss3 36510 | The class of cosets by ` R... |
| symrelcoss2 36511 | The class of cosets by ` R... |
| cossssid 36512 | Equivalent expressions for... |
| cossssid2 36513 | Equivalent expressions for... |
| cossssid3 36514 | Equivalent expressions for... |
| cossssid4 36515 | Equivalent expressions for... |
| cossssid5 36516 | Equivalent expressions for... |
| brcosscnv 36517 | ` A ` and ` B ` are cosets... |
| brcosscnv2 36518 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 36519 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 36520 | Cosets by the converse ran... |
| cosscnvssid3 36521 | Equivalent expressions for... |
| cosscnvssid4 36522 | Equivalent expressions for... |
| cosscnvssid5 36523 | Equivalent expressions for... |
| coss0 36524 | Cosets by the empty set ar... |
| cossid 36525 | Cosets by the identity rel... |
| cosscnvid 36526 | Cosets by the converse ide... |
| trcoss 36527 | Sufficient condition for t... |
| eleccossin 36528 | Two ways of saying that th... |
| trcoss2 36529 | Equivalent expressions for... |
| elrels2 36531 | The element of the relatio... |
| elrelsrel 36532 | The element of the relatio... |
| elrelsrelim 36533 | The element of the relatio... |
| elrels5 36534 | Equivalent expressions for... |
| elrels6 36535 | Equivalent expressions for... |
| elrelscnveq3 36536 | Two ways of saying a relat... |
| elrelscnveq 36537 | Two ways of saying a relat... |
| elrelscnveq2 36538 | Two ways of saying a relat... |
| elrelscnveq4 36539 | Two ways of saying a relat... |
| cnvelrels 36540 | The converse of a set is a... |
| cosselrels 36541 | Cosets of sets are element... |
| cosscnvelrels 36542 | Cosets of converse sets ar... |
| dfssr2 36544 | Alternate definition of th... |
| relssr 36545 | The subset relation is a r... |
| brssr 36546 | The subset relation and su... |
| brssrid 36547 | Any set is a subset of its... |
| issetssr 36548 | Two ways of expressing set... |
| brssrres 36549 | Restricted subset binary r... |
| br1cnvssrres 36550 | Restricted converse subset... |
| brcnvssr 36551 | The converse of a subset r... |
| brcnvssrid 36552 | Any set is a converse subs... |
| br1cossxrncnvssrres 36553 | ` <. B , C >. ` and ` <. D... |
| extssr 36554 | Property of subset relatio... |
| dfrefrels2 36558 | Alternate definition of th... |
| dfrefrels3 36559 | Alternate definition of th... |
| dfrefrel2 36560 | Alternate definition of th... |
| dfrefrel3 36561 | Alternate definition of th... |
| elrefrels2 36562 | Element of the class of re... |
| elrefrels3 36563 | Element of the class of re... |
| elrefrelsrel 36564 | For sets, being an element... |
| refreleq 36565 | Equality theorem for refle... |
| refrelid 36566 | Identity relation is refle... |
| refrelcoss 36567 | The class of cosets by ` R... |
| dfcnvrefrels2 36571 | Alternate definition of th... |
| dfcnvrefrels3 36572 | Alternate definition of th... |
| dfcnvrefrel2 36573 | Alternate definition of th... |
| dfcnvrefrel3 36574 | Alternate definition of th... |
| elcnvrefrels2 36575 | Element of the class of co... |
| elcnvrefrels3 36576 | Element of the class of co... |
| elcnvrefrelsrel 36577 | For sets, being an element... |
| cnvrefrelcoss2 36578 | Necessary and sufficient c... |
| cosselcnvrefrels2 36579 | Necessary and sufficient c... |
| cosselcnvrefrels3 36580 | Necessary and sufficient c... |
| cosselcnvrefrels4 36581 | Necessary and sufficient c... |
| cosselcnvrefrels5 36582 | Necessary and sufficient c... |
| dfsymrels2 36586 | Alternate definition of th... |
| dfsymrels3 36587 | Alternate definition of th... |
| dfsymrels4 36588 | Alternate definition of th... |
| dfsymrels5 36589 | Alternate definition of th... |
| dfsymrel2 36590 | Alternate definition of th... |
| dfsymrel3 36591 | Alternate definition of th... |
| dfsymrel4 36592 | Alternate definition of th... |
| dfsymrel5 36593 | Alternate definition of th... |
| elsymrels2 36594 | Element of the class of sy... |
| elsymrels3 36595 | Element of the class of sy... |
| elsymrels4 36596 | Element of the class of sy... |
| elsymrels5 36597 | Element of the class of sy... |
| elsymrelsrel 36598 | For sets, being an element... |
| symreleq 36599 | Equality theorem for symme... |
| symrelim 36600 | Symmetric relation implies... |
| symrelcoss 36601 | The class of cosets by ` R... |
| idsymrel 36602 | The identity relation is s... |
| epnsymrel 36603 | The membership (epsilon) r... |
| symrefref2 36604 | Symmetry is a sufficient c... |
| symrefref3 36605 | Symmetry is a sufficient c... |
| refsymrels2 36606 | Elements of the class of r... |
| refsymrels3 36607 | Elements of the class of r... |
| refsymrel2 36608 | A relation which is reflex... |
| refsymrel3 36609 | A relation which is reflex... |
| elrefsymrels2 36610 | Elements of the class of r... |
| elrefsymrels3 36611 | Elements of the class of r... |
| elrefsymrelsrel 36612 | For sets, being an element... |
| dftrrels2 36616 | Alternate definition of th... |
| dftrrels3 36617 | Alternate definition of th... |
| dftrrel2 36618 | Alternate definition of th... |
| dftrrel3 36619 | Alternate definition of th... |
| eltrrels2 36620 | Element of the class of tr... |
| eltrrels3 36621 | Element of the class of tr... |
| eltrrelsrel 36622 | For sets, being an element... |
| trreleq 36623 | Equality theorem for the t... |
| dfeqvrels2 36628 | Alternate definition of th... |
| dfeqvrels3 36629 | Alternate definition of th... |
| dfeqvrel2 36630 | Alternate definition of th... |
| dfeqvrel3 36631 | Alternate definition of th... |
| eleqvrels2 36632 | Element of the class of eq... |
| eleqvrels3 36633 | Element of the class of eq... |
| eleqvrelsrel 36634 | For sets, being an element... |
| elcoeleqvrels 36635 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 36636 | For sets, being an element... |
| eqvrelrel 36637 | An equivalence relation is... |
| eqvrelrefrel 36638 | An equivalence relation is... |
| eqvrelsymrel 36639 | An equivalence relation is... |
| eqvreltrrel 36640 | An equivalence relation is... |
| eqvrelim 36641 | Equivalence relation impli... |
| eqvreleq 36642 | Equality theorem for equiv... |
| eqvreleqi 36643 | Equality theorem for equiv... |
| eqvreleqd 36644 | Equality theorem for equiv... |
| eqvrelsym 36645 | An equivalence relation is... |
| eqvrelsymb 36646 | An equivalence relation is... |
| eqvreltr 36647 | An equivalence relation is... |
| eqvreltrd 36648 | A transitivity relation fo... |
| eqvreltr4d 36649 | A transitivity relation fo... |
| eqvrelref 36650 | An equivalence relation is... |
| eqvrelth 36651 | Basic property of equivale... |
| eqvrelcl 36652 | Elementhood in the field o... |
| eqvrelthi 36653 | Basic property of equivale... |
| eqvreldisj 36654 | Equivalence classes do not... |
| qsdisjALTV 36655 | Elements of a quotient set... |
| eqvrelqsel 36656 | If an element of a quotien... |
| eqvrelcoss 36657 | Two ways to express equiva... |
| eqvrelcoss3 36658 | Two ways to express equiva... |
| eqvrelcoss2 36659 | Two ways to express equiva... |
| eqvrelcoss4 36660 | Two ways to express equiva... |
| dfcoeleqvrels 36661 | Alternate definition of th... |
| dfcoeleqvrel 36662 | Alternate definition of th... |
| brredunds 36666 | Binary relation on the cla... |
| brredundsredund 36667 | For sets, binary relation ... |
| redundss3 36668 | Implication of redundancy ... |
| redundeq1 36669 | Equivalence of redundancy ... |
| redundpim3 36670 | Implication of redundancy ... |
| redundpbi1 36671 | Equivalence of redundancy ... |
| refrelsredund4 36672 | The naive version of the c... |
| refrelsredund2 36673 | The naive version of the c... |
| refrelsredund3 36674 | The naive version of the c... |
| refrelredund4 36675 | The naive version of the d... |
| refrelredund2 36676 | The naive version of the d... |
| refrelredund3 36677 | The naive version of the d... |
| dmqseq 36680 | Equality theorem for domai... |
| dmqseqi 36681 | Equality theorem for domai... |
| dmqseqd 36682 | Equality theorem for domai... |
| dmqseqeq1 36683 | Equality theorem for domai... |
| dmqseqeq1i 36684 | Equality theorem for domai... |
| dmqseqeq1d 36685 | Equality theorem for domai... |
| brdmqss 36686 | The domain quotient binary... |
| brdmqssqs 36687 | If ` A ` and ` R ` are set... |
| n0eldmqs 36688 | The empty set is not an el... |
| n0eldmqseq 36689 | The empty set is not an el... |
| n0el3 36690 | Two ways of expressing tha... |
| cnvepresdmqss 36691 | The domain quotient binary... |
| cnvepresdmqs 36692 | The domain quotient predic... |
| unidmqs 36693 | The range of a relation is... |
| unidmqseq 36694 | The union of the domain qu... |
| dmqseqim 36695 | If the domain quotient of ... |
| dmqseqim2 36696 | Lemma for ~ erim2 . (Cont... |
| releldmqs 36697 | Elementhood in the domain ... |
| eldmqs1cossres 36698 | Elementhood in the domain ... |
| releldmqscoss 36699 | Elementhood in the domain ... |
| dmqscoelseq 36700 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 36701 | Two ways to express the eq... |
| brers 36706 | Binary equivalence relatio... |
| dferALTV2 36707 | Equivalence relation with ... |
| erALTVeq1 36708 | Equality theorem for equiv... |
| erALTVeq1i 36709 | Equality theorem for equiv... |
| erALTVeq1d 36710 | Equality theorem for equiv... |
| dfmember 36711 | Alternate definition of th... |
| dfmember2 36712 | Alternate definition of th... |
| dfmember3 36713 | Alternate definition of th... |
| eqvreldmqs 36714 | Two ways to express member... |
| brerser 36715 | Binary equivalence relatio... |
| erim2 36716 | Equivalence relation on it... |
| erim 36717 | Equivalence relation on it... |
| dffunsALTV 36721 | Alternate definition of th... |
| dffunsALTV2 36722 | Alternate definition of th... |
| dffunsALTV3 36723 | Alternate definition of th... |
| dffunsALTV4 36724 | Alternate definition of th... |
| dffunsALTV5 36725 | Alternate definition of th... |
| dffunALTV2 36726 | Alternate definition of th... |
| dffunALTV3 36727 | Alternate definition of th... |
| dffunALTV4 36728 | Alternate definition of th... |
| dffunALTV5 36729 | Alternate definition of th... |
| elfunsALTV 36730 | Elementhood in the class o... |
| elfunsALTV2 36731 | Elementhood in the class o... |
| elfunsALTV3 36732 | Elementhood in the class o... |
| elfunsALTV4 36733 | Elementhood in the class o... |
| elfunsALTV5 36734 | Elementhood in the class o... |
| elfunsALTVfunALTV 36735 | The element of the class o... |
| funALTVfun 36736 | Our definition of the func... |
| funALTVss 36737 | Subclass theorem for funct... |
| funALTVeq 36738 | Equality theorem for funct... |
| funALTVeqi 36739 | Equality inference for the... |
| funALTVeqd 36740 | Equality deduction for the... |
| dfdisjs 36746 | Alternate definition of th... |
| dfdisjs2 36747 | Alternate definition of th... |
| dfdisjs3 36748 | Alternate definition of th... |
| dfdisjs4 36749 | Alternate definition of th... |
| dfdisjs5 36750 | Alternate definition of th... |
| dfdisjALTV 36751 | Alternate definition of th... |
| dfdisjALTV2 36752 | Alternate definition of th... |
| dfdisjALTV3 36753 | Alternate definition of th... |
| dfdisjALTV4 36754 | Alternate definition of th... |
| dfdisjALTV5 36755 | Alternate definition of th... |
| dfeldisj2 36756 | Alternate definition of th... |
| dfeldisj3 36757 | Alternate definition of th... |
| dfeldisj4 36758 | Alternate definition of th... |
| dfeldisj5 36759 | Alternate definition of th... |
| eldisjs 36760 | Elementhood in the class o... |
| eldisjs2 36761 | Elementhood in the class o... |
| eldisjs3 36762 | Elementhood in the class o... |
| eldisjs4 36763 | Elementhood in the class o... |
| eldisjs5 36764 | Elementhood in the class o... |
| eldisjsdisj 36765 | The element of the class o... |
| eleldisjs 36766 | Elementhood in the disjoin... |
| eleldisjseldisj 36767 | The element of the disjoin... |
| disjrel 36768 | Disjoint relation is a rel... |
| disjss 36769 | Subclass theorem for disjo... |
| disjssi 36770 | Subclass theorem for disjo... |
| disjssd 36771 | Subclass theorem for disjo... |
| disjeq 36772 | Equality theorem for disjo... |
| disjeqi 36773 | Equality theorem for disjo... |
| disjeqd 36774 | Equality theorem for disjo... |
| disjdmqseqeq1 36775 | Lemma for the equality the... |
| eldisjss 36776 | Subclass theorem for disjo... |
| eldisjssi 36777 | Subclass theorem for disjo... |
| eldisjssd 36778 | Subclass theorem for disjo... |
| eldisjeq 36779 | Equality theorem for disjo... |
| eldisjeqi 36780 | Equality theorem for disjo... |
| eldisjeqd 36781 | Equality theorem for disjo... |
| disjxrn 36782 | Two ways of saying that a ... |
| disjorimxrn 36783 | Disjointness condition for... |
| disjimxrn 36784 | Disjointness condition for... |
| disjimres 36785 | Disjointness condition for... |
| disjimin 36786 | Disjointness condition for... |
| disjiminres 36787 | Disjointness condition for... |
| disjimxrnres 36788 | Disjointness condition for... |
| disjALTV0 36789 | The null class is disjoint... |
| disjALTVid 36790 | The class of identity rela... |
| disjALTVidres 36791 | The class of identity rela... |
| disjALTVinidres 36792 | The intersection with rest... |
| disjALTVxrnidres 36793 | The class of range Cartesi... |
| prtlem60 36794 | Lemma for ~ prter3 . (Con... |
| bicomdd 36795 | Commute two sides of a bic... |
| jca2r 36796 | Inference conjoining the c... |
| jca3 36797 | Inference conjoining the c... |
| prtlem70 36798 | Lemma for ~ prter3 : a rea... |
| ibdr 36799 | Reverse of ~ ibd . (Contr... |
| prtlem100 36800 | Lemma for ~ prter3 . (Con... |
| prtlem5 36801 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 36802 | Lemma for ~ prter2 . (Con... |
| brabsb2 36803 | A closed form of ~ brabsb ... |
| eqbrrdv2 36804 | Other version of ~ eqbrrdi... |
| prtlem9 36805 | Lemma for ~ prter3 . (Con... |
| prtlem10 36806 | Lemma for ~ prter3 . (Con... |
| prtlem11 36807 | Lemma for ~ prter2 . (Con... |
| prtlem12 36808 | Lemma for ~ prtex and ~ pr... |
| prtlem13 36809 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 36810 | Lemma for ~ prtex , ~ prte... |
| prtlem400 36811 | Lemma for ~ prter2 and als... |
| erprt 36814 | The quotient set of an equ... |
| prtlem14 36815 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 36816 | Lemma for ~ prter1 and ~ p... |
| prtlem17 36817 | Lemma for ~ prter2 . (Con... |
| prtlem18 36818 | Lemma for ~ prter2 . (Con... |
| prtlem19 36819 | Lemma for ~ prter2 . (Con... |
| prter1 36820 | Every partition generates ... |
| prtex 36821 | The equivalence relation g... |
| prter2 36822 | The quotient set of the eq... |
| prter3 36823 | For every partition there ... |
| axc5 36834 | This theorem repeats ~ sp ... |
| ax4fromc4 36835 | Rederivation of Axiom ~ ax... |
| ax10fromc7 36836 | Rederivation of Axiom ~ ax... |
| ax6fromc10 36837 | Rederivation of Axiom ~ ax... |
| hba1-o 36838 | The setvar ` x ` is not fr... |
| axc4i-o 36839 | Inference version of ~ ax-... |
| equid1 36840 | Proof of ~ equid from our ... |
| equcomi1 36841 | Proof of ~ equcomi from ~ ... |
| aecom-o 36842 | Commutation law for identi... |
| aecoms-o 36843 | A commutation rule for ide... |
| hbae-o 36844 | All variables are effectiv... |
| dral1-o 36845 | Formula-building lemma for... |
| ax12fromc15 36846 | Rederivation of Axiom ~ ax... |
| ax13fromc9 36847 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 36848 | Axiom to quantify a variab... |
| sps-o 36849 | Generalization of antecede... |
| hbequid 36850 | Bound-variable hypothesis ... |
| nfequid-o 36851 | Bound-variable hypothesis ... |
| axc5c7 36852 | Proof of a single axiom th... |
| axc5c7toc5 36853 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 36854 | Rederivation of ~ ax-c7 fr... |
| axc711 36855 | Proof of a single axiom th... |
| nfa1-o 36856 | ` x ` is not free in ` A. ... |
| axc711toc7 36857 | Rederivation of ~ ax-c7 fr... |
| axc711to11 36858 | Rederivation of ~ ax-11 fr... |
| axc5c711 36859 | Proof of a single axiom th... |
| axc5c711toc5 36860 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 36861 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 36862 | Rederivation of ~ ax-11 fr... |
| equidqe 36863 | ~ equid with existential q... |
| axc5sp1 36864 | A special case of ~ ax-c5 ... |
| equidq 36865 | ~ equid with universal qua... |
| equid1ALT 36866 | Alternate proof of ~ equid... |
| axc11nfromc11 36867 | Rederivation of ~ ax-c11n ... |
| naecoms-o 36868 | A commutation rule for dis... |
| hbnae-o 36869 | All variables are effectiv... |
| dvelimf-o 36870 | Proof of ~ dvelimh that us... |
| dral2-o 36871 | Formula-building lemma for... |
| aev-o 36872 | A "distinctor elimination"... |
| ax5eq 36873 | Theorem to add distinct qu... |
| dveeq2-o 36874 | Quantifier introduction wh... |
| axc16g-o 36875 | A generalization of Axiom ... |
| dveeq1-o 36876 | Quantifier introduction wh... |
| dveeq1-o16 36877 | Version of ~ dveeq1 using ... |
| ax5el 36878 | Theorem to add distinct qu... |
| axc11n-16 36879 | This theorem shows that, g... |
| dveel2ALT 36880 | Alternate proof of ~ dveel... |
| ax12f 36881 | Basis step for constructin... |
| ax12eq 36882 | Basis step for constructin... |
| ax12el 36883 | Basis step for constructin... |
| ax12indn 36884 | Induction step for constru... |
| ax12indi 36885 | Induction step for constru... |
| ax12indalem 36886 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 36887 | Alternate proof of ~ ax12i... |
| ax12inda2 36888 | Induction step for constru... |
| ax12inda 36889 | Induction step for constru... |
| ax12v2-o 36890 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 36891 | Derive ~ ax-c15 from a hyp... |
| axc11-o 36892 | Show that ~ ax-c11 can be ... |
| fsumshftd 36893 | Index shift of a finite su... |
| riotaclbgBAD 36895 | Closure of restricted iota... |
| riotaclbBAD 36896 | Closure of restricted iota... |
| riotasvd 36897 | Deduction version of ~ rio... |
| riotasv2d 36898 | Value of description binde... |
| riotasv2s 36899 | The value of description b... |
| riotasv 36900 | Value of description binde... |
| riotasv3d 36901 | A property ` ch ` holding ... |
| elimhyps 36902 | A version of ~ elimhyp usi... |
| dedths 36903 | A version of weak deductio... |
| renegclALT 36904 | Closure law for negative o... |
| elimhyps2 36905 | Generalization of ~ elimhy... |
| dedths2 36906 | Generalization of ~ dedths... |
| nfcxfrdf 36907 | A utility lemma to transfe... |
| nfded 36908 | A deduction theorem that c... |
| nfded2 36909 | A deduction theorem that c... |
| nfunidALT2 36910 | Deduction version of ~ nfu... |
| nfunidALT 36911 | Deduction version of ~ nfu... |
| nfopdALT 36912 | Deduction version of bound... |
| cnaddcom 36913 | Recover the commutative la... |
| toycom 36914 | Show the commutative law f... |
| lshpset 36919 | The set of all hyperplanes... |
| islshp 36920 | The predicate "is a hyperp... |
| islshpsm 36921 | Hyperplane properties expr... |
| lshplss 36922 | A hyperplane is a subspace... |
| lshpne 36923 | A hyperplane is not equal ... |
| lshpnel 36924 | A hyperplane's generating ... |
| lshpnelb 36925 | The subspace sum of a hype... |
| lshpnel2N 36926 | Condition that determines ... |
| lshpne0 36927 | The member of the span in ... |
| lshpdisj 36928 | A hyperplane and the span ... |
| lshpcmp 36929 | If two hyperplanes are com... |
| lshpinN 36930 | The intersection of two di... |
| lsatset 36931 | The set of all 1-dim subsp... |
| islsat 36932 | The predicate "is a 1-dim ... |
| lsatlspsn2 36933 | The span of a nonzero sing... |
| lsatlspsn 36934 | The span of a nonzero sing... |
| islsati 36935 | A 1-dim subspace (atom) (o... |
| lsateln0 36936 | A 1-dim subspace (atom) (o... |
| lsatlss 36937 | The set of 1-dim subspaces... |
| lsatlssel 36938 | An atom is a subspace. (C... |
| lsatssv 36939 | An atom is a set of vector... |
| lsatn0 36940 | A 1-dim subspace (atom) of... |
| lsatspn0 36941 | The span of a vector is an... |
| lsator0sp 36942 | The span of a vector is ei... |
| lsatssn0 36943 | A subspace (or any class) ... |
| lsatcmp 36944 | If two atoms are comparabl... |
| lsatcmp2 36945 | If an atom is included in ... |
| lsatel 36946 | A nonzero vector in an ato... |
| lsatelbN 36947 | A nonzero vector in an ato... |
| lsat2el 36948 | Two atoms sharing a nonzer... |
| lsmsat 36949 | Convert comparison of atom... |
| lsatfixedN 36950 | Show equality with the spa... |
| lsmsatcv 36951 | Subspace sum has the cover... |
| lssatomic 36952 | The lattice of subspaces i... |
| lssats 36953 | The lattice of subspaces i... |
| lpssat 36954 | Two subspaces in a proper ... |
| lrelat 36955 | Subspaces are relatively a... |
| lssatle 36956 | The ordering of two subspa... |
| lssat 36957 | Two subspaces in a proper ... |
| islshpat 36958 | Hyperplane properties expr... |
| lcvfbr 36961 | The covers relation for a ... |
| lcvbr 36962 | The covers relation for a ... |
| lcvbr2 36963 | The covers relation for a ... |
| lcvbr3 36964 | The covers relation for a ... |
| lcvpss 36965 | The covers relation implie... |
| lcvnbtwn 36966 | The covers relation implie... |
| lcvntr 36967 | The covers relation is not... |
| lcvnbtwn2 36968 | The covers relation implie... |
| lcvnbtwn3 36969 | The covers relation implie... |
| lsmcv2 36970 | Subspace sum has the cover... |
| lcvat 36971 | If a subspace covers anoth... |
| lsatcv0 36972 | An atom covers the zero su... |
| lsatcveq0 36973 | A subspace covered by an a... |
| lsat0cv 36974 | A subspace is an atom iff ... |
| lcvexchlem1 36975 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 36976 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 36977 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 36978 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 36979 | Lemma for ~ lcvexch . (Co... |
| lcvexch 36980 | Subspaces satisfy the exch... |
| lcvp 36981 | Covering property of Defin... |
| lcv1 36982 | Covering property of a sub... |
| lcv2 36983 | Covering property of a sub... |
| lsatexch 36984 | The atom exchange property... |
| lsatnle 36985 | The meet of a subspace and... |
| lsatnem0 36986 | The meet of distinct atoms... |
| lsatexch1 36987 | The atom exch1ange propert... |
| lsatcv0eq 36988 | If the sum of two atoms co... |
| lsatcv1 36989 | Two atoms covering the zer... |
| lsatcvatlem 36990 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 36991 | A nonzero subspace less th... |
| lsatcvat2 36992 | A subspace covered by the ... |
| lsatcvat3 36993 | A condition implying that ... |
| islshpcv 36994 | Hyperplane properties expr... |
| l1cvpat 36995 | A subspace covered by the ... |
| l1cvat 36996 | Create an atom under an el... |
| lshpat 36997 | Create an atom under a hyp... |
| lflset 37000 | The set of linear function... |
| islfl 37001 | The predicate "is a linear... |
| lfli 37002 | Property of a linear funct... |
| islfld 37003 | Properties that determine ... |
| lflf 37004 | A linear functional is a f... |
| lflcl 37005 | A linear functional value ... |
| lfl0 37006 | A linear functional is zer... |
| lfladd 37007 | Property of a linear funct... |
| lflsub 37008 | Property of a linear funct... |
| lflmul 37009 | Property of a linear funct... |
| lfl0f 37010 | The zero function is a fun... |
| lfl1 37011 | A nonzero functional has a... |
| lfladdcl 37012 | Closure of addition of two... |
| lfladdcom 37013 | Commutativity of functiona... |
| lfladdass 37014 | Associativity of functiona... |
| lfladd0l 37015 | Functional addition with t... |
| lflnegcl 37016 | Closure of the negative of... |
| lflnegl 37017 | A functional plus its nega... |
| lflvscl 37018 | Closure of a scalar produc... |
| lflvsdi1 37019 | Distributive law for (righ... |
| lflvsdi2 37020 | Reverse distributive law f... |
| lflvsdi2a 37021 | Reverse distributive law f... |
| lflvsass 37022 | Associative law for (right... |
| lfl0sc 37023 | The (right vector space) s... |
| lflsc0N 37024 | The scalar product with th... |
| lfl1sc 37025 | The (right vector space) s... |
| lkrfval 37028 | The kernel of a functional... |
| lkrval 37029 | Value of the kernel of a f... |
| ellkr 37030 | Membership in the kernel o... |
| lkrval2 37031 | Value of the kernel of a f... |
| ellkr2 37032 | Membership in the kernel o... |
| lkrcl 37033 | A member of the kernel of ... |
| lkrf0 37034 | The value of a functional ... |
| lkr0f 37035 | The kernel of the zero fun... |
| lkrlss 37036 | The kernel of a linear fun... |
| lkrssv 37037 | The kernel of a linear fun... |
| lkrsc 37038 | The kernel of a nonzero sc... |
| lkrscss 37039 | The kernel of a scalar pro... |
| eqlkr 37040 | Two functionals with the s... |
| eqlkr2 37041 | Two functionals with the s... |
| eqlkr3 37042 | Two functionals with the s... |
| lkrlsp 37043 | The subspace sum of a kern... |
| lkrlsp2 37044 | The subspace sum of a kern... |
| lkrlsp3 37045 | The subspace sum of a kern... |
| lkrshp 37046 | The kernel of a nonzero fu... |
| lkrshp3 37047 | The kernels of nonzero fun... |
| lkrshpor 37048 | The kernel of a functional... |
| lkrshp4 37049 | A kernel is a hyperplane i... |
| lshpsmreu 37050 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 37051 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 37052 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 37053 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 37054 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 37055 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 37056 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 37057 | The set ` G ` defined by h... |
| lshpkr 37058 | The kernel of functional `... |
| lshpkrex 37059 | There exists a functional ... |
| lshpset2N 37060 | The set of all hyperplanes... |
| islshpkrN 37061 | The predicate "is a hyperp... |
| lfl1dim 37062 | Equivalent expressions for... |
| lfl1dim2N 37063 | Equivalent expressions for... |
| ldualset 37066 | Define the (left) dual of ... |
| ldualvbase 37067 | The vectors of a dual spac... |
| ldualelvbase 37068 | Utility theorem for conver... |
| ldualfvadd 37069 | Vector addition in the dua... |
| ldualvadd 37070 | Vector addition in the dua... |
| ldualvaddcl 37071 | The value of vector additi... |
| ldualvaddval 37072 | The value of the value of ... |
| ldualsca 37073 | The ring of scalars of the... |
| ldualsbase 37074 | Base set of scalar ring fo... |
| ldualsaddN 37075 | Scalar addition for the du... |
| ldualsmul 37076 | Scalar multiplication for ... |
| ldualfvs 37077 | Scalar product operation f... |
| ldualvs 37078 | Scalar product operation v... |
| ldualvsval 37079 | Value of scalar product op... |
| ldualvscl 37080 | The scalar product operati... |
| ldualvaddcom 37081 | Commutative law for vector... |
| ldualvsass 37082 | Associative law for scalar... |
| ldualvsass2 37083 | Associative law for scalar... |
| ldualvsdi1 37084 | Distributive law for scala... |
| ldualvsdi2 37085 | Reverse distributive law f... |
| ldualgrplem 37086 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 37087 | The dual of a vector space... |
| ldual0 37088 | The zero scalar of the dua... |
| ldual1 37089 | The unit scalar of the dua... |
| ldualneg 37090 | The negative of a scalar o... |
| ldual0v 37091 | The zero vector of the dua... |
| ldual0vcl 37092 | The dual zero vector is a ... |
| lduallmodlem 37093 | Lemma for ~ lduallmod . (... |
| lduallmod 37094 | The dual of a left module ... |
| lduallvec 37095 | The dual of a left vector ... |
| ldualvsub 37096 | The value of vector subtra... |
| ldualvsubcl 37097 | Closure of vector subtract... |
| ldualvsubval 37098 | The value of the value of ... |
| ldualssvscl 37099 | Closure of scalar product ... |
| ldualssvsubcl 37100 | Closure of vector subtract... |
| ldual0vs 37101 | Scalar zero times a functi... |
| lkr0f2 37102 | The kernel of the zero fun... |
| lduallkr3 37103 | The kernels of nonzero fun... |
| lkrpssN 37104 | Proper subset relation bet... |
| lkrin 37105 | Intersection of the kernel... |
| eqlkr4 37106 | Two functionals with the s... |
| ldual1dim 37107 | Equivalent expressions for... |
| ldualkrsc 37108 | The kernel of a nonzero sc... |
| lkrss 37109 | The kernel of a scalar pro... |
| lkrss2N 37110 | Two functionals with kerne... |
| lkreqN 37111 | Proportional functionals h... |
| lkrlspeqN 37112 | Condition for colinear fun... |
| isopos 37121 | The predicate "is an ortho... |
| opposet 37122 | Every orthoposet is a pose... |
| oposlem 37123 | Lemma for orthoposet prope... |
| op01dm 37124 | Conditions necessary for z... |
| op0cl 37125 | An orthoposet has a zero e... |
| op1cl 37126 | An orthoposet has a unit e... |
| op0le 37127 | Orthoposet zero is less th... |
| ople0 37128 | An element less than or eq... |
| opnlen0 37129 | An element not less than a... |
| lub0N 37130 | The least upper bound of t... |
| opltn0 37131 | A lattice element greater ... |
| ople1 37132 | Any element is less than t... |
| op1le 37133 | If the orthoposet unit is ... |
| glb0N 37134 | The greatest lower bound o... |
| opoccl 37135 | Closure of orthocomplement... |
| opococ 37136 | Double negative law for or... |
| opcon3b 37137 | Contraposition law for ort... |
| opcon2b 37138 | Orthocomplement contraposi... |
| opcon1b 37139 | Orthocomplement contraposi... |
| oplecon3 37140 | Contraposition law for ort... |
| oplecon3b 37141 | Contraposition law for ort... |
| oplecon1b 37142 | Contraposition law for str... |
| opoc1 37143 | Orthocomplement of orthopo... |
| opoc0 37144 | Orthocomplement of orthopo... |
| opltcon3b 37145 | Contraposition law for str... |
| opltcon1b 37146 | Contraposition law for str... |
| opltcon2b 37147 | Contraposition law for str... |
| opexmid 37148 | Law of excluded middle for... |
| opnoncon 37149 | Law of contradiction for o... |
| riotaocN 37150 | The orthocomplement of the... |
| cmtfvalN 37151 | Value of commutes relation... |
| cmtvalN 37152 | Equivalence for commutes r... |
| isolat 37153 | The predicate "is an ortho... |
| ollat 37154 | An ortholattice is a latti... |
| olop 37155 | An ortholattice is an orth... |
| olposN 37156 | An ortholattice is a poset... |
| isolatiN 37157 | Properties that determine ... |
| oldmm1 37158 | De Morgan's law for meet i... |
| oldmm2 37159 | De Morgan's law for meet i... |
| oldmm3N 37160 | De Morgan's law for meet i... |
| oldmm4 37161 | De Morgan's law for meet i... |
| oldmj1 37162 | De Morgan's law for join i... |
| oldmj2 37163 | De Morgan's law for join i... |
| oldmj3 37164 | De Morgan's law for join i... |
| oldmj4 37165 | De Morgan's law for join i... |
| olj01 37166 | An ortholattice element jo... |
| olj02 37167 | An ortholattice element jo... |
| olm11 37168 | The meet of an ortholattic... |
| olm12 37169 | The meet of an ortholattic... |
| latmassOLD 37170 | Ortholattice meet is assoc... |
| latm12 37171 | A rearrangement of lattice... |
| latm32 37172 | A rearrangement of lattice... |
| latmrot 37173 | Rotate lattice meet of 3 c... |
| latm4 37174 | Rearrangement of lattice m... |
| latmmdiN 37175 | Lattice meet distributes o... |
| latmmdir 37176 | Lattice meet distributes o... |
| olm01 37177 | Meet with lattice zero is ... |
| olm02 37178 | Meet with lattice zero is ... |
| isoml 37179 | The predicate "is an ortho... |
| isomliN 37180 | Properties that determine ... |
| omlol 37181 | An orthomodular lattice is... |
| omlop 37182 | An orthomodular lattice is... |
| omllat 37183 | An orthomodular lattice is... |
| omllaw 37184 | The orthomodular law. (Co... |
| omllaw2N 37185 | Variation of orthomodular ... |
| omllaw3 37186 | Orthomodular law equivalen... |
| omllaw4 37187 | Orthomodular law equivalen... |
| omllaw5N 37188 | The orthomodular law. Rem... |
| cmtcomlemN 37189 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 37190 | Commutation is symmetric. ... |
| cmt2N 37191 | Commutation with orthocomp... |
| cmt3N 37192 | Commutation with orthocomp... |
| cmt4N 37193 | Commutation with orthocomp... |
| cmtbr2N 37194 | Alternate definition of th... |
| cmtbr3N 37195 | Alternate definition for t... |
| cmtbr4N 37196 | Alternate definition for t... |
| lecmtN 37197 | Ordered elements commute. ... |
| cmtidN 37198 | Any element commutes with ... |
| omlfh1N 37199 | Foulis-Holland Theorem, pa... |
| omlfh3N 37200 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 37201 | Analogue of modular law ~ ... |
| omlspjN 37202 | Contraction of a Sasaki pr... |
| cvrfval 37209 | Value of covers relation "... |
| cvrval 37210 | Binary relation expressing... |
| cvrlt 37211 | The covers relation implie... |
| cvrnbtwn 37212 | There is no element betwee... |
| ncvr1 37213 | No element covers the latt... |
| cvrletrN 37214 | Property of an element abo... |
| cvrval2 37215 | Binary relation expressing... |
| cvrnbtwn2 37216 | The covers relation implie... |
| cvrnbtwn3 37217 | The covers relation implie... |
| cvrcon3b 37218 | Contraposition law for the... |
| cvrle 37219 | The covers relation implie... |
| cvrnbtwn4 37220 | The covers relation implie... |
| cvrnle 37221 | The covers relation implie... |
| cvrne 37222 | The covers relation implie... |
| cvrnrefN 37223 | The covers relation is not... |
| cvrcmp 37224 | If two lattice elements th... |
| cvrcmp2 37225 | If two lattice elements co... |
| pats 37226 | The set of atoms in a pose... |
| isat 37227 | The predicate "is an atom"... |
| isat2 37228 | The predicate "is an atom"... |
| atcvr0 37229 | An atom covers zero. ( ~ ... |
| atbase 37230 | An atom is a member of the... |
| atssbase 37231 | The set of atoms is a subs... |
| 0ltat 37232 | An atom is greater than ze... |
| leatb 37233 | A poset element less than ... |
| leat 37234 | A poset element less than ... |
| leat2 37235 | A nonzero poset element le... |
| leat3 37236 | A poset element less than ... |
| meetat 37237 | The meet of any element wi... |
| meetat2 37238 | The meet of any element wi... |
| isatl 37240 | The predicate "is an atomi... |
| atllat 37241 | An atomic lattice is a lat... |
| atlpos 37242 | An atomic lattice is a pos... |
| atl0dm 37243 | Condition necessary for ze... |
| atl0cl 37244 | An atomic lattice has a ze... |
| atl0le 37245 | Orthoposet zero is less th... |
| atlle0 37246 | An element less than or eq... |
| atlltn0 37247 | A lattice element greater ... |
| isat3 37248 | The predicate "is an atom"... |
| atn0 37249 | An atom is not zero. ( ~ ... |
| atnle0 37250 | An atom is not less than o... |
| atlen0 37251 | A lattice element is nonze... |
| atcmp 37252 | If two atoms are comparabl... |
| atncmp 37253 | Frequently-used variation ... |
| atnlt 37254 | Two atoms cannot satisfy t... |
| atcvreq0 37255 | An element covered by an a... |
| atncvrN 37256 | Two atoms cannot satisfy t... |
| atlex 37257 | Every nonzero element of a... |
| atnle 37258 | Two ways of expressing "an... |
| atnem0 37259 | The meet of distinct atoms... |
| atlatmstc 37260 | An atomic, complete, ortho... |
| atlatle 37261 | The ordering of two Hilber... |
| atlrelat1 37262 | An atomistic lattice with ... |
| iscvlat 37264 | The predicate "is an atomi... |
| iscvlat2N 37265 | The predicate "is an atomi... |
| cvlatl 37266 | An atomic lattice with the... |
| cvllat 37267 | An atomic lattice with the... |
| cvlposN 37268 | An atomic lattice with the... |
| cvlexch1 37269 | An atomic covering lattice... |
| cvlexch2 37270 | An atomic covering lattice... |
| cvlexchb1 37271 | An atomic covering lattice... |
| cvlexchb2 37272 | An atomic covering lattice... |
| cvlexch3 37273 | An atomic covering lattice... |
| cvlexch4N 37274 | An atomic covering lattice... |
| cvlatexchb1 37275 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 37276 | A version of ~ cvlexchb2 f... |
| cvlatexch1 37277 | Atom exchange property. (... |
| cvlatexch2 37278 | Atom exchange property. (... |
| cvlatexch3 37279 | Atom exchange property. (... |
| cvlcvr1 37280 | The covering property. Pr... |
| cvlcvrp 37281 | A Hilbert lattice satisfie... |
| cvlatcvr1 37282 | An atom is covered by its ... |
| cvlatcvr2 37283 | An atom is covered by its ... |
| cvlsupr2 37284 | Two equivalent ways of exp... |
| cvlsupr3 37285 | Two equivalent ways of exp... |
| cvlsupr4 37286 | Consequence of superpositi... |
| cvlsupr5 37287 | Consequence of superpositi... |
| cvlsupr6 37288 | Consequence of superpositi... |
| cvlsupr7 37289 | Consequence of superpositi... |
| cvlsupr8 37290 | Consequence of superpositi... |
| ishlat1 37293 | The predicate "is a Hilber... |
| ishlat2 37294 | The predicate "is a Hilber... |
| ishlat3N 37295 | The predicate "is a Hilber... |
| ishlatiN 37296 | Properties that determine ... |
| hlomcmcv 37297 | A Hilbert lattice is ortho... |
| hloml 37298 | A Hilbert lattice is ortho... |
| hlclat 37299 | A Hilbert lattice is compl... |
| hlcvl 37300 | A Hilbert lattice is an at... |
| hlatl 37301 | A Hilbert lattice is atomi... |
| hlol 37302 | A Hilbert lattice is an or... |
| hlop 37303 | A Hilbert lattice is an or... |
| hllat 37304 | A Hilbert lattice is a lat... |
| hllatd 37305 | Deduction form of ~ hllat ... |
| hlomcmat 37306 | A Hilbert lattice is ortho... |
| hlpos 37307 | A Hilbert lattice is a pos... |
| hlatjcl 37308 | Closure of join operation.... |
| hlatjcom 37309 | Commutatitivity of join op... |
| hlatjidm 37310 | Idempotence of join operat... |
| hlatjass 37311 | Lattice join is associativ... |
| hlatj12 37312 | Swap 1st and 2nd members o... |
| hlatj32 37313 | Swap 2nd and 3rd members o... |
| hlatjrot 37314 | Rotate lattice join of 3 c... |
| hlatj4 37315 | Rearrangement of lattice j... |
| hlatlej1 37316 | A join's first argument is... |
| hlatlej2 37317 | A join's second argument i... |
| glbconN 37318 | De Morgan's law for GLB an... |
| glbconxN 37319 | De Morgan's law for GLB an... |
| atnlej1 37320 | If an atom is not less tha... |
| atnlej2 37321 | If an atom is not less tha... |
| hlsuprexch 37322 | A Hilbert lattice has the ... |
| hlexch1 37323 | A Hilbert lattice has the ... |
| hlexch2 37324 | A Hilbert lattice has the ... |
| hlexchb1 37325 | A Hilbert lattice has the ... |
| hlexchb2 37326 | A Hilbert lattice has the ... |
| hlsupr 37327 | A Hilbert lattice has the ... |
| hlsupr2 37328 | A Hilbert lattice has the ... |
| hlhgt4 37329 | A Hilbert lattice has a he... |
| hlhgt2 37330 | A Hilbert lattice has a he... |
| hl0lt1N 37331 | Lattice 0 is less than lat... |
| hlexch3 37332 | A Hilbert lattice has the ... |
| hlexch4N 37333 | A Hilbert lattice has the ... |
| hlatexchb1 37334 | A version of ~ hlexchb1 fo... |
| hlatexchb2 37335 | A version of ~ hlexchb2 fo... |
| hlatexch1 37336 | Atom exchange property. (... |
| hlatexch2 37337 | Atom exchange property. (... |
| hlatmstcOLDN 37338 | An atomic, complete, ortho... |
| hlatle 37339 | The ordering of two Hilber... |
| hlateq 37340 | The equality of two Hilber... |
| hlrelat1 37341 | An atomistic lattice with ... |
| hlrelat5N 37342 | An atomistic lattice with ... |
| hlrelat 37343 | A Hilbert lattice is relat... |
| hlrelat2 37344 | A consequence of relative ... |
| exatleN 37345 | A condition for an atom to... |
| hl2at 37346 | A Hilbert lattice has at l... |
| atex 37347 | At least one atom exists. ... |
| intnatN 37348 | If the intersection with a... |
| 2llnne2N 37349 | Condition implying that tw... |
| 2llnneN 37350 | Condition implying that tw... |
| cvr1 37351 | A Hilbert lattice has the ... |
| cvr2N 37352 | Less-than and covers equiv... |
| hlrelat3 37353 | The Hilbert lattice is rel... |
| cvrval3 37354 | Binary relation expressing... |
| cvrval4N 37355 | Binary relation expressing... |
| cvrval5 37356 | Binary relation expressing... |
| cvrp 37357 | A Hilbert lattice satisfie... |
| atcvr1 37358 | An atom is covered by its ... |
| atcvr2 37359 | An atom is covered by its ... |
| cvrexchlem 37360 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 37361 | A Hilbert lattice satisfie... |
| cvratlem 37362 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 37363 | A nonzero Hilbert lattice ... |
| ltltncvr 37364 | A chained strong ordering ... |
| ltcvrntr 37365 | Non-transitive condition f... |
| cvrntr 37366 | The covers relation is not... |
| atcvr0eq 37367 | The covers relation is not... |
| lnnat 37368 | A line (the join of two di... |
| atcvrj0 37369 | Two atoms covering the zer... |
| cvrat2 37370 | A Hilbert lattice element ... |
| atcvrneN 37371 | Inequality derived from at... |
| atcvrj1 37372 | Condition for an atom to b... |
| atcvrj2b 37373 | Condition for an atom to b... |
| atcvrj2 37374 | Condition for an atom to b... |
| atleneN 37375 | Inequality derived from at... |
| atltcvr 37376 | An equivalence of less-tha... |
| atle 37377 | Any nonzero element has an... |
| atlt 37378 | Two atoms are unequal iff ... |
| atlelt 37379 | Transfer less-than relatio... |
| 2atlt 37380 | Given an atom less than an... |
| atexchcvrN 37381 | Atom exchange property. V... |
| atexchltN 37382 | Atom exchange property. V... |
| cvrat3 37383 | A condition implying that ... |
| cvrat4 37384 | A condition implying exist... |
| cvrat42 37385 | Commuted version of ~ cvra... |
| 2atjm 37386 | The meet of a line (expres... |
| atbtwn 37387 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 37388 | There exists a 3rd atom ` ... |
| atbtwnex 37389 | Given atoms ` P ` in ` X `... |
| 3noncolr2 37390 | Two ways to express 3 non-... |
| 3noncolr1N 37391 | Two ways to express 3 non-... |
| hlatcon3 37392 | Atom exchange combined wit... |
| hlatcon2 37393 | Atom exchange combined wit... |
| 4noncolr3 37394 | A way to express 4 non-col... |
| 4noncolr2 37395 | A way to express 4 non-col... |
| 4noncolr1 37396 | A way to express 4 non-col... |
| athgt 37397 | A Hilbert lattice, whose h... |
| 3dim0 37398 | There exists a 3-dimension... |
| 3dimlem1 37399 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 37400 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 37401 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 37402 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 37403 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 37404 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 37405 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 37406 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 37407 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 37408 | Construct a 3-dimensional ... |
| 3dim2 37409 | Construct 2 new layers on ... |
| 3dim3 37410 | Construct a new layer on t... |
| 2dim 37411 | Generate a height-3 elemen... |
| 1dimN 37412 | An atom is covered by a he... |
| 1cvrco 37413 | The orthocomplement of an ... |
| 1cvratex 37414 | There exists an atom less ... |
| 1cvratlt 37415 | An atom less than or equal... |
| 1cvrjat 37416 | An element covered by the ... |
| 1cvrat 37417 | Create an atom under an el... |
| ps-1 37418 | The join of two atoms ` R ... |
| ps-2 37419 | Lattice analogue for the p... |
| 2atjlej 37420 | Two atoms are different if... |
| hlatexch3N 37421 | Rearrange join of atoms in... |
| hlatexch4 37422 | Exchange 2 atoms. (Contri... |
| ps-2b 37423 | Variation of projective ge... |
| 3atlem1 37424 | Lemma for ~ 3at . (Contri... |
| 3atlem2 37425 | Lemma for ~ 3at . (Contri... |
| 3atlem3 37426 | Lemma for ~ 3at . (Contri... |
| 3atlem4 37427 | Lemma for ~ 3at . (Contri... |
| 3atlem5 37428 | Lemma for ~ 3at . (Contri... |
| 3atlem6 37429 | Lemma for ~ 3at . (Contri... |
| 3atlem7 37430 | Lemma for ~ 3at . (Contri... |
| 3at 37431 | Any three non-colinear ato... |
| llnset 37446 | The set of lattice lines i... |
| islln 37447 | The predicate "is a lattic... |
| islln4 37448 | The predicate "is a lattic... |
| llni 37449 | Condition implying a latti... |
| llnbase 37450 | A lattice line is a lattic... |
| islln3 37451 | The predicate "is a lattic... |
| islln2 37452 | The predicate "is a lattic... |
| llni2 37453 | The join of two different ... |
| llnnleat 37454 | An atom cannot majorize a ... |
| llnneat 37455 | A lattice line is not an a... |
| 2atneat 37456 | The join of two distinct a... |
| llnn0 37457 | A lattice line is nonzero.... |
| islln2a 37458 | The predicate "is a lattic... |
| llnle 37459 | Any element greater than 0... |
| atcvrlln2 37460 | An atom under a line is co... |
| atcvrlln 37461 | An element covering an ato... |
| llnexatN 37462 | Given an atom on a line, t... |
| llncmp 37463 | If two lattice lines are c... |
| llnnlt 37464 | Two lattice lines cannot s... |
| 2llnmat 37465 | Two intersecting lines int... |
| 2at0mat0 37466 | Special case of ~ 2atmat0 ... |
| 2atmat0 37467 | The meet of two unequal li... |
| 2atm 37468 | An atom majorized by two d... |
| ps-2c 37469 | Variation of projective ge... |
| lplnset 37470 | The set of lattice planes ... |
| islpln 37471 | The predicate "is a lattic... |
| islpln4 37472 | The predicate "is a lattic... |
| lplni 37473 | Condition implying a latti... |
| islpln3 37474 | The predicate "is a lattic... |
| lplnbase 37475 | A lattice plane is a latti... |
| islpln5 37476 | The predicate "is a lattic... |
| islpln2 37477 | The predicate "is a lattic... |
| lplni2 37478 | The join of 3 different at... |
| lvolex3N 37479 | There is an atom outside o... |
| llnmlplnN 37480 | The intersection of a line... |
| lplnle 37481 | Any element greater than 0... |
| lplnnle2at 37482 | A lattice line (or atom) c... |
| lplnnleat 37483 | A lattice plane cannot maj... |
| lplnnlelln 37484 | A lattice plane is not les... |
| 2atnelpln 37485 | The join of two atoms is n... |
| lplnneat 37486 | No lattice plane is an ato... |
| lplnnelln 37487 | No lattice plane is a latt... |
| lplnn0N 37488 | A lattice plane is nonzero... |
| islpln2a 37489 | The predicate "is a lattic... |
| islpln2ah 37490 | The predicate "is a lattic... |
| lplnriaN 37491 | Property of a lattice plan... |
| lplnribN 37492 | Property of a lattice plan... |
| lplnric 37493 | Property of a lattice plan... |
| lplnri1 37494 | Property of a lattice plan... |
| lplnri2N 37495 | Property of a lattice plan... |
| lplnri3N 37496 | Property of a lattice plan... |
| lplnllnneN 37497 | Two lattice lines defined ... |
| llncvrlpln2 37498 | A lattice line under a lat... |
| llncvrlpln 37499 | An element covering a latt... |
| 2lplnmN 37500 | If the join of two lattice... |
| 2llnmj 37501 | The meet of two lattice li... |
| 2atmat 37502 | The meet of two intersecti... |
| lplncmp 37503 | If two lattice planes are ... |
| lplnexatN 37504 | Given a lattice line on a ... |
| lplnexllnN 37505 | Given an atom on a lattice... |
| lplnnlt 37506 | Two lattice planes cannot ... |
| 2llnjaN 37507 | The join of two different ... |
| 2llnjN 37508 | The join of two different ... |
| 2llnm2N 37509 | The meet of two different ... |
| 2llnm3N 37510 | Two lattice lines in a lat... |
| 2llnm4 37511 | Two lattice lines that maj... |
| 2llnmeqat 37512 | An atom equals the interse... |
| lvolset 37513 | The set of 3-dim lattice v... |
| islvol 37514 | The predicate "is a 3-dim ... |
| islvol4 37515 | The predicate "is a 3-dim ... |
| lvoli 37516 | Condition implying a 3-dim... |
| islvol3 37517 | The predicate "is a 3-dim ... |
| lvoli3 37518 | Condition implying a 3-dim... |
| lvolbase 37519 | A 3-dim lattice volume is ... |
| islvol5 37520 | The predicate "is a 3-dim ... |
| islvol2 37521 | The predicate "is a 3-dim ... |
| lvoli2 37522 | The join of 4 different at... |
| lvolnle3at 37523 | A lattice plane (or lattic... |
| lvolnleat 37524 | An atom cannot majorize a ... |
| lvolnlelln 37525 | A lattice line cannot majo... |
| lvolnlelpln 37526 | A lattice plane cannot maj... |
| 3atnelvolN 37527 | The join of 3 atoms is not... |
| 2atnelvolN 37528 | The join of two atoms is n... |
| lvolneatN 37529 | No lattice volume is an at... |
| lvolnelln 37530 | No lattice volume is a lat... |
| lvolnelpln 37531 | No lattice volume is a lat... |
| lvoln0N 37532 | A lattice volume is nonzer... |
| islvol2aN 37533 | The predicate "is a lattic... |
| 4atlem0a 37534 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 37535 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 37536 | Lemma for ~ 4at . (Contri... |
| 4atlem3 37537 | Lemma for ~ 4at . Break i... |
| 4atlem3a 37538 | Lemma for ~ 4at . Break i... |
| 4atlem3b 37539 | Lemma for ~ 4at . Break i... |
| 4atlem4a 37540 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 37541 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 37542 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 37543 | Lemma for ~ 4at . Frequen... |
| 4atlem9 37544 | Lemma for ~ 4at . Substit... |
| 4atlem10a 37545 | Lemma for ~ 4at . Substit... |
| 4atlem10b 37546 | Lemma for ~ 4at . Substit... |
| 4atlem10 37547 | Lemma for ~ 4at . Combine... |
| 4atlem11a 37548 | Lemma for ~ 4at . Substit... |
| 4atlem11b 37549 | Lemma for ~ 4at . Substit... |
| 4atlem11 37550 | Lemma for ~ 4at . Combine... |
| 4atlem12a 37551 | Lemma for ~ 4at . Substit... |
| 4atlem12b 37552 | Lemma for ~ 4at . Substit... |
| 4atlem12 37553 | Lemma for ~ 4at . Combine... |
| 4at 37554 | Four atoms determine a lat... |
| 4at2 37555 | Four atoms determine a lat... |
| lplncvrlvol2 37556 | A lattice line under a lat... |
| lplncvrlvol 37557 | An element covering a latt... |
| lvolcmp 37558 | If two lattice planes are ... |
| lvolnltN 37559 | Two lattice volumes cannot... |
| 2lplnja 37560 | The join of two different ... |
| 2lplnj 37561 | The join of two different ... |
| 2lplnm2N 37562 | The meet of two different ... |
| 2lplnmj 37563 | The meet of two lattice pl... |
| dalemkehl 37564 | Lemma for ~ dath . Freque... |
| dalemkelat 37565 | Lemma for ~ dath . Freque... |
| dalemkeop 37566 | Lemma for ~ dath . Freque... |
| dalempea 37567 | Lemma for ~ dath . Freque... |
| dalemqea 37568 | Lemma for ~ dath . Freque... |
| dalemrea 37569 | Lemma for ~ dath . Freque... |
| dalemsea 37570 | Lemma for ~ dath . Freque... |
| dalemtea 37571 | Lemma for ~ dath . Freque... |
| dalemuea 37572 | Lemma for ~ dath . Freque... |
| dalemyeo 37573 | Lemma for ~ dath . Freque... |
| dalemzeo 37574 | Lemma for ~ dath . Freque... |
| dalemclpjs 37575 | Lemma for ~ dath . Freque... |
| dalemclqjt 37576 | Lemma for ~ dath . Freque... |
| dalemclrju 37577 | Lemma for ~ dath . Freque... |
| dalem-clpjq 37578 | Lemma for ~ dath . Freque... |
| dalemceb 37579 | Lemma for ~ dath . Freque... |
| dalempeb 37580 | Lemma for ~ dath . Freque... |
| dalemqeb 37581 | Lemma for ~ dath . Freque... |
| dalemreb 37582 | Lemma for ~ dath . Freque... |
| dalemseb 37583 | Lemma for ~ dath . Freque... |
| dalemteb 37584 | Lemma for ~ dath . Freque... |
| dalemueb 37585 | Lemma for ~ dath . Freque... |
| dalempjqeb 37586 | Lemma for ~ dath . Freque... |
| dalemsjteb 37587 | Lemma for ~ dath . Freque... |
| dalemtjueb 37588 | Lemma for ~ dath . Freque... |
| dalemqrprot 37589 | Lemma for ~ dath . Freque... |
| dalemyeb 37590 | Lemma for ~ dath . Freque... |
| dalemcnes 37591 | Lemma for ~ dath . Freque... |
| dalempnes 37592 | Lemma for ~ dath . Freque... |
| dalemqnet 37593 | Lemma for ~ dath . Freque... |
| dalempjsen 37594 | Lemma for ~ dath . Freque... |
| dalemply 37595 | Lemma for ~ dath . Freque... |
| dalemsly 37596 | Lemma for ~ dath . Freque... |
| dalemswapyz 37597 | Lemma for ~ dath . Swap t... |
| dalemrot 37598 | Lemma for ~ dath . Rotate... |
| dalemrotyz 37599 | Lemma for ~ dath . Rotate... |
| dalem1 37600 | Lemma for ~ dath . Show t... |
| dalemcea 37601 | Lemma for ~ dath . Freque... |
| dalem2 37602 | Lemma for ~ dath . Show t... |
| dalemdea 37603 | Lemma for ~ dath . Freque... |
| dalemeea 37604 | Lemma for ~ dath . Freque... |
| dalem3 37605 | Lemma for ~ dalemdnee . (... |
| dalem4 37606 | Lemma for ~ dalemdnee . (... |
| dalemdnee 37607 | Lemma for ~ dath . Axis o... |
| dalem5 37608 | Lemma for ~ dath . Atom `... |
| dalem6 37609 | Lemma for ~ dath . Analog... |
| dalem7 37610 | Lemma for ~ dath . Analog... |
| dalem8 37611 | Lemma for ~ dath . Plane ... |
| dalem-cly 37612 | Lemma for ~ dalem9 . Cent... |
| dalem9 37613 | Lemma for ~ dath . Since ... |
| dalem10 37614 | Lemma for ~ dath . Atom `... |
| dalem11 37615 | Lemma for ~ dath . Analog... |
| dalem12 37616 | Lemma for ~ dath . Analog... |
| dalem13 37617 | Lemma for ~ dalem14 . (Co... |
| dalem14 37618 | Lemma for ~ dath . Planes... |
| dalem15 37619 | Lemma for ~ dath . The ax... |
| dalem16 37620 | Lemma for ~ dath . The at... |
| dalem17 37621 | Lemma for ~ dath . When p... |
| dalem18 37622 | Lemma for ~ dath . Show t... |
| dalem19 37623 | Lemma for ~ dath . Show t... |
| dalemccea 37624 | Lemma for ~ dath . Freque... |
| dalemddea 37625 | Lemma for ~ dath . Freque... |
| dalem-ccly 37626 | Lemma for ~ dath . Freque... |
| dalem-ddly 37627 | Lemma for ~ dath . Freque... |
| dalemccnedd 37628 | Lemma for ~ dath . Freque... |
| dalemclccjdd 37629 | Lemma for ~ dath . Freque... |
| dalemcceb 37630 | Lemma for ~ dath . Freque... |
| dalemswapyzps 37631 | Lemma for ~ dath . Swap t... |
| dalemrotps 37632 | Lemma for ~ dath . Rotate... |
| dalemcjden 37633 | Lemma for ~ dath . Show t... |
| dalem20 37634 | Lemma for ~ dath . Show t... |
| dalem21 37635 | Lemma for ~ dath . Show t... |
| dalem22 37636 | Lemma for ~ dath . Show t... |
| dalem23 37637 | Lemma for ~ dath . Show t... |
| dalem24 37638 | Lemma for ~ dath . Show t... |
| dalem25 37639 | Lemma for ~ dath . Show t... |
| dalem27 37640 | Lemma for ~ dath . Show t... |
| dalem28 37641 | Lemma for ~ dath . Lemma ... |
| dalem29 37642 | Lemma for ~ dath . Analog... |
| dalem30 37643 | Lemma for ~ dath . Analog... |
| dalem31N 37644 | Lemma for ~ dath . Analog... |
| dalem32 37645 | Lemma for ~ dath . Analog... |
| dalem33 37646 | Lemma for ~ dath . Analog... |
| dalem34 37647 | Lemma for ~ dath . Analog... |
| dalem35 37648 | Lemma for ~ dath . Analog... |
| dalem36 37649 | Lemma for ~ dath . Analog... |
| dalem37 37650 | Lemma for ~ dath . Analog... |
| dalem38 37651 | Lemma for ~ dath . Plane ... |
| dalem39 37652 | Lemma for ~ dath . Auxili... |
| dalem40 37653 | Lemma for ~ dath . Analog... |
| dalem41 37654 | Lemma for ~ dath . (Contr... |
| dalem42 37655 | Lemma for ~ dath . Auxili... |
| dalem43 37656 | Lemma for ~ dath . Planes... |
| dalem44 37657 | Lemma for ~ dath . Dummy ... |
| dalem45 37658 | Lemma for ~ dath . Dummy ... |
| dalem46 37659 | Lemma for ~ dath . Analog... |
| dalem47 37660 | Lemma for ~ dath . Analog... |
| dalem48 37661 | Lemma for ~ dath . Analog... |
| dalem49 37662 | Lemma for ~ dath . Analog... |
| dalem50 37663 | Lemma for ~ dath . Analog... |
| dalem51 37664 | Lemma for ~ dath . Constr... |
| dalem52 37665 | Lemma for ~ dath . Lines ... |
| dalem53 37666 | Lemma for ~ dath . The au... |
| dalem54 37667 | Lemma for ~ dath . Line `... |
| dalem55 37668 | Lemma for ~ dath . Lines ... |
| dalem56 37669 | Lemma for ~ dath . Analog... |
| dalem57 37670 | Lemma for ~ dath . Axis o... |
| dalem58 37671 | Lemma for ~ dath . Analog... |
| dalem59 37672 | Lemma for ~ dath . Analog... |
| dalem60 37673 | Lemma for ~ dath . ` B ` i... |
| dalem61 37674 | Lemma for ~ dath . Show t... |
| dalem62 37675 | Lemma for ~ dath . Elimin... |
| dalem63 37676 | Lemma for ~ dath . Combin... |
| dath 37677 | Desargues's theorem of pro... |
| dath2 37678 | Version of Desargues's the... |
| lineset 37679 | The set of lines in a Hilb... |
| isline 37680 | The predicate "is a line".... |
| islinei 37681 | Condition implying "is a l... |
| pointsetN 37682 | The set of points in a Hil... |
| ispointN 37683 | The predicate "is a point"... |
| atpointN 37684 | The singleton of an atom i... |
| psubspset 37685 | The set of projective subs... |
| ispsubsp 37686 | The predicate "is a projec... |
| ispsubsp2 37687 | The predicate "is a projec... |
| psubspi 37688 | Property of a projective s... |
| psubspi2N 37689 | Property of a projective s... |
| 0psubN 37690 | The empty set is a project... |
| snatpsubN 37691 | The singleton of an atom i... |
| pointpsubN 37692 | A point (singleton of an a... |
| linepsubN 37693 | A line is a projective sub... |
| atpsubN 37694 | The set of all atoms is a ... |
| psubssat 37695 | A projective subspace cons... |
| psubatN 37696 | A member of a projective s... |
| pmapfval 37697 | The projective map of a Hi... |
| pmapval 37698 | Value of the projective ma... |
| elpmap 37699 | Member of a projective map... |
| pmapssat 37700 | The projective map of a Hi... |
| pmapssbaN 37701 | A weakening of ~ pmapssat ... |
| pmaple 37702 | The projective map of a Hi... |
| pmap11 37703 | The projective map of a Hi... |
| pmapat 37704 | The projective map of an a... |
| elpmapat 37705 | Member of the projective m... |
| pmap0 37706 | Value of the projective ma... |
| pmapeq0 37707 | A projective map value is ... |
| pmap1N 37708 | Value of the projective ma... |
| pmapsub 37709 | The projective map of a Hi... |
| pmapglbx 37710 | The projective map of the ... |
| pmapglb 37711 | The projective map of the ... |
| pmapglb2N 37712 | The projective map of the ... |
| pmapglb2xN 37713 | The projective map of the ... |
| pmapmeet 37714 | The projective map of a me... |
| isline2 37715 | Definition of line in term... |
| linepmap 37716 | A line described with a pr... |
| isline3 37717 | Definition of line in term... |
| isline4N 37718 | Definition of line in term... |
| lneq2at 37719 | A line equals the join of ... |
| lnatexN 37720 | There is an atom in a line... |
| lnjatN 37721 | Given an atom in a line, t... |
| lncvrelatN 37722 | A lattice element covered ... |
| lncvrat 37723 | A line covers the atoms it... |
| lncmp 37724 | If two lines are comparabl... |
| 2lnat 37725 | Two intersecting lines int... |
| 2atm2atN 37726 | Two joins with a common at... |
| 2llnma1b 37727 | Generalization of ~ 2llnma... |
| 2llnma1 37728 | Two different intersecting... |
| 2llnma3r 37729 | Two different intersecting... |
| 2llnma2 37730 | Two different intersecting... |
| 2llnma2rN 37731 | Two different intersecting... |
| cdlema1N 37732 | A condition for required f... |
| cdlema2N 37733 | A condition for required f... |
| cdlemblem 37734 | Lemma for ~ cdlemb . (Con... |
| cdlemb 37735 | Given two atoms not less t... |
| paddfval 37738 | Projective subspace sum op... |
| paddval 37739 | Projective subspace sum op... |
| elpadd 37740 | Member of a projective sub... |
| elpaddn0 37741 | Member of projective subsp... |
| paddvaln0N 37742 | Projective subspace sum op... |
| elpaddri 37743 | Condition implying members... |
| elpaddatriN 37744 | Condition implying members... |
| elpaddat 37745 | Membership in a projective... |
| elpaddatiN 37746 | Consequence of membership ... |
| elpadd2at 37747 | Membership in a projective... |
| elpadd2at2 37748 | Membership in a projective... |
| paddunssN 37749 | Projective subspace sum in... |
| elpadd0 37750 | Member of projective subsp... |
| paddval0 37751 | Projective subspace sum wi... |
| padd01 37752 | Projective subspace sum wi... |
| padd02 37753 | Projective subspace sum wi... |
| paddcom 37754 | Projective subspace sum co... |
| paddssat 37755 | A projective subspace sum ... |
| sspadd1 37756 | A projective subspace sum ... |
| sspadd2 37757 | A projective subspace sum ... |
| paddss1 37758 | Subset law for projective ... |
| paddss2 37759 | Subset law for projective ... |
| paddss12 37760 | Subset law for projective ... |
| paddasslem1 37761 | Lemma for ~ paddass . (Co... |
| paddasslem2 37762 | Lemma for ~ paddass . (Co... |
| paddasslem3 37763 | Lemma for ~ paddass . Res... |
| paddasslem4 37764 | Lemma for ~ paddass . Com... |
| paddasslem5 37765 | Lemma for ~ paddass . Sho... |
| paddasslem6 37766 | Lemma for ~ paddass . (Co... |
| paddasslem7 37767 | Lemma for ~ paddass . Com... |
| paddasslem8 37768 | Lemma for ~ paddass . (Co... |
| paddasslem9 37769 | Lemma for ~ paddass . Com... |
| paddasslem10 37770 | Lemma for ~ paddass . Use... |
| paddasslem11 37771 | Lemma for ~ paddass . The... |
| paddasslem12 37772 | Lemma for ~ paddass . The... |
| paddasslem13 37773 | Lemma for ~ paddass . The... |
| paddasslem14 37774 | Lemma for ~ paddass . Rem... |
| paddasslem15 37775 | Lemma for ~ paddass . Use... |
| paddasslem16 37776 | Lemma for ~ paddass . Use... |
| paddasslem17 37777 | Lemma for ~ paddass . The... |
| paddasslem18 37778 | Lemma for ~ paddass . Com... |
| paddass 37779 | Projective subspace sum is... |
| padd12N 37780 | Commutative/associative la... |
| padd4N 37781 | Rearrangement of 4 terms i... |
| paddidm 37782 | Projective subspace sum is... |
| paddclN 37783 | The projective sum of two ... |
| paddssw1 37784 | Subset law for projective ... |
| paddssw2 37785 | Subset law for projective ... |
| paddss 37786 | Subset law for projective ... |
| pmodlem1 37787 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 37788 | Lemma for ~ pmod1i . (Con... |
| pmod1i 37789 | The modular law holds in a... |
| pmod2iN 37790 | Dual of the modular law. ... |
| pmodN 37791 | The modular law for projec... |
| pmodl42N 37792 | Lemma derived from modular... |
| pmapjoin 37793 | The projective map of the ... |
| pmapjat1 37794 | The projective map of the ... |
| pmapjat2 37795 | The projective map of the ... |
| pmapjlln1 37796 | The projective map of the ... |
| hlmod1i 37797 | A version of the modular l... |
| atmod1i1 37798 | Version of modular law ~ p... |
| atmod1i1m 37799 | Version of modular law ~ p... |
| atmod1i2 37800 | Version of modular law ~ p... |
| llnmod1i2 37801 | Version of modular law ~ p... |
| atmod2i1 37802 | Version of modular law ~ p... |
| atmod2i2 37803 | Version of modular law ~ p... |
| llnmod2i2 37804 | Version of modular law ~ p... |
| atmod3i1 37805 | Version of modular law tha... |
| atmod3i2 37806 | Version of modular law tha... |
| atmod4i1 37807 | Version of modular law tha... |
| atmod4i2 37808 | Version of modular law tha... |
| llnexchb2lem 37809 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 37810 | Line exchange property (co... |
| llnexch2N 37811 | Line exchange property (co... |
| dalawlem1 37812 | Lemma for ~ dalaw . Speci... |
| dalawlem2 37813 | Lemma for ~ dalaw . Utili... |
| dalawlem3 37814 | Lemma for ~ dalaw . First... |
| dalawlem4 37815 | Lemma for ~ dalaw . Secon... |
| dalawlem5 37816 | Lemma for ~ dalaw . Speci... |
| dalawlem6 37817 | Lemma for ~ dalaw . First... |
| dalawlem7 37818 | Lemma for ~ dalaw . Secon... |
| dalawlem8 37819 | Lemma for ~ dalaw . Speci... |
| dalawlem9 37820 | Lemma for ~ dalaw . Speci... |
| dalawlem10 37821 | Lemma for ~ dalaw . Combi... |
| dalawlem11 37822 | Lemma for ~ dalaw . First... |
| dalawlem12 37823 | Lemma for ~ dalaw . Secon... |
| dalawlem13 37824 | Lemma for ~ dalaw . Speci... |
| dalawlem14 37825 | Lemma for ~ dalaw . Combi... |
| dalawlem15 37826 | Lemma for ~ dalaw . Swap ... |
| dalaw 37827 | Desargues's law, derived f... |
| pclfvalN 37830 | The projective subspace cl... |
| pclvalN 37831 | Value of the projective su... |
| pclclN 37832 | Closure of the projective ... |
| elpclN 37833 | Membership in the projecti... |
| elpcliN 37834 | Implication of membership ... |
| pclssN 37835 | Ordering is preserved by s... |
| pclssidN 37836 | A set of atoms is included... |
| pclidN 37837 | The projective subspace cl... |
| pclbtwnN 37838 | A projective subspace sand... |
| pclunN 37839 | The projective subspace cl... |
| pclun2N 37840 | The projective subspace cl... |
| pclfinN 37841 | The projective subspace cl... |
| pclcmpatN 37842 | The set of projective subs... |
| polfvalN 37845 | The projective subspace po... |
| polvalN 37846 | Value of the projective su... |
| polval2N 37847 | Alternate expression for v... |
| polsubN 37848 | The polarity of a set of a... |
| polssatN 37849 | The polarity of a set of a... |
| pol0N 37850 | The polarity of the empty ... |
| pol1N 37851 | The polarity of the whole ... |
| 2pol0N 37852 | The closed subspace closur... |
| polpmapN 37853 | The polarity of a projecti... |
| 2polpmapN 37854 | Double polarity of a proje... |
| 2polvalN 37855 | Value of double polarity. ... |
| 2polssN 37856 | A set of atoms is a subset... |
| 3polN 37857 | Triple polarity cancels to... |
| polcon3N 37858 | Contraposition law for pol... |
| 2polcon4bN 37859 | Contraposition law for pol... |
| polcon2N 37860 | Contraposition law for pol... |
| polcon2bN 37861 | Contraposition law for pol... |
| pclss2polN 37862 | The projective subspace cl... |
| pcl0N 37863 | The projective subspace cl... |
| pcl0bN 37864 | The projective subspace cl... |
| pmaplubN 37865 | The LUB of a projective ma... |
| sspmaplubN 37866 | A set of atoms is a subset... |
| 2pmaplubN 37867 | Double projective map of a... |
| paddunN 37868 | The closure of the project... |
| poldmj1N 37869 | De Morgan's law for polari... |
| pmapj2N 37870 | The projective map of the ... |
| pmapocjN 37871 | The projective map of the ... |
| polatN 37872 | The polarity of the single... |
| 2polatN 37873 | Double polarity of the sin... |
| pnonsingN 37874 | The intersection of a set ... |
| psubclsetN 37877 | The set of closed projecti... |
| ispsubclN 37878 | The predicate "is a closed... |
| psubcliN 37879 | Property of a closed proje... |
| psubcli2N 37880 | Property of a closed proje... |
| psubclsubN 37881 | A closed projective subspa... |
| psubclssatN 37882 | A closed projective subspa... |
| pmapidclN 37883 | Projective map of the LUB ... |
| 0psubclN 37884 | The empty set is a closed ... |
| 1psubclN 37885 | The set of all atoms is a ... |
| atpsubclN 37886 | A point (singleton of an a... |
| pmapsubclN 37887 | A projective map value is ... |
| ispsubcl2N 37888 | Alternate predicate for "i... |
| psubclinN 37889 | The intersection of two cl... |
| paddatclN 37890 | The projective sum of a cl... |
| pclfinclN 37891 | The projective subspace cl... |
| linepsubclN 37892 | A line is a closed project... |
| polsubclN 37893 | A polarity is a closed pro... |
| poml4N 37894 | Orthomodular law for proje... |
| poml5N 37895 | Orthomodular law for proje... |
| poml6N 37896 | Orthomodular law for proje... |
| osumcllem1N 37897 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 37898 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 37899 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 37900 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 37901 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 37902 | Lemma for ~ osumclN . Use... |
| osumcllem7N 37903 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 37904 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 37905 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 37906 | Lemma for ~ osumclN . Con... |
| osumcllem11N 37907 | Lemma for ~ osumclN . (Co... |
| osumclN 37908 | Closure of orthogonal sum.... |
| pmapojoinN 37909 | For orthogonal elements, p... |
| pexmidN 37910 | Excluded middle law for cl... |
| pexmidlem1N 37911 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 37912 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 37913 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 37914 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 37915 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 37916 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 37917 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 37918 | Lemma for ~ pexmidN . The... |
| pexmidALTN 37919 | Excluded middle law for cl... |
| pl42lem1N 37920 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 37921 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 37922 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 37923 | Lemma for ~ pl42N . (Cont... |
| pl42N 37924 | Law holding in a Hilbert l... |
| watfvalN 37933 | The W atoms function. (Co... |
| watvalN 37934 | Value of the W atoms funct... |
| iswatN 37935 | The predicate "is a W atom... |
| lhpset 37936 | The set of co-atoms (latti... |
| islhp 37937 | The predicate "is a co-ato... |
| islhp2 37938 | The predicate "is a co-ato... |
| lhpbase 37939 | A co-atom is a member of t... |
| lhp1cvr 37940 | The lattice unit covers a ... |
| lhplt 37941 | An atom under a co-atom is... |
| lhp2lt 37942 | The join of two atoms unde... |
| lhpexlt 37943 | There exists an atom less ... |
| lhp0lt 37944 | A co-atom is greater than ... |
| lhpn0 37945 | A co-atom is nonzero. TOD... |
| lhpexle 37946 | There exists an atom under... |
| lhpexnle 37947 | There exists an atom not u... |
| lhpexle1lem 37948 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 37949 | There exists an atom under... |
| lhpexle2lem 37950 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 37951 | There exists atom under a ... |
| lhpexle3lem 37952 | There exists atom under a ... |
| lhpexle3 37953 | There exists atom under a ... |
| lhpex2leN 37954 | There exist at least two d... |
| lhpoc 37955 | The orthocomplement of a c... |
| lhpoc2N 37956 | The orthocomplement of an ... |
| lhpocnle 37957 | The orthocomplement of a c... |
| lhpocat 37958 | The orthocomplement of a c... |
| lhpocnel 37959 | The orthocomplement of a c... |
| lhpocnel2 37960 | The orthocomplement of a c... |
| lhpjat1 37961 | The join of a co-atom (hyp... |
| lhpjat2 37962 | The join of a co-atom (hyp... |
| lhpj1 37963 | The join of a co-atom (hyp... |
| lhpmcvr 37964 | The meet of a lattice hype... |
| lhpmcvr2 37965 | Alternate way to express t... |
| lhpmcvr3 37966 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 37967 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 37968 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 37969 | Specialization of ~ lhpmcv... |
| lhpm0atN 37970 | If the meet of a lattice h... |
| lhpmat 37971 | An element covered by the ... |
| lhpmatb 37972 | An element covered by the ... |
| lhp2at0 37973 | Join and meet with differe... |
| lhp2atnle 37974 | Inequality for 2 different... |
| lhp2atne 37975 | Inequality for joins with ... |
| lhp2at0nle 37976 | Inequality for 2 different... |
| lhp2at0ne 37977 | Inequality for joins with ... |
| lhpelim 37978 | Eliminate an atom not unde... |
| lhpmod2i2 37979 | Modular law for hyperplane... |
| lhpmod6i1 37980 | Modular law for hyperplane... |
| lhprelat3N 37981 | The Hilbert lattice is rel... |
| cdlemb2 37982 | Given two atoms not under ... |
| lhple 37983 | Property of a lattice elem... |
| lhpat 37984 | Create an atom under a co-... |
| lhpat4N 37985 | Property of an atom under ... |
| lhpat2 37986 | Create an atom under a co-... |
| lhpat3 37987 | There is only one atom und... |
| 4atexlemk 37988 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 37989 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 37990 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 37991 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 37992 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 37993 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 37994 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 37995 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 37996 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 37997 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 37998 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 37999 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 38000 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 38001 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 38002 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 38003 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 38004 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 38005 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 38006 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 38007 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 38008 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 38009 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 38010 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 38011 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 38012 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 38013 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 38014 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 38015 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 38016 | Whenever there are at leas... |
| 4atex 38017 | Whenever there are at leas... |
| 4atex2 38018 | More general version of ~ ... |
| 4atex2-0aOLDN 38019 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 38020 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 38021 | Same as ~ 4atex2 except th... |
| 4atex3 38022 | More general version of ~ ... |
| lautset 38023 | The set of lattice automor... |
| islaut 38024 | The predicate "is a lattic... |
| lautle 38025 | Less-than or equal propert... |
| laut1o 38026 | A lattice automorphism is ... |
| laut11 38027 | One-to-one property of a l... |
| lautcl 38028 | A lattice automorphism val... |
| lautcnvclN 38029 | Reverse closure of a latti... |
| lautcnvle 38030 | Less-than or equal propert... |
| lautcnv 38031 | The converse of a lattice ... |
| lautlt 38032 | Less-than property of a la... |
| lautcvr 38033 | Covering property of a lat... |
| lautj 38034 | Meet property of a lattice... |
| lautm 38035 | Meet property of a lattice... |
| lauteq 38036 | A lattice automorphism arg... |
| idlaut 38037 | The identity function is a... |
| lautco 38038 | The composition of two lat... |
| pautsetN 38039 | The set of projective auto... |
| ispautN 38040 | The predicate "is a projec... |
| ldilfset 38049 | The mapping from fiducial ... |
| ldilset 38050 | The set of lattice dilatio... |
| isldil 38051 | The predicate "is a lattic... |
| ldillaut 38052 | A lattice dilation is an a... |
| ldil1o 38053 | A lattice dilation is a on... |
| ldilval 38054 | Value of a lattice dilatio... |
| idldil 38055 | The identity function is a... |
| ldilcnv 38056 | The converse of a lattice ... |
| ldilco 38057 | The composition of two lat... |
| ltrnfset 38058 | The set of all lattice tra... |
| ltrnset 38059 | The set of lattice transla... |
| isltrn 38060 | The predicate "is a lattic... |
| isltrn2N 38061 | The predicate "is a lattic... |
| ltrnu 38062 | Uniqueness property of a l... |
| ltrnldil 38063 | A lattice translation is a... |
| ltrnlaut 38064 | A lattice translation is a... |
| ltrn1o 38065 | A lattice translation is a... |
| ltrncl 38066 | Closure of a lattice trans... |
| ltrn11 38067 | One-to-one property of a l... |
| ltrncnvnid 38068 | If a translation is differ... |
| ltrncoidN 38069 | Two translations are equal... |
| ltrnle 38070 | Less-than or equal propert... |
| ltrncnvleN 38071 | Less-than or equal propert... |
| ltrnm 38072 | Lattice translation of a m... |
| ltrnj 38073 | Lattice translation of a m... |
| ltrncvr 38074 | Covering property of a lat... |
| ltrnval1 38075 | Value of a lattice transla... |
| ltrnid 38076 | A lattice translation is t... |
| ltrnnid 38077 | If a lattice translation i... |
| ltrnatb 38078 | The lattice translation of... |
| ltrncnvatb 38079 | The converse of the lattic... |
| ltrnel 38080 | The lattice translation of... |
| ltrnat 38081 | The lattice translation of... |
| ltrncnvat 38082 | The converse of the lattic... |
| ltrncnvel 38083 | The converse of the lattic... |
| ltrncoelN 38084 | Composition of lattice tra... |
| ltrncoat 38085 | Composition of lattice tra... |
| ltrncoval 38086 | Two ways to express value ... |
| ltrncnv 38087 | The converse of a lattice ... |
| ltrn11at 38088 | Frequently used one-to-one... |
| ltrneq2 38089 | The equality of two transl... |
| ltrneq 38090 | The equality of two transl... |
| idltrn 38091 | The identity function is a... |
| ltrnmw 38092 | Property of lattice transl... |
| dilfsetN 38093 | The mapping from fiducial ... |
| dilsetN 38094 | The set of dilations for a... |
| isdilN 38095 | The predicate "is a dilati... |
| trnfsetN 38096 | The mapping from fiducial ... |
| trnsetN 38097 | The set of translations fo... |
| istrnN 38098 | The predicate "is a transl... |
| trlfset 38101 | The set of all traces of l... |
| trlset 38102 | The set of traces of latti... |
| trlval 38103 | The value of the trace of ... |
| trlval2 38104 | The value of the trace of ... |
| trlcl 38105 | Closure of the trace of a ... |
| trlcnv 38106 | The trace of the converse ... |
| trljat1 38107 | The value of a translation... |
| trljat2 38108 | The value of a translation... |
| trljat3 38109 | The value of a translation... |
| trlat 38110 | If an atom differs from it... |
| trl0 38111 | If an atom not under the f... |
| trlator0 38112 | The trace of a lattice tra... |
| trlatn0 38113 | The trace of a lattice tra... |
| trlnidat 38114 | The trace of a lattice tra... |
| ltrnnidn 38115 | If a lattice translation i... |
| ltrnideq 38116 | Property of the identity l... |
| trlid0 38117 | The trace of the identity ... |
| trlnidatb 38118 | A lattice translation is n... |
| trlid0b 38119 | A lattice translation is t... |
| trlnid 38120 | Different translations wit... |
| ltrn2ateq 38121 | Property of the equality o... |
| ltrnateq 38122 | If any atom (under ` W ` )... |
| ltrnatneq 38123 | If any atom (under ` W ` )... |
| ltrnatlw 38124 | If the value of an atom eq... |
| trlle 38125 | The trace of a lattice tra... |
| trlne 38126 | The trace of a lattice tra... |
| trlnle 38127 | The atom not under the fid... |
| trlval3 38128 | The value of the trace of ... |
| trlval4 38129 | The value of the trace of ... |
| trlval5 38130 | The value of the trace of ... |
| arglem1N 38131 | Lemma for Desargues's law.... |
| cdlemc1 38132 | Part of proof of Lemma C i... |
| cdlemc2 38133 | Part of proof of Lemma C i... |
| cdlemc3 38134 | Part of proof of Lemma C i... |
| cdlemc4 38135 | Part of proof of Lemma C i... |
| cdlemc5 38136 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 38137 | Lemma for ~ cdlemc . (Con... |
| cdlemc 38138 | Lemma C in [Crawley] p. 11... |
| cdlemd1 38139 | Part of proof of Lemma D i... |
| cdlemd2 38140 | Part of proof of Lemma D i... |
| cdlemd3 38141 | Part of proof of Lemma D i... |
| cdlemd4 38142 | Part of proof of Lemma D i... |
| cdlemd5 38143 | Part of proof of Lemma D i... |
| cdlemd6 38144 | Part of proof of Lemma D i... |
| cdlemd7 38145 | Part of proof of Lemma D i... |
| cdlemd8 38146 | Part of proof of Lemma D i... |
| cdlemd9 38147 | Part of proof of Lemma D i... |
| cdlemd 38148 | If two translations agree ... |
| ltrneq3 38149 | Two translations agree at ... |
| cdleme00a 38150 | Part of proof of Lemma E i... |
| cdleme0aa 38151 | Part of proof of Lemma E i... |
| cdleme0a 38152 | Part of proof of Lemma E i... |
| cdleme0b 38153 | Part of proof of Lemma E i... |
| cdleme0c 38154 | Part of proof of Lemma E i... |
| cdleme0cp 38155 | Part of proof of Lemma E i... |
| cdleme0cq 38156 | Part of proof of Lemma E i... |
| cdleme0dN 38157 | Part of proof of Lemma E i... |
| cdleme0e 38158 | Part of proof of Lemma E i... |
| cdleme0fN 38159 | Part of proof of Lemma E i... |
| cdleme0gN 38160 | Part of proof of Lemma E i... |
| cdlemeulpq 38161 | Part of proof of Lemma E i... |
| cdleme01N 38162 | Part of proof of Lemma E i... |
| cdleme02N 38163 | Part of proof of Lemma E i... |
| cdleme0ex1N 38164 | Part of proof of Lemma E i... |
| cdleme0ex2N 38165 | Part of proof of Lemma E i... |
| cdleme0moN 38166 | Part of proof of Lemma E i... |
| cdleme1b 38167 | Part of proof of Lemma E i... |
| cdleme1 38168 | Part of proof of Lemma E i... |
| cdleme2 38169 | Part of proof of Lemma E i... |
| cdleme3b 38170 | Part of proof of Lemma E i... |
| cdleme3c 38171 | Part of proof of Lemma E i... |
| cdleme3d 38172 | Part of proof of Lemma E i... |
| cdleme3e 38173 | Part of proof of Lemma E i... |
| cdleme3fN 38174 | Part of proof of Lemma E i... |
| cdleme3g 38175 | Part of proof of Lemma E i... |
| cdleme3h 38176 | Part of proof of Lemma E i... |
| cdleme3fa 38177 | Part of proof of Lemma E i... |
| cdleme3 38178 | Part of proof of Lemma E i... |
| cdleme4 38179 | Part of proof of Lemma E i... |
| cdleme4a 38180 | Part of proof of Lemma E i... |
| cdleme5 38181 | Part of proof of Lemma E i... |
| cdleme6 38182 | Part of proof of Lemma E i... |
| cdleme7aa 38183 | Part of proof of Lemma E i... |
| cdleme7a 38184 | Part of proof of Lemma E i... |
| cdleme7b 38185 | Part of proof of Lemma E i... |
| cdleme7c 38186 | Part of proof of Lemma E i... |
| cdleme7d 38187 | Part of proof of Lemma E i... |
| cdleme7e 38188 | Part of proof of Lemma E i... |
| cdleme7ga 38189 | Part of proof of Lemma E i... |
| cdleme7 38190 | Part of proof of Lemma E i... |
| cdleme8 38191 | Part of proof of Lemma E i... |
| cdleme9a 38192 | Part of proof of Lemma E i... |
| cdleme9b 38193 | Utility lemma for Lemma E ... |
| cdleme9 38194 | Part of proof of Lemma E i... |
| cdleme10 38195 | Part of proof of Lemma E i... |
| cdleme8tN 38196 | Part of proof of Lemma E i... |
| cdleme9taN 38197 | Part of proof of Lemma E i... |
| cdleme9tN 38198 | Part of proof of Lemma E i... |
| cdleme10tN 38199 | Part of proof of Lemma E i... |
| cdleme16aN 38200 | Part of proof of Lemma E i... |
| cdleme11a 38201 | Part of proof of Lemma E i... |
| cdleme11c 38202 | Part of proof of Lemma E i... |
| cdleme11dN 38203 | Part of proof of Lemma E i... |
| cdleme11e 38204 | Part of proof of Lemma E i... |
| cdleme11fN 38205 | Part of proof of Lemma E i... |
| cdleme11g 38206 | Part of proof of Lemma E i... |
| cdleme11h 38207 | Part of proof of Lemma E i... |
| cdleme11j 38208 | Part of proof of Lemma E i... |
| cdleme11k 38209 | Part of proof of Lemma E i... |
| cdleme11l 38210 | Part of proof of Lemma E i... |
| cdleme11 38211 | Part of proof of Lemma E i... |
| cdleme12 38212 | Part of proof of Lemma E i... |
| cdleme13 38213 | Part of proof of Lemma E i... |
| cdleme14 38214 | Part of proof of Lemma E i... |
| cdleme15a 38215 | Part of proof of Lemma E i... |
| cdleme15b 38216 | Part of proof of Lemma E i... |
| cdleme15c 38217 | Part of proof of Lemma E i... |
| cdleme15d 38218 | Part of proof of Lemma E i... |
| cdleme15 38219 | Part of proof of Lemma E i... |
| cdleme16b 38220 | Part of proof of Lemma E i... |
| cdleme16c 38221 | Part of proof of Lemma E i... |
| cdleme16d 38222 | Part of proof of Lemma E i... |
| cdleme16e 38223 | Part of proof of Lemma E i... |
| cdleme16f 38224 | Part of proof of Lemma E i... |
| cdleme16g 38225 | Part of proof of Lemma E i... |
| cdleme16 38226 | Part of proof of Lemma E i... |
| cdleme17a 38227 | Part of proof of Lemma E i... |
| cdleme17b 38228 | Lemma leading to ~ cdleme1... |
| cdleme17c 38229 | Part of proof of Lemma E i... |
| cdleme17d1 38230 | Part of proof of Lemma E i... |
| cdleme0nex 38231 | Part of proof of Lemma E i... |
| cdleme18a 38232 | Part of proof of Lemma E i... |
| cdleme18b 38233 | Part of proof of Lemma E i... |
| cdleme18c 38234 | Part of proof of Lemma E i... |
| cdleme22gb 38235 | Utility lemma for Lemma E ... |
| cdleme18d 38236 | Part of proof of Lemma E i... |
| cdlemesner 38237 | Part of proof of Lemma E i... |
| cdlemedb 38238 | Part of proof of Lemma E i... |
| cdlemeda 38239 | Part of proof of Lemma E i... |
| cdlemednpq 38240 | Part of proof of Lemma E i... |
| cdlemednuN 38241 | Part of proof of Lemma E i... |
| cdleme20zN 38242 | Part of proof of Lemma E i... |
| cdleme20y 38243 | Part of proof of Lemma E i... |
| cdleme19a 38244 | Part of proof of Lemma E i... |
| cdleme19b 38245 | Part of proof of Lemma E i... |
| cdleme19c 38246 | Part of proof of Lemma E i... |
| cdleme19d 38247 | Part of proof of Lemma E i... |
| cdleme19e 38248 | Part of proof of Lemma E i... |
| cdleme19f 38249 | Part of proof of Lemma E i... |
| cdleme20aN 38250 | Part of proof of Lemma E i... |
| cdleme20bN 38251 | Part of proof of Lemma E i... |
| cdleme20c 38252 | Part of proof of Lemma E i... |
| cdleme20d 38253 | Part of proof of Lemma E i... |
| cdleme20e 38254 | Part of proof of Lemma E i... |
| cdleme20f 38255 | Part of proof of Lemma E i... |
| cdleme20g 38256 | Part of proof of Lemma E i... |
| cdleme20h 38257 | Part of proof of Lemma E i... |
| cdleme20i 38258 | Part of proof of Lemma E i... |
| cdleme20j 38259 | Part of proof of Lemma E i... |
| cdleme20k 38260 | Part of proof of Lemma E i... |
| cdleme20l1 38261 | Part of proof of Lemma E i... |
| cdleme20l2 38262 | Part of proof of Lemma E i... |
| cdleme20l 38263 | Part of proof of Lemma E i... |
| cdleme20m 38264 | Part of proof of Lemma E i... |
| cdleme20 38265 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 38266 | Part of proof of Lemma E i... |
| cdleme21b 38267 | Part of proof of Lemma E i... |
| cdleme21c 38268 | Part of proof of Lemma E i... |
| cdleme21at 38269 | Part of proof of Lemma E i... |
| cdleme21ct 38270 | Part of proof of Lemma E i... |
| cdleme21d 38271 | Part of proof of Lemma E i... |
| cdleme21e 38272 | Part of proof of Lemma E i... |
| cdleme21f 38273 | Part of proof of Lemma E i... |
| cdleme21g 38274 | Part of proof of Lemma E i... |
| cdleme21h 38275 | Part of proof of Lemma E i... |
| cdleme21i 38276 | Part of proof of Lemma E i... |
| cdleme21j 38277 | Combine ~ cdleme20 and ~ c... |
| cdleme21 38278 | Part of proof of Lemma E i... |
| cdleme21k 38279 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 38280 | Part of proof of Lemma E i... |
| cdleme22a 38281 | Part of proof of Lemma E i... |
| cdleme22b 38282 | Part of proof of Lemma E i... |
| cdleme22cN 38283 | Part of proof of Lemma E i... |
| cdleme22d 38284 | Part of proof of Lemma E i... |
| cdleme22e 38285 | Part of proof of Lemma E i... |
| cdleme22eALTN 38286 | Part of proof of Lemma E i... |
| cdleme22f 38287 | Part of proof of Lemma E i... |
| cdleme22f2 38288 | Part of proof of Lemma E i... |
| cdleme22g 38289 | Part of proof of Lemma E i... |
| cdleme23a 38290 | Part of proof of Lemma E i... |
| cdleme23b 38291 | Part of proof of Lemma E i... |
| cdleme23c 38292 | Part of proof of Lemma E i... |
| cdleme24 38293 | Quantified version of ~ cd... |
| cdleme25a 38294 | Lemma for ~ cdleme25b . (... |
| cdleme25b 38295 | Transform ~ cdleme24 . TO... |
| cdleme25c 38296 | Transform ~ cdleme25b . (... |
| cdleme25dN 38297 | Transform ~ cdleme25c . (... |
| cdleme25cl 38298 | Show closure of the unique... |
| cdleme25cv 38299 | Change bound variables in ... |
| cdleme26e 38300 | Part of proof of Lemma E i... |
| cdleme26ee 38301 | Part of proof of Lemma E i... |
| cdleme26eALTN 38302 | Part of proof of Lemma E i... |
| cdleme26fALTN 38303 | Part of proof of Lemma E i... |
| cdleme26f 38304 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 38305 | Part of proof of Lemma E i... |
| cdleme26f2 38306 | Part of proof of Lemma E i... |
| cdleme27cl 38307 | Part of proof of Lemma E i... |
| cdleme27a 38308 | Part of proof of Lemma E i... |
| cdleme27b 38309 | Lemma for ~ cdleme27N . (... |
| cdleme27N 38310 | Part of proof of Lemma E i... |
| cdleme28a 38311 | Lemma for ~ cdleme25b . T... |
| cdleme28b 38312 | Lemma for ~ cdleme25b . T... |
| cdleme28c 38313 | Part of proof of Lemma E i... |
| cdleme28 38314 | Quantified version of ~ cd... |
| cdleme29ex 38315 | Lemma for ~ cdleme29b . (... |
| cdleme29b 38316 | Transform ~ cdleme28 . (C... |
| cdleme29c 38317 | Transform ~ cdleme28b . (... |
| cdleme29cl 38318 | Show closure of the unique... |
| cdleme30a 38319 | Part of proof of Lemma E i... |
| cdleme31so 38320 | Part of proof of Lemma E i... |
| cdleme31sn 38321 | Part of proof of Lemma E i... |
| cdleme31sn1 38322 | Part of proof of Lemma E i... |
| cdleme31se 38323 | Part of proof of Lemma D i... |
| cdleme31se2 38324 | Part of proof of Lemma D i... |
| cdleme31sc 38325 | Part of proof of Lemma E i... |
| cdleme31sde 38326 | Part of proof of Lemma D i... |
| cdleme31snd 38327 | Part of proof of Lemma D i... |
| cdleme31sdnN 38328 | Part of proof of Lemma E i... |
| cdleme31sn1c 38329 | Part of proof of Lemma E i... |
| cdleme31sn2 38330 | Part of proof of Lemma E i... |
| cdleme31fv 38331 | Part of proof of Lemma E i... |
| cdleme31fv1 38332 | Part of proof of Lemma E i... |
| cdleme31fv1s 38333 | Part of proof of Lemma E i... |
| cdleme31fv2 38334 | Part of proof of Lemma E i... |
| cdleme31id 38335 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 38336 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 38337 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 38338 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 38339 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 38340 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 38341 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 38342 | Part of proof of Lemma E i... |
| cdlemefr29exN 38343 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 38344 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 38345 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 38346 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 38347 | TODO fix comment. (Contri... |
| cdlemefr29clN 38348 | Show closure of the unique... |
| cdleme43frv1snN 38349 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 38350 | Part of proof of Lemma E i... |
| cdlemefr32fva1 38351 | Part of proof of Lemma E i... |
| cdlemefr31fv1 38352 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 38353 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 38354 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 38355 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 38356 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 38357 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 38358 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 38359 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 38360 | Show closure of the unique... |
| cdleme43fsv1snlem 38361 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 38362 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 38363 | Part of proof of Lemma E i... |
| cdlemefs32fva1 38364 | Part of proof of Lemma E i... |
| cdlemefs31fv1 38365 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 38366 | Value of f(r) when r is an... |
| cdlemefs44 38367 | Value of f_s(r) when r is ... |
| cdlemefr45 38368 | Value of f(r) when r is an... |
| cdlemefr45e 38369 | Explicit expansion of ~ cd... |
| cdlemefs45 38370 | Value of f_s(r) when r is ... |
| cdlemefs45ee 38371 | Explicit expansion of ~ cd... |
| cdlemefs45eN 38372 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 38373 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 38374 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 38375 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 38376 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 38377 | Show closure of ` [_ R / s... |
| cdleme32fva 38378 | Part of proof of Lemma D i... |
| cdleme32fva1 38379 | Part of proof of Lemma D i... |
| cdleme32fvaw 38380 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 38381 | Part of proof of Lemma D i... |
| cdleme32a 38382 | Part of proof of Lemma D i... |
| cdleme32b 38383 | Part of proof of Lemma D i... |
| cdleme32c 38384 | Part of proof of Lemma D i... |
| cdleme32d 38385 | Part of proof of Lemma D i... |
| cdleme32e 38386 | Part of proof of Lemma D i... |
| cdleme32f 38387 | Part of proof of Lemma D i... |
| cdleme32le 38388 | Part of proof of Lemma D i... |
| cdleme35a 38389 | Part of proof of Lemma E i... |
| cdleme35fnpq 38390 | Part of proof of Lemma E i... |
| cdleme35b 38391 | Part of proof of Lemma E i... |
| cdleme35c 38392 | Part of proof of Lemma E i... |
| cdleme35d 38393 | Part of proof of Lemma E i... |
| cdleme35e 38394 | Part of proof of Lemma E i... |
| cdleme35f 38395 | Part of proof of Lemma E i... |
| cdleme35g 38396 | Part of proof of Lemma E i... |
| cdleme35h 38397 | Part of proof of Lemma E i... |
| cdleme35h2 38398 | Part of proof of Lemma E i... |
| cdleme35sn2aw 38399 | Part of proof of Lemma E i... |
| cdleme35sn3a 38400 | Part of proof of Lemma E i... |
| cdleme36a 38401 | Part of proof of Lemma E i... |
| cdleme36m 38402 | Part of proof of Lemma E i... |
| cdleme37m 38403 | Part of proof of Lemma E i... |
| cdleme38m 38404 | Part of proof of Lemma E i... |
| cdleme38n 38405 | Part of proof of Lemma E i... |
| cdleme39a 38406 | Part of proof of Lemma E i... |
| cdleme39n 38407 | Part of proof of Lemma E i... |
| cdleme40m 38408 | Part of proof of Lemma E i... |
| cdleme40n 38409 | Part of proof of Lemma E i... |
| cdleme40v 38410 | Part of proof of Lemma E i... |
| cdleme40w 38411 | Part of proof of Lemma E i... |
| cdleme42a 38412 | Part of proof of Lemma E i... |
| cdleme42c 38413 | Part of proof of Lemma E i... |
| cdleme42d 38414 | Part of proof of Lemma E i... |
| cdleme41sn3aw 38415 | Part of proof of Lemma E i... |
| cdleme41sn4aw 38416 | Part of proof of Lemma E i... |
| cdleme41snaw 38417 | Part of proof of Lemma E i... |
| cdleme41fva11 38418 | Part of proof of Lemma E i... |
| cdleme42b 38419 | Part of proof of Lemma E i... |
| cdleme42e 38420 | Part of proof of Lemma E i... |
| cdleme42f 38421 | Part of proof of Lemma E i... |
| cdleme42g 38422 | Part of proof of Lemma E i... |
| cdleme42h 38423 | Part of proof of Lemma E i... |
| cdleme42i 38424 | Part of proof of Lemma E i... |
| cdleme42k 38425 | Part of proof of Lemma E i... |
| cdleme42ke 38426 | Part of proof of Lemma E i... |
| cdleme42keg 38427 | Part of proof of Lemma E i... |
| cdleme42mN 38428 | Part of proof of Lemma E i... |
| cdleme42mgN 38429 | Part of proof of Lemma E i... |
| cdleme43aN 38430 | Part of proof of Lemma E i... |
| cdleme43bN 38431 | Lemma for Lemma E in [Craw... |
| cdleme43cN 38432 | Part of proof of Lemma E i... |
| cdleme43dN 38433 | Part of proof of Lemma E i... |
| cdleme46f2g2 38434 | Conversion for ` G ` to re... |
| cdleme46f2g1 38435 | Conversion for ` G ` to re... |
| cdleme17d2 38436 | Part of proof of Lemma E i... |
| cdleme17d3 38437 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 38438 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 38439 | Part of proof of Lemma E i... |
| cdleme48fv 38440 | Part of proof of Lemma D i... |
| cdleme48fvg 38441 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 38442 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 38443 | TODO: fix comment. TODO: ... |
| cdleme48b 38444 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 38445 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 38446 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 38447 | TODO: fix comment. (Contr... |
| cdleme4gfv 38448 | Part of proof of Lemma D i... |
| cdlemeg47b 38449 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 38450 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 38451 | Value of g_s(r) when r is ... |
| cdlemeg49le 38452 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 38453 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 38454 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 38455 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 38456 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 38457 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 38458 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 38459 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 38460 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 38461 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 38462 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 38463 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 38464 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 38465 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 38466 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 38467 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 38468 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 38469 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 38470 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 38471 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 38472 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 38473 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 38474 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 38475 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 38476 | TODO: fix comment. (Contr... |
| cdleme48gfv1 38477 | TODO: fix comment. (Contr... |
| cdleme48gfv 38478 | TODO: fix comment. (Contr... |
| cdleme48fgv 38479 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 38480 | Part of proof of Lemma D i... |
| cdleme50lebi 38481 | Part of proof of Lemma D i... |
| cdleme50eq 38482 | Part of proof of Lemma D i... |
| cdleme50f 38483 | Part of proof of Lemma D i... |
| cdleme50f1 38484 | Part of proof of Lemma D i... |
| cdleme50rnlem 38485 | Part of proof of Lemma D i... |
| cdleme50rn 38486 | Part of proof of Lemma D i... |
| cdleme50f1o 38487 | Part of proof of Lemma D i... |
| cdleme50laut 38488 | Part of proof of Lemma D i... |
| cdleme50ldil 38489 | Part of proof of Lemma D i... |
| cdleme50trn1 38490 | Part of proof that ` F ` i... |
| cdleme50trn2a 38491 | Part of proof that ` F ` i... |
| cdleme50trn2 38492 | Part of proof that ` F ` i... |
| cdleme50trn12 38493 | Part of proof that ` F ` i... |
| cdleme50trn3 38494 | Part of proof that ` F ` i... |
| cdleme50trn123 38495 | Part of proof that ` F ` i... |
| cdleme51finvfvN 38496 | Part of proof of Lemma E i... |
| cdleme51finvN 38497 | Part of proof of Lemma E i... |
| cdleme50ltrn 38498 | Part of proof of Lemma E i... |
| cdleme51finvtrN 38499 | Part of proof of Lemma E i... |
| cdleme50ex 38500 | Part of Lemma E in [Crawle... |
| cdleme 38501 | Lemma E in [Crawley] p. 11... |
| cdlemf1 38502 | Part of Lemma F in [Crawle... |
| cdlemf2 38503 | Part of Lemma F in [Crawle... |
| cdlemf 38504 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 38505 | ~ cdlemf with additional c... |
| cdlemftr3 38506 | Special case of ~ cdlemf s... |
| cdlemftr2 38507 | Special case of ~ cdlemf s... |
| cdlemftr1 38508 | Part of proof of Lemma G o... |
| cdlemftr0 38509 | Special case of ~ cdlemf s... |
| trlord 38510 | The ordering of two Hilber... |
| cdlemg1a 38511 | Shorter expression for ` G... |
| cdlemg1b2 38512 | This theorem can be used t... |
| cdlemg1idlemN 38513 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 38514 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 38515 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 38516 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 38517 | This theorem can be used t... |
| cdlemg1idN 38518 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 38519 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 38520 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 38521 | Version of ~ cdleme51finvt... |
| ltrniotaval 38522 | Value of the unique transl... |
| ltrniotacnvval 38523 | Converse value of the uniq... |
| ltrniotaidvalN 38524 | Value of the unique transl... |
| ltrniotavalbN 38525 | Value of the unique transl... |
| cdlemeiota 38526 | A translation is uniquely ... |
| cdlemg1ci2 38527 | Any function of the form o... |
| cdlemg1cN 38528 | Any translation belongs to... |
| cdlemg1cex 38529 | Any translation is one of ... |
| cdlemg2cN 38530 | Any translation belongs to... |
| cdlemg2dN 38531 | This theorem can be used t... |
| cdlemg2cex 38532 | Any translation is one of ... |
| cdlemg2ce 38533 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 38534 | Part of proof of Lemma E i... |
| cdlemg2fvlem 38535 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 38536 | ~ cdleme42keg with simpler... |
| cdlemg2idN 38537 | Version of ~ cdleme31id wi... |
| cdlemg3a 38538 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 38539 | TODO: Replace this with ~... |
| cdlemg2fv 38540 | Value of a translation in ... |
| cdlemg2fv2 38541 | Value of a translation in ... |
| cdlemg2k 38542 | ~ cdleme42keg with simpler... |
| cdlemg2kq 38543 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 38544 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 38545 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 38546 | TODO: Is there a simpler ... |
| cdlemb3 38547 | Given two atoms not under ... |
| cdlemg7fvbwN 38548 | Properties of a translatio... |
| cdlemg4a 38549 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 38550 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 38551 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 38552 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 38553 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 38554 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 38555 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 38556 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 38557 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 38558 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 38559 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 38560 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 38561 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 38562 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 38563 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 38564 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 38565 | Value of a translation com... |
| cdlemg7aN 38566 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 38567 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 38568 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 38569 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 38570 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 38571 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 38572 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 38573 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 38574 | The triples ` <. P , ( F `... |
| cdlemg9 38575 | The triples ` <. P , ( F `... |
| cdlemg10b 38576 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 38577 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 38578 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 38579 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 38580 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 38581 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 38582 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 38583 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 38584 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 38585 | The triples ` <. P , ( F `... |
| cdlemg12c 38586 | The triples ` <. P , ( F `... |
| cdlemg12d 38587 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 38588 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 38589 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 38590 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 38591 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 38592 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 38593 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 38594 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 38595 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 38596 | Eliminate the ` ( F `` P )... |
| cdlemg15 38597 | Eliminate the ` ( (... |
| cdlemg16 38598 | Part of proof of Lemma G o... |
| cdlemg16ALTN 38599 | This version of ~ cdlemg16... |
| cdlemg16z 38600 | Eliminate ` ( ( F `... |
| cdlemg16zz 38601 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 38602 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 38603 | Part of proof of Lemma G i... |
| cdlemg17dN 38604 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 38605 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 38606 | TODO: fix comment. (Contr... |
| cdlemg17f 38607 | TODO: fix comment. (Contr... |
| cdlemg17g 38608 | TODO: fix comment. (Contr... |
| cdlemg17h 38609 | TODO: fix comment. (Contr... |
| cdlemg17i 38610 | TODO: fix comment. (Contr... |
| cdlemg17ir 38611 | TODO: fix comment. (Contr... |
| cdlemg17j 38612 | TODO: fix comment. (Contr... |
| cdlemg17pq 38613 | Utility theorem for swappi... |
| cdlemg17bq 38614 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 38615 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 38616 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 38617 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 38618 | Part of Lemma G of [Crawle... |
| cdlemg18a 38619 | Show two lines are differe... |
| cdlemg18b 38620 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 38621 | Show two lines intersect a... |
| cdlemg18d 38622 | Show two lines intersect a... |
| cdlemg18 38623 | Show two lines intersect a... |
| cdlemg19a 38624 | Show two lines intersect a... |
| cdlemg19 38625 | Show two lines intersect a... |
| cdlemg20 38626 | Show two lines intersect a... |
| cdlemg21 38627 | Version of cdlemg19 with `... |
| cdlemg22 38628 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 38629 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 38630 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 38631 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 38632 | ~ cdlemg16zz restated for ... |
| cdlemg27a 38633 | For use with case when ` (... |
| cdlemg28a 38634 | Part of proof of Lemma G o... |
| cdlemg31b0N 38635 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 38636 | TODO: Fix comment. (Cont... |
| cdlemg27b 38637 | TODO: Fix comment. (Cont... |
| cdlemg31a 38638 | TODO: fix comment. (Contr... |
| cdlemg31b 38639 | TODO: fix comment. (Contr... |
| cdlemg31c 38640 | Show that when ` N ` is an... |
| cdlemg31d 38641 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 38642 | TODO: Fix comment. (Cont... |
| cdlemg33c0 38643 | TODO: Fix comment. (Cont... |
| cdlemg28b 38644 | Part of proof of Lemma G o... |
| cdlemg28 38645 | Part of proof of Lemma G o... |
| cdlemg29 38646 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 38647 | TODO: Fix comment. (Cont... |
| cdlemg33b 38648 | TODO: Fix comment. (Cont... |
| cdlemg33c 38649 | TODO: Fix comment. (Cont... |
| cdlemg33d 38650 | TODO: Fix comment. (Cont... |
| cdlemg33e 38651 | TODO: Fix comment. (Cont... |
| cdlemg33 38652 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 38653 | Use cdlemg33 to eliminate ... |
| cdlemg35 38654 | TODO: Fix comment. TODO:... |
| cdlemg36 38655 | Use cdlemg35 to eliminate ... |
| cdlemg38 38656 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 38657 | Eliminate ` =/= ` conditio... |
| cdlemg40 38658 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 38659 | Convert ~ cdlemg40 to func... |
| ltrnco 38660 | The composition of two tra... |
| trlcocnv 38661 | Swap the arguments of the ... |
| trlcoabs 38662 | Absorption into a composit... |
| trlcoabs2N 38663 | Absorption of the trace of... |
| trlcoat 38664 | The trace of a composition... |
| trlcocnvat 38665 | Commonly used special case... |
| trlconid 38666 | The composition of two dif... |
| trlcolem 38667 | Lemma for ~ trlco . (Cont... |
| trlco 38668 | The trace of a composition... |
| trlcone 38669 | If two translations have d... |
| cdlemg42 38670 | Part of proof of Lemma G o... |
| cdlemg43 38671 | Part of proof of Lemma G o... |
| cdlemg44a 38672 | Part of proof of Lemma G o... |
| cdlemg44b 38673 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 38674 | Part of proof of Lemma G o... |
| cdlemg47a 38675 | TODO: fix comment. TODO: ... |
| cdlemg46 38676 | Part of proof of Lemma G o... |
| cdlemg47 38677 | Part of proof of Lemma G o... |
| cdlemg48 38678 | Eliminate ` h ` from ~ cdl... |
| ltrncom 38679 | Composition is commutative... |
| ltrnco4 38680 | Rearrange a composition of... |
| trljco 38681 | Trace joined with trace of... |
| trljco2 38682 | Trace joined with trace of... |
| tgrpfset 38685 | The translation group maps... |
| tgrpset 38686 | The translation group for ... |
| tgrpbase 38687 | The base set of the transl... |
| tgrpopr 38688 | The group operation of the... |
| tgrpov 38689 | The group operation value ... |
| tgrpgrplem 38690 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 38691 | The translation group is a... |
| tgrpabl 38692 | The translation group is a... |
| tendofset 38699 | The set of all trace-prese... |
| tendoset 38700 | The set of trace-preservin... |
| istendo 38701 | The predicate "is a trace-... |
| tendotp 38702 | Trace-preserving property ... |
| istendod 38703 | Deduce the predicate "is a... |
| tendof 38704 | Functionality of a trace-p... |
| tendoeq1 38705 | Condition determining equa... |
| tendovalco 38706 | Value of composition of tr... |
| tendocoval 38707 | Value of composition of en... |
| tendocl 38708 | Closure of a trace-preserv... |
| tendoco2 38709 | Distribution of compositio... |
| tendoidcl 38710 | The identity is a trace-pr... |
| tendo1mul 38711 | Multiplicative identity mu... |
| tendo1mulr 38712 | Multiplicative identity mu... |
| tendococl 38713 | The composition of two tra... |
| tendoid 38714 | The identity value of a tr... |
| tendoeq2 38715 | Condition determining equa... |
| tendoplcbv 38716 | Define sum operation for t... |
| tendopl 38717 | Value of endomorphism sum ... |
| tendopl2 38718 | Value of result of endomor... |
| tendoplcl2 38719 | Value of result of endomor... |
| tendoplco2 38720 | Value of result of endomor... |
| tendopltp 38721 | Trace-preserving property ... |
| tendoplcl 38722 | Endomorphism sum is a trac... |
| tendoplcom 38723 | The endomorphism sum opera... |
| tendoplass 38724 | The endomorphism sum opera... |
| tendodi1 38725 | Endomorphism composition d... |
| tendodi2 38726 | Endomorphism composition d... |
| tendo0cbv 38727 | Define additive identity f... |
| tendo02 38728 | Value of additive identity... |
| tendo0co2 38729 | The additive identity trac... |
| tendo0tp 38730 | Trace-preserving property ... |
| tendo0cl 38731 | The additive identity is a... |
| tendo0pl 38732 | Property of the additive i... |
| tendo0plr 38733 | Property of the additive i... |
| tendoicbv 38734 | Define inverse function fo... |
| tendoi 38735 | Value of inverse endomorph... |
| tendoi2 38736 | Value of additive inverse ... |
| tendoicl 38737 | Closure of the additive in... |
| tendoipl 38738 | Property of the additive i... |
| tendoipl2 38739 | Property of the additive i... |
| erngfset 38740 | The division rings on trac... |
| erngset 38741 | The division ring on trace... |
| erngbase 38742 | The base set of the divisi... |
| erngfplus 38743 | Ring addition operation. ... |
| erngplus 38744 | Ring addition operation. ... |
| erngplus2 38745 | Ring addition operation. ... |
| erngfmul 38746 | Ring multiplication operat... |
| erngmul 38747 | Ring addition operation. ... |
| erngfset-rN 38748 | The division rings on trac... |
| erngset-rN 38749 | The division ring on trace... |
| erngbase-rN 38750 | The base set of the divisi... |
| erngfplus-rN 38751 | Ring addition operation. ... |
| erngplus-rN 38752 | Ring addition operation. ... |
| erngplus2-rN 38753 | Ring addition operation. ... |
| erngfmul-rN 38754 | Ring multiplication operat... |
| erngmul-rN 38755 | Ring addition operation. ... |
| cdlemh1 38756 | Part of proof of Lemma H o... |
| cdlemh2 38757 | Part of proof of Lemma H o... |
| cdlemh 38758 | Lemma H of [Crawley] p. 11... |
| cdlemi1 38759 | Part of proof of Lemma I o... |
| cdlemi2 38760 | Part of proof of Lemma I o... |
| cdlemi 38761 | Lemma I of [Crawley] p. 11... |
| cdlemj1 38762 | Part of proof of Lemma J o... |
| cdlemj2 38763 | Part of proof of Lemma J o... |
| cdlemj3 38764 | Part of proof of Lemma J o... |
| tendocan 38765 | Cancellation law: if the v... |
| tendoid0 38766 | A trace-preserving endomor... |
| tendo0mul 38767 | Additive identity multipli... |
| tendo0mulr 38768 | Additive identity multipli... |
| tendo1ne0 38769 | The identity (unity) is no... |
| tendoconid 38770 | The composition (product) ... |
| tendotr 38771 | The trace of the value of ... |
| cdlemk1 38772 | Part of proof of Lemma K o... |
| cdlemk2 38773 | Part of proof of Lemma K o... |
| cdlemk3 38774 | Part of proof of Lemma K o... |
| cdlemk4 38775 | Part of proof of Lemma K o... |
| cdlemk5a 38776 | Part of proof of Lemma K o... |
| cdlemk5 38777 | Part of proof of Lemma K o... |
| cdlemk6 38778 | Part of proof of Lemma K o... |
| cdlemk8 38779 | Part of proof of Lemma K o... |
| cdlemk9 38780 | Part of proof of Lemma K o... |
| cdlemk9bN 38781 | Part of proof of Lemma K o... |
| cdlemki 38782 | Part of proof of Lemma K o... |
| cdlemkvcl 38783 | Part of proof of Lemma K o... |
| cdlemk10 38784 | Part of proof of Lemma K o... |
| cdlemksv 38785 | Part of proof of Lemma K o... |
| cdlemksel 38786 | Part of proof of Lemma K o... |
| cdlemksat 38787 | Part of proof of Lemma K o... |
| cdlemksv2 38788 | Part of proof of Lemma K o... |
| cdlemk7 38789 | Part of proof of Lemma K o... |
| cdlemk11 38790 | Part of proof of Lemma K o... |
| cdlemk12 38791 | Part of proof of Lemma K o... |
| cdlemkoatnle 38792 | Utility lemma. (Contribut... |
| cdlemk13 38793 | Part of proof of Lemma K o... |
| cdlemkole 38794 | Utility lemma. (Contribut... |
| cdlemk14 38795 | Part of proof of Lemma K o... |
| cdlemk15 38796 | Part of proof of Lemma K o... |
| cdlemk16a 38797 | Part of proof of Lemma K o... |
| cdlemk16 38798 | Part of proof of Lemma K o... |
| cdlemk17 38799 | Part of proof of Lemma K o... |
| cdlemk1u 38800 | Part of proof of Lemma K o... |
| cdlemk5auN 38801 | Part of proof of Lemma K o... |
| cdlemk5u 38802 | Part of proof of Lemma K o... |
| cdlemk6u 38803 | Part of proof of Lemma K o... |
| cdlemkj 38804 | Part of proof of Lemma K o... |
| cdlemkuvN 38805 | Part of proof of Lemma K o... |
| cdlemkuel 38806 | Part of proof of Lemma K o... |
| cdlemkuat 38807 | Part of proof of Lemma K o... |
| cdlemkuv2 38808 | Part of proof of Lemma K o... |
| cdlemk18 38809 | Part of proof of Lemma K o... |
| cdlemk19 38810 | Part of proof of Lemma K o... |
| cdlemk7u 38811 | Part of proof of Lemma K o... |
| cdlemk11u 38812 | Part of proof of Lemma K o... |
| cdlemk12u 38813 | Part of proof of Lemma K o... |
| cdlemk21N 38814 | Part of proof of Lemma K o... |
| cdlemk20 38815 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 38816 | Utility lemma. (Contribut... |
| cdlemk13-2N 38817 | Part of proof of Lemma K o... |
| cdlemkole-2N 38818 | Utility lemma. (Contribut... |
| cdlemk14-2N 38819 | Part of proof of Lemma K o... |
| cdlemk15-2N 38820 | Part of proof of Lemma K o... |
| cdlemk16-2N 38821 | Part of proof of Lemma K o... |
| cdlemk17-2N 38822 | Part of proof of Lemma K o... |
| cdlemkj-2N 38823 | Part of proof of Lemma K o... |
| cdlemkuv-2N 38824 | Part of proof of Lemma K o... |
| cdlemkuel-2N 38825 | Part of proof of Lemma K o... |
| cdlemkuv2-2 38826 | Part of proof of Lemma K o... |
| cdlemk18-2N 38827 | Part of proof of Lemma K o... |
| cdlemk19-2N 38828 | Part of proof of Lemma K o... |
| cdlemk7u-2N 38829 | Part of proof of Lemma K o... |
| cdlemk11u-2N 38830 | Part of proof of Lemma K o... |
| cdlemk12u-2N 38831 | Part of proof of Lemma K o... |
| cdlemk21-2N 38832 | Part of proof of Lemma K o... |
| cdlemk20-2N 38833 | Part of proof of Lemma K o... |
| cdlemk22 38834 | Part of proof of Lemma K o... |
| cdlemk30 38835 | Part of proof of Lemma K o... |
| cdlemkuu 38836 | Convert between function a... |
| cdlemk31 38837 | Part of proof of Lemma K o... |
| cdlemk32 38838 | Part of proof of Lemma K o... |
| cdlemkuel-3 38839 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 38840 | Part of proof of Lemma K o... |
| cdlemk18-3N 38841 | Part of proof of Lemma K o... |
| cdlemk22-3 38842 | Part of proof of Lemma K o... |
| cdlemk23-3 38843 | Part of proof of Lemma K o... |
| cdlemk24-3 38844 | Part of proof of Lemma K o... |
| cdlemk25-3 38845 | Part of proof of Lemma K o... |
| cdlemk26b-3 38846 | Part of proof of Lemma K o... |
| cdlemk26-3 38847 | Part of proof of Lemma K o... |
| cdlemk27-3 38848 | Part of proof of Lemma K o... |
| cdlemk28-3 38849 | Part of proof of Lemma K o... |
| cdlemk33N 38850 | Part of proof of Lemma K o... |
| cdlemk34 38851 | Part of proof of Lemma K o... |
| cdlemk29-3 38852 | Part of proof of Lemma K o... |
| cdlemk35 38853 | Part of proof of Lemma K o... |
| cdlemk36 38854 | Part of proof of Lemma K o... |
| cdlemk37 38855 | Part of proof of Lemma K o... |
| cdlemk38 38856 | Part of proof of Lemma K o... |
| cdlemk39 38857 | Part of proof of Lemma K o... |
| cdlemk40 38858 | TODO: fix comment. (Contr... |
| cdlemk40t 38859 | TODO: fix comment. (Contr... |
| cdlemk40f 38860 | TODO: fix comment. (Contr... |
| cdlemk41 38861 | Part of proof of Lemma K o... |
| cdlemkfid1N 38862 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 38863 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 38864 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 38865 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 38866 | TODO: is this useful or sh... |
| cdlemky 38867 | Part of proof of Lemma K o... |
| cdlemkyu 38868 | Convert between function a... |
| cdlemkyuu 38869 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 38870 | Part of proof of Lemma K o... |
| cdlemk19ylem 38871 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 38872 | Part of proof of Lemma K o... |
| cdlemk19y 38873 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 38874 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 38875 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 38876 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 38877 | The value of the tau funct... |
| cdlemk35s 38878 | Substitution version of ~ ... |
| cdlemk35s-id 38879 | Substitution version of ~ ... |
| cdlemk39s 38880 | Substitution version of ~ ... |
| cdlemk39s-id 38881 | Substitution version of ~ ... |
| cdlemk42 38882 | Part of proof of Lemma K o... |
| cdlemk19xlem 38883 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 38884 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 38885 | Part of proof of Lemma K o... |
| cdlemk11tc 38886 | Part of proof of Lemma K o... |
| cdlemk11t 38887 | Part of proof of Lemma K o... |
| cdlemk45 38888 | Part of proof of Lemma K o... |
| cdlemk46 38889 | Part of proof of Lemma K o... |
| cdlemk47 38890 | Part of proof of Lemma K o... |
| cdlemk48 38891 | Part of proof of Lemma K o... |
| cdlemk49 38892 | Part of proof of Lemma K o... |
| cdlemk50 38893 | Part of proof of Lemma K o... |
| cdlemk51 38894 | Part of proof of Lemma K o... |
| cdlemk52 38895 | Part of proof of Lemma K o... |
| cdlemk53a 38896 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 38897 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 38898 | Part of proof of Lemma K o... |
| cdlemk54 38899 | Part of proof of Lemma K o... |
| cdlemk55a 38900 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 38901 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 38902 | Part of proof of Lemma K o... |
| cdlemkyyN 38903 | Part of proof of Lemma K o... |
| cdlemk43N 38904 | Part of proof of Lemma K o... |
| cdlemk35u 38905 | Substitution version of ~ ... |
| cdlemk55u1 38906 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 38907 | Part of proof of Lemma K o... |
| cdlemk39u1 38908 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 38909 | Part of proof of Lemma K o... |
| cdlemk19u1 38910 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 38911 | Part of Lemma K of [Crawle... |
| cdlemk56 38912 | Part of Lemma K of [Crawle... |
| cdlemk19w 38913 | Use a fixed element to eli... |
| cdlemk56w 38914 | Use a fixed element to eli... |
| cdlemk 38915 | Lemma K of [Crawley] p. 11... |
| tendoex 38916 | Generalization of Lemma K ... |
| cdleml1N 38917 | Part of proof of Lemma L o... |
| cdleml2N 38918 | Part of proof of Lemma L o... |
| cdleml3N 38919 | Part of proof of Lemma L o... |
| cdleml4N 38920 | Part of proof of Lemma L o... |
| cdleml5N 38921 | Part of proof of Lemma L o... |
| cdleml6 38922 | Part of proof of Lemma L o... |
| cdleml7 38923 | Part of proof of Lemma L o... |
| cdleml8 38924 | Part of proof of Lemma L o... |
| cdleml9 38925 | Part of proof of Lemma L o... |
| dva1dim 38926 | Two expressions for the 1-... |
| dvhb1dimN 38927 | Two expressions for the 1-... |
| erng1lem 38928 | Value of the endomorphism ... |
| erngdvlem1 38929 | Lemma for ~ eringring . (... |
| erngdvlem2N 38930 | Lemma for ~ eringring . (... |
| erngdvlem3 38931 | Lemma for ~ eringring . (... |
| erngdvlem4 38932 | Lemma for ~ erngdv . (Con... |
| eringring 38933 | An endomorphism ring is a ... |
| erngdv 38934 | An endomorphism ring is a ... |
| erng0g 38935 | The division ring zero of ... |
| erng1r 38936 | The division ring unit of ... |
| erngdvlem1-rN 38937 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 38938 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 38939 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 38940 | Lemma for ~ erngdv . (Con... |
| erngring-rN 38941 | An endomorphism ring is a ... |
| erngdv-rN 38942 | An endomorphism ring is a ... |
| dvafset 38945 | The constructed partial ve... |
| dvaset 38946 | The constructed partial ve... |
| dvasca 38947 | The ring base set of the c... |
| dvabase 38948 | The ring base set of the c... |
| dvafplusg 38949 | Ring addition operation fo... |
| dvaplusg 38950 | Ring addition operation fo... |
| dvaplusgv 38951 | Ring addition operation fo... |
| dvafmulr 38952 | Ring multiplication operat... |
| dvamulr 38953 | Ring multiplication operat... |
| dvavbase 38954 | The vectors (vector base s... |
| dvafvadd 38955 | The vector sum operation f... |
| dvavadd 38956 | Ring addition operation fo... |
| dvafvsca 38957 | Ring addition operation fo... |
| dvavsca 38958 | Ring addition operation fo... |
| tendospcl 38959 | Closure of endomorphism sc... |
| tendospass 38960 | Associative law for endomo... |
| tendospdi1 38961 | Forward distributive law f... |
| tendocnv 38962 | Converse of a trace-preser... |
| tendospdi2 38963 | Reverse distributive law f... |
| tendospcanN 38964 | Cancellation law for trace... |
| dvaabl 38965 | The constructed partial ve... |
| dvalveclem 38966 | Lemma for ~ dvalvec . (Co... |
| dvalvec 38967 | The constructed partial ve... |
| dva0g 38968 | The zero vector of partial... |
| diaffval 38971 | The partial isomorphism A ... |
| diafval 38972 | The partial isomorphism A ... |
| diaval 38973 | The partial isomorphism A ... |
| diaelval 38974 | Member of the partial isom... |
| diafn 38975 | Functionality and domain o... |
| diadm 38976 | Domain of the partial isom... |
| diaeldm 38977 | Member of domain of the pa... |
| diadmclN 38978 | A member of domain of the ... |
| diadmleN 38979 | A member of domain of the ... |
| dian0 38980 | The value of the partial i... |
| dia0eldmN 38981 | The lattice zero belongs t... |
| dia1eldmN 38982 | The fiducial hyperplane (t... |
| diass 38983 | The value of the partial i... |
| diael 38984 | A member of the value of t... |
| diatrl 38985 | Trace of a member of the p... |
| diaelrnN 38986 | Any value of the partial i... |
| dialss 38987 | The value of partial isomo... |
| diaord 38988 | The partial isomorphism A ... |
| dia11N 38989 | The partial isomorphism A ... |
| diaf11N 38990 | The partial isomorphism A ... |
| diaclN 38991 | Closure of partial isomorp... |
| diacnvclN 38992 | Closure of partial isomorp... |
| dia0 38993 | The value of the partial i... |
| dia1N 38994 | The value of the partial i... |
| dia1elN 38995 | The largest subspace in th... |
| diaglbN 38996 | Partial isomorphism A of a... |
| diameetN 38997 | Partial isomorphism A of a... |
| diainN 38998 | Inverse partial isomorphis... |
| diaintclN 38999 | The intersection of partia... |
| diasslssN 39000 | The partial isomorphism A ... |
| diassdvaN 39001 | The partial isomorphism A ... |
| dia1dim 39002 | Two expressions for the 1-... |
| dia1dim2 39003 | Two expressions for a 1-di... |
| dia1dimid 39004 | A vector (translation) bel... |
| dia2dimlem1 39005 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 39006 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 39007 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 39008 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 39009 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 39010 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 39011 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 39012 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 39013 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 39014 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 39015 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 39016 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 39017 | Lemma for ~ dia2dim . Eli... |
| dia2dim 39018 | A two-dimensional subspace... |
| dvhfset 39021 | The constructed full vecto... |
| dvhset 39022 | The constructed full vecto... |
| dvhsca 39023 | The ring of scalars of the... |
| dvhbase 39024 | The ring base set of the c... |
| dvhfplusr 39025 | Ring addition operation fo... |
| dvhfmulr 39026 | Ring multiplication operat... |
| dvhmulr 39027 | Ring multiplication operat... |
| dvhvbase 39028 | The vectors (vector base s... |
| dvhelvbasei 39029 | Vector membership in the c... |
| dvhvaddcbv 39030 | Change bound variables to ... |
| dvhvaddval 39031 | The vector sum operation f... |
| dvhfvadd 39032 | The vector sum operation f... |
| dvhvadd 39033 | The vector sum operation f... |
| dvhopvadd 39034 | The vector sum operation f... |
| dvhopvadd2 39035 | The vector sum operation f... |
| dvhvaddcl 39036 | Closure of the vector sum ... |
| dvhvaddcomN 39037 | Commutativity of vector su... |
| dvhvaddass 39038 | Associativity of vector su... |
| dvhvscacbv 39039 | Change bound variables to ... |
| dvhvscaval 39040 | The scalar product operati... |
| dvhfvsca 39041 | Scalar product operation f... |
| dvhvsca 39042 | Scalar product operation f... |
| dvhopvsca 39043 | Scalar product operation f... |
| dvhvscacl 39044 | Closure of the scalar prod... |
| tendoinvcl 39045 | Closure of multiplicative ... |
| tendolinv 39046 | Left multiplicative invers... |
| tendorinv 39047 | Right multiplicative inver... |
| dvhgrp 39048 | The full vector space ` U ... |
| dvhlveclem 39049 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 39050 | The full vector space ` U ... |
| dvhlmod 39051 | The full vector space ` U ... |
| dvh0g 39052 | The zero vector of vector ... |
| dvheveccl 39053 | Properties of a unit vecto... |
| dvhopclN 39054 | Closure of a ` DVecH ` vec... |
| dvhopaddN 39055 | Sum of ` DVecH ` vectors e... |
| dvhopspN 39056 | Scalar product of ` DVecH ... |
| dvhopN 39057 | Decompose a ` DVecH ` vect... |
| dvhopellsm 39058 | Ordered pair membership in... |
| cdlemm10N 39059 | The image of the map ` G `... |
| docaffvalN 39062 | Subspace orthocomplement f... |
| docafvalN 39063 | Subspace orthocomplement f... |
| docavalN 39064 | Subspace orthocomplement f... |
| docaclN 39065 | Closure of subspace orthoc... |
| diaocN 39066 | Value of partial isomorphi... |
| doca2N 39067 | Double orthocomplement of ... |
| doca3N 39068 | Double orthocomplement of ... |
| dvadiaN 39069 | Any closed subspace is a m... |
| diarnN 39070 | Partial isomorphism A maps... |
| diaf1oN 39071 | The partial isomorphism A ... |
| djaffvalN 39074 | Subspace join for ` DVecA ... |
| djafvalN 39075 | Subspace join for ` DVecA ... |
| djavalN 39076 | Subspace join for ` DVecA ... |
| djaclN 39077 | Closure of subspace join f... |
| djajN 39078 | Transfer lattice join to `... |
| dibffval 39081 | The partial isomorphism B ... |
| dibfval 39082 | The partial isomorphism B ... |
| dibval 39083 | The partial isomorphism B ... |
| dibopelvalN 39084 | Member of the partial isom... |
| dibval2 39085 | Value of the partial isomo... |
| dibopelval2 39086 | Member of the partial isom... |
| dibval3N 39087 | Value of the partial isomo... |
| dibelval3 39088 | Member of the partial isom... |
| dibopelval3 39089 | Member of the partial isom... |
| dibelval1st 39090 | Membership in value of the... |
| dibelval1st1 39091 | Membership in value of the... |
| dibelval1st2N 39092 | Membership in value of the... |
| dibelval2nd 39093 | Membership in value of the... |
| dibn0 39094 | The value of the partial i... |
| dibfna 39095 | Functionality and domain o... |
| dibdiadm 39096 | Domain of the partial isom... |
| dibfnN 39097 | Functionality and domain o... |
| dibdmN 39098 | Domain of the partial isom... |
| dibeldmN 39099 | Member of domain of the pa... |
| dibord 39100 | The isomorphism B for a la... |
| dib11N 39101 | The isomorphism B for a la... |
| dibf11N 39102 | The partial isomorphism A ... |
| dibclN 39103 | Closure of partial isomorp... |
| dibvalrel 39104 | The value of partial isomo... |
| dib0 39105 | The value of partial isomo... |
| dib1dim 39106 | Two expressions for the 1-... |
| dibglbN 39107 | Partial isomorphism B of a... |
| dibintclN 39108 | The intersection of partia... |
| dib1dim2 39109 | Two expressions for a 1-di... |
| dibss 39110 | The partial isomorphism B ... |
| diblss 39111 | The value of partial isomo... |
| diblsmopel 39112 | Membership in subspace sum... |
| dicffval 39115 | The partial isomorphism C ... |
| dicfval 39116 | The partial isomorphism C ... |
| dicval 39117 | The partial isomorphism C ... |
| dicopelval 39118 | Membership in value of the... |
| dicelvalN 39119 | Membership in value of the... |
| dicval2 39120 | The partial isomorphism C ... |
| dicelval3 39121 | Member of the partial isom... |
| dicopelval2 39122 | Membership in value of the... |
| dicelval2N 39123 | Membership in value of the... |
| dicfnN 39124 | Functionality and domain o... |
| dicdmN 39125 | Domain of the partial isom... |
| dicvalrelN 39126 | The value of partial isomo... |
| dicssdvh 39127 | The partial isomorphism C ... |
| dicelval1sta 39128 | Membership in value of the... |
| dicelval1stN 39129 | Membership in value of the... |
| dicelval2nd 39130 | Membership in value of the... |
| dicvaddcl 39131 | Membership in value of the... |
| dicvscacl 39132 | Membership in value of the... |
| dicn0 39133 | The value of the partial i... |
| diclss 39134 | The value of partial isomo... |
| diclspsn 39135 | The value of isomorphism C... |
| cdlemn2 39136 | Part of proof of Lemma N o... |
| cdlemn2a 39137 | Part of proof of Lemma N o... |
| cdlemn3 39138 | Part of proof of Lemma N o... |
| cdlemn4 39139 | Part of proof of Lemma N o... |
| cdlemn4a 39140 | Part of proof of Lemma N o... |
| cdlemn5pre 39141 | Part of proof of Lemma N o... |
| cdlemn5 39142 | Part of proof of Lemma N o... |
| cdlemn6 39143 | Part of proof of Lemma N o... |
| cdlemn7 39144 | Part of proof of Lemma N o... |
| cdlemn8 39145 | Part of proof of Lemma N o... |
| cdlemn9 39146 | Part of proof of Lemma N o... |
| cdlemn10 39147 | Part of proof of Lemma N o... |
| cdlemn11a 39148 | Part of proof of Lemma N o... |
| cdlemn11b 39149 | Part of proof of Lemma N o... |
| cdlemn11c 39150 | Part of proof of Lemma N o... |
| cdlemn11pre 39151 | Part of proof of Lemma N o... |
| cdlemn11 39152 | Part of proof of Lemma N o... |
| cdlemn 39153 | Lemma N of [Crawley] p. 12... |
| dihordlem6 39154 | Part of proof of Lemma N o... |
| dihordlem7 39155 | Part of proof of Lemma N o... |
| dihordlem7b 39156 | Part of proof of Lemma N o... |
| dihjustlem 39157 | Part of proof after Lemma ... |
| dihjust 39158 | Part of proof after Lemma ... |
| dihord1 39159 | Part of proof after Lemma ... |
| dihord2a 39160 | Part of proof after Lemma ... |
| dihord2b 39161 | Part of proof after Lemma ... |
| dihord2cN 39162 | Part of proof after Lemma ... |
| dihord11b 39163 | Part of proof after Lemma ... |
| dihord10 39164 | Part of proof after Lemma ... |
| dihord11c 39165 | Part of proof after Lemma ... |
| dihord2pre 39166 | Part of proof after Lemma ... |
| dihord2pre2 39167 | Part of proof after Lemma ... |
| dihord2 39168 | Part of proof after Lemma ... |
| dihffval 39171 | The isomorphism H for a la... |
| dihfval 39172 | Isomorphism H for a lattic... |
| dihval 39173 | Value of isomorphism H for... |
| dihvalc 39174 | Value of isomorphism H for... |
| dihlsscpre 39175 | Closure of isomorphism H f... |
| dihvalcqpre 39176 | Value of isomorphism H for... |
| dihvalcq 39177 | Value of isomorphism H for... |
| dihvalb 39178 | Value of isomorphism H for... |
| dihopelvalbN 39179 | Ordered pair member of the... |
| dihvalcqat 39180 | Value of isomorphism H for... |
| dih1dimb 39181 | Two expressions for a 1-di... |
| dih1dimb2 39182 | Isomorphism H at an atom u... |
| dih1dimc 39183 | Isomorphism H at an atom n... |
| dib2dim 39184 | Extend ~ dia2dim to partia... |
| dih2dimb 39185 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 39186 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 39187 | Ordered pair member of the... |
| dihvalcq2 39188 | Value of isomorphism H for... |
| dihopelvalcpre 39189 | Member of value of isomorp... |
| dihopelvalc 39190 | Member of value of isomorp... |
| dihlss 39191 | The value of isomorphism H... |
| dihss 39192 | The value of isomorphism H... |
| dihssxp 39193 | An isomorphism H value is ... |
| dihopcl 39194 | Closure of an ordered pair... |
| xihopellsmN 39195 | Ordered pair membership in... |
| dihopellsm 39196 | Ordered pair membership in... |
| dihord6apre 39197 | Part of proof that isomorp... |
| dihord3 39198 | The isomorphism H for a la... |
| dihord4 39199 | The isomorphism H for a la... |
| dihord5b 39200 | Part of proof that isomorp... |
| dihord6b 39201 | Part of proof that isomorp... |
| dihord6a 39202 | Part of proof that isomorp... |
| dihord5apre 39203 | Part of proof that isomorp... |
| dihord5a 39204 | Part of proof that isomorp... |
| dihord 39205 | The isomorphism H is order... |
| dih11 39206 | The isomorphism H is one-t... |
| dihf11lem 39207 | Functionality of the isomo... |
| dihf11 39208 | The isomorphism H for a la... |
| dihfn 39209 | Functionality and domain o... |
| dihdm 39210 | Domain of isomorphism H. (... |
| dihcl 39211 | Closure of isomorphism H. ... |
| dihcnvcl 39212 | Closure of isomorphism H c... |
| dihcnvid1 39213 | The converse isomorphism o... |
| dihcnvid2 39214 | The isomorphism of a conve... |
| dihcnvord 39215 | Ordering property for conv... |
| dihcnv11 39216 | The converse of isomorphis... |
| dihsslss 39217 | The isomorphism H maps to ... |
| dihrnlss 39218 | The isomorphism H maps to ... |
| dihrnss 39219 | The isomorphism H maps to ... |
| dihvalrel 39220 | The value of isomorphism H... |
| dih0 39221 | The value of isomorphism H... |
| dih0bN 39222 | A lattice element is zero ... |
| dih0vbN 39223 | A vector is zero iff its s... |
| dih0cnv 39224 | The isomorphism H converse... |
| dih0rn 39225 | The zero subspace belongs ... |
| dih0sb 39226 | A subspace is zero iff the... |
| dih1 39227 | The value of isomorphism H... |
| dih1rn 39228 | The full vector space belo... |
| dih1cnv 39229 | The isomorphism H converse... |
| dihwN 39230 | Value of isomorphism H at ... |
| dihmeetlem1N 39231 | Isomorphism H of a conjunc... |
| dihglblem5apreN 39232 | A conjunction property of ... |
| dihglblem5aN 39233 | A conjunction property of ... |
| dihglblem2aN 39234 | Lemma for isomorphism H of... |
| dihglblem2N 39235 | The GLB of a set of lattic... |
| dihglblem3N 39236 | Isomorphism H of a lattice... |
| dihglblem3aN 39237 | Isomorphism H of a lattice... |
| dihglblem4 39238 | Isomorphism H of a lattice... |
| dihglblem5 39239 | Isomorphism H of a lattice... |
| dihmeetlem2N 39240 | Isomorphism H of a conjunc... |
| dihglbcpreN 39241 | Isomorphism H of a lattice... |
| dihglbcN 39242 | Isomorphism H of a lattice... |
| dihmeetcN 39243 | Isomorphism H of a lattice... |
| dihmeetbN 39244 | Isomorphism H of a lattice... |
| dihmeetbclemN 39245 | Lemma for isomorphism H of... |
| dihmeetlem3N 39246 | Lemma for isomorphism H of... |
| dihmeetlem4preN 39247 | Lemma for isomorphism H of... |
| dihmeetlem4N 39248 | Lemma for isomorphism H of... |
| dihmeetlem5 39249 | Part of proof that isomorp... |
| dihmeetlem6 39250 | Lemma for isomorphism H of... |
| dihmeetlem7N 39251 | Lemma for isomorphism H of... |
| dihjatc1 39252 | Lemma for isomorphism H of... |
| dihjatc2N 39253 | Isomorphism H of join with... |
| dihjatc3 39254 | Isomorphism H of join with... |
| dihmeetlem8N 39255 | Lemma for isomorphism H of... |
| dihmeetlem9N 39256 | Lemma for isomorphism H of... |
| dihmeetlem10N 39257 | Lemma for isomorphism H of... |
| dihmeetlem11N 39258 | Lemma for isomorphism H of... |
| dihmeetlem12N 39259 | Lemma for isomorphism H of... |
| dihmeetlem13N 39260 | Lemma for isomorphism H of... |
| dihmeetlem14N 39261 | Lemma for isomorphism H of... |
| dihmeetlem15N 39262 | Lemma for isomorphism H of... |
| dihmeetlem16N 39263 | Lemma for isomorphism H of... |
| dihmeetlem17N 39264 | Lemma for isomorphism H of... |
| dihmeetlem18N 39265 | Lemma for isomorphism H of... |
| dihmeetlem19N 39266 | Lemma for isomorphism H of... |
| dihmeetlem20N 39267 | Lemma for isomorphism H of... |
| dihmeetALTN 39268 | Isomorphism H of a lattice... |
| dih1dimatlem0 39269 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 39270 | Lemma for ~ dih1dimat . (... |
| dih1dimat 39271 | Any 1-dimensional subspace... |
| dihlsprn 39272 | The span of a vector belon... |
| dihlspsnssN 39273 | A subspace included in a 1... |
| dihlspsnat 39274 | The inverse isomorphism H ... |
| dihatlat 39275 | The isomorphism H of an at... |
| dihat 39276 | There exists at least one ... |
| dihpN 39277 | The value of isomorphism H... |
| dihlatat 39278 | The reverse isomorphism H ... |
| dihatexv 39279 | There is a nonzero vector ... |
| dihatexv2 39280 | There is a nonzero vector ... |
| dihglblem6 39281 | Isomorphism H of a lattice... |
| dihglb 39282 | Isomorphism H of a lattice... |
| dihglb2 39283 | Isomorphism H of a lattice... |
| dihmeet 39284 | Isomorphism H of a lattice... |
| dihintcl 39285 | The intersection of closed... |
| dihmeetcl 39286 | Closure of closed subspace... |
| dihmeet2 39287 | Reverse isomorphism H of a... |
| dochffval 39290 | Subspace orthocomplement f... |
| dochfval 39291 | Subspace orthocomplement f... |
| dochval 39292 | Subspace orthocomplement f... |
| dochval2 39293 | Subspace orthocomplement f... |
| dochcl 39294 | Closure of subspace orthoc... |
| dochlss 39295 | A subspace orthocomplement... |
| dochssv 39296 | A subspace orthocomplement... |
| dochfN 39297 | Domain and codomain of the... |
| dochvalr 39298 | Orthocomplement of a close... |
| doch0 39299 | Orthocomplement of the zer... |
| doch1 39300 | Orthocomplement of the uni... |
| dochoc0 39301 | The zero subspace is close... |
| dochoc1 39302 | The unit subspace (all vec... |
| dochvalr2 39303 | Orthocomplement of a close... |
| dochvalr3 39304 | Orthocomplement of a close... |
| doch2val2 39305 | Double orthocomplement for... |
| dochss 39306 | Subset law for orthocomple... |
| dochocss 39307 | Double negative law for or... |
| dochoc 39308 | Double negative law for or... |
| dochsscl 39309 | If a set of vectors is inc... |
| dochoccl 39310 | A set of vectors is closed... |
| dochord 39311 | Ordering law for orthocomp... |
| dochord2N 39312 | Ordering law for orthocomp... |
| dochord3 39313 | Ordering law for orthocomp... |
| doch11 39314 | Orthocomplement is one-to-... |
| dochsordN 39315 | Strict ordering law for or... |
| dochn0nv 39316 | An orthocomplement is nonz... |
| dihoml4c 39317 | Version of ~ dihoml4 with ... |
| dihoml4 39318 | Orthomodular law for const... |
| dochspss 39319 | The span of a set of vecto... |
| dochocsp 39320 | The span of an orthocomple... |
| dochspocN 39321 | The span of an orthocomple... |
| dochocsn 39322 | The double orthocomplement... |
| dochsncom 39323 | Swap vectors in an orthoco... |
| dochsat 39324 | The double orthocomplement... |
| dochshpncl 39325 | If a hyperplane is not clo... |
| dochlkr 39326 | Equivalent conditions for ... |
| dochkrshp 39327 | The closure of a kernel is... |
| dochkrshp2 39328 | Properties of the closure ... |
| dochkrshp3 39329 | Properties of the closure ... |
| dochkrshp4 39330 | Properties of the closure ... |
| dochdmj1 39331 | De Morgan-like law for sub... |
| dochnoncon 39332 | Law of noncontradiction. ... |
| dochnel2 39333 | A nonzero member of a subs... |
| dochnel 39334 | A nonzero vector doesn't b... |
| djhffval 39337 | Subspace join for ` DVecH ... |
| djhfval 39338 | Subspace join for ` DVecH ... |
| djhval 39339 | Subspace join for ` DVecH ... |
| djhval2 39340 | Value of subspace join for... |
| djhcl 39341 | Closure of subspace join f... |
| djhlj 39342 | Transfer lattice join to `... |
| djhljjN 39343 | Lattice join in terms of `... |
| djhjlj 39344 | ` DVecH ` vector space clo... |
| djhj 39345 | ` DVecH ` vector space clo... |
| djhcom 39346 | Subspace join commutes. (... |
| djhspss 39347 | Subspace span of union is ... |
| djhsumss 39348 | Subspace sum is a subset o... |
| dihsumssj 39349 | The subspace sum of two is... |
| djhunssN 39350 | Subspace union is a subset... |
| dochdmm1 39351 | De Morgan-like law for clo... |
| djhexmid 39352 | Excluded middle property o... |
| djh01 39353 | Closed subspace join with ... |
| djh02 39354 | Closed subspace join with ... |
| djhlsmcl 39355 | A closed subspace sum equa... |
| djhcvat42 39356 | A covering property. ( ~ ... |
| dihjatb 39357 | Isomorphism H of lattice j... |
| dihjatc 39358 | Isomorphism H of lattice j... |
| dihjatcclem1 39359 | Lemma for isomorphism H of... |
| dihjatcclem2 39360 | Lemma for isomorphism H of... |
| dihjatcclem3 39361 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 39362 | Lemma for isomorphism H of... |
| dihjatcc 39363 | Isomorphism H of lattice j... |
| dihjat 39364 | Isomorphism H of lattice j... |
| dihprrnlem1N 39365 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 39366 | Lemma for ~ dihprrn . (Co... |
| dihprrn 39367 | The span of a vector pair ... |
| djhlsmat 39368 | The sum of two subspace at... |
| dihjat1lem 39369 | Subspace sum of a closed s... |
| dihjat1 39370 | Subspace sum of a closed s... |
| dihsmsprn 39371 | Subspace sum of a closed s... |
| dihjat2 39372 | The subspace sum of a clos... |
| dihjat3 39373 | Isomorphism H of lattice j... |
| dihjat4 39374 | Transfer the subspace sum ... |
| dihjat6 39375 | Transfer the subspace sum ... |
| dihsmsnrn 39376 | The subspace sum of two si... |
| dihsmatrn 39377 | The subspace sum of a clos... |
| dihjat5N 39378 | Transfer lattice join with... |
| dvh4dimat 39379 | There is an atom that is o... |
| dvh3dimatN 39380 | There is an atom that is o... |
| dvh2dimatN 39381 | Given an atom, there exist... |
| dvh1dimat 39382 | There exists an atom. (Co... |
| dvh1dim 39383 | There exists a nonzero vec... |
| dvh4dimlem 39384 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 39385 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 39386 | There is a vector that is ... |
| dvh3dim 39387 | There is a vector that is ... |
| dvh4dimN 39388 | There is a vector that is ... |
| dvh3dim2 39389 | There is a vector that is ... |
| dvh3dim3N 39390 | There is a vector that is ... |
| dochsnnz 39391 | The orthocomplement of a s... |
| dochsatshp 39392 | The orthocomplement of a s... |
| dochsatshpb 39393 | The orthocomplement of a s... |
| dochsnshp 39394 | The orthocomplement of a n... |
| dochshpsat 39395 | A hyperplane is closed iff... |
| dochkrsat 39396 | The orthocomplement of a k... |
| dochkrsat2 39397 | The orthocomplement of a k... |
| dochsat0 39398 | The orthocomplement of a k... |
| dochkrsm 39399 | The subspace sum of a clos... |
| dochexmidat 39400 | Special case of excluded m... |
| dochexmidlem1 39401 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 39402 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 39403 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 39404 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 39405 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 39406 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 39407 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 39408 | Lemma for ~ dochexmid . T... |
| dochexmid 39409 | Excluded middle law for cl... |
| dochsnkrlem1 39410 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 39411 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 39412 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 39413 | A (closed) kernel expresse... |
| dochsnkr2 39414 | Kernel of the explicit fun... |
| dochsnkr2cl 39415 | The ` X ` determining func... |
| dochflcl 39416 | Closure of the explicit fu... |
| dochfl1 39417 | The value of the explicit ... |
| dochfln0 39418 | The value of a functional ... |
| dochkr1 39419 | A nonzero functional has a... |
| dochkr1OLDN 39420 | A nonzero functional has a... |
| lpolsetN 39423 | The set of polarities of a... |
| islpolN 39424 | The predicate "is a polari... |
| islpoldN 39425 | Properties that determine ... |
| lpolfN 39426 | Functionality of a polarit... |
| lpolvN 39427 | The polarity of the whole ... |
| lpolconN 39428 | Contraposition property of... |
| lpolsatN 39429 | The polarity of an atomic ... |
| lpolpolsatN 39430 | Property of a polarity. (... |
| dochpolN 39431 | The subspace orthocompleme... |
| lcfl1lem 39432 | Property of a functional w... |
| lcfl1 39433 | Property of a functional w... |
| lcfl2 39434 | Property of a functional w... |
| lcfl3 39435 | Property of a functional w... |
| lcfl4N 39436 | Property of a functional w... |
| lcfl5 39437 | Property of a functional w... |
| lcfl5a 39438 | Property of a functional w... |
| lcfl6lem 39439 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 39440 | Lemma for ~ lcfl7N . If t... |
| lcfl6 39441 | Property of a functional w... |
| lcfl7N 39442 | Property of a functional w... |
| lcfl8 39443 | Property of a functional w... |
| lcfl8a 39444 | Property of a functional w... |
| lcfl8b 39445 | Property of a nonzero func... |
| lcfl9a 39446 | Property implying that a f... |
| lclkrlem1 39447 | The set of functionals hav... |
| lclkrlem2a 39448 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 39449 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 39450 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 39451 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 39452 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 39453 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 39454 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 39455 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 39456 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 39457 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 39458 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 39459 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 39460 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 39461 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 39462 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 39463 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 39464 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 39465 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 39466 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 39467 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 39468 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 39469 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 39470 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 39471 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 39472 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 39473 | The set of functionals hav... |
| lclkr 39474 | The set of functionals wit... |
| lcfls1lem 39475 | Property of a functional w... |
| lcfls1N 39476 | Property of a functional w... |
| lcfls1c 39477 | Property of a functional w... |
| lclkrslem1 39478 | The set of functionals hav... |
| lclkrslem2 39479 | The set of functionals hav... |
| lclkrs 39480 | The set of functionals hav... |
| lclkrs2 39481 | The set of functionals wit... |
| lcfrvalsnN 39482 | Reconstruction from the du... |
| lcfrlem1 39483 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 39484 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 39485 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 39486 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 39487 | Lemma for ~ lcfr . The se... |
| lcfrlem6 39488 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 39489 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 39490 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 39491 | Lemma for ~ lcf1o . (This... |
| lcf1o 39492 | Define a function ` J ` th... |
| lcfrlem10 39493 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 39494 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 39495 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 39496 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 39497 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 39498 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 39499 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 39500 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 39501 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 39502 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 39503 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 39504 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 39505 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 39506 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 39507 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 39508 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 39509 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 39510 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 39511 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 39512 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 39513 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 39514 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 39515 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 39516 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 39517 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 39518 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 39519 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 39520 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 39521 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 39522 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 39523 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 39524 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 39525 | Lemma for ~ lcfr . Elimin... |
| lcfr 39526 | Reconstruction of a subspa... |
| lcdfval 39529 | Dual vector space of funct... |
| lcdval 39530 | Dual vector space of funct... |
| lcdval2 39531 | Dual vector space of funct... |
| lcdlvec 39532 | The dual vector space of f... |
| lcdlmod 39533 | The dual vector space of f... |
| lcdvbase 39534 | Vector base set of a dual ... |
| lcdvbasess 39535 | The vector base set of the... |
| lcdvbaselfl 39536 | A vector in the base set o... |
| lcdvbasecl 39537 | Closure of the value of a ... |
| lcdvadd 39538 | Vector addition for the cl... |
| lcdvaddval 39539 | The value of the value of ... |
| lcdsca 39540 | The ring of scalars of the... |
| lcdsbase 39541 | Base set of scalar ring fo... |
| lcdsadd 39542 | Scalar addition for the cl... |
| lcdsmul 39543 | Scalar multiplication for ... |
| lcdvs 39544 | Scalar product for the clo... |
| lcdvsval 39545 | Value of scalar product op... |
| lcdvscl 39546 | The scalar product operati... |
| lcdlssvscl 39547 | Closure of scalar product ... |
| lcdvsass 39548 | Associative law for scalar... |
| lcd0 39549 | The zero scalar of the clo... |
| lcd1 39550 | The unit scalar of the clo... |
| lcdneg 39551 | The unit scalar of the clo... |
| lcd0v 39552 | The zero functional in the... |
| lcd0v2 39553 | The zero functional in the... |
| lcd0vvalN 39554 | Value of the zero function... |
| lcd0vcl 39555 | Closure of the zero functi... |
| lcd0vs 39556 | A scalar zero times a func... |
| lcdvs0N 39557 | A scalar times the zero fu... |
| lcdvsub 39558 | The value of vector subtra... |
| lcdvsubval 39559 | The value of the value of ... |
| lcdlss 39560 | Subspaces of a dual vector... |
| lcdlss2N 39561 | Subspaces of a dual vector... |
| lcdlsp 39562 | Span in the set of functio... |
| lcdlkreqN 39563 | Colinear functionals have ... |
| lcdlkreq2N 39564 | Colinear functionals have ... |
| mapdffval 39567 | Projectivity from vector s... |
| mapdfval 39568 | Projectivity from vector s... |
| mapdval 39569 | Value of projectivity from... |
| mapdvalc 39570 | Value of projectivity from... |
| mapdval2N 39571 | Value of projectivity from... |
| mapdval3N 39572 | Value of projectivity from... |
| mapdval4N 39573 | Value of projectivity from... |
| mapdval5N 39574 | Value of projectivity from... |
| mapdordlem1a 39575 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 39576 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 39577 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 39578 | Lemma for ~ mapdord . Ord... |
| mapdord 39579 | Ordering property of the m... |
| mapd11 39580 | The map defined by ~ df-ma... |
| mapddlssN 39581 | The mapping of a subspace ... |
| mapdsn 39582 | Value of the map defined b... |
| mapdsn2 39583 | Value of the map defined b... |
| mapdsn3 39584 | Value of the map defined b... |
| mapd1dim2lem1N 39585 | Value of the map defined b... |
| mapdrvallem2 39586 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 39587 | Lemma for ~ mapdrval . (C... |
| mapdrval 39588 | Given a dual subspace ` R ... |
| mapd1o 39589 | The map defined by ~ df-ma... |
| mapdrn 39590 | Range of the map defined b... |
| mapdunirnN 39591 | Union of the range of the ... |
| mapdrn2 39592 | Range of the map defined b... |
| mapdcnvcl 39593 | Closure of the converse of... |
| mapdcl 39594 | Closure the value of the m... |
| mapdcnvid1N 39595 | Converse of the value of t... |
| mapdsord 39596 | Strong ordering property o... |
| mapdcl2 39597 | The mapping of a subspace ... |
| mapdcnvid2 39598 | Value of the converse of t... |
| mapdcnvordN 39599 | Ordering property of the c... |
| mapdcnv11N 39600 | The converse of the map de... |
| mapdcv 39601 | Covering property of the c... |
| mapdincl 39602 | Closure of dual subspace i... |
| mapdin 39603 | Subspace intersection is p... |
| mapdlsmcl 39604 | Closure of dual subspace s... |
| mapdlsm 39605 | Subspace sum is preserved ... |
| mapd0 39606 | Projectivity map of the ze... |
| mapdcnvatN 39607 | Atoms are preserved by the... |
| mapdat 39608 | Atoms are preserved by the... |
| mapdspex 39609 | The map of a span equals t... |
| mapdn0 39610 | Transfer nonzero property ... |
| mapdncol 39611 | Transfer non-colinearity f... |
| mapdindp 39612 | Transfer (part of) vector ... |
| mapdpglem1 39613 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 39614 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 39615 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 39616 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 39617 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 39618 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 39619 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 39620 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 39621 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 39622 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 39623 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 39624 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 39625 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 39626 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 39627 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 39628 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 39629 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 39630 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 39631 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 39632 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 39633 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 39634 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 39635 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 39636 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 39637 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 39638 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 39639 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 39640 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 39641 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 39642 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 39643 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 39644 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 39645 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 39646 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 39647 | Part 1 of proof of the fir... |
| baerlem3lem1 39648 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 39649 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 39650 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 39651 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 39652 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 39653 | Lemma for ~ baerlem5b . (... |
| baerlem3 39654 | An equality that holds whe... |
| baerlem5a 39655 | An equality that holds whe... |
| baerlem5b 39656 | An equality that holds whe... |
| baerlem5amN 39657 | An equality that holds whe... |
| baerlem5bmN 39658 | An equality that holds whe... |
| baerlem5abmN 39659 | An equality that holds whe... |
| mapdindp0 39660 | Vector independence lemma.... |
| mapdindp1 39661 | Vector independence lemma.... |
| mapdindp2 39662 | Vector independence lemma.... |
| mapdindp3 39663 | Vector independence lemma.... |
| mapdindp4 39664 | Vector independence lemma.... |
| mapdhval 39665 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 39666 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 39667 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 39668 | Lemmma for ~~? mapdh . (C... |
| mapdheq 39669 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 39670 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 39671 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 39672 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 39673 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 39674 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 39675 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 39676 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 39677 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 39678 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 39679 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 39680 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 39681 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 39682 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 39683 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 39684 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 39685 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 39686 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 39687 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 39688 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 39689 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 39690 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 39691 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 39692 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 39693 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 39694 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 39695 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 39696 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 39699 | Map from nonzero vectors t... |
| hvmapfval 39700 | Map from nonzero vectors t... |
| hvmapval 39701 | Value of map from nonzero ... |
| hvmapvalvalN 39702 | Value of value of map (i.e... |
| hvmapidN 39703 | The value of the vector to... |
| hvmap1o 39704 | The vector to functional m... |
| hvmapclN 39705 | Closure of the vector to f... |
| hvmap1o2 39706 | The vector to functional m... |
| hvmapcl2 39707 | Closure of the vector to f... |
| hvmaplfl 39708 | The vector to functional m... |
| hvmaplkr 39709 | Kernel of the vector to fu... |
| mapdhvmap 39710 | Relationship between ` map... |
| lspindp5 39711 | Obtain an independent vect... |
| hdmaplem1 39712 | Lemma to convert a frequen... |
| hdmaplem2N 39713 | Lemma to convert a frequen... |
| hdmaplem3 39714 | Lemma to convert a frequen... |
| hdmaplem4 39715 | Lemma to convert a frequen... |
| mapdh8a 39716 | Part of Part (8) in [Baer]... |
| mapdh8aa 39717 | Part of Part (8) in [Baer]... |
| mapdh8ab 39718 | Part of Part (8) in [Baer]... |
| mapdh8ac 39719 | Part of Part (8) in [Baer]... |
| mapdh8ad 39720 | Part of Part (8) in [Baer]... |
| mapdh8b 39721 | Part of Part (8) in [Baer]... |
| mapdh8c 39722 | Part of Part (8) in [Baer]... |
| mapdh8d0N 39723 | Part of Part (8) in [Baer]... |
| mapdh8d 39724 | Part of Part (8) in [Baer]... |
| mapdh8e 39725 | Part of Part (8) in [Baer]... |
| mapdh8g 39726 | Part of Part (8) in [Baer]... |
| mapdh8i 39727 | Part of Part (8) in [Baer]... |
| mapdh8j 39728 | Part of Part (8) in [Baer]... |
| mapdh8 39729 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 39730 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 39731 | Lemma for part (9) in [Bae... |
| hdmap1ffval 39736 | Preliminary map from vecto... |
| hdmap1fval 39737 | Preliminary map from vecto... |
| hdmap1vallem 39738 | Value of preliminary map f... |
| hdmap1val 39739 | Value of preliminary map f... |
| hdmap1val0 39740 | Value of preliminary map f... |
| hdmap1val2 39741 | Value of preliminary map f... |
| hdmap1eq 39742 | The defining equation for ... |
| hdmap1cbv 39743 | Frequently used lemma to c... |
| hdmap1valc 39744 | Connect the value of the p... |
| hdmap1cl 39745 | Convert closure theorem ~ ... |
| hdmap1eq2 39746 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 39747 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 39748 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 39749 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 39750 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 39751 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 39752 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 39753 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 39754 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 39755 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 39756 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 39757 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 39758 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 39759 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 39760 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 39761 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 39762 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 39763 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 39764 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 39765 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 39766 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 39767 | Map from vectors to functi... |
| hdmapfval 39768 | Map from vectors to functi... |
| hdmapval 39769 | Value of map from vectors ... |
| hdmapfnN 39770 | Functionality of map from ... |
| hdmapcl 39771 | Closure of map from vector... |
| hdmapval2lem 39772 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 39773 | Value of map from vectors ... |
| hdmapval0 39774 | Value of map from vectors ... |
| hdmapeveclem 39775 | Lemma for ~ hdmapevec . T... |
| hdmapevec 39776 | Value of map from vectors ... |
| hdmapevec2 39777 | The inner product of the r... |
| hdmapval3lemN 39778 | Value of map from vectors ... |
| hdmapval3N 39779 | Value of map from vectors ... |
| hdmap10lem 39780 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 39781 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 39782 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 39783 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 39784 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 39785 | Part of proof of part 12 i... |
| hdmapnzcl 39786 | Nonzero vector closure of ... |
| hdmapneg 39787 | Part of proof of part 12 i... |
| hdmapsub 39788 | Part of proof of part 12 i... |
| hdmap11 39789 | Part of proof of part 12 i... |
| hdmaprnlem1N 39790 | Part of proof of part 12 i... |
| hdmaprnlem3N 39791 | Part of proof of part 12 i... |
| hdmaprnlem3uN 39792 | Part of proof of part 12 i... |
| hdmaprnlem4tN 39793 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 39794 | Part of proof of part 12 i... |
| hdmaprnlem6N 39795 | Part of proof of part 12 i... |
| hdmaprnlem7N 39796 | Part of proof of part 12 i... |
| hdmaprnlem8N 39797 | Part of proof of part 12 i... |
| hdmaprnlem9N 39798 | Part of proof of part 12 i... |
| hdmaprnlem3eN 39799 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 39800 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 39801 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 39802 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 39803 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 39804 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 39805 | Part of proof of part 12 i... |
| hdmapf1oN 39806 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 39807 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 39808 | Prior to part 14 in [Baer]... |
| hdmap14lem1 39809 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 39810 | Prior to part 14 in [Baer]... |
| hdmap14lem3 39811 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 39812 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 39813 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 39814 | Case where ` F ` is zero. ... |
| hdmap14lem7 39815 | Combine cases of ` F ` . ... |
| hdmap14lem8 39816 | Part of proof of part 14 i... |
| hdmap14lem9 39817 | Part of proof of part 14 i... |
| hdmap14lem10 39818 | Part of proof of part 14 i... |
| hdmap14lem11 39819 | Part of proof of part 14 i... |
| hdmap14lem12 39820 | Lemma for proof of part 14... |
| hdmap14lem13 39821 | Lemma for proof of part 14... |
| hdmap14lem14 39822 | Part of proof of part 14 i... |
| hdmap14lem15 39823 | Part of proof of part 14 i... |
| hgmapffval 39826 | Map from the scalar divisi... |
| hgmapfval 39827 | Map from the scalar divisi... |
| hgmapval 39828 | Value of map from the scal... |
| hgmapfnN 39829 | Functionality of scalar si... |
| hgmapcl 39830 | Closure of scalar sigma ma... |
| hgmapdcl 39831 | Closure of the vector spac... |
| hgmapvs 39832 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 39833 | Value of the scalar sigma ... |
| hgmapval1 39834 | Value of the scalar sigma ... |
| hgmapadd 39835 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 39836 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 39837 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 39838 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 39839 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 39840 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 39841 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 39842 | Part of proof of part 16 i... |
| hgmap11 39843 | The scalar sigma map is on... |
| hgmapf1oN 39844 | The scalar sigma map is a ... |
| hgmapeq0 39845 | The scalar sigma map is ze... |
| hdmapipcl 39846 | The inner product (Hermiti... |
| hdmapln1 39847 | Linearity property that wi... |
| hdmaplna1 39848 | Additive property of first... |
| hdmaplns1 39849 | Subtraction property of fi... |
| hdmaplnm1 39850 | Multiplicative property of... |
| hdmaplna2 39851 | Additive property of secon... |
| hdmapglnm2 39852 | g-linear property of secon... |
| hdmapgln2 39853 | g-linear property that wil... |
| hdmaplkr 39854 | Kernel of the vector to du... |
| hdmapellkr 39855 | Membership in the kernel (... |
| hdmapip0 39856 | Zero property that will be... |
| hdmapip1 39857 | Construct a proportional v... |
| hdmapip0com 39858 | Commutation property of Ba... |
| hdmapinvlem1 39859 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 39860 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 39861 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 39862 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 39863 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 39864 | Involution property of sca... |
| hgmapvvlem2 39865 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 39866 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 39867 | Value of a double involuti... |
| hdmapglem7a 39868 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 39869 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 39870 | Lemma for ~ hdmapg . Line... |
| hdmapg 39871 | Apply the scalar sigma fun... |
| hdmapoc 39872 | Express our constructed or... |
| hlhilset 39875 | The final Hilbert space co... |
| hlhilsca 39876 | The scalar of the final co... |
| hlhilbase 39877 | The base set of the final ... |
| hlhilplus 39878 | The vector addition for th... |
| hlhilslem 39879 | Lemma for ~ hlhilsbase etc... |
| hlhilslemOLD 39880 | Obsolete version of ~ hlhi... |
| hlhilsbase 39881 | The scalar base set of the... |
| hlhilsbaseOLD 39882 | Obsolete version of ~ hlhi... |
| hlhilsplus 39883 | Scalar addition for the fi... |
| hlhilsplusOLD 39884 | Obsolete version of ~ hlhi... |
| hlhilsmul 39885 | Scalar multiplication for ... |
| hlhilsmulOLD 39886 | Obsolete version of ~ hlhi... |
| hlhilsbase2 39887 | The scalar base set of the... |
| hlhilsplus2 39888 | Scalar addition for the fi... |
| hlhilsmul2 39889 | Scalar multiplication for ... |
| hlhils0 39890 | The scalar ring zero for t... |
| hlhils1N 39891 | The scalar ring unity for ... |
| hlhilvsca 39892 | The scalar product for the... |
| hlhilip 39893 | Inner product operation fo... |
| hlhilipval 39894 | Value of inner product ope... |
| hlhilnvl 39895 | The involution operation o... |
| hlhillvec 39896 | The final constructed Hilb... |
| hlhildrng 39897 | The star division ring for... |
| hlhilsrnglem 39898 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 39899 | The star division ring for... |
| hlhil0 39900 | The zero vector for the fi... |
| hlhillsm 39901 | The vector sum operation f... |
| hlhilocv 39902 | The orthocomplement for th... |
| hlhillcs 39903 | The closed subspaces of th... |
| hlhilphllem 39904 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 39905 | Lemma for ~ hlhil . (Cont... |
| hlathil 39906 | Construction of a Hilbert ... |
| leexp1ad 39907 | Weak base ordering relatio... |
| relogbcld 39908 | Closure of the general log... |
| relogbexpd 39909 | Identity law for general l... |
| relogbzexpd 39910 | Power law for the general ... |
| logblebd 39911 | The general logarithm is m... |
| fzindd 39912 | Induction on the integers ... |
| uzindd 39913 | Induction on the upper int... |
| fzadd2d 39914 | Membership of a sum in a f... |
| zltlem1d 39915 | Integer ordering relation,... |
| zltp1led 39916 | Integer ordering relation,... |
| fzne2d 39917 | Elementhood in a finite se... |
| eqfnfv2d2 39918 | Equality of functions is d... |
| fzsplitnd 39919 | Split a finite interval of... |
| fzsplitnr 39920 | Split a finite interval of... |
| addassnni 39921 | Associative law for additi... |
| addcomnni 39922 | Commutative law for additi... |
| mulassnni 39923 | Associative law for multip... |
| mulcomnni 39924 | Commutative law for multip... |
| gcdcomnni 39925 | Commutative law for gcd. ... |
| gcdnegnni 39926 | Negation invariance for gc... |
| neggcdnni 39927 | Negation invariance for gc... |
| bccl2d 39928 | Closure of the binomial co... |
| recbothd 39929 | Take reciprocal on both si... |
| gcdmultiplei 39930 | The GCD of a multiple of a... |
| gcdaddmzz2nni 39931 | Adding a multiple of one o... |
| gcdaddmzz2nncomi 39932 | Adding a multiple of one o... |
| gcdnncli 39933 | Closure of the gcd operato... |
| muldvds1d 39934 | If a product divides an in... |
| muldvds2d 39935 | If a product divides an in... |
| nndivdvdsd 39936 | A positive integer divides... |
| nnproddivdvdsd 39937 | A product of natural numbe... |
| coprmdvds2d 39938 | If an integer is divisible... |
| 12gcd5e1 39939 | The gcd of 12 and 5 is 1. ... |
| 60gcd6e6 39940 | The gcd of 60 and 6 is 6. ... |
| 60gcd7e1 39941 | The gcd of 60 and 7 is 1. ... |
| 420gcd8e4 39942 | The gcd of 420 and 8 is 4.... |
| lcmeprodgcdi 39943 | Calculate the least common... |
| 12lcm5e60 39944 | The lcm of 12 and 5 is 60.... |
| 60lcm6e60 39945 | The lcm of 60 and 6 is 60.... |
| 60lcm7e420 39946 | The lcm of 60 and 7 is 420... |
| 420lcm8e840 39947 | The lcm of 420 and 8 is 84... |
| lcmfunnnd 39948 | Useful equation to calcula... |
| lcm1un 39949 | Least common multiple of n... |
| lcm2un 39950 | Least common multiple of n... |
| lcm3un 39951 | Least common multiple of n... |
| lcm4un 39952 | Least common multiple of n... |
| lcm5un 39953 | Least common multiple of n... |
| lcm6un 39954 | Least common multiple of n... |
| lcm7un 39955 | Least common multiple of n... |
| lcm8un 39956 | Least common multiple of n... |
| 3factsumint1 39957 | Move constants out of inte... |
| 3factsumint2 39958 | Move constants out of inte... |
| 3factsumint3 39959 | Move constants out of inte... |
| 3factsumint4 39960 | Move constants out of inte... |
| 3factsumint 39961 | Helpful equation for lcm i... |
| resopunitintvd 39962 | Restrict continuous functi... |
| resclunitintvd 39963 | Restrict continuous functi... |
| resdvopclptsd 39964 | Restrict derivative on uni... |
| lcmineqlem1 39965 | Part of lcm inequality lem... |
| lcmineqlem2 39966 | Part of lcm inequality lem... |
| lcmineqlem3 39967 | Part of lcm inequality lem... |
| lcmineqlem4 39968 | Part of lcm inequality lem... |
| lcmineqlem5 39969 | Technical lemma for recipr... |
| lcmineqlem6 39970 | Part of lcm inequality lem... |
| lcmineqlem7 39971 | Derivative of 1-x for chai... |
| lcmineqlem8 39972 | Derivative of (1-x)^(N-M).... |
| lcmineqlem9 39973 | (1-x)^(N-M) is continuous.... |
| lcmineqlem10 39974 | Induction step of ~ lcmine... |
| lcmineqlem11 39975 | Induction step, continuati... |
| lcmineqlem12 39976 | Base case for induction. ... |
| lcmineqlem13 39977 | Induction proof for lcm in... |
| lcmineqlem14 39978 | Technical lemma for inequa... |
| lcmineqlem15 39979 | F times the least common m... |
| lcmineqlem16 39980 | Technical divisibility lem... |
| lcmineqlem17 39981 | Inequality of 2^{2n}. (Co... |
| lcmineqlem18 39982 | Technical lemma to shift f... |
| lcmineqlem19 39983 | Dividing implies inequalit... |
| lcmineqlem20 39984 | Inequality for lcm lemma. ... |
| lcmineqlem21 39985 | The lcm inequality lemma w... |
| lcmineqlem22 39986 | The lcm inequality lemma w... |
| lcmineqlem23 39987 | Penultimate step to the lc... |
| lcmineqlem 39988 | The least common multiple ... |
| 3exp7 39989 | 3 to the power of 7 equals... |
| 3lexlogpow5ineq1 39990 | First inequality in inequa... |
| 3lexlogpow5ineq2 39991 | Second inequality in inequ... |
| 3lexlogpow5ineq4 39992 | Sharper logarithm inequali... |
| 3lexlogpow5ineq3 39993 | Combined inequality chain ... |
| 3lexlogpow2ineq1 39994 | Result for bound in AKS in... |
| 3lexlogpow2ineq2 39995 | Result for bound in AKS in... |
| 3lexlogpow5ineq5 39996 | Result for bound in AKS in... |
| intlewftc 39997 | Inequality inference by in... |
| aks4d1lem1 39998 | Technical lemma to reduce ... |
| aks4d1p1p1 39999 | Exponential law for finite... |
| dvrelog2 40000 | The derivative of the loga... |
| dvrelog3 40001 | The derivative of the loga... |
| dvrelog2b 40002 | Derivative of the binary l... |
| 0nonelalab 40003 | Technical lemma for open i... |
| dvrelogpow2b 40004 | Derivative of the power of... |
| aks4d1p1p3 40005 | Bound of a ceiling of the ... |
| aks4d1p1p2 40006 | Rewrite ` A ` in more suit... |
| aks4d1p1p4 40007 | Technical step for inequal... |
| dvle2 40008 | Collapsed ~ dvle . (Contr... |
| aks4d1p1p6 40009 | Inequality lift to differe... |
| aks4d1p1p7 40010 | Bound of intermediary of i... |
| aks4d1p1p5 40011 | Show inequality for existe... |
| aks4d1p1 40012 | Show inequality for existe... |
| aks4d1p2 40013 | Technical lemma for existe... |
| aks4d1p3 40014 | There exists a small enoug... |
| aks4d1p4 40015 | There exists a small enoug... |
| aks4d1p5 40016 | Show that ` N ` and ` R ` ... |
| aks4d1p6 40017 | The maximal prime power ex... |
| aks4d1p7d1 40018 | Technical step in AKS lemm... |
| aks4d1p7 40019 | Technical step in AKS lemm... |
| aks4d1p8d1 40020 | If a prime divides one num... |
| aks4d1p8d2 40021 | Any prime power dividing a... |
| aks4d1p8d3 40022 | The remainder of a divisio... |
| aks4d1p8 40023 | Show that ` N ` and ` R ` ... |
| aks4d1p9 40024 | Show that the order is bou... |
| aks4d1 40025 | Lemma 4.1 from ~ https://w... |
| 5bc2eq10 40026 | The value of 5 choose 2. ... |
| facp2 40027 | The factorial of a success... |
| 2np3bcnp1 40028 | Part of induction step for... |
| 2ap1caineq 40029 | Inequality for Theorem 6.6... |
| sticksstones1 40030 | Different strictly monoton... |
| sticksstones2 40031 | The range function on stri... |
| sticksstones3 40032 | The range function on stri... |
| sticksstones4 40033 | Equinumerosity lemma for s... |
| sticksstones5 40034 | Count the number of strict... |
| sticksstones6 40035 | Function induces an order ... |
| sticksstones7 40036 | Closure property of sticks... |
| sticksstones8 40037 | Establish mapping between ... |
| sticksstones9 40038 | Establish mapping between ... |
| sticksstones10 40039 | Establish mapping between ... |
| sticksstones11 40040 | Establish bijective mappin... |
| sticksstones12a 40041 | Establish bijective mappin... |
| sticksstones12 40042 | Establish bijective mappin... |
| sticksstones13 40043 | Establish bijective mappin... |
| sticksstones14 40044 | Sticks and stones with def... |
| sticksstones15 40045 | Sticks and stones with alm... |
| sticksstones16 40046 | Sticks and stones with col... |
| sticksstones17 40047 | Extend sticks and stones t... |
| sticksstones18 40048 | Extend sticks and stones t... |
| sticksstones19 40049 | Extend sticks and stones t... |
| sticksstones20 40050 | Lift sticks and stones to ... |
| sticksstones21 40051 | Lift sticks and stones to ... |
| sticksstones22 40052 | Non-exhaustive sticks and ... |
| metakunt1 40053 | A is an endomapping. (Con... |
| metakunt2 40054 | A is an endomapping. (Con... |
| metakunt3 40055 | Value of A. (Contributed b... |
| metakunt4 40056 | Value of A. (Contributed b... |
| metakunt5 40057 | C is the left inverse for ... |
| metakunt6 40058 | C is the left inverse for ... |
| metakunt7 40059 | C is the left inverse for ... |
| metakunt8 40060 | C is the left inverse for ... |
| metakunt9 40061 | C is the left inverse for ... |
| metakunt10 40062 | C is the right inverse for... |
| metakunt11 40063 | C is the right inverse for... |
| metakunt12 40064 | C is the right inverse for... |
| metakunt13 40065 | C is the right inverse for... |
| metakunt14 40066 | A is a primitive permutati... |
| metakunt15 40067 | Construction of another pe... |
| metakunt16 40068 | Construction of another pe... |
| metakunt17 40069 | The union of three disjoin... |
| metakunt18 40070 | Disjoint domains and codom... |
| metakunt19 40071 | Domains on restrictions of... |
| metakunt20 40072 | Show that B coincides on t... |
| metakunt21 40073 | Show that B coincides on t... |
| metakunt22 40074 | Show that B coincides on t... |
| metakunt23 40075 | B coincides on the union o... |
| metakunt24 40076 | Technical condition such t... |
| metakunt25 40077 | B is a permutation. (Cont... |
| metakunt26 40078 | Construction of one soluti... |
| metakunt27 40079 | Construction of one soluti... |
| metakunt28 40080 | Construction of one soluti... |
| metakunt29 40081 | Construction of one soluti... |
| metakunt30 40082 | Construction of one soluti... |
| metakunt31 40083 | Construction of one soluti... |
| metakunt32 40084 | Construction of one soluti... |
| metakunt33 40085 | Construction of one soluti... |
| metakunt34 40086 | ` D ` is a permutation. (... |
| andiff 40087 | Adding biconditional when ... |
| fac2xp3 40088 | Factorial of 2x+3, sublemm... |
| prodsplit 40089 | Product split into two fac... |
| 2xp3dxp2ge1d 40090 | 2x+3 is greater than or eq... |
| factwoffsmonot 40091 | A factorial with offset is... |
| bicomdALT 40092 | Alternate proof of ~ bicom... |
| elabgw 40093 | Membership in a class abst... |
| elab2gw 40094 | Membership in a class abst... |
| elrab2w 40095 | Membership in a restricted... |
| ruvALT 40096 | Alternate proof of ~ ruv w... |
| sn-wcdeq 40097 | Alternative to ~ wcdeq and... |
| acos1half 40098 | The arccosine of ` 1 / 2 `... |
| isdomn5 40099 | The right conjunct in the ... |
| isdomn4 40100 | A ring is a domain iff it ... |
| ioin9i8 40101 | Miscellaneous inference cr... |
| jaodd 40102 | Double deduction form of ~... |
| syl3an12 40103 | A double syllogism inferen... |
| sbtd 40104 | A true statement is true u... |
| sbor2 40105 | One direction of ~ sbor , ... |
| 19.9dev 40106 | ~ 19.9d in the case of an ... |
| rspcedvdw 40107 | Version of ~ rspcedvd wher... |
| 2rspcedvdw 40108 | Double application of ~ rs... |
| 3rspcedvdw 40109 | Triple application of ~ rs... |
| 3rspcedvd 40110 | Triple application of ~ rs... |
| eqimssd 40111 | Equality implies inclusion... |
| rabdif 40112 | Move difference in and out... |
| sn-axrep5v 40113 | A condensed form of ~ axre... |
| sn-axprlem3 40114 | ~ axprlem3 using only Tars... |
| sn-el 40115 | A version of ~ el with an ... |
| sn-dtru 40116 | ~ dtru without ~ ax-8 or ~... |
| sn-iotalem 40117 | An unused lemma showing th... |
| sn-iotalemcor 40118 | Corollary of ~ sn-iotalem ... |
| sn-iotaval 40119 | Version of ~ iotaval using... |
| sn-iotauni 40120 | Version of ~ iotauni using... |
| sn-iotanul 40121 | Version of ~ iotanul using... |
| sn-iotassuni 40122 | ~ iotassuni without ~ ax-1... |
| sn-iotaex 40123 | ~ iotaex without ~ ax-10 ,... |
| brif1 40124 | Move a relation inside and... |
| brif2 40125 | Move a relation inside and... |
| brif12 40126 | Move a relation inside and... |
| pssexg 40127 | The proper subset of a set... |
| pssn0 40128 | A proper superset is nonem... |
| psspwb 40129 | Classes are proper subclas... |
| xppss12 40130 | Proper subset theorem for ... |
| elpwbi 40131 | Membership in a power set,... |
| opelxpii 40132 | Ordered pair membership in... |
| imaopab 40133 | The image of a class of or... |
| fnsnbt 40134 | A function's domain is a s... |
| fnimasnd 40135 | The image of a function by... |
| fvmptd4 40136 | Deduction version of ~ fvm... |
| ofun 40137 | A function operation of un... |
| dfqs2 40138 | Alternate definition of qu... |
| dfqs3 40139 | Alternate definition of qu... |
| qseq12d 40140 | Equality theorem for quoti... |
| qsalrel 40141 | The quotient set is equal ... |
| elmapdd 40142 | Deduction associated with ... |
| isfsuppd 40143 | Deduction form of ~ isfsup... |
| fzosumm1 40144 | Separate out the last term... |
| ccatcan2d 40145 | Cancellation law for conca... |
| nelsubginvcld 40146 | The inverse of a non-subgr... |
| nelsubgcld 40147 | A non-subgroup-member plus... |
| nelsubgsubcld 40148 | A non-subgroup-member minu... |
| rnasclg 40149 | The set of injected scalar... |
| selvval2lem1 40150 | ` T ` is an associative al... |
| selvval2lem2 40151 | ` D ` is a ring homomorphi... |
| selvval2lem3 40152 | The third argument passed ... |
| selvval2lemn 40153 | A lemma to illustrate the ... |
| selvval2lem4 40154 | The fourth argument passed... |
| selvval2lem5 40155 | The fifth argument passed ... |
| selvcl 40156 | Closure of the "variable s... |
| frlmfielbas 40157 | The vectors of a finite fr... |
| frlmfzwrd 40158 | A vector of a module with ... |
| frlmfzowrd 40159 | A vector of a module with ... |
| frlmfzolen 40160 | The dimension of a vector ... |
| frlmfzowrdb 40161 | The vectors of a module wi... |
| frlmfzoccat 40162 | The concatenation of two v... |
| frlmvscadiccat 40163 | Scalar multiplication dist... |
| ismhmd 40164 | Deduction version of ~ ism... |
| ablcmnd 40165 | An Abelian group is a comm... |
| ringcld 40166 | Closure of the multiplicat... |
| ringassd 40167 | Associative law for multip... |
| ringlidmd 40168 | The unit element of a ring... |
| ringridmd 40169 | The unit element of a ring... |
| ringabld 40170 | A ring is an Abelian group... |
| ringcmnd 40171 | A ring is a commutative mo... |
| drngringd 40172 | A division ring is a ring.... |
| drnggrpd 40173 | A division ring is a group... |
| drnginvrcld 40174 | Closure of the multiplicat... |
| drnginvrn0d 40175 | A multiplicative inverse i... |
| drnginvrld 40176 | Property of the multiplica... |
| drnginvrrd 40177 | Property of the multiplica... |
| drngmulcanad 40178 | Cancellation of a nonzero ... |
| drngmulcan2ad 40179 | Cancellation of a nonzero ... |
| drnginvmuld 40180 | Inverse of a nonzero produ... |
| lmodgrpd 40181 | A left module is a group. ... |
| lvecgrp 40182 | A vector space is a group.... |
| lveclmodd 40183 | A vector space is a left m... |
| lvecgrpd 40184 | A vector space is a group.... |
| lvecring 40185 | The scalar component of a ... |
| lmhmlvec 40186 | The property for modules t... |
| frlm0vald 40187 | All coordinates of the zer... |
| frlmsnic 40188 | Given a free module with a... |
| uvccl 40189 | A unit vector is a vector.... |
| uvcn0 40190 | A unit vector is nonzero. ... |
| pwselbasr 40191 | The reverse direction of ~... |
| pwspjmhmmgpd 40192 | The projection given by ~ ... |
| pwsexpg 40193 | Value of a group exponenti... |
| pwsgprod 40194 | Finite products in a power... |
| evlsval3 40195 | Give a formula for the pol... |
| evlsscaval 40196 | Polynomial evaluation buil... |
| evlsvarval 40197 | Polynomial evaluation buil... |
| evlsbagval 40198 | Polynomial evaluation buil... |
| evlsexpval 40199 | Polynomial evaluation buil... |
| evlsaddval 40200 | Polynomial evaluation buil... |
| evlsmulval 40201 | Polynomial evaluation buil... |
| fsuppind 40202 | Induction on functions ` F... |
| fsuppssindlem1 40203 | Lemma for ~ fsuppssind . ... |
| fsuppssindlem2 40204 | Lemma for ~ fsuppssind . ... |
| fsuppssind 40205 | Induction on functions ` F... |
| mhpind 40206 | The homogeneous polynomial... |
| mhphflem 40207 | Lemma for ~ mhphf . Add s... |
| mhphf 40208 | A homogeneous polynomial d... |
| mhphf2 40209 | A homogeneous polynomial d... |
| c0exALT 40210 | Alternate proof of ~ c0ex ... |
| 0cnALT3 40211 | Alternate proof of ~ 0cn u... |
| elre0re 40212 | Specialized version of ~ 0... |
| 1t1e1ALT 40213 | Alternate proof of ~ 1t1e1... |
| remulcan2d 40214 | ~ mulcan2d for real number... |
| readdid1addid2d 40215 | Given some real number ` B... |
| sn-1ne2 40216 | A proof of ~ 1ne2 without ... |
| nnn1suc 40217 | A positive integer that is... |
| nnadd1com 40218 | Addition with 1 is commuta... |
| nnaddcom 40219 | Addition is commutative fo... |
| nnaddcomli 40220 | Version of ~ addcomli for ... |
| nnadddir 40221 | Right-distributivity for n... |
| nnmul1com 40222 | Multiplication with 1 is c... |
| nnmulcom 40223 | Multiplication is commutat... |
| mvrrsubd 40224 | Move a subtraction in the ... |
| laddrotrd 40225 | Rotate the variables right... |
| raddcom12d 40226 | Swap the first two variabl... |
| lsubrotld 40227 | Rotate the variables left ... |
| lsubcom23d 40228 | Swap the second and third ... |
| addsubeq4com 40229 | Relation between sums and ... |
| sqsumi 40230 | A sum squared. (Contribut... |
| negn0nposznnd 40231 | Lemma for ~ dffltz . (Con... |
| sqmid3api 40232 | Value of the square of the... |
| decaddcom 40233 | Commute ones place in addi... |
| sqn5i 40234 | The square of a number end... |
| sqn5ii 40235 | The square of a number end... |
| decpmulnc 40236 | Partial products algorithm... |
| decpmul 40237 | Partial products algorithm... |
| sqdeccom12 40238 | The square of a number in ... |
| sq3deccom12 40239 | Variant of ~ sqdeccom12 wi... |
| 235t711 40240 | Calculate a product by lon... |
| ex-decpmul 40241 | Example usage of ~ decpmul... |
| oexpreposd 40242 | Lemma for ~ dffltz . TODO... |
| ltexp1d 40243 | ~ ltmul1d for exponentiati... |
| ltexp1dd 40244 | Raising both sides of 'les... |
| exp11nnd 40245 | ~ sq11d for positive real ... |
| exp11d 40246 | ~ exp11nnd for nonzero int... |
| 0dvds0 40247 | 0 divides 0. (Contributed... |
| absdvdsabsb 40248 | Divisibility is invariant ... |
| dvdsexpim 40249 | ~ dvdssqim generalized to ... |
| gcdnn0id 40250 | The ` gcd ` of a nonnegati... |
| gcdle1d 40251 | The greatest common diviso... |
| gcdle2d 40252 | The greatest common diviso... |
| dvdsexpad 40253 | Deduction associated with ... |
| nn0rppwr 40254 | If ` A ` and ` B ` are rel... |
| expgcd 40255 | Exponentiation distributes... |
| nn0expgcd 40256 | Exponentiation distributes... |
| zexpgcd 40257 | Exponentiation distributes... |
| numdenexp 40258 | ~ numdensq extended to non... |
| numexp 40259 | ~ numsq extended to nonneg... |
| denexp 40260 | ~ densq extended to nonneg... |
| dvdsexpnn 40261 | ~ dvdssqlem generalized to... |
| dvdsexpnn0 40262 | ~ dvdsexpnn generalized to... |
| dvdsexpb 40263 | ~ dvdssq generalized to po... |
| posqsqznn 40264 | When a positive rational s... |
| cxpgt0d 40265 | A positive real raised to ... |
| zrtelqelz 40266 | ~ zsqrtelqelz generalized ... |
| zrtdvds 40267 | A positive integer root di... |
| rtprmirr 40268 | The root of a prime number... |
| resubval 40271 | Value of real subtraction,... |
| renegeulemv 40272 | Lemma for ~ renegeu and si... |
| renegeulem 40273 | Lemma for ~ renegeu and si... |
| renegeu 40274 | Existential uniqueness of ... |
| rernegcl 40275 | Closure law for negative r... |
| renegadd 40276 | Relationship between real ... |
| renegid 40277 | Addition of a real number ... |
| reneg0addid2 40278 | Negative zero is a left ad... |
| resubeulem1 40279 | Lemma for ~ resubeu . A v... |
| resubeulem2 40280 | Lemma for ~ resubeu . A v... |
| resubeu 40281 | Existential uniqueness of ... |
| rersubcl 40282 | Closure for real subtracti... |
| resubadd 40283 | Relation between real subt... |
| resubaddd 40284 | Relationship between subtr... |
| resubf 40285 | Real subtraction is an ope... |
| repncan2 40286 | Addition and subtraction o... |
| repncan3 40287 | Addition and subtraction o... |
| readdsub 40288 | Law for addition and subtr... |
| reladdrsub 40289 | Move LHS of a sum into RHS... |
| reltsub1 40290 | Subtraction from both side... |
| reltsubadd2 40291 | 'Less than' relationship b... |
| resubcan2 40292 | Cancellation law for real ... |
| resubsub4 40293 | Law for double subtraction... |
| rennncan2 40294 | Cancellation law for real ... |
| renpncan3 40295 | Cancellation law for real ... |
| repnpcan 40296 | Cancellation law for addit... |
| reppncan 40297 | Cancellation law for mixed... |
| resubidaddid1lem 40298 | Lemma for ~ resubidaddid1 ... |
| resubidaddid1 40299 | Any real number subtracted... |
| resubdi 40300 | Distribution of multiplica... |
| re1m1e0m0 40301 | Equality of two left-addit... |
| sn-00idlem1 40302 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 40303 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 40304 | Lemma for ~ sn-00id . (Co... |
| sn-00id 40305 | ~ 00id proven without ~ ax... |
| re0m0e0 40306 | Real number version of ~ 0... |
| readdid2 40307 | Real number version of ~ a... |
| sn-addid2 40308 | ~ addid2 without ~ ax-mulc... |
| remul02 40309 | Real number version of ~ m... |
| sn-0ne2 40310 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 40311 | Real number version of ~ m... |
| resubid 40312 | Subtraction of a real numb... |
| readdid1 40313 | Real number version of ~ a... |
| resubid1 40314 | Real number version of ~ s... |
| renegneg 40315 | A real number is equal to ... |
| readdcan2 40316 | Commuted version of ~ read... |
| renegid2 40317 | Commuted version of ~ rene... |
| sn-it0e0 40318 | Proof of ~ it0e0 without ~... |
| sn-negex12 40319 | A combination of ~ cnegex ... |
| sn-negex 40320 | Proof of ~ cnegex without ... |
| sn-negex2 40321 | Proof of ~ cnegex2 without... |
| sn-addcand 40322 | ~ addcand without ~ ax-mul... |
| sn-addid1 40323 | ~ addid1 without ~ ax-mulc... |
| sn-addcan2d 40324 | ~ addcan2d without ~ ax-mu... |
| reixi 40325 | ~ ixi without ~ ax-mulcom ... |
| rei4 40326 | ~ i4 without ~ ax-mulcom .... |
| sn-addid0 40327 | A number that sums to itse... |
| sn-mul01 40328 | ~ mul01 without ~ ax-mulco... |
| sn-subeu 40329 | ~ negeu without ~ ax-mulco... |
| sn-subcl 40330 | ~ subcl without ~ ax-mulco... |
| sn-subf 40331 | ~ subf without ~ ax-mulcom... |
| resubeqsub 40332 | Equivalence between real s... |
| subresre 40333 | Subtraction restricted to ... |
| addinvcom 40334 | A number commutes with its... |
| remulinvcom 40335 | A left multiplicative inve... |
| remulid2 40336 | Commuted version of ~ ax-1... |
| sn-1ticom 40337 | Lemma for ~ sn-mulid2 and ... |
| sn-mulid2 40338 | ~ mulid2 without ~ ax-mulc... |
| it1ei 40339 | ` 1 ` is a multiplicative ... |
| ipiiie0 40340 | The multiplicative inverse... |
| remulcand 40341 | Commuted version of ~ remu... |
| sn-0tie0 40342 | Lemma for ~ sn-mul02 . Co... |
| sn-mul02 40343 | ~ mul02 without ~ ax-mulco... |
| sn-ltaddpos 40344 | ~ ltaddpos without ~ ax-mu... |
| reposdif 40345 | Comparison of two numbers ... |
| relt0neg1 40346 | Comparison of a real and i... |
| relt0neg2 40347 | Comparison of a real and i... |
| mulgt0con1dlem 40348 | Lemma for ~ mulgt0con1d . ... |
| mulgt0con1d 40349 | Counterpart to ~ mulgt0con... |
| mulgt0con2d 40350 | Lemma for ~ mulgt0b2d and ... |
| mulgt0b2d 40351 | Biconditional, deductive f... |
| sn-ltmul2d 40352 | ~ ltmul2d without ~ ax-mul... |
| sn-0lt1 40353 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 40354 | ~ ltp1 without ~ ax-mulcom... |
| reneg1lt0 40355 | Lemma for ~ sn-inelr . (C... |
| sn-inelr 40356 | ~ inelr without ~ ax-mulco... |
| itrere 40357 | ` _i ` times a real is rea... |
| retire 40358 | Commuted version of ~ itre... |
| cnreeu 40359 | The reals in the expressio... |
| sn-sup2 40360 | ~ sup2 with exactly the sa... |
| prjspval 40363 | Value of the projective sp... |
| prjsprel 40364 | Utility theorem regarding ... |
| prjspertr 40365 | The relation in ` PrjSp ` ... |
| prjsperref 40366 | The relation in ` PrjSp ` ... |
| prjspersym 40367 | The relation in ` PrjSp ` ... |
| prjsper 40368 | The relation used to defin... |
| prjspreln0 40369 | Two nonzero vectors are eq... |
| prjspvs 40370 | A nonzero multiple of a ve... |
| prjsprellsp 40371 | Two vectors are equivalent... |
| prjspeclsp 40372 | The vectors equivalent to ... |
| prjspval2 40373 | Alternate definition of pr... |
| prjspnval 40376 | Value of the n-dimensional... |
| prjspnerlem 40377 | A lemma showing that the e... |
| prjspnval2 40378 | Value of the n-dimensional... |
| prjspner 40379 | The relation used to defin... |
| prjspnvs 40380 | A nonzero multiple of a ve... |
| 0prjspnlem 40381 | Lemma for ~ 0prjspn . The... |
| prjspnfv01 40382 | Any vector is equivalent t... |
| prjspner01 40383 | Any vector is equivalent t... |
| prjspner1 40384 | Two vectors whose zeroth c... |
| 0prjspnrel 40385 | In the zero-dimensional pr... |
| 0prjspn 40386 | A zero-dimensional project... |
| dffltz 40387 | Fermat's Last Theorem (FLT... |
| fltmul 40388 | A counterexample to FLT st... |
| fltdiv 40389 | A counterexample to FLT st... |
| flt0 40390 | A counterexample for FLT d... |
| fltdvdsabdvdsc 40391 | Any factor of both ` A ` a... |
| fltabcoprmex 40392 | A counterexample to FLT im... |
| fltaccoprm 40393 | A counterexample to FLT wi... |
| fltbccoprm 40394 | A counterexample to FLT wi... |
| fltabcoprm 40395 | A counterexample to FLT wi... |
| infdesc 40396 | Infinite descent. The hyp... |
| fltne 40397 | If a counterexample to FLT... |
| flt4lem 40398 | Raising a number to the fo... |
| flt4lem1 40399 | Satisfy the antecedent use... |
| flt4lem2 40400 | If ` A ` is even, ` B ` is... |
| flt4lem3 40401 | Equivalent to ~ pythagtrip... |
| flt4lem4 40402 | If the product of two copr... |
| flt4lem5 40403 | In the context of the lemm... |
| flt4lem5elem 40404 | Version of ~ fltaccoprm an... |
| flt4lem5a 40405 | Part 1 of Equation 1 of ... |
| flt4lem5b 40406 | Part 2 of Equation 1 of ... |
| flt4lem5c 40407 | Part 2 of Equation 2 of ... |
| flt4lem5d 40408 | Part 3 of Equation 2 of ... |
| flt4lem5e 40409 | Satisfy the hypotheses of ... |
| flt4lem5f 40410 | Final equation of ~... |
| flt4lem6 40411 | Remove shared factors in a... |
| flt4lem7 40412 | Convert ~ flt4lem5f into a... |
| nna4b4nsq 40413 | Strengthening of Fermat's ... |
| fltltc 40414 | ` ( C ^ N ) ` is the large... |
| fltnltalem 40415 | Lemma for ~ fltnlta . A l... |
| fltnlta 40416 | In a Fermat counterexample... |
| binom2d 40417 | Deduction form of binom2. ... |
| cu3addd 40418 | Cube of sum of three numbe... |
| sqnegd 40419 | The square of the negative... |
| negexpidd 40420 | The sum of a real number t... |
| rexlimdv3d 40421 | An extended version of ~ r... |
| 3cubeslem1 40422 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 40423 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 40424 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 40425 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 40426 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 40427 | Lemma for ~ 3cubes . This... |
| 3cubes 40428 | Every rational number is a... |
| rntrclfvOAI 40429 | The range of the transitiv... |
| moxfr 40430 | Transfer at-most-one betwe... |
| imaiinfv 40431 | Indexed intersection of an... |
| elrfi 40432 | Elementhood in a set of re... |
| elrfirn 40433 | Elementhood in a set of re... |
| elrfirn2 40434 | Elementhood in a set of re... |
| cmpfiiin 40435 | In a compact topology, a s... |
| ismrcd1 40436 | Any function from the subs... |
| ismrcd2 40437 | Second half of ~ ismrcd1 .... |
| istopclsd 40438 | A closure function which s... |
| ismrc 40439 | A function is a Moore clos... |
| isnacs 40442 | Expand definition of Noeth... |
| nacsfg 40443 | In a Noetherian-type closu... |
| isnacs2 40444 | Express Noetherian-type cl... |
| mrefg2 40445 | Slight variation on finite... |
| mrefg3 40446 | Slight variation on finite... |
| nacsacs 40447 | A closure system of Noethe... |
| isnacs3 40448 | A choice-free order equiva... |
| incssnn0 40449 | Transitivity induction of ... |
| nacsfix 40450 | An increasing sequence of ... |
| constmap 40451 | A constant (represented wi... |
| mapco2g 40452 | Renaming indices in a tupl... |
| mapco2 40453 | Post-composition (renaming... |
| mapfzcons 40454 | Extending a one-based mapp... |
| mapfzcons1 40455 | Recover prefix mapping fro... |
| mapfzcons1cl 40456 | A nonempty mapping has a p... |
| mapfzcons2 40457 | Recover added element from... |
| mptfcl 40458 | Interpret range of a maps-... |
| mzpclval 40463 | Substitution lemma for ` m... |
| elmzpcl 40464 | Double substitution lemma ... |
| mzpclall 40465 | The set of all functions w... |
| mzpcln0 40466 | Corollary of ~ mzpclall : ... |
| mzpcl1 40467 | Defining property 1 of a p... |
| mzpcl2 40468 | Defining property 2 of a p... |
| mzpcl34 40469 | Defining properties 3 and ... |
| mzpval 40470 | Value of the ` mzPoly ` fu... |
| dmmzp 40471 | ` mzPoly ` is defined for ... |
| mzpincl 40472 | Polynomial closedness is a... |
| mzpconst 40473 | Constant functions are pol... |
| mzpf 40474 | A polynomial function is a... |
| mzpproj 40475 | A projection function is p... |
| mzpadd 40476 | The pointwise sum of two p... |
| mzpmul 40477 | The pointwise product of t... |
| mzpconstmpt 40478 | A constant function expres... |
| mzpaddmpt 40479 | Sum of polynomial function... |
| mzpmulmpt 40480 | Product of polynomial func... |
| mzpsubmpt 40481 | The difference of two poly... |
| mzpnegmpt 40482 | Negation of a polynomial f... |
| mzpexpmpt 40483 | Raise a polynomial functio... |
| mzpindd 40484 | "Structural" induction to ... |
| mzpmfp 40485 | Relationship between multi... |
| mzpsubst 40486 | Substituting polynomials f... |
| mzprename 40487 | Simplified version of ~ mz... |
| mzpresrename 40488 | A polynomial is a polynomi... |
| mzpcompact2lem 40489 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 40490 | Polynomials are finitary o... |
| coeq0i 40491 | ~ coeq0 but without explic... |
| fzsplit1nn0 40492 | Split a finite 1-based set... |
| eldiophb 40495 | Initial expression of Diop... |
| eldioph 40496 | Condition for a set to be ... |
| diophrw 40497 | Renaming and adding unused... |
| eldioph2lem1 40498 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 40499 | Lemma for ~ eldioph2 . Co... |
| eldioph2 40500 | Construct a Diophantine se... |
| eldioph2b 40501 | While Diophantine sets wer... |
| eldiophelnn0 40502 | Remove antecedent on ` B `... |
| eldioph3b 40503 | Define Diophantine sets in... |
| eldioph3 40504 | Inference version of ~ eld... |
| ellz1 40505 | Membership in a lower set ... |
| lzunuz 40506 | The union of a lower set o... |
| fz1eqin 40507 | Express a one-based finite... |
| lzenom 40508 | Lower integers are countab... |
| elmapresaunres2 40509 | ~ fresaunres2 transposed t... |
| diophin 40510 | If two sets are Diophantin... |
| diophun 40511 | If two sets are Diophantin... |
| eldiophss 40512 | Diophantine sets are sets ... |
| diophrex 40513 | Projecting a Diophantine s... |
| eq0rabdioph 40514 | This is the first of a num... |
| eqrabdioph 40515 | Diophantine set builder fo... |
| 0dioph 40516 | The null set is Diophantin... |
| vdioph 40517 | The "universal" set (as la... |
| anrabdioph 40518 | Diophantine set builder fo... |
| orrabdioph 40519 | Diophantine set builder fo... |
| 3anrabdioph 40520 | Diophantine set builder fo... |
| 3orrabdioph 40521 | Diophantine set builder fo... |
| 2sbcrex 40522 | Exchange an existential qu... |
| sbcrexgOLD 40523 | Interchange class substitu... |
| 2sbcrexOLD 40524 | Exchange an existential qu... |
| sbc2rex 40525 | Exchange a substitution wi... |
| sbc2rexgOLD 40526 | Exchange a substitution wi... |
| sbc4rex 40527 | Exchange a substitution wi... |
| sbc4rexgOLD 40528 | Exchange a substitution wi... |
| sbcrot3 40529 | Rotate a sequence of three... |
| sbcrot5 40530 | Rotate a sequence of five ... |
| sbccomieg 40531 | Commute two explicit subst... |
| rexrabdioph 40532 | Diophantine set builder fo... |
| rexfrabdioph 40533 | Diophantine set builder fo... |
| 2rexfrabdioph 40534 | Diophantine set builder fo... |
| 3rexfrabdioph 40535 | Diophantine set builder fo... |
| 4rexfrabdioph 40536 | Diophantine set builder fo... |
| 6rexfrabdioph 40537 | Diophantine set builder fo... |
| 7rexfrabdioph 40538 | Diophantine set builder fo... |
| rabdiophlem1 40539 | Lemma for arithmetic dioph... |
| rabdiophlem2 40540 | Lemma for arithmetic dioph... |
| elnn0rabdioph 40541 | Diophantine set builder fo... |
| rexzrexnn0 40542 | Rewrite an existential qua... |
| lerabdioph 40543 | Diophantine set builder fo... |
| eluzrabdioph 40544 | Diophantine set builder fo... |
| elnnrabdioph 40545 | Diophantine set builder fo... |
| ltrabdioph 40546 | Diophantine set builder fo... |
| nerabdioph 40547 | Diophantine set builder fo... |
| dvdsrabdioph 40548 | Divisibility is a Diophant... |
| eldioph4b 40549 | Membership in ` Dioph ` ex... |
| eldioph4i 40550 | Forward-only version of ~ ... |
| diophren 40551 | Change variables in a Diop... |
| rabrenfdioph 40552 | Change variable numbers in... |
| rabren3dioph 40553 | Change variable numbers in... |
| fphpd 40554 | Pigeonhole principle expre... |
| fphpdo 40555 | Pigeonhole principle for s... |
| ctbnfien 40556 | An infinite subset of a co... |
| fiphp3d 40557 | Infinite pigeonhole princi... |
| rencldnfilem 40558 | Lemma for ~ rencldnfi . (... |
| rencldnfi 40559 | A set of real numbers whic... |
| irrapxlem1 40560 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 40561 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 40562 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 40563 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 40564 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 40565 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 40566 | Dirichlet's approximation ... |
| pellexlem1 40567 | Lemma for ~ pellex . Arit... |
| pellexlem2 40568 | Lemma for ~ pellex . Arit... |
| pellexlem3 40569 | Lemma for ~ pellex . To e... |
| pellexlem4 40570 | Lemma for ~ pellex . Invo... |
| pellexlem5 40571 | Lemma for ~ pellex . Invo... |
| pellexlem6 40572 | Lemma for ~ pellex . Doin... |
| pellex 40573 | Every Pell equation has a ... |
| pell1qrval 40584 | Value of the set of first-... |
| elpell1qr 40585 | Membership in a first-quad... |
| pell14qrval 40586 | Value of the set of positi... |
| elpell14qr 40587 | Membership in the set of p... |
| pell1234qrval 40588 | Value of the set of genera... |
| elpell1234qr 40589 | Membership in the set of g... |
| pell1234qrre 40590 | General Pell solutions are... |
| pell1234qrne0 40591 | No solution to a Pell equa... |
| pell1234qrreccl 40592 | General solutions of the P... |
| pell1234qrmulcl 40593 | General solutions of the P... |
| pell14qrss1234 40594 | A positive Pell solution i... |
| pell14qrre 40595 | A positive Pell solution i... |
| pell14qrne0 40596 | A positive Pell solution i... |
| pell14qrgt0 40597 | A positive Pell solution i... |
| pell14qrrp 40598 | A positive Pell solution i... |
| pell1234qrdich 40599 | A general Pell solution is... |
| elpell14qr2 40600 | A number is a positive Pel... |
| pell14qrmulcl 40601 | Positive Pell solutions ar... |
| pell14qrreccl 40602 | Positive Pell solutions ar... |
| pell14qrdivcl 40603 | Positive Pell solutions ar... |
| pell14qrexpclnn0 40604 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 40605 | Positive Pell solutions ar... |
| pell1qrss14 40606 | First-quadrant Pell soluti... |
| pell14qrdich 40607 | A positive Pell solution i... |
| pell1qrge1 40608 | A Pell solution in the fir... |
| pell1qr1 40609 | 1 is a Pell solution and i... |
| elpell1qr2 40610 | The first quadrant solutio... |
| pell1qrgaplem 40611 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 40612 | First-quadrant Pell soluti... |
| pell14qrgap 40613 | Positive Pell solutions ar... |
| pell14qrgapw 40614 | Positive Pell solutions ar... |
| pellqrexplicit 40615 | Condition for a calculated... |
| infmrgelbi 40616 | Any lower bound of a nonem... |
| pellqrex 40617 | There is a nontrivial solu... |
| pellfundval 40618 | Value of the fundamental s... |
| pellfundre 40619 | The fundamental solution o... |
| pellfundge 40620 | Lower bound on the fundame... |
| pellfundgt1 40621 | Weak lower bound on the Pe... |
| pellfundlb 40622 | A nontrivial first quadran... |
| pellfundglb 40623 | If a real is larger than t... |
| pellfundex 40624 | The fundamental solution a... |
| pellfund14gap 40625 | There are no solutions bet... |
| pellfundrp 40626 | The fundamental Pell solut... |
| pellfundne1 40627 | The fundamental Pell solut... |
| reglogcl 40628 | General logarithm is a rea... |
| reglogltb 40629 | General logarithm preserve... |
| reglogleb 40630 | General logarithm preserve... |
| reglogmul 40631 | Multiplication law for gen... |
| reglogexp 40632 | Power law for general log.... |
| reglogbas 40633 | General log of the base is... |
| reglog1 40634 | General log of 1 is 0. (C... |
| reglogexpbas 40635 | General log of a power of ... |
| pellfund14 40636 | Every positive Pell soluti... |
| pellfund14b 40637 | The positive Pell solution... |
| rmxfval 40642 | Value of the X sequence. ... |
| rmyfval 40643 | Value of the Y sequence. ... |
| rmspecsqrtnq 40644 | The discriminant used to d... |
| rmspecnonsq 40645 | The discriminant used to d... |
| qirropth 40646 | This lemma implements the ... |
| rmspecfund 40647 | The base of exponent used ... |
| rmxyelqirr 40648 | The solutions used to cons... |
| rmxypairf1o 40649 | The function used to extra... |
| rmxyelxp 40650 | Lemma for ~ frmx and ~ frm... |
| frmx 40651 | The X sequence is a nonneg... |
| frmy 40652 | The Y sequence is an integ... |
| rmxyval 40653 | Main definition of the X a... |
| rmspecpos 40654 | The discriminant used to d... |
| rmxycomplete 40655 | The X and Y sequences take... |
| rmxynorm 40656 | The X and Y sequences defi... |
| rmbaserp 40657 | The base of exponentiation... |
| rmxyneg 40658 | Negation law for X and Y s... |
| rmxyadd 40659 | Addition formula for X and... |
| rmxy1 40660 | Value of the X and Y seque... |
| rmxy0 40661 | Value of the X and Y seque... |
| rmxneg 40662 | Negation law (even functio... |
| rmx0 40663 | Value of X sequence at 0. ... |
| rmx1 40664 | Value of X sequence at 1. ... |
| rmxadd 40665 | Addition formula for X seq... |
| rmyneg 40666 | Negation formula for Y seq... |
| rmy0 40667 | Value of Y sequence at 0. ... |
| rmy1 40668 | Value of Y sequence at 1. ... |
| rmyadd 40669 | Addition formula for Y seq... |
| rmxp1 40670 | Special addition-of-1 form... |
| rmyp1 40671 | Special addition of 1 form... |
| rmxm1 40672 | Subtraction of 1 formula f... |
| rmym1 40673 | Subtraction of 1 formula f... |
| rmxluc 40674 | The X sequence is a Lucas ... |
| rmyluc 40675 | The Y sequence is a Lucas ... |
| rmyluc2 40676 | Lucas sequence property of... |
| rmxdbl 40677 | "Double-angle formula" for... |
| rmydbl 40678 | "Double-angle formula" for... |
| monotuz 40679 | A function defined on an u... |
| monotoddzzfi 40680 | A function which is odd an... |
| monotoddzz 40681 | A function (given implicit... |
| oddcomabszz 40682 | An odd function which take... |
| 2nn0ind 40683 | Induction on nonnegative i... |
| zindbi 40684 | Inductively transfer a pro... |
| rmxypos 40685 | For all nonnegative indice... |
| ltrmynn0 40686 | The Y-sequence is strictly... |
| ltrmxnn0 40687 | The X-sequence is strictly... |
| lermxnn0 40688 | The X-sequence is monotoni... |
| rmxnn 40689 | The X-sequence is defined ... |
| ltrmy 40690 | The Y-sequence is strictly... |
| rmyeq0 40691 | Y is zero only at zero. (... |
| rmyeq 40692 | Y is one-to-one. (Contrib... |
| lermy 40693 | Y is monotonic (non-strict... |
| rmynn 40694 | ` rmY ` is positive for po... |
| rmynn0 40695 | ` rmY ` is nonnegative for... |
| rmyabs 40696 | ` rmY ` commutes with ` ab... |
| jm2.24nn 40697 | X(n) is strictly greater t... |
| jm2.17a 40698 | First half of lemma 2.17 o... |
| jm2.17b 40699 | Weak form of the second ha... |
| jm2.17c 40700 | Second half of lemma 2.17 ... |
| jm2.24 40701 | Lemma 2.24 of [JonesMatija... |
| rmygeid 40702 | Y(n) increases faster than... |
| congtr 40703 | A wff of the form ` A || (... |
| congadd 40704 | If two pairs of numbers ar... |
| congmul 40705 | If two pairs of numbers ar... |
| congsym 40706 | Congruence mod ` A ` is a ... |
| congneg 40707 | If two integers are congru... |
| congsub 40708 | If two pairs of numbers ar... |
| congid 40709 | Every integer is congruent... |
| mzpcong 40710 | Polynomials commute with c... |
| congrep 40711 | Every integer is congruent... |
| congabseq 40712 | If two integers are congru... |
| acongid 40713 | A wff like that in this th... |
| acongsym 40714 | Symmetry of alternating co... |
| acongneg2 40715 | Negate right side of alter... |
| acongtr 40716 | Transitivity of alternatin... |
| acongeq12d 40717 | Substitution deduction for... |
| acongrep 40718 | Every integer is alternati... |
| fzmaxdif 40719 | Bound on the difference be... |
| fzneg 40720 | Reflection of a finite ran... |
| acongeq 40721 | Two numbers in the fundame... |
| dvdsacongtr 40722 | Alternating congruence pas... |
| coprmdvdsb 40723 | Multiplication by a coprim... |
| modabsdifz 40724 | Divisibility in terms of m... |
| dvdsabsmod0 40725 | Divisibility in terms of m... |
| jm2.18 40726 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 40727 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 40728 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 40729 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 40730 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 40731 | Lemma 2.19 of [JonesMatija... |
| jm2.21 40732 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 40733 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 40734 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 40735 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 40736 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 40737 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 40738 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 40739 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 40740 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 40741 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 40742 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 40743 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 40744 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 40745 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 40746 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 40747 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 40748 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 40749 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 40750 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 40751 | Lemma for ~ rmydioph . Us... |
| rmydioph 40752 | ~ jm2.27 restated in terms... |
| rmxdiophlem 40753 | X can be expressed in term... |
| rmxdioph 40754 | X is a Diophantine functio... |
| jm3.1lem1 40755 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 40756 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 40757 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 40758 | Diophantine expression for... |
| expdiophlem1 40759 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 40760 | Lemma for ~ expdioph . Ex... |
| expdioph 40761 | The exponential function i... |
| setindtr 40762 | Set induction for sets con... |
| setindtrs 40763 | Set induction scheme witho... |
| dford3lem1 40764 | Lemma for ~ dford3 . (Con... |
| dford3lem2 40765 | Lemma for ~ dford3 . (Con... |
| dford3 40766 | Ordinals are precisely the... |
| dford4 40767 | ~ dford3 expressed in prim... |
| wopprc 40768 | Unrelated: Wiener pairs t... |
| rpnnen3lem 40769 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 40770 | Dedekind cut injection of ... |
| axac10 40771 | Characterization of choice... |
| harinf 40772 | The Hartogs number of an i... |
| wdom2d2 40773 | Deduction for weak dominan... |
| ttac 40774 | Tarski's theorem about cho... |
| pw2f1ocnv 40775 | Define a bijection between... |
| pw2f1o2 40776 | Define a bijection between... |
| pw2f1o2val 40777 | Function value of the ~ pw... |
| pw2f1o2val2 40778 | Membership in a mapped set... |
| soeq12d 40779 | Equality deduction for tot... |
| freq12d 40780 | Equality deduction for fou... |
| weeq12d 40781 | Equality deduction for wel... |
| limsuc2 40782 | Limit ordinals in the sens... |
| wepwsolem 40783 | Transfer an ordering on ch... |
| wepwso 40784 | A well-ordering induces a ... |
| dnnumch1 40785 | Define an enumeration of a... |
| dnnumch2 40786 | Define an enumeration (wea... |
| dnnumch3lem 40787 | Value of the ordinal injec... |
| dnnumch3 40788 | Define an injection from a... |
| dnwech 40789 | Define a well-ordering fro... |
| fnwe2val 40790 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 40791 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 40792 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 40793 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 40794 | A well-ordering can be con... |
| aomclem1 40795 | Lemma for ~ dfac11 . This... |
| aomclem2 40796 | Lemma for ~ dfac11 . Succ... |
| aomclem3 40797 | Lemma for ~ dfac11 . Succ... |
| aomclem4 40798 | Lemma for ~ dfac11 . Limi... |
| aomclem5 40799 | Lemma for ~ dfac11 . Comb... |
| aomclem6 40800 | Lemma for ~ dfac11 . Tran... |
| aomclem7 40801 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 40802 | Lemma for ~ dfac11 . Perf... |
| dfac11 40803 | The right-hand side of thi... |
| kelac1 40804 | Kelley's choice, basic for... |
| kelac2lem 40805 | Lemma for ~ kelac2 and ~ d... |
| kelac2 40806 | Kelley's choice, most comm... |
| dfac21 40807 | Tychonoff's theorem is a c... |
| islmodfg 40810 | Property of a finitely gen... |
| islssfg 40811 | Property of a finitely gen... |
| islssfg2 40812 | Property of a finitely gen... |
| islssfgi 40813 | Finitely spanned subspaces... |
| fglmod 40814 | Finitely generated left mo... |
| lsmfgcl 40815 | The sum of two finitely ge... |
| islnm 40818 | Property of being a Noethe... |
| islnm2 40819 | Property of being a Noethe... |
| lnmlmod 40820 | A Noetherian left module i... |
| lnmlssfg 40821 | A submodule of Noetherian ... |
| lnmlsslnm 40822 | All submodules of a Noethe... |
| lnmfg 40823 | A Noetherian left module i... |
| kercvrlsm 40824 | The domain of a linear fun... |
| lmhmfgima 40825 | A homomorphism maps finite... |
| lnmepi 40826 | Epimorphic images of Noeth... |
| lmhmfgsplit 40827 | If the kernel and range of... |
| lmhmlnmsplit 40828 | If the kernel and range of... |
| lnmlmic 40829 | Noetherian is an invariant... |
| pwssplit4 40830 | Splitting for structure po... |
| filnm 40831 | Finite left modules are No... |
| pwslnmlem0 40832 | Zeroeth powers are Noether... |
| pwslnmlem1 40833 | First powers are Noetheria... |
| pwslnmlem2 40834 | A sum of powers is Noether... |
| pwslnm 40835 | Finite powers of Noetheria... |
| unxpwdom3 40836 | Weaker version of ~ unxpwd... |
| pwfi2f1o 40837 | The ~ pw2f1o bijection rel... |
| pwfi2en 40838 | Finitely supported indicat... |
| frlmpwfi 40839 | Formal linear combinations... |
| gicabl 40840 | Being Abelian is a group i... |
| imasgim 40841 | A relabeling of the elemen... |
| isnumbasgrplem1 40842 | A set which is equipollent... |
| harn0 40843 | The Hartogs number of a se... |
| numinfctb 40844 | A numerable infinite set c... |
| isnumbasgrplem2 40845 | If the (to be thought of a... |
| isnumbasgrplem3 40846 | Every nonempty numerable s... |
| isnumbasabl 40847 | A set is numerable iff it ... |
| isnumbasgrp 40848 | A set is numerable iff it ... |
| dfacbasgrp 40849 | A choice equivalent in abs... |
| islnr 40852 | Property of a left-Noether... |
| lnrring 40853 | Left-Noetherian rings are ... |
| lnrlnm 40854 | Left-Noetherian rings have... |
| islnr2 40855 | Property of being a left-N... |
| islnr3 40856 | Relate left-Noetherian rin... |
| lnr2i 40857 | Given an ideal in a left-N... |
| lpirlnr 40858 | Left principal ideal rings... |
| lnrfrlm 40859 | Finite-dimensional free mo... |
| lnrfg 40860 | Finitely-generated modules... |
| lnrfgtr 40861 | A submodule of a finitely ... |
| hbtlem1 40864 | Value of the leading coeff... |
| hbtlem2 40865 | Leading coefficient ideals... |
| hbtlem7 40866 | Functionality of leading c... |
| hbtlem4 40867 | The leading ideal function... |
| hbtlem3 40868 | The leading ideal function... |
| hbtlem5 40869 | The leading ideal function... |
| hbtlem6 40870 | There is a finite set of p... |
| hbt 40871 | The Hilbert Basis Theorem ... |
| dgrsub2 40876 | Subtracting two polynomial... |
| elmnc 40877 | Property of a monic polyno... |
| mncply 40878 | A monic polynomial is a po... |
| mnccoe 40879 | A monic polynomial has lea... |
| mncn0 40880 | A monic polynomial is not ... |
| dgraaval 40885 | Value of the degree functi... |
| dgraalem 40886 | Properties of the degree o... |
| dgraacl 40887 | Closure of the degree func... |
| dgraaf 40888 | Degree function on algebra... |
| dgraaub 40889 | Upper bound on degree of a... |
| dgraa0p 40890 | A rational polynomial of d... |
| mpaaeu 40891 | An algebraic number has ex... |
| mpaaval 40892 | Value of the minimal polyn... |
| mpaalem 40893 | Properties of the minimal ... |
| mpaacl 40894 | Minimal polynomial is a po... |
| mpaadgr 40895 | Minimal polynomial has deg... |
| mpaaroot 40896 | The minimal polynomial of ... |
| mpaamn 40897 | Minimal polynomial is moni... |
| itgoval 40902 | Value of the integral-over... |
| aaitgo 40903 | The standard algebraic num... |
| itgoss 40904 | An integral element is int... |
| itgocn 40905 | All integral elements are ... |
| cnsrexpcl 40906 | Exponentiation is closed i... |
| fsumcnsrcl 40907 | Finite sums are closed in ... |
| cnsrplycl 40908 | Polynomials are closed in ... |
| rgspnval 40909 | Value of the ring-span of ... |
| rgspncl 40910 | The ring-span of a set is ... |
| rgspnssid 40911 | The ring-span of a set con... |
| rgspnmin 40912 | The ring-span is contained... |
| rgspnid 40913 | The span of a subring is i... |
| rngunsnply 40914 | Adjoining one element to a... |
| flcidc 40915 | Finite linear combinations... |
| algstr 40918 | Lemma to shorten proofs of... |
| algbase 40919 | The base set of a construc... |
| algaddg 40920 | The additive operation of ... |
| algmulr 40921 | The multiplicative operati... |
| algsca 40922 | The set of scalars of a co... |
| algvsca 40923 | The scalar product operati... |
| mendval 40924 | Value of the module endomo... |
| mendbas 40925 | Base set of the module end... |
| mendplusgfval 40926 | Addition in the module end... |
| mendplusg 40927 | A specific addition in the... |
| mendmulrfval 40928 | Multiplication in the modu... |
| mendmulr 40929 | A specific multiplication ... |
| mendsca 40930 | The module endomorphism al... |
| mendvscafval 40931 | Scalar multiplication in t... |
| mendvsca 40932 | A specific scalar multipli... |
| mendring 40933 | The module endomorphism al... |
| mendlmod 40934 | The module endomorphism al... |
| mendassa 40935 | The module endomorphism al... |
| idomrootle 40936 | No element of an integral ... |
| idomodle 40937 | Limit on the number of ` N... |
| fiuneneq 40938 | Two finite sets of equal s... |
| idomsubgmo 40939 | The units of an integral d... |
| proot1mul 40940 | Any primitive ` N ` -th ro... |
| proot1hash 40941 | If an integral domain has ... |
| proot1ex 40942 | The complex field has prim... |
| isdomn3 40945 | Nonzero elements form a mu... |
| mon1pid 40946 | Monicity and degree of the... |
| mon1psubm 40947 | Monic polynomials are a mu... |
| deg1mhm 40948 | Homomorphic property of th... |
| cytpfn 40949 | Functionality of the cyclo... |
| cytpval 40950 | Substitutions for the Nth ... |
| fgraphopab 40951 | Express a function as a su... |
| fgraphxp 40952 | Express a function as a su... |
| hausgraph 40953 | The graph of a continuous ... |
| iocunico 40958 | Split an open interval int... |
| iocinico 40959 | The intersection of two se... |
| iocmbl 40960 | An open-below, closed-abov... |
| cnioobibld 40961 | A bounded, continuous func... |
| arearect 40962 | The area of a rectangle wh... |
| areaquad 40963 | The area of a quadrilatera... |
| ifpan123g 40964 | Conjunction of conditional... |
| ifpan23 40965 | Conjunction of conditional... |
| ifpdfor2 40966 | Define or in terms of cond... |
| ifporcor 40967 | Corollary of commutation o... |
| ifpdfan2 40968 | Define and with conditiona... |
| ifpancor 40969 | Corollary of commutation o... |
| ifpdfor 40970 | Define or in terms of cond... |
| ifpdfan 40971 | Define and with conditiona... |
| ifpbi2 40972 | Equivalence theorem for co... |
| ifpbi3 40973 | Equivalence theorem for co... |
| ifpim1 40974 | Restate implication as con... |
| ifpnot 40975 | Restate negated wff as con... |
| ifpid2 40976 | Restate wff as conditional... |
| ifpim2 40977 | Restate implication as con... |
| ifpbi23 40978 | Equivalence theorem for co... |
| ifpbiidcor 40979 | Restatement of ~ biid . (... |
| ifpbicor 40980 | Corollary of commutation o... |
| ifpxorcor 40981 | Corollary of commutation o... |
| ifpbi1 40982 | Equivalence theorem for co... |
| ifpnot23 40983 | Negation of conditional lo... |
| ifpnotnotb 40984 | Factor conditional logic o... |
| ifpnorcor 40985 | Corollary of commutation o... |
| ifpnancor 40986 | Corollary of commutation o... |
| ifpnot23b 40987 | Negation of conditional lo... |
| ifpbiidcor2 40988 | Restatement of ~ biid . (... |
| ifpnot23c 40989 | Negation of conditional lo... |
| ifpnot23d 40990 | Negation of conditional lo... |
| ifpdfnan 40991 | Define nand as conditional... |
| ifpdfxor 40992 | Define xor as conditional ... |
| ifpbi12 40993 | Equivalence theorem for co... |
| ifpbi13 40994 | Equivalence theorem for co... |
| ifpbi123 40995 | Equivalence theorem for co... |
| ifpidg 40996 | Restate wff as conditional... |
| ifpid3g 40997 | Restate wff as conditional... |
| ifpid2g 40998 | Restate wff as conditional... |
| ifpid1g 40999 | Restate wff as conditional... |
| ifpim23g 41000 | Restate implication as con... |
| ifpim3 41001 | Restate implication as con... |
| ifpnim1 41002 | Restate negated implicatio... |
| ifpim4 41003 | Restate implication as con... |
| ifpnim2 41004 | Restate negated implicatio... |
| ifpim123g 41005 | Implication of conditional... |
| ifpim1g 41006 | Implication of conditional... |
| ifp1bi 41007 | Substitute the first eleme... |
| ifpbi1b 41008 | When the first variable is... |
| ifpimimb 41009 | Factor conditional logic o... |
| ifpororb 41010 | Factor conditional logic o... |
| ifpananb 41011 | Factor conditional logic o... |
| ifpnannanb 41012 | Factor conditional logic o... |
| ifpor123g 41013 | Disjunction of conditional... |
| ifpimim 41014 | Consequnce of implication.... |
| ifpbibib 41015 | Factor conditional logic o... |
| ifpxorxorb 41016 | Factor conditional logic o... |
| rp-fakeimass 41017 | A special case where impli... |
| rp-fakeanorass 41018 | A special case where a mix... |
| rp-fakeoranass 41019 | A special case where a mix... |
| rp-fakeinunass 41020 | A special case where a mix... |
| rp-fakeuninass 41021 | A special case where a mix... |
| rp-isfinite5 41022 | A set is said to be finite... |
| rp-isfinite6 41023 | A set is said to be finite... |
| intabssd 41024 | When for each element ` y ... |
| eu0 41025 | There is only one empty se... |
| epelon2 41026 | Over the ordinal numbers, ... |
| ontric3g 41027 | For all ` x , y e. On ` , ... |
| dfsucon 41028 | ` A ` is called a successo... |
| snen1g 41029 | A singleton is equinumerou... |
| snen1el 41030 | A singleton is equinumerou... |
| sn1dom 41031 | A singleton is dominated b... |
| pr2dom 41032 | An unordered pair is domin... |
| tr3dom 41033 | An unordered triple is dom... |
| ensucne0 41034 | A class equinumerous to a ... |
| ensucne0OLD 41035 | A class equinumerous to a ... |
| dfom6 41036 | Let ` _om ` be defined to ... |
| infordmin 41037 | ` _om ` is the smallest in... |
| iscard4 41038 | Two ways to express the pr... |
| iscard5 41039 | Two ways to express the pr... |
| elrncard 41040 | Let us define a cardinal n... |
| harval3 41041 | ` ( har `` A ) ` is the le... |
| harval3on 41042 | For any ordinal number ` A... |
| en2pr 41043 | A class is equinumerous to... |
| pr2cv 41044 | If an unordered pair is eq... |
| pr2el1 41045 | If an unordered pair is eq... |
| pr2cv1 41046 | If an unordered pair is eq... |
| pr2el2 41047 | If an unordered pair is eq... |
| pr2cv2 41048 | If an unordered pair is eq... |
| pren2 41049 | An unordered pair is equin... |
| pr2eldif1 41050 | If an unordered pair is eq... |
| pr2eldif2 41051 | If an unordered pair is eq... |
| pren2d 41052 | A pair of two distinct set... |
| aleph1min 41053 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 41054 | ` aleph ` is a strictly or... |
| alephiso3 41055 | ` aleph ` is a strictly or... |
| pwelg 41056 | The powerclass is an eleme... |
| pwinfig 41057 | The powerclass of an infin... |
| pwinfi2 41058 | The powerclass of an infin... |
| pwinfi3 41059 | The powerclass of an infin... |
| pwinfi 41060 | The powerclass of an infin... |
| fipjust 41061 | A definition of the finite... |
| cllem0 41062 | The class of all sets with... |
| superficl 41063 | The class of all supersets... |
| superuncl 41064 | The class of all supersets... |
| ssficl 41065 | The class of all subsets o... |
| ssuncl 41066 | The class of all subsets o... |
| ssdifcl 41067 | The class of all subsets o... |
| sssymdifcl 41068 | The class of all subsets o... |
| fiinfi 41069 | If two classes have the fi... |
| rababg 41070 | Condition when restricted ... |
| elintabg 41071 | Two ways of saying a set i... |
| elinintab 41072 | Two ways of saying a set i... |
| elmapintrab 41073 | Two ways to say a set is a... |
| elinintrab 41074 | Two ways of saying a set i... |
| inintabss 41075 | Upper bound on intersectio... |
| inintabd 41076 | Value of the intersection ... |
| xpinintabd 41077 | Value of the intersection ... |
| relintabex 41078 | If the intersection of a c... |
| elcnvcnvintab 41079 | Two ways of saying a set i... |
| relintab 41080 | Value of the intersection ... |
| nonrel 41081 | A non-relation is equal to... |
| elnonrel 41082 | Only an ordered pair where... |
| cnvssb 41083 | Subclass theorem for conve... |
| relnonrel 41084 | The non-relation part of a... |
| cnvnonrel 41085 | The converse of the non-re... |
| brnonrel 41086 | A non-relation cannot rela... |
| dmnonrel 41087 | The domain of the non-rela... |
| rnnonrel 41088 | The range of the non-relat... |
| resnonrel 41089 | A restriction of the non-r... |
| imanonrel 41090 | An image under the non-rel... |
| cononrel1 41091 | Composition with the non-r... |
| cononrel2 41092 | Composition with the non-r... |
| elmapintab 41093 | Two ways to say a set is a... |
| fvnonrel 41094 | The function value of any ... |
| elinlem 41095 | Two ways to say a set is a... |
| elcnvcnvlem 41096 | Two ways to say a set is a... |
| cnvcnvintabd 41097 | Value of the relationship ... |
| elcnvlem 41098 | Two ways to say a set is a... |
| elcnvintab 41099 | Two ways of saying a set i... |
| cnvintabd 41100 | Value of the converse of t... |
| undmrnresiss 41101 | Two ways of saying the ide... |
| reflexg 41102 | Two ways of saying a relat... |
| cnvssco 41103 | A condition weaker than re... |
| refimssco 41104 | Reflexive relations are su... |
| cleq2lem 41105 | Equality implies bijection... |
| cbvcllem 41106 | Change of bound variable i... |
| clublem 41107 | If a superset ` Y ` of ` X... |
| clss2lem 41108 | The closure of a property ... |
| dfid7 41109 | Definition of identity rel... |
| mptrcllem 41110 | Show two versions of a clo... |
| cotrintab 41111 | The intersection of a clas... |
| rclexi 41112 | The reflexive closure of a... |
| rtrclexlem 41113 | Existence of relation impl... |
| rtrclex 41114 | The reflexive-transitive c... |
| trclubgNEW 41115 | If a relation exists then ... |
| trclubNEW 41116 | If a relation exists then ... |
| trclexi 41117 | The transitive closure of ... |
| rtrclexi 41118 | The reflexive-transitive c... |
| clrellem 41119 | When the property ` ps ` h... |
| clcnvlem 41120 | When ` A ` , an upper boun... |
| cnvtrucl0 41121 | The converse of the trivia... |
| cnvrcl0 41122 | The converse of the reflex... |
| cnvtrcl0 41123 | The converse of the transi... |
| dmtrcl 41124 | The domain of the transiti... |
| rntrcl 41125 | The range of the transitiv... |
| dfrtrcl5 41126 | Definition of reflexive-tr... |
| trcleq2lemRP 41127 | Equality implies bijection... |
| sqrtcvallem1 41128 | Two ways of saying a compl... |
| reabsifneg 41129 | Alternate expression for t... |
| reabsifnpos 41130 | Alternate expression for t... |
| reabsifpos 41131 | Alternate expression for t... |
| reabsifnneg 41132 | Alternate expression for t... |
| reabssgn 41133 | Alternate expression for t... |
| sqrtcvallem2 41134 | Equivalent to saying that ... |
| sqrtcvallem3 41135 | Equivalent to saying that ... |
| sqrtcvallem4 41136 | Equivalent to saying that ... |
| sqrtcvallem5 41137 | Equivalent to saying that ... |
| sqrtcval 41138 | Explicit formula for the c... |
| sqrtcval2 41139 | Explicit formula for the c... |
| resqrtval 41140 | Real part of the complex s... |
| imsqrtval 41141 | Imaginary part of the comp... |
| resqrtvalex 41142 | Example for ~ resqrtval . ... |
| imsqrtvalex 41143 | Example for ~ imsqrtval . ... |
| al3im 41144 | Version of ~ ax-4 for a ne... |
| intima0 41145 | Two ways of expressing the... |
| elimaint 41146 | Element of image of inters... |
| cnviun 41147 | Converse of indexed union.... |
| imaiun1 41148 | The image of an indexed un... |
| coiun1 41149 | Composition with an indexe... |
| elintima 41150 | Element of intersection of... |
| intimass 41151 | The image under the inters... |
| intimass2 41152 | The image under the inters... |
| intimag 41153 | Requirement for the image ... |
| intimasn 41154 | Two ways to express the im... |
| intimasn2 41155 | Two ways to express the im... |
| ss2iundf 41156 | Subclass theorem for index... |
| ss2iundv 41157 | Subclass theorem for index... |
| cbviuneq12df 41158 | Rule used to change the bo... |
| cbviuneq12dv 41159 | Rule used to change the bo... |
| conrel1d 41160 | Deduction about compositio... |
| conrel2d 41161 | Deduction about compositio... |
| trrelind 41162 | The intersection of transi... |
| xpintrreld 41163 | The intersection of a tran... |
| restrreld 41164 | The restriction of a trans... |
| trrelsuperreldg 41165 | Concrete construction of a... |
| trficl 41166 | The class of all transitiv... |
| cnvtrrel 41167 | The converse of a transiti... |
| trrelsuperrel2dg 41168 | Concrete construction of a... |
| dfrcl2 41171 | Reflexive closure of a rel... |
| dfrcl3 41172 | Reflexive closure of a rel... |
| dfrcl4 41173 | Reflexive closure of a rel... |
| relexp2 41174 | A set operated on by the r... |
| relexpnul 41175 | If the domain and range of... |
| eliunov2 41176 | Membership in the indexed ... |
| eltrclrec 41177 | Membership in the indexed ... |
| elrtrclrec 41178 | Membership in the indexed ... |
| briunov2 41179 | Two classes related by the... |
| brmptiunrelexpd 41180 | If two elements are connec... |
| fvmptiunrelexplb0d 41181 | If the indexed union range... |
| fvmptiunrelexplb0da 41182 | If the indexed union range... |
| fvmptiunrelexplb1d 41183 | If the indexed union range... |
| brfvid 41184 | If two elements are connec... |
| brfvidRP 41185 | If two elements are connec... |
| fvilbd 41186 | A set is a subset of its i... |
| fvilbdRP 41187 | A set is a subset of its i... |
| brfvrcld 41188 | If two elements are connec... |
| brfvrcld2 41189 | If two elements are connec... |
| fvrcllb0d 41190 | A restriction of the ident... |
| fvrcllb0da 41191 | A restriction of the ident... |
| fvrcllb1d 41192 | A set is a subset of its i... |
| brtrclrec 41193 | Two classes related by the... |
| brrtrclrec 41194 | Two classes related by the... |
| briunov2uz 41195 | Two classes related by the... |
| eliunov2uz 41196 | Membership in the indexed ... |
| ov2ssiunov2 41197 | Any particular operator va... |
| relexp0eq 41198 | The zeroth power of relati... |
| iunrelexp0 41199 | Simplification of zeroth p... |
| relexpxpnnidm 41200 | Any positive power of a Ca... |
| relexpiidm 41201 | Any power of any restricti... |
| relexpss1d 41202 | The relational power of a ... |
| comptiunov2i 41203 | The composition two indexe... |
| corclrcl 41204 | The reflexive closure is i... |
| iunrelexpmin1 41205 | The indexed union of relat... |
| relexpmulnn 41206 | With exponents limited to ... |
| relexpmulg 41207 | With ordered exponents, th... |
| trclrelexplem 41208 | The union of relational po... |
| iunrelexpmin2 41209 | The indexed union of relat... |
| relexp01min 41210 | With exponents limited to ... |
| relexp1idm 41211 | Repeated raising a relatio... |
| relexp0idm 41212 | Repeated raising a relatio... |
| relexp0a 41213 | Absorbtion law for zeroth ... |
| relexpxpmin 41214 | The composition of powers ... |
| relexpaddss 41215 | The composition of two pow... |
| iunrelexpuztr 41216 | The indexed union of relat... |
| dftrcl3 41217 | Transitive closure of a re... |
| brfvtrcld 41218 | If two elements are connec... |
| fvtrcllb1d 41219 | A set is a subset of its i... |
| trclfvcom 41220 | The transitive closure of ... |
| cnvtrclfv 41221 | The converse of the transi... |
| cotrcltrcl 41222 | The transitive closure is ... |
| trclimalb2 41223 | Lower bound for image unde... |
| brtrclfv2 41224 | Two ways to indicate two e... |
| trclfvdecomr 41225 | The transitive closure of ... |
| trclfvdecoml 41226 | The transitive closure of ... |
| dmtrclfvRP 41227 | The domain of the transiti... |
| rntrclfvRP 41228 | The range of the transitiv... |
| rntrclfv 41229 | The range of the transitiv... |
| dfrtrcl3 41230 | Reflexive-transitive closu... |
| brfvrtrcld 41231 | If two elements are connec... |
| fvrtrcllb0d 41232 | A restriction of the ident... |
| fvrtrcllb0da 41233 | A restriction of the ident... |
| fvrtrcllb1d 41234 | A set is a subset of its i... |
| dfrtrcl4 41235 | Reflexive-transitive closu... |
| corcltrcl 41236 | The composition of the ref... |
| cortrcltrcl 41237 | Composition with the refle... |
| corclrtrcl 41238 | Composition with the refle... |
| cotrclrcl 41239 | The composition of the ref... |
| cortrclrcl 41240 | Composition with the refle... |
| cotrclrtrcl 41241 | Composition with the refle... |
| cortrclrtrcl 41242 | The reflexive-transitive c... |
| frege77d 41243 | If the images of both ` { ... |
| frege81d 41244 | If the image of ` U ` is a... |
| frege83d 41245 | If the image of the union ... |
| frege96d 41246 | If ` C ` follows ` A ` in ... |
| frege87d 41247 | If the images of both ` { ... |
| frege91d 41248 | If ` B ` follows ` A ` in ... |
| frege97d 41249 | If ` A ` contains all elem... |
| frege98d 41250 | If ` C ` follows ` A ` and... |
| frege102d 41251 | If either ` A ` and ` C ` ... |
| frege106d 41252 | If ` B ` follows ` A ` in ... |
| frege108d 41253 | If either ` A ` and ` C ` ... |
| frege109d 41254 | If ` A ` contains all elem... |
| frege114d 41255 | If either ` R ` relates ` ... |
| frege111d 41256 | If either ` A ` and ` C ` ... |
| frege122d 41257 | If ` F ` is a function, ` ... |
| frege124d 41258 | If ` F ` is a function, ` ... |
| frege126d 41259 | If ` F ` is a function, ` ... |
| frege129d 41260 | If ` F ` is a function and... |
| frege131d 41261 | If ` F ` is a function and... |
| frege133d 41262 | If ` F ` is a function and... |
| dfxor4 41263 | Express exclusive-or in te... |
| dfxor5 41264 | Express exclusive-or in te... |
| df3or2 41265 | Express triple-or in terms... |
| df3an2 41266 | Express triple-and in term... |
| nev 41267 | Express that not every set... |
| 0pssin 41268 | Express that an intersecti... |
| dfhe2 41271 | The property of relation `... |
| dfhe3 41272 | The property of relation `... |
| heeq12 41273 | Equality law for relations... |
| heeq1 41274 | Equality law for relations... |
| heeq2 41275 | Equality law for relations... |
| sbcheg 41276 | Distribute proper substitu... |
| hess 41277 | Subclass law for relations... |
| xphe 41278 | Any Cartesian product is h... |
| 0he 41279 | The empty relation is here... |
| 0heALT 41280 | The empty relation is here... |
| he0 41281 | Any relation is hereditary... |
| unhe1 41282 | The union of two relations... |
| snhesn 41283 | Any singleton is hereditar... |
| idhe 41284 | The identity relation is h... |
| psshepw 41285 | The relation between sets ... |
| sshepw 41286 | The relation between sets ... |
| rp-simp2-frege 41289 | Simplification of triple c... |
| rp-simp2 41290 | Simplification of triple c... |
| rp-frege3g 41291 | Add antecedent to ~ ax-fre... |
| frege3 41292 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 41293 | Double-use of ~ ax-frege2 ... |
| rp-frege24 41294 | Introducing an embedded an... |
| rp-frege4g 41295 | Deduction related to distr... |
| frege4 41296 | Special case of closed for... |
| frege5 41297 | A closed form of ~ syl . ... |
| rp-7frege 41298 | Distribute antecedent and ... |
| rp-4frege 41299 | Elimination of a nested an... |
| rp-6frege 41300 | Elimination of a nested an... |
| rp-8frege 41301 | Eliminate antecedent when ... |
| rp-frege25 41302 | Closed form for ~ a1dd . ... |
| frege6 41303 | A closed form of ~ imim2d ... |
| axfrege8 41304 | Swap antecedents. Identic... |
| frege7 41305 | A closed form of ~ syl6 . ... |
| frege26 41307 | Identical to ~ idd . Prop... |
| frege27 41308 | We cannot (at the same tim... |
| frege9 41309 | Closed form of ~ syl with ... |
| frege12 41310 | A closed form of ~ com23 .... |
| frege11 41311 | Elimination of a nested an... |
| frege24 41312 | Closed form for ~ a1d . D... |
| frege16 41313 | A closed form of ~ com34 .... |
| frege25 41314 | Closed form for ~ a1dd . ... |
| frege18 41315 | Closed form of a syllogism... |
| frege22 41316 | A closed form of ~ com45 .... |
| frege10 41317 | Result commuting anteceden... |
| frege17 41318 | A closed form of ~ com3l .... |
| frege13 41319 | A closed form of ~ com3r .... |
| frege14 41320 | Closed form of a deduction... |
| frege19 41321 | A closed form of ~ syl6 . ... |
| frege23 41322 | Syllogism followed by rota... |
| frege15 41323 | A closed form of ~ com4r .... |
| frege21 41324 | Replace antecedent in ante... |
| frege20 41325 | A closed form of ~ syl8 . ... |
| axfrege28 41326 | Contraposition. Identical... |
| frege29 41328 | Closed form of ~ con3d . ... |
| frege30 41329 | Commuted, closed form of ~... |
| axfrege31 41330 | Identical to ~ notnotr . ... |
| frege32 41332 | Deduce ~ con1 from ~ con3 ... |
| frege33 41333 | If ` ph ` or ` ps ` takes ... |
| frege34 41334 | If as a conseqence of the ... |
| frege35 41335 | Commuted, closed form of ~... |
| frege36 41336 | The case in which ` ps ` i... |
| frege37 41337 | If ` ch ` is a necessary c... |
| frege38 41338 | Identical to ~ pm2.21 . P... |
| frege39 41339 | Syllogism between ~ pm2.18... |
| frege40 41340 | Anything implies ~ pm2.18 ... |
| axfrege41 41341 | Identical to ~ notnot . A... |
| frege42 41343 | Not not ~ id . Propositio... |
| frege43 41344 | If there is a choice only ... |
| frege44 41345 | Similar to a commuted ~ pm... |
| frege45 41346 | Deduce ~ pm2.6 from ~ con1... |
| frege46 41347 | If ` ps ` holds when ` ph ... |
| frege47 41348 | Deduce consequence follows... |
| frege48 41349 | Closed form of syllogism w... |
| frege49 41350 | Closed form of deduction w... |
| frege50 41351 | Closed form of ~ jaoi . P... |
| frege51 41352 | Compare with ~ jaod . Pro... |
| axfrege52a 41353 | Justification for ~ ax-fre... |
| frege52aid 41355 | The case when the content ... |
| frege53aid 41356 | Specialization of ~ frege5... |
| frege53a 41357 | Lemma for ~ frege55a . Pr... |
| axfrege54a 41358 | Justification for ~ ax-fre... |
| frege54cor0a 41360 | Synonym for logical equiva... |
| frege54cor1a 41361 | Reflexive equality. (Cont... |
| frege55aid 41362 | Lemma for ~ frege57aid . ... |
| frege55lem1a 41363 | Necessary deduction regard... |
| frege55lem2a 41364 | Core proof of Proposition ... |
| frege55a 41365 | Proposition 55 of [Frege18... |
| frege55cor1a 41366 | Proposition 55 of [Frege18... |
| frege56aid 41367 | Lemma for ~ frege57aid . ... |
| frege56a 41368 | Proposition 56 of [Frege18... |
| frege57aid 41369 | This is the all imporant f... |
| frege57a 41370 | Analogue of ~ frege57aid .... |
| axfrege58a 41371 | Identical to ~ anifp . Ju... |
| frege58acor 41373 | Lemma for ~ frege59a . (C... |
| frege59a 41374 | A kind of Aristotelian inf... |
| frege60a 41375 | Swap antecedents of ~ ax-f... |
| frege61a 41376 | Lemma for ~ frege65a . Pr... |
| frege62a 41377 | A kind of Aristotelian inf... |
| frege63a 41378 | Proposition 63 of [Frege18... |
| frege64a 41379 | Lemma for ~ frege65a . Pr... |
| frege65a 41380 | A kind of Aristotelian inf... |
| frege66a 41381 | Swap antecedents of ~ freg... |
| frege67a 41382 | Lemma for ~ frege68a . Pr... |
| frege68a 41383 | Combination of applying a ... |
| axfrege52c 41384 | Justification for ~ ax-fre... |
| frege52b 41386 | The case when the content ... |
| frege53b 41387 | Lemma for frege102 (via ~ ... |
| axfrege54c 41388 | Reflexive equality of clas... |
| frege54b 41390 | Reflexive equality of sets... |
| frege54cor1b 41391 | Reflexive equality. (Cont... |
| frege55lem1b 41392 | Necessary deduction regard... |
| frege55lem2b 41393 | Lemma for ~ frege55b . Co... |
| frege55b 41394 | Lemma for ~ frege57b . Pr... |
| frege56b 41395 | Lemma for ~ frege57b . Pr... |
| frege57b 41396 | Analogue of ~ frege57aid .... |
| axfrege58b 41397 | If ` A. x ph ` is affirmed... |
| frege58bid 41399 | If ` A. x ph ` is affirmed... |
| frege58bcor 41400 | Lemma for ~ frege59b . (C... |
| frege59b 41401 | A kind of Aristotelian inf... |
| frege60b 41402 | Swap antecedents of ~ ax-f... |
| frege61b 41403 | Lemma for ~ frege65b . Pr... |
| frege62b 41404 | A kind of Aristotelian inf... |
| frege63b 41405 | Lemma for ~ frege91 . Pro... |
| frege64b 41406 | Lemma for ~ frege65b . Pr... |
| frege65b 41407 | A kind of Aristotelian inf... |
| frege66b 41408 | Swap antecedents of ~ freg... |
| frege67b 41409 | Lemma for ~ frege68b . Pr... |
| frege68b 41410 | Combination of applying a ... |
| frege53c 41411 | Proposition 53 of [Frege18... |
| frege54cor1c 41412 | Reflexive equality. (Cont... |
| frege55lem1c 41413 | Necessary deduction regard... |
| frege55lem2c 41414 | Core proof of Proposition ... |
| frege55c 41415 | Proposition 55 of [Frege18... |
| frege56c 41416 | Lemma for ~ frege57c . Pr... |
| frege57c 41417 | Swap order of implication ... |
| frege58c 41418 | Principle related to ~ sp ... |
| frege59c 41419 | A kind of Aristotelian inf... |
| frege60c 41420 | Swap antecedents of ~ freg... |
| frege61c 41421 | Lemma for ~ frege65c . Pr... |
| frege62c 41422 | A kind of Aristotelian inf... |
| frege63c 41423 | Analogue of ~ frege63b . ... |
| frege64c 41424 | Lemma for ~ frege65c . Pr... |
| frege65c 41425 | A kind of Aristotelian inf... |
| frege66c 41426 | Swap antecedents of ~ freg... |
| frege67c 41427 | Lemma for ~ frege68c . Pr... |
| frege68c 41428 | Combination of applying a ... |
| dffrege69 41429 | If from the proposition th... |
| frege70 41430 | Lemma for ~ frege72 . Pro... |
| frege71 41431 | Lemma for ~ frege72 . Pro... |
| frege72 41432 | If property ` A ` is hered... |
| frege73 41433 | Lemma for ~ frege87 . Pro... |
| frege74 41434 | If ` X ` has a property ` ... |
| frege75 41435 | If from the proposition th... |
| dffrege76 41436 | If from the two propositio... |
| frege77 41437 | If ` Y ` follows ` X ` in ... |
| frege78 41438 | Commuted form of of ~ freg... |
| frege79 41439 | Distributed form of ~ freg... |
| frege80 41440 | Add additional condition t... |
| frege81 41441 | If ` X ` has a property ` ... |
| frege82 41442 | Closed-form deduction base... |
| frege83 41443 | Apply commuted form of ~ f... |
| frege84 41444 | Commuted form of ~ frege81... |
| frege85 41445 | Commuted form of ~ frege77... |
| frege86 41446 | Conclusion about element o... |
| frege87 41447 | If ` Z ` is a result of an... |
| frege88 41448 | Commuted form of ~ frege87... |
| frege89 41449 | One direction of ~ dffrege... |
| frege90 41450 | Add antecedent to ~ frege8... |
| frege91 41451 | Every result of an applica... |
| frege92 41452 | Inference from ~ frege91 .... |
| frege93 41453 | Necessary condition for tw... |
| frege94 41454 | Looking one past a pair re... |
| frege95 41455 | Looking one past a pair re... |
| frege96 41456 | Every result of an applica... |
| frege97 41457 | The property of following ... |
| frege98 41458 | If ` Y ` follows ` X ` and... |
| dffrege99 41459 | If ` Z ` is identical with... |
| frege100 41460 | One direction of ~ dffrege... |
| frege101 41461 | Lemma for ~ frege102 . Pr... |
| frege102 41462 | If ` Z ` belongs to the ` ... |
| frege103 41463 | Proposition 103 of [Frege1... |
| frege104 41464 | Proposition 104 of [Frege1... |
| frege105 41465 | Proposition 105 of [Frege1... |
| frege106 41466 | Whatever follows ` X ` in ... |
| frege107 41467 | Proposition 107 of [Frege1... |
| frege108 41468 | If ` Y ` belongs to the ` ... |
| frege109 41469 | The property of belonging ... |
| frege110 41470 | Proposition 110 of [Frege1... |
| frege111 41471 | If ` Y ` belongs to the ` ... |
| frege112 41472 | Identity implies belonging... |
| frege113 41473 | Proposition 113 of [Frege1... |
| frege114 41474 | If ` X ` belongs to the ` ... |
| dffrege115 41475 | If from the circumstance t... |
| frege116 41476 | One direction of ~ dffrege... |
| frege117 41477 | Lemma for ~ frege118 . Pr... |
| frege118 41478 | Simplified application of ... |
| frege119 41479 | Lemma for ~ frege120 . Pr... |
| frege120 41480 | Simplified application of ... |
| frege121 41481 | Lemma for ~ frege122 . Pr... |
| frege122 41482 | If ` X ` is a result of an... |
| frege123 41483 | Lemma for ~ frege124 . Pr... |
| frege124 41484 | If ` X ` is a result of an... |
| frege125 41485 | Lemma for ~ frege126 . Pr... |
| frege126 41486 | If ` M ` follows ` Y ` in ... |
| frege127 41487 | Communte antecedents of ~ ... |
| frege128 41488 | Lemma for ~ frege129 . Pr... |
| frege129 41489 | If the procedure ` R ` is ... |
| frege130 41490 | Lemma for ~ frege131 . Pr... |
| frege131 41491 | If the procedure ` R ` is ... |
| frege132 41492 | Lemma for ~ frege133 . Pr... |
| frege133 41493 | If the procedure ` R ` is ... |
| enrelmap 41494 | The set of all possible re... |
| enrelmapr 41495 | The set of all possible re... |
| enmappw 41496 | The set of all mappings fr... |
| enmappwid 41497 | The set of all mappings fr... |
| rfovd 41498 | Value of the operator, ` (... |
| rfovfvd 41499 | Value of the operator, ` (... |
| rfovfvfvd 41500 | Value of the operator, ` (... |
| rfovcnvf1od 41501 | Properties of the operator... |
| rfovcnvd 41502 | Value of the converse of t... |
| rfovf1od 41503 | The value of the operator,... |
| rfovcnvfvd 41504 | Value of the converse of t... |
| fsovd 41505 | Value of the operator, ` (... |
| fsovrfovd 41506 | The operator which gives a... |
| fsovfvd 41507 | Value of the operator, ` (... |
| fsovfvfvd 41508 | Value of the operator, ` (... |
| fsovfd 41509 | The operator, ` ( A O B ) ... |
| fsovcnvlem 41510 | The ` O ` operator, which ... |
| fsovcnvd 41511 | The value of the converse ... |
| fsovcnvfvd 41512 | The value of the converse ... |
| fsovf1od 41513 | The value of ` ( A O B ) `... |
| dssmapfvd 41514 | Value of the duality opera... |
| dssmapfv2d 41515 | Value of the duality opera... |
| dssmapfv3d 41516 | Value of the duality opera... |
| dssmapnvod 41517 | For any base set ` B ` the... |
| dssmapf1od 41518 | For any base set ` B ` the... |
| dssmap2d 41519 | For any base set ` B ` the... |
| or3or 41520 | Decompose disjunction into... |
| andi3or 41521 | Distribute over triple dis... |
| uneqsn 41522 | If a union of classes is e... |
| df3o2 41523 | Ordinal 3 is the unordered... |
| df3o3 41524 | Ordinal 3, fully expanded.... |
| brfvimex 41525 | If a binary relation holds... |
| brovmptimex 41526 | If a binary relation holds... |
| brovmptimex1 41527 | If a binary relation holds... |
| brovmptimex2 41528 | If a binary relation holds... |
| brcoffn 41529 | Conditions allowing the de... |
| brcofffn 41530 | Conditions allowing the de... |
| brco2f1o 41531 | Conditions allowing the de... |
| brco3f1o 41532 | Conditions allowing the de... |
| ntrclsbex 41533 | If (pseudo-)interior and (... |
| ntrclsrcomplex 41534 | The relative complement of... |
| neik0imk0p 41535 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 41536 | If an interior function is... |
| ntrkbimka 41537 | If the interiors of disjoi... |
| ntrk0kbimka 41538 | If the interiors of disjoi... |
| clsk3nimkb 41539 | If the base set is not emp... |
| clsk1indlem0 41540 | The ansatz closure functio... |
| clsk1indlem2 41541 | The ansatz closure functio... |
| clsk1indlem3 41542 | The ansatz closure functio... |
| clsk1indlem4 41543 | The ansatz closure functio... |
| clsk1indlem1 41544 | The ansatz closure functio... |
| clsk1independent 41545 | For generalized closure fu... |
| neik0pk1imk0 41546 | Kuratowski's K0' and K1 ax... |
| isotone1 41547 | Two different ways to say ... |
| isotone2 41548 | Two different ways to say ... |
| ntrk1k3eqk13 41549 | An interior function is bo... |
| ntrclsf1o 41550 | If (pseudo-)interior and (... |
| ntrclsnvobr 41551 | If (pseudo-)interior and (... |
| ntrclsiex 41552 | If (pseudo-)interior and (... |
| ntrclskex 41553 | If (pseudo-)interior and (... |
| ntrclsfv1 41554 | If (pseudo-)interior and (... |
| ntrclsfv2 41555 | If (pseudo-)interior and (... |
| ntrclselnel1 41556 | If (pseudo-)interior and (... |
| ntrclselnel2 41557 | If (pseudo-)interior and (... |
| ntrclsfv 41558 | The value of the interior ... |
| ntrclsfveq1 41559 | If interior and closure fu... |
| ntrclsfveq2 41560 | If interior and closure fu... |
| ntrclsfveq 41561 | If interior and closure fu... |
| ntrclsss 41562 | If interior and closure fu... |
| ntrclsneine0lem 41563 | If (pseudo-)interior and (... |
| ntrclsneine0 41564 | If (pseudo-)interior and (... |
| ntrclscls00 41565 | If (pseudo-)interior and (... |
| ntrclsiso 41566 | If (pseudo-)interior and (... |
| ntrclsk2 41567 | An interior function is co... |
| ntrclskb 41568 | The interiors of disjoint ... |
| ntrclsk3 41569 | The intersection of interi... |
| ntrclsk13 41570 | The interior of the inters... |
| ntrclsk4 41571 | Idempotence of the interio... |
| ntrneibex 41572 | If (pseudo-)interior and (... |
| ntrneircomplex 41573 | The relative complement of... |
| ntrneif1o 41574 | If (pseudo-)interior and (... |
| ntrneiiex 41575 | If (pseudo-)interior and (... |
| ntrneinex 41576 | If (pseudo-)interior and (... |
| ntrneicnv 41577 | If (pseudo-)interior and (... |
| ntrneifv1 41578 | If (pseudo-)interior and (... |
| ntrneifv2 41579 | If (pseudo-)interior and (... |
| ntrneiel 41580 | If (pseudo-)interior and (... |
| ntrneifv3 41581 | The value of the neighbors... |
| ntrneineine0lem 41582 | If (pseudo-)interior and (... |
| ntrneineine1lem 41583 | If (pseudo-)interior and (... |
| ntrneifv4 41584 | The value of the interior ... |
| ntrneiel2 41585 | Membership in iterated int... |
| ntrneineine0 41586 | If (pseudo-)interior and (... |
| ntrneineine1 41587 | If (pseudo-)interior and (... |
| ntrneicls00 41588 | If (pseudo-)interior and (... |
| ntrneicls11 41589 | If (pseudo-)interior and (... |
| ntrneiiso 41590 | If (pseudo-)interior and (... |
| ntrneik2 41591 | An interior function is co... |
| ntrneix2 41592 | An interior (closure) func... |
| ntrneikb 41593 | The interiors of disjoint ... |
| ntrneixb 41594 | The interiors (closures) o... |
| ntrneik3 41595 | The intersection of interi... |
| ntrneix3 41596 | The closure of the union o... |
| ntrneik13 41597 | The interior of the inters... |
| ntrneix13 41598 | The closure of the union o... |
| ntrneik4w 41599 | Idempotence of the interio... |
| ntrneik4 41600 | Idempotence of the interio... |
| clsneibex 41601 | If (pseudo-)closure and (p... |
| clsneircomplex 41602 | The relative complement of... |
| clsneif1o 41603 | If a (pseudo-)closure func... |
| clsneicnv 41604 | If a (pseudo-)closure func... |
| clsneikex 41605 | If closure and neighborhoo... |
| clsneinex 41606 | If closure and neighborhoo... |
| clsneiel1 41607 | If a (pseudo-)closure func... |
| clsneiel2 41608 | If a (pseudo-)closure func... |
| clsneifv3 41609 | Value of the neighborhoods... |
| clsneifv4 41610 | Value of the closure (inte... |
| neicvgbex 41611 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 41612 | The relative complement of... |
| neicvgf1o 41613 | If neighborhood and conver... |
| neicvgnvo 41614 | If neighborhood and conver... |
| neicvgnvor 41615 | If neighborhood and conver... |
| neicvgmex 41616 | If the neighborhoods and c... |
| neicvgnex 41617 | If the neighborhoods and c... |
| neicvgel1 41618 | A subset being an element ... |
| neicvgel2 41619 | The complement of a subset... |
| neicvgfv 41620 | The value of the neighborh... |
| ntrrn 41621 | The range of the interior ... |
| ntrf 41622 | The interior function of a... |
| ntrf2 41623 | The interior function is a... |
| ntrelmap 41624 | The interior function is a... |
| clsf2 41625 | The closure function is a ... |
| clselmap 41626 | The closure function is a ... |
| dssmapntrcls 41627 | The interior and closure o... |
| dssmapclsntr 41628 | The closure and interior o... |
| gneispa 41629 | Each point ` p ` of the ne... |
| gneispb 41630 | Given a neighborhood ` N `... |
| gneispace2 41631 | The predicate that ` F ` i... |
| gneispace3 41632 | The predicate that ` F ` i... |
| gneispace 41633 | The predicate that ` F ` i... |
| gneispacef 41634 | A generic neighborhood spa... |
| gneispacef2 41635 | A generic neighborhood spa... |
| gneispacefun 41636 | A generic neighborhood spa... |
| gneispacern 41637 | A generic neighborhood spa... |
| gneispacern2 41638 | A generic neighborhood spa... |
| gneispace0nelrn 41639 | A generic neighborhood spa... |
| gneispace0nelrn2 41640 | A generic neighborhood spa... |
| gneispace0nelrn3 41641 | A generic neighborhood spa... |
| gneispaceel 41642 | Every neighborhood of a po... |
| gneispaceel2 41643 | Every neighborhood of a po... |
| gneispacess 41644 | All supersets of a neighbo... |
| gneispacess2 41645 | All supersets of a neighbo... |
| k0004lem1 41646 | Application of ~ ssin to r... |
| k0004lem2 41647 | A mapping with a particula... |
| k0004lem3 41648 | When the value of a mappin... |
| k0004val 41649 | The topological simplex of... |
| k0004ss1 41650 | The topological simplex of... |
| k0004ss2 41651 | The topological simplex of... |
| k0004ss3 41652 | The topological simplex of... |
| k0004val0 41653 | The topological simplex of... |
| inductionexd 41654 | Simple induction example. ... |
| wwlemuld 41655 | Natural deduction form of ... |
| leeq1d 41656 | Specialization of ~ breq1d... |
| leeq2d 41657 | Specialization of ~ breq2d... |
| absmulrposd 41658 | Specialization of absmuld ... |
| imadisjld 41659 | Natural dduction form of o... |
| imadisjlnd 41660 | Natural deduction form of ... |
| wnefimgd 41661 | The image of a mapping fro... |
| fco2d 41662 | Natural deduction form of ... |
| wfximgfd 41663 | The value of a function on... |
| extoimad 41664 | If |f(x)| <= C for all x t... |
| imo72b2lem0 41665 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 41666 | Natural deduction form of ... |
| imo72b2lem2 41667 | Lemma for ~ imo72b2 . (Co... |
| suprlubrd 41668 | Natural deduction form of ... |
| imo72b2lem1 41669 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 41670 | 'Less than or equal to' re... |
| lemuldiv4d 41671 | 'Less than or equal to' re... |
| imo72b2 41672 | IMO 1972 B2. (14th Intern... |
| int-addcomd 41673 | AdditionCommutativity gene... |
| int-addassocd 41674 | AdditionAssociativity gene... |
| int-addsimpd 41675 | AdditionSimplification gen... |
| int-mulcomd 41676 | MultiplicationCommutativit... |
| int-mulassocd 41677 | MultiplicationAssociativit... |
| int-mulsimpd 41678 | MultiplicationSimplificati... |
| int-leftdistd 41679 | AdditionMultiplicationLeft... |
| int-rightdistd 41680 | AdditionMultiplicationRigh... |
| int-sqdefd 41681 | SquareDefinition generator... |
| int-mul11d 41682 | First MultiplicationOne ge... |
| int-mul12d 41683 | Second MultiplicationOne g... |
| int-add01d 41684 | First AdditionZero generat... |
| int-add02d 41685 | Second AdditionZero genera... |
| int-sqgeq0d 41686 | SquareGEQZero generator ru... |
| int-eqprincd 41687 | PrincipleOfEquality genera... |
| int-eqtransd 41688 | EqualityTransitivity gener... |
| int-eqmvtd 41689 | EquMoveTerm generator rule... |
| int-eqineqd 41690 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 41691 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 41692 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 41693 | SecondPrincipleOfInequalit... |
| int-ineqtransd 41694 | InequalityTransitivity gen... |
| unitadd 41695 | Theorem used in conjunctio... |
| gsumws3 41696 | Valuation of a length 3 wo... |
| gsumws4 41697 | Valuation of a length 4 wo... |
| amgm2d 41698 | Arithmetic-geometric mean ... |
| amgm3d 41699 | Arithmetic-geometric mean ... |
| amgm4d 41700 | Arithmetic-geometric mean ... |
| spALT 41701 | ~ sp can be proven from th... |
| elnelneqd 41702 | Two classes are not equal ... |
| elnelneq2d 41703 | Two classes are not equal ... |
| rr-spce 41704 | Prove an existential. (Co... |
| rexlimdvaacbv 41705 | Unpack a restricted existe... |
| rexlimddvcbvw 41706 | Unpack a restricted existe... |
| rexlimddvcbv 41707 | Unpack a restricted existe... |
| rr-elrnmpt3d 41708 | Elementhood in an image se... |
| finnzfsuppd 41709 | If a function is zero outs... |
| rr-phpd 41710 | Equivalent of ~ php withou... |
| suceqd 41711 | Deduction associated with ... |
| tfindsd 41712 | Deduction associated with ... |
| mnringvald 41715 | Value of the monoid ring f... |
| mnringnmulrd 41716 | Components of a monoid rin... |
| mnringnmulrdOLD 41717 | Obsolete version of ~ mnri... |
| mnringbased 41718 | The base set of a monoid r... |
| mnringbasedOLD 41719 | Obsolete version of ~ mnri... |
| mnringbaserd 41720 | The base set of a monoid r... |
| mnringelbased 41721 | Membership in the base set... |
| mnringbasefd 41722 | Elements of a monoid ring ... |
| mnringbasefsuppd 41723 | Elements of a monoid ring ... |
| mnringaddgd 41724 | The additive operation of ... |
| mnringaddgdOLD 41725 | Obsolete version of ~ mnri... |
| mnring0gd 41726 | The additive identity of a... |
| mnring0g2d 41727 | The additive identity of a... |
| mnringmulrd 41728 | The ring product of a mono... |
| mnringscad 41729 | The scalar ring of a monoi... |
| mnringscadOLD 41730 | Obsolete version of ~ mnri... |
| mnringvscad 41731 | The scalar product of a mo... |
| mnringvscadOLD 41732 | Obsolete version of ~ mnri... |
| mnringlmodd 41733 | Monoid rings are left modu... |
| mnringmulrvald 41734 | Value of multiplication in... |
| mnringmulrcld 41735 | Monoid rings are closed un... |
| gru0eld 41736 | A nonempty Grothendieck un... |
| grusucd 41737 | Grothendieck universes are... |
| r1rankcld 41738 | Any rank of the cumulative... |
| grur1cld 41739 | Grothendieck universes are... |
| grurankcld 41740 | Grothendieck universes are... |
| grurankrcld 41741 | If a Grothendieck universe... |
| scotteqd 41744 | Equality theorem for the S... |
| scotteq 41745 | Closed form of ~ scotteqd ... |
| nfscott 41746 | Bound-variable hypothesis ... |
| scottabf 41747 | Value of the Scott operati... |
| scottab 41748 | Value of the Scott operati... |
| scottabes 41749 | Value of the Scott operati... |
| scottss 41750 | Scott's trick produces a s... |
| elscottab 41751 | An element of the output o... |
| scottex2 41752 | ~ scottex expressed using ... |
| scotteld 41753 | The Scott operation sends ... |
| scottelrankd 41754 | Property of a Scott's tric... |
| scottrankd 41755 | Rank of a nonempty Scott's... |
| gruscottcld 41756 | If a Grothendieck universe... |
| dfcoll2 41759 | Alternate definition of th... |
| colleq12d 41760 | Equality theorem for the c... |
| colleq1 41761 | Equality theorem for the c... |
| colleq2 41762 | Equality theorem for the c... |
| nfcoll 41763 | Bound-variable hypothesis ... |
| collexd 41764 | The output of the collecti... |
| cpcolld 41765 | Property of the collection... |
| cpcoll2d 41766 | ~ cpcolld with an extra ex... |
| grucollcld 41767 | A Grothendieck universe co... |
| ismnu 41768 | The hypothesis of this the... |
| mnuop123d 41769 | Operations of a minimal un... |
| mnussd 41770 | Minimal universes are clos... |
| mnuss2d 41771 | ~ mnussd with arguments pr... |
| mnu0eld 41772 | A nonempty minimal univers... |
| mnuop23d 41773 | Second and third operation... |
| mnupwd 41774 | Minimal universes are clos... |
| mnusnd 41775 | Minimal universes are clos... |
| mnuprssd 41776 | A minimal universe contain... |
| mnuprss2d 41777 | Special case of ~ mnuprssd... |
| mnuop3d 41778 | Third operation of a minim... |
| mnuprdlem1 41779 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 41780 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 41781 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 41782 | Lemma for ~ mnuprd . Gene... |
| mnuprd 41783 | Minimal universes are clos... |
| mnuunid 41784 | Minimal universes are clos... |
| mnuund 41785 | Minimal universes are clos... |
| mnutrcld 41786 | Minimal universes contain ... |
| mnutrd 41787 | Minimal universes are tran... |
| mnurndlem1 41788 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 41789 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 41790 | Minimal universes contain ... |
| mnugrud 41791 | Minimal universes are Grot... |
| grumnudlem 41792 | Lemma for ~ grumnud . (Co... |
| grumnud 41793 | Grothendieck universes are... |
| grumnueq 41794 | The class of Grothendieck ... |
| expandan 41795 | Expand conjunction to prim... |
| expandexn 41796 | Expand an existential quan... |
| expandral 41797 | Expand a restricted univer... |
| expandrexn 41798 | Expand a restricted existe... |
| expandrex 41799 | Expand a restricted existe... |
| expanduniss 41800 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 41801 | Express the predicate on `... |
| rr-grothprimbi 41802 | Express "every set is cont... |
| inagrud 41803 | Inaccessible levels of the... |
| inaex 41804 | Assuming the Tarski-Grothe... |
| gruex 41805 | Assuming the Tarski-Grothe... |
| rr-groth 41806 | An equivalent of ~ ax-grot... |
| rr-grothprim 41807 | An equivalent of ~ ax-grot... |
| ismnushort 41808 | Express the predicate on `... |
| dfuniv2 41809 | Alternative definition of ... |
| rr-grothshortbi 41810 | Express "every set is cont... |
| rr-grothshort 41811 | A shorter equivalent of ~ ... |
| nanorxor 41812 | 'nand' is equivalent to th... |
| undisjrab 41813 | Union of two disjoint rest... |
| iso0 41814 | The empty set is an ` R , ... |
| ssrecnpr 41815 | ` RR ` is a subset of both... |
| seff 41816 | Let set ` S ` be the real ... |
| sblpnf 41817 | The infinity ball in the a... |
| prmunb2 41818 | The primes are unbounded. ... |
| dvgrat 41819 | Ratio test for divergence ... |
| cvgdvgrat 41820 | Ratio test for convergence... |
| radcnvrat 41821 | Let ` L ` be the limit, if... |
| reldvds 41822 | The divides relation is in... |
| nznngen 41823 | All positive integers in t... |
| nzss 41824 | The set of multiples of _m... |
| nzin 41825 | The intersection of the se... |
| nzprmdif 41826 | Subtract one prime's multi... |
| hashnzfz 41827 | Special case of ~ hashdvds... |
| hashnzfz2 41828 | Special case of ~ hashnzfz... |
| hashnzfzclim 41829 | As the upper bound ` K ` o... |
| caofcan 41830 | Transfer a cancellation la... |
| ofsubid 41831 | Function analogue of ~ sub... |
| ofmul12 41832 | Function analogue of ~ mul... |
| ofdivrec 41833 | Function analogue of ~ div... |
| ofdivcan4 41834 | Function analogue of ~ div... |
| ofdivdiv2 41835 | Function analogue of ~ div... |
| lhe4.4ex1a 41836 | Example of the Fundamental... |
| dvsconst 41837 | Derivative of a constant f... |
| dvsid 41838 | Derivative of the identity... |
| dvsef 41839 | Derivative of the exponent... |
| expgrowthi 41840 | Exponential growth and dec... |
| dvconstbi 41841 | The derivative of a functi... |
| expgrowth 41842 | Exponential growth and dec... |
| bccval 41845 | Value of the generalized b... |
| bcccl 41846 | Closure of the generalized... |
| bcc0 41847 | The generalized binomial c... |
| bccp1k 41848 | Generalized binomial coeff... |
| bccm1k 41849 | Generalized binomial coeff... |
| bccn0 41850 | Generalized binomial coeff... |
| bccn1 41851 | Generalized binomial coeff... |
| bccbc 41852 | The binomial coefficient a... |
| uzmptshftfval 41853 | When ` F ` is a maps-to fu... |
| dvradcnv2 41854 | The radius of convergence ... |
| binomcxplemwb 41855 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 41856 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 41857 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 41858 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 41859 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 41860 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 41861 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 41862 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 41863 | Lemma for ~ binomcxp . Wh... |
| binomcxp 41864 | Generalize the binomial th... |
| pm10.12 41865 | Theorem *10.12 in [Whitehe... |
| pm10.14 41866 | Theorem *10.14 in [Whitehe... |
| pm10.251 41867 | Theorem *10.251 in [Whiteh... |
| pm10.252 41868 | Theorem *10.252 in [Whiteh... |
| pm10.253 41869 | Theorem *10.253 in [Whiteh... |
| albitr 41870 | Theorem *10.301 in [Whiteh... |
| pm10.42 41871 | Theorem *10.42 in [Whitehe... |
| pm10.52 41872 | Theorem *10.52 in [Whitehe... |
| pm10.53 41873 | Theorem *10.53 in [Whitehe... |
| pm10.541 41874 | Theorem *10.541 in [Whiteh... |
| pm10.542 41875 | Theorem *10.542 in [Whiteh... |
| pm10.55 41876 | Theorem *10.55 in [Whitehe... |
| pm10.56 41877 | Theorem *10.56 in [Whitehe... |
| pm10.57 41878 | Theorem *10.57 in [Whitehe... |
| 2alanimi 41879 | Removes two universal quan... |
| 2al2imi 41880 | Removes two universal quan... |
| pm11.11 41881 | Theorem *11.11 in [Whitehe... |
| pm11.12 41882 | Theorem *11.12 in [Whitehe... |
| 19.21vv 41883 | Compare Theorem *11.3 in [... |
| 2alim 41884 | Theorem *11.32 in [Whitehe... |
| 2albi 41885 | Theorem *11.33 in [Whitehe... |
| 2exim 41886 | Theorem *11.34 in [Whitehe... |
| 2exbi 41887 | Theorem *11.341 in [Whiteh... |
| spsbce-2 41888 | Theorem *11.36 in [Whitehe... |
| 19.33-2 41889 | Theorem *11.421 in [Whiteh... |
| 19.36vv 41890 | Theorem *11.43 in [Whitehe... |
| 19.31vv 41891 | Theorem *11.44 in [Whitehe... |
| 19.37vv 41892 | Theorem *11.46 in [Whitehe... |
| 19.28vv 41893 | Theorem *11.47 in [Whitehe... |
| pm11.52 41894 | Theorem *11.52 in [Whitehe... |
| aaanv 41895 | Theorem *11.56 in [Whitehe... |
| pm11.57 41896 | Theorem *11.57 in [Whitehe... |
| pm11.58 41897 | Theorem *11.58 in [Whitehe... |
| pm11.59 41898 | Theorem *11.59 in [Whitehe... |
| pm11.6 41899 | Theorem *11.6 in [Whitehea... |
| pm11.61 41900 | Theorem *11.61 in [Whitehe... |
| pm11.62 41901 | Theorem *11.62 in [Whitehe... |
| pm11.63 41902 | Theorem *11.63 in [Whitehe... |
| pm11.7 41903 | Theorem *11.7 in [Whitehea... |
| pm11.71 41904 | Theorem *11.71 in [Whitehe... |
| sbeqal1 41905 | If ` x = y ` always implie... |
| sbeqal1i 41906 | Suppose you know ` x = y `... |
| sbeqal2i 41907 | If ` x = y ` implies ` x =... |
| axc5c4c711 41908 | Proof of a theorem that ca... |
| axc5c4c711toc5 41909 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 41910 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 41911 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 41912 | Rederivation of ~ ax-11 fr... |
| axc11next 41913 | This theorem shows that, g... |
| pm13.13a 41914 | One result of theorem *13.... |
| pm13.13b 41915 | Theorem *13.13 in [Whitehe... |
| pm13.14 41916 | Theorem *13.14 in [Whitehe... |
| pm13.192 41917 | Theorem *13.192 in [Whiteh... |
| pm13.193 41918 | Theorem *13.193 in [Whiteh... |
| pm13.194 41919 | Theorem *13.194 in [Whiteh... |
| pm13.195 41920 | Theorem *13.195 in [Whiteh... |
| pm13.196a 41921 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 41922 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 41923 | Theorem *13.22 in [Whitehe... |
| iotain 41924 | Equivalence between two di... |
| iotaexeu 41925 | The iota class exists. Th... |
| iotasbc 41926 | Definition *14.01 in [Whit... |
| iotasbc2 41927 | Theorem *14.111 in [Whiteh... |
| pm14.12 41928 | Theorem *14.12 in [Whitehe... |
| pm14.122a 41929 | Theorem *14.122 in [Whiteh... |
| pm14.122b 41930 | Theorem *14.122 in [Whiteh... |
| pm14.122c 41931 | Theorem *14.122 in [Whiteh... |
| pm14.123a 41932 | Theorem *14.123 in [Whiteh... |
| pm14.123b 41933 | Theorem *14.123 in [Whiteh... |
| pm14.123c 41934 | Theorem *14.123 in [Whiteh... |
| pm14.18 41935 | Theorem *14.18 in [Whitehe... |
| iotaequ 41936 | Theorem *14.2 in [Whitehea... |
| iotavalb 41937 | Theorem *14.202 in [Whiteh... |
| iotasbc5 41938 | Theorem *14.205 in [Whiteh... |
| pm14.24 41939 | Theorem *14.24 in [Whitehe... |
| iotavalsb 41940 | Theorem *14.242 in [Whiteh... |
| sbiota1 41941 | Theorem *14.25 in [Whitehe... |
| sbaniota 41942 | Theorem *14.26 in [Whitehe... |
| eubiOLD 41943 | Obsolete proof of ~ eubi a... |
| iotasbcq 41944 | Theorem *14.272 in [Whiteh... |
| elnev 41945 | Any set that contains one ... |
| rusbcALT 41946 | A version of Russell's par... |
| compeq 41947 | Equality between two ways ... |
| compne 41948 | The complement of ` A ` is... |
| compab 41949 | Two ways of saying "the co... |
| conss2 41950 | Contrapositive law for sub... |
| conss1 41951 | Contrapositive law for sub... |
| ralbidar 41952 | More general form of ~ ral... |
| rexbidar 41953 | More general form of ~ rex... |
| dropab1 41954 | Theorem to aid use of the ... |
| dropab2 41955 | Theorem to aid use of the ... |
| ipo0 41956 | If the identity relation p... |
| ifr0 41957 | A class that is founded by... |
| ordpss 41958 | ~ ordelpss with an anteced... |
| fvsb 41959 | Explicit substitution of a... |
| fveqsb 41960 | Implicit substitution of a... |
| xpexb 41961 | A Cartesian product exists... |
| trelpss 41962 | An element of a transitive... |
| addcomgi 41963 | Generalization of commutat... |
| addrval 41973 | Value of the operation of ... |
| subrval 41974 | Value of the operation of ... |
| mulvval 41975 | Value of the operation of ... |
| addrfv 41976 | Vector addition at a value... |
| subrfv 41977 | Vector subtraction at a va... |
| mulvfv 41978 | Scalar multiplication at a... |
| addrfn 41979 | Vector addition produces a... |
| subrfn 41980 | Vector subtraction produce... |
| mulvfn 41981 | Scalar multiplication prod... |
| addrcom 41982 | Vector addition is commuta... |
| idiALT 41986 | Placeholder for ~ idi . T... |
| exbir 41987 | Exportation implication al... |
| 3impexpbicom 41988 | Version of ~ 3impexp where... |
| 3impexpbicomi 41989 | Inference associated with ... |
| bi1imp 41990 | Importation inference simi... |
| bi2imp 41991 | Importation inference simi... |
| bi3impb 41992 | Similar to ~ 3impb with im... |
| bi3impa 41993 | Similar to ~ 3impa with im... |
| bi23impib 41994 | ~ 3impib with the inner im... |
| bi13impib 41995 | ~ 3impib with the outer im... |
| bi123impib 41996 | ~ 3impib with the implicat... |
| bi13impia 41997 | ~ 3impia with the outer im... |
| bi123impia 41998 | ~ 3impia with the implicat... |
| bi33imp12 41999 | ~ 3imp with innermost impl... |
| bi23imp13 42000 | ~ 3imp with middle implica... |
| bi13imp23 42001 | ~ 3imp with outermost impl... |
| bi13imp2 42002 | Similar to ~ 3imp except t... |
| bi12imp3 42003 | Similar to ~ 3imp except a... |
| bi23imp1 42004 | Similar to ~ 3imp except a... |
| bi123imp0 42005 | Similar to ~ 3imp except a... |
| 4animp1 42006 | A single hypothesis unific... |
| 4an31 42007 | A rearrangement of conjunc... |
| 4an4132 42008 | A rearrangement of conjunc... |
| expcomdg 42009 | Biconditional form of ~ ex... |
| iidn3 42010 | ~ idn3 without virtual ded... |
| ee222 42011 | ~ e222 without virtual ded... |
| ee3bir 42012 | Right-biconditional form o... |
| ee13 42013 | ~ e13 without virtual dedu... |
| ee121 42014 | ~ e121 without virtual ded... |
| ee122 42015 | ~ e122 without virtual ded... |
| ee333 42016 | ~ e333 without virtual ded... |
| ee323 42017 | ~ e323 without virtual ded... |
| 3ornot23 42018 | If the second and third di... |
| orbi1r 42019 | ~ orbi1 with order of disj... |
| 3orbi123 42020 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 42021 | Closed form of ~ syl5 . D... |
| impexpd 42022 | The following User's Proof... |
| com3rgbi 42023 | The following User's Proof... |
| impexpdcom 42024 | The following User's Proof... |
| ee1111 42025 | Non-virtual deduction form... |
| pm2.43bgbi 42026 | Logical equivalence of a 2... |
| pm2.43cbi 42027 | Logical equivalence of a 3... |
| ee233 42028 | Non-virtual deduction form... |
| imbi13 42029 | Join three logical equival... |
| ee33 42030 | Non-virtual deduction form... |
| con5 42031 | Biconditional contrapositi... |
| con5i 42032 | Inference form of ~ con5 .... |
| exlimexi 42033 | Inference similar to Theor... |
| sb5ALT 42034 | Equivalence for substituti... |
| eexinst01 42035 | ~ exinst01 without virtual... |
| eexinst11 42036 | ~ exinst11 without virtual... |
| vk15.4j 42037 | Excercise 4j of Unit 15 of... |
| notnotrALT 42038 | Converse of double negatio... |
| con3ALT2 42039 | Contraposition. Alternate... |
| ssralv2 42040 | Quantification restricted ... |
| sbc3or 42041 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 42042 | Closed form of ~ alrimi wi... |
| rspsbc2 42043 | ~ rspsbc with two quantify... |
| sbcoreleleq 42044 | Substitution of a setvar v... |
| tratrb 42045 | If a class is transitive a... |
| ordelordALT 42046 | An element of an ordinal c... |
| sbcim2g 42047 | Distribution of class subs... |
| sbcbi 42048 | Implication form of ~ sbcb... |
| trsbc 42049 | Formula-building inference... |
| truniALT 42050 | The union of a class of tr... |
| onfrALTlem5 42051 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 42052 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 42053 | Lemma for ~ onfrALT . (Co... |
| ggen31 42054 | ~ gen31 without virtual de... |
| onfrALTlem2 42055 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 42056 | A theorem pertaining to th... |
| onfrALTlem1 42057 | Lemma for ~ onfrALT . (Co... |
| onfrALT 42058 | The membership relation is... |
| 19.41rg 42059 | Closed form of right-to-le... |
| opelopab4 42060 | Ordered pair membership in... |
| 2pm13.193 42061 | ~ pm13.193 for two variabl... |
| hbntal 42062 | A closed form of ~ hbn . ~... |
| hbimpg 42063 | A closed form of ~ hbim . ... |
| hbalg 42064 | Closed form of ~ hbal . D... |
| hbexg 42065 | Closed form of ~ nfex . D... |
| ax6e2eq 42066 | Alternate form of ~ ax6e f... |
| ax6e2nd 42067 | If at least two sets exist... |
| ax6e2ndeq 42068 | "At least two sets exist" ... |
| 2sb5nd 42069 | Equivalence for double sub... |
| 2uasbanh 42070 | Distribute the unabbreviat... |
| 2uasban 42071 | Distribute the unabbreviat... |
| e2ebind 42072 | Absorption of an existenti... |
| elpwgded 42073 | ~ elpwgdedVD in convention... |
| trelded 42074 | Deduction form of ~ trel .... |
| jaoded 42075 | Deduction form of ~ jao . ... |
| sbtT 42076 | A substitution into a theo... |
| not12an2impnot1 42077 | If a double conjunction is... |
| in1 42080 | Inference form of ~ df-vd1... |
| iin1 42081 | ~ in1 without virtual dedu... |
| dfvd1ir 42082 | Inference form of ~ df-vd1... |
| idn1 42083 | Virtual deduction identity... |
| dfvd1imp 42084 | Left-to-right part of defi... |
| dfvd1impr 42085 | Right-to-left part of defi... |
| dfvd2 42088 | Definition of a 2-hypothes... |
| dfvd2an 42091 | Definition of a 2-hypothes... |
| dfvd2ani 42092 | Inference form of ~ dfvd2a... |
| dfvd2anir 42093 | Right-to-left inference fo... |
| dfvd2i 42094 | Inference form of ~ dfvd2 ... |
| dfvd2ir 42095 | Right-to-left inference fo... |
| dfvd3 42100 | Definition of a 3-hypothes... |
| dfvd3i 42101 | Inference form of ~ dfvd3 ... |
| dfvd3ir 42102 | Right-to-left inference fo... |
| dfvd3an 42103 | Definition of a 3-hypothes... |
| dfvd3ani 42104 | Inference form of ~ dfvd3a... |
| dfvd3anir 42105 | Right-to-left inference fo... |
| vd01 42106 | A virtual hypothesis virtu... |
| vd02 42107 | Two virtual hypotheses vir... |
| vd03 42108 | A theorem is virtually inf... |
| vd12 42109 | A virtual deduction with 1... |
| vd13 42110 | A virtual deduction with 1... |
| vd23 42111 | A virtual deduction with 2... |
| dfvd2imp 42112 | The virtual deduction form... |
| dfvd2impr 42113 | A 2-antecedent nested impl... |
| in2 42114 | The virtual deduction intr... |
| int2 42115 | The virtual deduction intr... |
| iin2 42116 | ~ in2 without virtual dedu... |
| in2an 42117 | The virtual deduction intr... |
| in3 42118 | The virtual deduction intr... |
| iin3 42119 | ~ in3 without virtual dedu... |
| in3an 42120 | The virtual deduction intr... |
| int3 42121 | The virtual deduction intr... |
| idn2 42122 | Virtual deduction identity... |
| iden2 42123 | Virtual deduction identity... |
| idn3 42124 | Virtual deduction identity... |
| gen11 42125 | Virtual deduction generali... |
| gen11nv 42126 | Virtual deduction generali... |
| gen12 42127 | Virtual deduction generali... |
| gen21 42128 | Virtual deduction generali... |
| gen21nv 42129 | Virtual deduction form of ... |
| gen31 42130 | Virtual deduction generali... |
| gen22 42131 | Virtual deduction generali... |
| ggen22 42132 | ~ gen22 without virtual de... |
| exinst 42133 | Existential Instantiation.... |
| exinst01 42134 | Existential Instantiation.... |
| exinst11 42135 | Existential Instantiation.... |
| e1a 42136 | A Virtual deduction elimin... |
| el1 42137 | A Virtual deduction elimin... |
| e1bi 42138 | Biconditional form of ~ e1... |
| e1bir 42139 | Right biconditional form o... |
| e2 42140 | A virtual deduction elimin... |
| e2bi 42141 | Biconditional form of ~ e2... |
| e2bir 42142 | Right biconditional form o... |
| ee223 42143 | ~ e223 without virtual ded... |
| e223 42144 | A virtual deduction elimin... |
| e222 42145 | A virtual deduction elimin... |
| e220 42146 | A virtual deduction elimin... |
| ee220 42147 | ~ e220 without virtual ded... |
| e202 42148 | A virtual deduction elimin... |
| ee202 42149 | ~ e202 without virtual ded... |
| e022 42150 | A virtual deduction elimin... |
| ee022 42151 | ~ e022 without virtual ded... |
| e002 42152 | A virtual deduction elimin... |
| ee002 42153 | ~ e002 without virtual ded... |
| e020 42154 | A virtual deduction elimin... |
| ee020 42155 | ~ e020 without virtual ded... |
| e200 42156 | A virtual deduction elimin... |
| ee200 42157 | ~ e200 without virtual ded... |
| e221 42158 | A virtual deduction elimin... |
| ee221 42159 | ~ e221 without virtual ded... |
| e212 42160 | A virtual deduction elimin... |
| ee212 42161 | ~ e212 without virtual ded... |
| e122 42162 | A virtual deduction elimin... |
| e112 42163 | A virtual deduction elimin... |
| ee112 42164 | ~ e112 without virtual ded... |
| e121 42165 | A virtual deduction elimin... |
| e211 42166 | A virtual deduction elimin... |
| ee211 42167 | ~ e211 without virtual ded... |
| e210 42168 | A virtual deduction elimin... |
| ee210 42169 | ~ e210 without virtual ded... |
| e201 42170 | A virtual deduction elimin... |
| ee201 42171 | ~ e201 without virtual ded... |
| e120 42172 | A virtual deduction elimin... |
| ee120 42173 | Virtual deduction rule ~ e... |
| e021 42174 | A virtual deduction elimin... |
| ee021 42175 | ~ e021 without virtual ded... |
| e012 42176 | A virtual deduction elimin... |
| ee012 42177 | ~ e012 without virtual ded... |
| e102 42178 | A virtual deduction elimin... |
| ee102 42179 | ~ e102 without virtual ded... |
| e22 42180 | A virtual deduction elimin... |
| e22an 42181 | Conjunction form of ~ e22 ... |
| ee22an 42182 | ~ e22an without virtual de... |
| e111 42183 | A virtual deduction elimin... |
| e1111 42184 | A virtual deduction elimin... |
| e110 42185 | A virtual deduction elimin... |
| ee110 42186 | ~ e110 without virtual ded... |
| e101 42187 | A virtual deduction elimin... |
| ee101 42188 | ~ e101 without virtual ded... |
| e011 42189 | A virtual deduction elimin... |
| ee011 42190 | ~ e011 without virtual ded... |
| e100 42191 | A virtual deduction elimin... |
| ee100 42192 | ~ e100 without virtual ded... |
| e010 42193 | A virtual deduction elimin... |
| ee010 42194 | ~ e010 without virtual ded... |
| e001 42195 | A virtual deduction elimin... |
| ee001 42196 | ~ e001 without virtual ded... |
| e11 42197 | A virtual deduction elimin... |
| e11an 42198 | Conjunction form of ~ e11 ... |
| ee11an 42199 | ~ e11an without virtual de... |
| e01 42200 | A virtual deduction elimin... |
| e01an 42201 | Conjunction form of ~ e01 ... |
| ee01an 42202 | ~ e01an without virtual de... |
| e10 42203 | A virtual deduction elimin... |
| e10an 42204 | Conjunction form of ~ e10 ... |
| ee10an 42205 | ~ e10an without virtual de... |
| e02 42206 | A virtual deduction elimin... |
| e02an 42207 | Conjunction form of ~ e02 ... |
| ee02an 42208 | ~ e02an without virtual de... |
| eel021old 42209 | ~ el021old without virtual... |
| el021old 42210 | A virtual deduction elimin... |
| eel132 42211 | ~ syl2an with antecedents ... |
| eel000cT 42212 | An elimination deduction. ... |
| eel0TT 42213 | An elimination deduction. ... |
| eelT00 42214 | An elimination deduction. ... |
| eelTTT 42215 | An elimination deduction. ... |
| eelT11 42216 | An elimination deduction. ... |
| eelT1 42217 | Syllogism inference combin... |
| eelT12 42218 | An elimination deduction. ... |
| eelTT1 42219 | An elimination deduction. ... |
| eelT01 42220 | An elimination deduction. ... |
| eel0T1 42221 | An elimination deduction. ... |
| eel12131 42222 | An elimination deduction. ... |
| eel2131 42223 | ~ syl2an with antecedents ... |
| eel3132 42224 | ~ syl2an with antecedents ... |
| eel0321old 42225 | ~ el0321old without virtua... |
| el0321old 42226 | A virtual deduction elimin... |
| eel2122old 42227 | ~ el2122old without virtua... |
| el2122old 42228 | A virtual deduction elimin... |
| eel0000 42229 | Elimination rule similar t... |
| eel00001 42230 | An elimination deduction. ... |
| eel00000 42231 | Elimination rule similar ~... |
| eel11111 42232 | Five-hypothesis eliminatio... |
| e12 42233 | A virtual deduction elimin... |
| e12an 42234 | Conjunction form of ~ e12 ... |
| el12 42235 | Virtual deduction form of ... |
| e20 42236 | A virtual deduction elimin... |
| e20an 42237 | Conjunction form of ~ e20 ... |
| ee20an 42238 | ~ e20an without virtual de... |
| e21 42239 | A virtual deduction elimin... |
| e21an 42240 | Conjunction form of ~ e21 ... |
| ee21an 42241 | ~ e21an without virtual de... |
| e333 42242 | A virtual deduction elimin... |
| e33 42243 | A virtual deduction elimin... |
| e33an 42244 | Conjunction form of ~ e33 ... |
| ee33an 42245 | ~ e33an without virtual de... |
| e3 42246 | Meta-connective form of ~ ... |
| e3bi 42247 | Biconditional form of ~ e3... |
| e3bir 42248 | Right biconditional form o... |
| e03 42249 | A virtual deduction elimin... |
| ee03 42250 | ~ e03 without virtual dedu... |
| e03an 42251 | Conjunction form of ~ e03 ... |
| ee03an 42252 | Conjunction form of ~ ee03... |
| e30 42253 | A virtual deduction elimin... |
| ee30 42254 | ~ e30 without virtual dedu... |
| e30an 42255 | A virtual deduction elimin... |
| ee30an 42256 | Conjunction form of ~ ee30... |
| e13 42257 | A virtual deduction elimin... |
| e13an 42258 | A virtual deduction elimin... |
| ee13an 42259 | ~ e13an without virtual de... |
| e31 42260 | A virtual deduction elimin... |
| ee31 42261 | ~ e31 without virtual dedu... |
| e31an 42262 | A virtual deduction elimin... |
| ee31an 42263 | ~ e31an without virtual de... |
| e23 42264 | A virtual deduction elimin... |
| e23an 42265 | A virtual deduction elimin... |
| ee23an 42266 | ~ e23an without virtual de... |
| e32 42267 | A virtual deduction elimin... |
| ee32 42268 | ~ e32 without virtual dedu... |
| e32an 42269 | A virtual deduction elimin... |
| ee32an 42270 | ~ e33an without virtual de... |
| e123 42271 | A virtual deduction elimin... |
| ee123 42272 | ~ e123 without virtual ded... |
| el123 42273 | A virtual deduction elimin... |
| e233 42274 | A virtual deduction elimin... |
| e323 42275 | A virtual deduction elimin... |
| e000 42276 | A virtual deduction elimin... |
| e00 42277 | Elimination rule identical... |
| e00an 42278 | Elimination rule identical... |
| eel00cT 42279 | An elimination deduction. ... |
| eelTT 42280 | An elimination deduction. ... |
| e0a 42281 | Elimination rule identical... |
| eelT 42282 | An elimination deduction. ... |
| eel0cT 42283 | An elimination deduction. ... |
| eelT0 42284 | An elimination deduction. ... |
| e0bi 42285 | Elimination rule identical... |
| e0bir 42286 | Elimination rule identical... |
| uun0.1 42287 | Convention notation form o... |
| un0.1 42288 | ` T. ` is the constant tru... |
| uunT1 42289 | A deduction unionizing a n... |
| uunT1p1 42290 | A deduction unionizing a n... |
| uunT21 42291 | A deduction unionizing a n... |
| uun121 42292 | A deduction unionizing a n... |
| uun121p1 42293 | A deduction unionizing a n... |
| uun132 42294 | A deduction unionizing a n... |
| uun132p1 42295 | A deduction unionizing a n... |
| anabss7p1 42296 | A deduction unionizing a n... |
| un10 42297 | A unionizing deduction. (... |
| un01 42298 | A unionizing deduction. (... |
| un2122 42299 | A deduction unionizing a n... |
| uun2131 42300 | A deduction unionizing a n... |
| uun2131p1 42301 | A deduction unionizing a n... |
| uunTT1 42302 | A deduction unionizing a n... |
| uunTT1p1 42303 | A deduction unionizing a n... |
| uunTT1p2 42304 | A deduction unionizing a n... |
| uunT11 42305 | A deduction unionizing a n... |
| uunT11p1 42306 | A deduction unionizing a n... |
| uunT11p2 42307 | A deduction unionizing a n... |
| uunT12 42308 | A deduction unionizing a n... |
| uunT12p1 42309 | A deduction unionizing a n... |
| uunT12p2 42310 | A deduction unionizing a n... |
| uunT12p3 42311 | A deduction unionizing a n... |
| uunT12p4 42312 | A deduction unionizing a n... |
| uunT12p5 42313 | A deduction unionizing a n... |
| uun111 42314 | A deduction unionizing a n... |
| 3anidm12p1 42315 | A deduction unionizing a n... |
| 3anidm12p2 42316 | A deduction unionizing a n... |
| uun123 42317 | A deduction unionizing a n... |
| uun123p1 42318 | A deduction unionizing a n... |
| uun123p2 42319 | A deduction unionizing a n... |
| uun123p3 42320 | A deduction unionizing a n... |
| uun123p4 42321 | A deduction unionizing a n... |
| uun2221 42322 | A deduction unionizing a n... |
| uun2221p1 42323 | A deduction unionizing a n... |
| uun2221p2 42324 | A deduction unionizing a n... |
| 3impdirp1 42325 | A deduction unionizing a n... |
| 3impcombi 42326 | A 1-hypothesis proposition... |
| trsspwALT 42327 | Virtual deduction proof of... |
| trsspwALT2 42328 | Virtual deduction proof of... |
| trsspwALT3 42329 | Short predicate calculus p... |
| sspwtr 42330 | Virtual deduction proof of... |
| sspwtrALT 42331 | Virtual deduction proof of... |
| sspwtrALT2 42332 | Short predicate calculus p... |
| pwtrVD 42333 | Virtual deduction proof of... |
| pwtrrVD 42334 | Virtual deduction proof of... |
| suctrALT 42335 | The successor of a transit... |
| snssiALTVD 42336 | Virtual deduction proof of... |
| snssiALT 42337 | If a class is an element o... |
| snsslVD 42338 | Virtual deduction proof of... |
| snssl 42339 | If a singleton is a subcla... |
| snelpwrVD 42340 | Virtual deduction proof of... |
| unipwrVD 42341 | Virtual deduction proof of... |
| unipwr 42342 | A class is a subclass of t... |
| sstrALT2VD 42343 | Virtual deduction proof of... |
| sstrALT2 42344 | Virtual deduction proof of... |
| suctrALT2VD 42345 | Virtual deduction proof of... |
| suctrALT2 42346 | Virtual deduction proof of... |
| elex2VD 42347 | Virtual deduction proof of... |
| elex22VD 42348 | Virtual deduction proof of... |
| eqsbc2VD 42349 | Virtual deduction proof of... |
| zfregs2VD 42350 | Virtual deduction proof of... |
| tpid3gVD 42351 | Virtual deduction proof of... |
| en3lplem1VD 42352 | Virtual deduction proof of... |
| en3lplem2VD 42353 | Virtual deduction proof of... |
| en3lpVD 42354 | Virtual deduction proof of... |
| simplbi2VD 42355 | Virtual deduction proof of... |
| 3ornot23VD 42356 | Virtual deduction proof of... |
| orbi1rVD 42357 | Virtual deduction proof of... |
| bitr3VD 42358 | Virtual deduction proof of... |
| 3orbi123VD 42359 | Virtual deduction proof of... |
| sbc3orgVD 42360 | Virtual deduction proof of... |
| 19.21a3con13vVD 42361 | Virtual deduction proof of... |
| exbirVD 42362 | Virtual deduction proof of... |
| exbiriVD 42363 | Virtual deduction proof of... |
| rspsbc2VD 42364 | Virtual deduction proof of... |
| 3impexpVD 42365 | Virtual deduction proof of... |
| 3impexpbicomVD 42366 | Virtual deduction proof of... |
| 3impexpbicomiVD 42367 | Virtual deduction proof of... |
| sbcoreleleqVD 42368 | Virtual deduction proof of... |
| hbra2VD 42369 | Virtual deduction proof of... |
| tratrbVD 42370 | Virtual deduction proof of... |
| al2imVD 42371 | Virtual deduction proof of... |
| syl5impVD 42372 | Virtual deduction proof of... |
| idiVD 42373 | Virtual deduction proof of... |
| ancomstVD 42374 | Closed form of ~ ancoms . ... |
| ssralv2VD 42375 | Quantification restricted ... |
| ordelordALTVD 42376 | An element of an ordinal c... |
| equncomVD 42377 | If a class equals the unio... |
| equncomiVD 42378 | Inference form of ~ equnco... |
| sucidALTVD 42379 | A set belongs to its succe... |
| sucidALT 42380 | A set belongs to its succe... |
| sucidVD 42381 | A set belongs to its succe... |
| imbi12VD 42382 | Implication form of ~ imbi... |
| imbi13VD 42383 | Join three logical equival... |
| sbcim2gVD 42384 | Distribution of class subs... |
| sbcbiVD 42385 | Implication form of ~ sbcb... |
| trsbcVD 42386 | Formula-building inference... |
| truniALTVD 42387 | The union of a class of tr... |
| ee33VD 42388 | Non-virtual deduction form... |
| trintALTVD 42389 | The intersection of a clas... |
| trintALT 42390 | The intersection of a clas... |
| undif3VD 42391 | The first equality of Exer... |
| sbcssgVD 42392 | Virtual deduction proof of... |
| csbingVD 42393 | Virtual deduction proof of... |
| onfrALTlem5VD 42394 | Virtual deduction proof of... |
| onfrALTlem4VD 42395 | Virtual deduction proof of... |
| onfrALTlem3VD 42396 | Virtual deduction proof of... |
| simplbi2comtVD 42397 | Virtual deduction proof of... |
| onfrALTlem2VD 42398 | Virtual deduction proof of... |
| onfrALTlem1VD 42399 | Virtual deduction proof of... |
| onfrALTVD 42400 | Virtual deduction proof of... |
| csbeq2gVD 42401 | Virtual deduction proof of... |
| csbsngVD 42402 | Virtual deduction proof of... |
| csbxpgVD 42403 | Virtual deduction proof of... |
| csbresgVD 42404 | Virtual deduction proof of... |
| csbrngVD 42405 | Virtual deduction proof of... |
| csbima12gALTVD 42406 | Virtual deduction proof of... |
| csbunigVD 42407 | Virtual deduction proof of... |
| csbfv12gALTVD 42408 | Virtual deduction proof of... |
| con5VD 42409 | Virtual deduction proof of... |
| relopabVD 42410 | Virtual deduction proof of... |
| 19.41rgVD 42411 | Virtual deduction proof of... |
| 2pm13.193VD 42412 | Virtual deduction proof of... |
| hbimpgVD 42413 | Virtual deduction proof of... |
| hbalgVD 42414 | Virtual deduction proof of... |
| hbexgVD 42415 | Virtual deduction proof of... |
| ax6e2eqVD 42416 | The following User's Proof... |
| ax6e2ndVD 42417 | The following User's Proof... |
| ax6e2ndeqVD 42418 | The following User's Proof... |
| 2sb5ndVD 42419 | The following User's Proof... |
| 2uasbanhVD 42420 | The following User's Proof... |
| e2ebindVD 42421 | The following User's Proof... |
| sb5ALTVD 42422 | The following User's Proof... |
| vk15.4jVD 42423 | The following User's Proof... |
| notnotrALTVD 42424 | The following User's Proof... |
| con3ALTVD 42425 | The following User's Proof... |
| elpwgdedVD 42426 | Membership in a power clas... |
| sspwimp 42427 | If a class is a subclass o... |
| sspwimpVD 42428 | The following User's Proof... |
| sspwimpcf 42429 | If a class is a subclass o... |
| sspwimpcfVD 42430 | The following User's Proof... |
| suctrALTcf 42431 | The sucessor of a transiti... |
| suctrALTcfVD 42432 | The following User's Proof... |
| suctrALT3 42433 | The successor of a transit... |
| sspwimpALT 42434 | If a class is a subclass o... |
| unisnALT 42435 | A set equals the union of ... |
| notnotrALT2 42436 | Converse of double negatio... |
| sspwimpALT2 42437 | If a class is a subclass o... |
| e2ebindALT 42438 | Absorption of an existenti... |
| ax6e2ndALT 42439 | If at least two sets exist... |
| ax6e2ndeqALT 42440 | "At least two sets exist" ... |
| 2sb5ndALT 42441 | Equivalence for double sub... |
| chordthmALT 42442 | The intersecting chords th... |
| isosctrlem1ALT 42443 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 42444 | The indexed union of conne... |
| iunconnALT 42445 | The indexed union of conne... |
| sineq0ALT 42446 | A complex number whose sin... |
| evth2f 42447 | A version of ~ evth2 using... |
| elunif 42448 | A version of ~ eluni using... |
| rzalf 42449 | A version of ~ rzal using ... |
| fvelrnbf 42450 | A version of ~ fvelrnb usi... |
| rfcnpre1 42451 | If F is a continuous funct... |
| ubelsupr 42452 | If U belongs to A and U is... |
| fsumcnf 42453 | A finite sum of functions ... |
| mulltgt0 42454 | The product of a negative ... |
| rspcegf 42455 | A version of ~ rspcev usin... |
| rabexgf 42456 | A version of ~ rabexg usin... |
| fcnre 42457 | A function continuous with... |
| sumsnd 42458 | A sum of a singleton is th... |
| evthf 42459 | A version of ~ evth using ... |
| cnfex 42460 | The class of continuous fu... |
| fnchoice 42461 | For a finite set, a choice... |
| refsumcn 42462 | A finite sum of continuous... |
| rfcnpre2 42463 | If ` F ` is a continuous f... |
| cncmpmax 42464 | When the hypothesis for th... |
| rfcnpre3 42465 | If F is a continuous funct... |
| rfcnpre4 42466 | If F is a continuous funct... |
| sumpair 42467 | Sum of two distinct comple... |
| rfcnnnub 42468 | Given a real continuous fu... |
| refsum2cnlem1 42469 | This is the core Lemma for... |
| refsum2cn 42470 | The sum of two continuus r... |
| elunnel2 42471 | A member of a union that i... |
| adantlllr 42472 | Deduction adding a conjunc... |
| 3adantlr3 42473 | Deduction adding a conjunc... |
| nnxrd 42474 | A natural number is an ext... |
| 3adantll2 42475 | Deduction adding a conjunc... |
| 3adantll3 42476 | Deduction adding a conjunc... |
| ssnel 42477 | If not element of a set, t... |
| elabrexg 42478 | Elementhood in an image se... |
| sncldre 42479 | A singleton is closed w.r.... |
| n0p 42480 | A polynomial with a nonzer... |
| pm2.65ni 42481 | Inference rule for proof b... |
| pwssfi 42482 | Every element of the power... |
| iuneq2df 42483 | Equality deduction for ind... |
| nnfoctb 42484 | There exists a mapping fro... |
| ssinss1d 42485 | Intersection preserves sub... |
| elpwinss 42486 | An element of the powerset... |
| unidmex 42487 | If ` F ` is a set, then ` ... |
| ndisj2 42488 | A non-disjointness conditi... |
| zenom 42489 | The set of integer numbers... |
| uzwo4 42490 | Well-ordering principle: a... |
| unisn0 42491 | The union of the singleton... |
| ssin0 42492 | If two classes are disjoin... |
| inabs3 42493 | Absorption law for interse... |
| pwpwuni 42494 | Relationship between power... |
| disjiun2 42495 | In a disjoint collection, ... |
| 0pwfi 42496 | The empty set is in any po... |
| ssinss2d 42497 | Intersection preserves sub... |
| zct 42498 | The set of integer numbers... |
| pwfin0 42499 | A finite set always belong... |
| uzct 42500 | An upper integer set is co... |
| iunxsnf 42501 | A singleton index picks ou... |
| fiiuncl 42502 | If a set is closed under t... |
| iunp1 42503 | The addition of the next s... |
| fiunicl 42504 | If a set is closed under t... |
| ixpeq2d 42505 | Equality theorem for infin... |
| disjxp1 42506 | The sets of a cartesian pr... |
| disjsnxp 42507 | The sets in the cartesian ... |
| eliind 42508 | Membership in indexed inte... |
| rspcef 42509 | Restricted existential spe... |
| inn0f 42510 | A nonempty intersection. ... |
| ixpssmapc 42511 | An infinite Cartesian prod... |
| inn0 42512 | A nonempty intersection. ... |
| elintd 42513 | Membership in class inters... |
| ssdf 42514 | A sufficient condition for... |
| brneqtrd 42515 | Substitution of equal clas... |
| ssnct 42516 | A set containing an uncoun... |
| ssuniint 42517 | Sufficient condition for b... |
| elintdv 42518 | Membership in class inters... |
| ssd 42519 | A sufficient condition for... |
| ralimralim 42520 | Introducing any antecedent... |
| snelmap 42521 | Membership of the element ... |
| xrnmnfpnf 42522 | An extended real that is n... |
| nelrnmpt 42523 | Non-membership in the rang... |
| iuneq1i 42524 | Equality theorem for index... |
| nssrex 42525 | Negation of subclass relat... |
| ssinc 42526 | Inclusion relation for a m... |
| ssdec 42527 | Inclusion relation for a m... |
| elixpconstg 42528 | Membership in an infinite ... |
| iineq1d 42529 | Equality theorem for index... |
| metpsmet 42530 | A metric is a pseudometric... |
| ixpssixp 42531 | Subclass theorem for infin... |
| ballss3 42532 | A sufficient condition for... |
| iunincfi 42533 | Given a sequence of increa... |
| nsstr 42534 | If it's not a subclass, it... |
| rexanuz3 42535 | Combine two different uppe... |
| cbvmpo2 42536 | Rule to change the second ... |
| cbvmpo1 42537 | Rule to change the first b... |
| eliuniin 42538 | Indexed union of indexed i... |
| ssabf 42539 | Subclass of a class abstra... |
| pssnssi 42540 | A proper subclass does not... |
| rabidim2 42541 | Membership in a restricted... |
| eluni2f 42542 | Membership in class union.... |
| eliin2f 42543 | Membership in indexed inte... |
| nssd 42544 | Negation of subclass relat... |
| iineq12dv 42545 | Equality deduction for ind... |
| supxrcld 42546 | The supremum of an arbitra... |
| elrestd 42547 | A sufficient condition for... |
| eliuniincex 42548 | Counterexample to show tha... |
| eliincex 42549 | Counterexample to show tha... |
| eliinid 42550 | Membership in an indexed i... |
| abssf 42551 | Class abstraction in a sub... |
| supxrubd 42552 | A member of a set of exten... |
| ssrabf 42553 | Subclass of a restricted c... |
| eliin2 42554 | Membership in indexed inte... |
| ssrab2f 42555 | Subclass relation for a re... |
| restuni3 42556 | The underlying set of a su... |
| rabssf 42557 | Restricted class abstracti... |
| eliuniin2 42558 | Indexed union of indexed i... |
| restuni4 42559 | The underlying set of a su... |
| restuni6 42560 | The underlying set of a su... |
| restuni5 42561 | The underlying set of a su... |
| unirestss 42562 | The union of an elementwis... |
| iniin1 42563 | Indexed intersection of in... |
| iniin2 42564 | Indexed intersection of in... |
| cbvrabv2 42565 | A more general version of ... |
| cbvrabv2w 42566 | A more general version of ... |
| iinssiin 42567 | Subset implication for an ... |
| eliind2 42568 | Membership in indexed inte... |
| iinssd 42569 | Subset implication for an ... |
| rabbida2 42570 | Equivalent wff's yield equ... |
| iinexd 42571 | The existence of an indexe... |
| rabexf 42572 | Separation Scheme in terms... |
| rabbida3 42573 | Equivalent wff's yield equ... |
| r19.36vf 42574 | Restricted quantifier vers... |
| raleqd 42575 | Equality deduction for res... |
| iinssf 42576 | Subset implication for an ... |
| iinssdf 42577 | Subset implication for an ... |
| resabs2i 42578 | Absorption law for restric... |
| ssdf2 42579 | A sufficient condition for... |
| rabssd 42580 | Restricted class abstracti... |
| rexnegd 42581 | Minus a real number. (Con... |
| rexlimd3 42582 | * Inference from Theorem 1... |
| resabs1i 42583 | Absorption law for restric... |
| nel1nelin 42584 | Membership in an intersect... |
| nel2nelin 42585 | Membership in an intersect... |
| nel1nelini 42586 | Membership in an intersect... |
| nel2nelini 42587 | Membership in an intersect... |
| eliunid 42588 | Membership in indexed unio... |
| reximddv3 42589 | Deduction from Theorem 19.... |
| reximdd 42590 | Deduction from Theorem 19.... |
| unfid 42591 | The union of two finite se... |
| feq1dd 42592 | Equality deduction for fun... |
| fnresdmss 42593 | A function does not change... |
| fmptsnxp 42594 | Maps-to notation and Carte... |
| fvmpt2bd 42595 | Value of a function given ... |
| rnmptfi 42596 | The range of a function wi... |
| fresin2 42597 | Restriction of a function ... |
| ffi 42598 | A function with finite dom... |
| suprnmpt 42599 | An explicit bound for the ... |
| rnffi 42600 | The range of a function wi... |
| mptelpm 42601 | A function in maps-to nota... |
| rnmptpr 42602 | Range of a function define... |
| resmpti 42603 | Restriction of the mapping... |
| founiiun 42604 | Union expressed as an inde... |
| rnresun 42605 | Distribution law for range... |
| dffo3f 42606 | An onto mapping expressed ... |
| elrnmptf 42607 | The range of a function in... |
| rnmptssrn 42608 | Inclusion relation for two... |
| disjf1 42609 | A 1 to 1 mapping built fro... |
| rnsnf 42610 | The range of a function wh... |
| wessf1ornlem 42611 | Given a function ` F ` on ... |
| wessf1orn 42612 | Given a function ` F ` on ... |
| foelrnf 42613 | Property of a surjective f... |
| nelrnres 42614 | If ` A ` is not in the ran... |
| disjrnmpt2 42615 | Disjointness of the range ... |
| elrnmpt1sf 42616 | Elementhood in an image se... |
| founiiun0 42617 | Union expressed as an inde... |
| disjf1o 42618 | A bijection built from dis... |
| fompt 42619 | Express being onto for a m... |
| disjinfi 42620 | Only a finite number of di... |
| fvovco 42621 | Value of the composition o... |
| ssnnf1octb 42622 | There exists a bijection b... |
| nnf1oxpnn 42623 | There is a bijection betwe... |
| rnmptssd 42624 | The range of an operation ... |
| projf1o 42625 | A biijection from a set to... |
| fvmap 42626 | Function value for a membe... |
| fvixp2 42627 | Projection of a factor of ... |
| fidmfisupp 42628 | A function with a finite d... |
| choicefi 42629 | For a finite set, a choice... |
| mpct 42630 | The exponentiation of a co... |
| cnmetcoval 42631 | Value of the distance func... |
| fcomptss 42632 | Express composition of two... |
| elmapsnd 42633 | Membership in a set expone... |
| mapss2 42634 | Subset inheritance for set... |
| fsneq 42635 | Equality condition for two... |
| difmap 42636 | Difference of two sets exp... |
| unirnmap 42637 | Given a subset of a set ex... |
| inmap 42638 | Intersection of two sets e... |
| fcoss 42639 | Composition of two mapping... |
| fsneqrn 42640 | Equality condition for two... |
| difmapsn 42641 | Difference of two sets exp... |
| mapssbi 42642 | Subset inheritance for set... |
| unirnmapsn 42643 | Equality theorem for a sub... |
| iunmapss 42644 | The indexed union of set e... |
| ssmapsn 42645 | A subset ` C ` of a set ex... |
| iunmapsn 42646 | The indexed union of set e... |
| absfico 42647 | Mapping domain and codomai... |
| icof 42648 | The set of left-closed rig... |
| elpmrn 42649 | The range of a partial fun... |
| imaexi 42650 | The image of a set is a se... |
| axccdom 42651 | Relax the constraint on ax... |
| dmmptdf 42652 | The domain of the mapping ... |
| elpmi2 42653 | The domain of a partial fu... |
| dmrelrnrel 42654 | A relation preserving func... |
| fvcod 42655 | Value of a function compos... |
| elrnmpoid 42656 | Membership in the range of... |
| axccd 42657 | An alternative version of ... |
| axccd2 42658 | An alternative version of ... |
| funimassd 42659 | Sufficient condition for t... |
| fimassd 42660 | The image of a class is a ... |
| feqresmptf 42661 | Express a restricted funct... |
| elrnmpt1d 42662 | Elementhood in an image se... |
| dmresss 42663 | The domain of a restrictio... |
| dmmptssf 42664 | The domain of a mapping is... |
| dmmptdf2 42665 | The domain of the mapping ... |
| dmuz 42666 | Domain of the upper intege... |
| fmptd2f 42667 | Domain and codomain of the... |
| mpteq1df 42668 | An equality theorem for th... |
| mpteq1dfOLD 42669 | Obsolete version of ~ mpte... |
| mptexf 42670 | If the domain of a functio... |
| fvmpt4 42671 | Value of a function given ... |
| fmptf 42672 | Functionality of the mappi... |
| resimass 42673 | The image of a restriction... |
| mptssid 42674 | The mapping operation expr... |
| mptfnd 42675 | The maps-to notation defin... |
| mpteq12daOLD 42676 | Obsolete version of ~ mpte... |
| rnmptlb 42677 | Boundness below of the ran... |
| rnmptbddlem 42678 | Boundness of the range of ... |
| rnmptbdd 42679 | Boundness of the range of ... |
| mptima2 42680 | Image of a function in map... |
| funimaeq 42681 | Membership relation for th... |
| rnmptssf 42682 | The range of an operation ... |
| rnmptbd2lem 42683 | Boundness below of the ran... |
| rnmptbd2 42684 | Boundness below of the ran... |
| infnsuprnmpt 42685 | The indexed infimum of rea... |
| suprclrnmpt 42686 | Closure of the indexed sup... |
| suprubrnmpt2 42687 | A member of a nonempty ind... |
| suprubrnmpt 42688 | A member of a nonempty ind... |
| rnmptssdf 42689 | The range of an operation ... |
| rnmptbdlem 42690 | Boundness above of the ran... |
| rnmptbd 42691 | Boundness above of the ran... |
| rnmptss2 42692 | The range of an operation ... |
| elmptima 42693 | The image of a function in... |
| ralrnmpt3 42694 | A restricted quantifier ov... |
| fvelima2 42695 | Function value in an image... |
| rnmptssbi 42696 | The range of an operation ... |
| fnfvelrnd 42697 | A function's value belongs... |
| imass2d 42698 | Subset theorem for image. ... |
| imassmpt 42699 | Membership relation for th... |
| fpmd 42700 | A total function is a part... |
| fconst7 42701 | An alternative way to expr... |
| sub2times 42702 | Subtracting from a number,... |
| abssubrp 42703 | The distance of two distin... |
| elfzfzo 42704 | Relationship between membe... |
| oddfl 42705 | Odd number representation ... |
| abscosbd 42706 | Bound for the absolute val... |
| mul13d 42707 | Commutative/associative la... |
| negpilt0 42708 | Negative ` _pi ` is negati... |
| dstregt0 42709 | A complex number ` A ` tha... |
| subadd4b 42710 | Rearrangement of 4 terms i... |
| xrlttri5d 42711 | Not equal and not larger i... |
| neglt 42712 | The negative of a positive... |
| zltlesub 42713 | If an integer ` N ` is les... |
| divlt0gt0d 42714 | The ratio of a negative nu... |
| subsub23d 42715 | Swap subtrahend and result... |
| 2timesgt 42716 | Double of a positive real ... |
| reopn 42717 | The reals are open with re... |
| elfzop1le2 42718 | A member in a half-open in... |
| sub31 42719 | Swap the first and third t... |
| nnne1ge2 42720 | A positive integer which i... |
| lefldiveq 42721 | A closed enough, smaller r... |
| negsubdi3d 42722 | Distribution of negative o... |
| ltdiv2dd 42723 | Division of a positive num... |
| abssinbd 42724 | Bound for the absolute val... |
| halffl 42725 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 42726 | Ordering relation for a st... |
| hashssle 42727 | The size of a subset of a ... |
| lttri5d 42728 | Not equal and not larger i... |
| fzisoeu 42729 | A finite ordered set has a... |
| lt3addmuld 42730 | If three real numbers are ... |
| absnpncan2d 42731 | Triangular inequality, com... |
| fperiodmullem 42732 | A function with period ` T... |
| fperiodmul 42733 | A function with period T i... |
| upbdrech 42734 | Choice of an upper bound f... |
| lt4addmuld 42735 | If four real numbers are l... |
| absnpncan3d 42736 | Triangular inequality, com... |
| upbdrech2 42737 | Choice of an upper bound f... |
| ssfiunibd 42738 | A finite union of bounded ... |
| fzdifsuc2 42739 | Remove a successor from th... |
| fzsscn 42740 | A finite sequence of integ... |
| divcan8d 42741 | A cancellation law for div... |
| dmmcand 42742 | Cancellation law for divis... |
| fzssre 42743 | A finite sequence of integ... |
| bccld 42744 | A binomial coefficient, in... |
| leadd12dd 42745 | Addition to both sides of ... |
| fzssnn0 42746 | A finite set of sequential... |
| xreqle 42747 | Equality implies 'less tha... |
| xaddid2d 42748 | ` 0 ` is a left identity f... |
| xadd0ge 42749 | A number is less than or e... |
| elfzolem1 42750 | A member in a half-open in... |
| xrgtned 42751 | 'Greater than' implies not... |
| xrleneltd 42752 | 'Less than or equal to' an... |
| xaddcomd 42753 | The extended real addition... |
| supxrre3 42754 | The supremum of a nonempty... |
| uzfissfz 42755 | For any finite subset of t... |
| xleadd2d 42756 | Addition of extended reals... |
| suprltrp 42757 | The supremum of a nonempty... |
| xleadd1d 42758 | Addition of extended reals... |
| xreqled 42759 | Equality implies 'less tha... |
| xrgepnfd 42760 | An extended real greater t... |
| xrge0nemnfd 42761 | A nonnegative extended rea... |
| supxrgere 42762 | If a real number can be ap... |
| iuneqfzuzlem 42763 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 42764 | If two unions indexed by u... |
| xle2addd 42765 | Adding both side of two in... |
| supxrgelem 42766 | If an extended real number... |
| supxrge 42767 | If an extended real number... |
| suplesup 42768 | If any element of ` A ` ca... |
| infxrglb 42769 | The infimum of a set of ex... |
| xadd0ge2 42770 | A number is less than or e... |
| nepnfltpnf 42771 | An extended real that is n... |
| ltadd12dd 42772 | Addition to both sides of ... |
| nemnftgtmnft 42773 | An extended real that is n... |
| xrgtso 42774 | 'Greater than' is a strict... |
| rpex 42775 | The positive reals form a ... |
| xrge0ge0 42776 | A nonnegative extended rea... |
| xrssre 42777 | A subset of extended reals... |
| ssuzfz 42778 | A finite subset of the upp... |
| absfun 42779 | The absolute value is a fu... |
| infrpge 42780 | The infimum of a nonempty,... |
| xrlexaddrp 42781 | If an extended real number... |
| supsubc 42782 | The supremum function dist... |
| xralrple2 42783 | Show that ` A ` is less th... |
| nnuzdisj 42784 | The first ` N ` elements o... |
| ltdivgt1 42785 | Divsion by a number greate... |
| xrltned 42786 | 'Less than' implies not eq... |
| nnsplit 42787 | Express the set of positiv... |
| divdiv3d 42788 | Division into a fraction. ... |
| abslt2sqd 42789 | Comparison of the square o... |
| qenom 42790 | The set of rational number... |
| qct 42791 | The set of rational number... |
| xrltnled 42792 | 'Less than' in terms of 'l... |
| lenlteq 42793 | 'less than or equal to' bu... |
| xrred 42794 | An extended real that is n... |
| rr2sscn2 42795 | The cartesian square of ` ... |
| infxr 42796 | The infimum of a set of ex... |
| infxrunb2 42797 | The infimum of an unbounde... |
| infxrbnd2 42798 | The infimum of a bounded-b... |
| infleinflem1 42799 | Lemma for ~ infleinf , cas... |
| infleinflem2 42800 | Lemma for ~ infleinf , whe... |
| infleinf 42801 | If any element of ` B ` ca... |
| xralrple4 42802 | Show that ` A ` is less th... |
| xralrple3 42803 | Show that ` A ` is less th... |
| eluzelzd 42804 | A member of an upper set o... |
| suplesup2 42805 | If any element of ` A ` is... |
| recnnltrp 42806 | ` N ` is a natural number ... |
| nnn0 42807 | The set of positive intege... |
| fzct 42808 | A finite set of sequential... |
| rpgtrecnn 42809 | Any positive real number i... |
| fzossuz 42810 | A half-open integer interv... |
| infxrrefi 42811 | The real and extended real... |
| xrralrecnnle 42812 | Show that ` A ` is less th... |
| fzoct 42813 | A finite set of sequential... |
| frexr 42814 | A function taking real val... |
| nnrecrp 42815 | The reciprocal of a positi... |
| reclt0d 42816 | The reciprocal of a negati... |
| lt0neg1dd 42817 | If a number is negative, i... |
| mnfled 42818 | Minus infinity is less tha... |
| infxrcld 42819 | The infimum of an arbitrar... |
| xrralrecnnge 42820 | Show that ` A ` is less th... |
| reclt0 42821 | The reciprocal of a negati... |
| ltmulneg 42822 | Multiplying by a negative ... |
| allbutfi 42823 | For all but finitely many.... |
| ltdiv23neg 42824 | Swap denominator with othe... |
| xreqnltd 42825 | A consequence of trichotom... |
| mnfnre2 42826 | Minus infinity is not a re... |
| zssxr 42827 | The integers are a subset ... |
| fisupclrnmpt 42828 | A nonempty finite indexed ... |
| supxrunb3 42829 | The supremum of an unbound... |
| elfzod 42830 | Membership in a half-open ... |
| fimaxre4 42831 | A nonempty finite set of r... |
| ren0 42832 | The set of reals is nonemp... |
| eluzelz2 42833 | A member of an upper set o... |
| resabs2d 42834 | Absorption law for restric... |
| uzid2 42835 | Membership of the least me... |
| supxrleubrnmpt 42836 | The supremum of a nonempty... |
| uzssre2 42837 | An upper set of integers i... |
| uzssd 42838 | Subset relationship for tw... |
| eluzd 42839 | Membership in an upper set... |
| infxrlbrnmpt2 42840 | A member of a nonempty ind... |
| xrre4 42841 | An extended real is real i... |
| uz0 42842 | The upper integers functio... |
| eluzelz2d 42843 | A member of an upper set o... |
| infleinf2 42844 | If any element in ` B ` is... |
| unb2ltle 42845 | "Unbounded below" expresse... |
| uzidd2 42846 | Membership of the least me... |
| uzssd2 42847 | Subset relationship for tw... |
| rexabslelem 42848 | An indexed set of absolute... |
| rexabsle 42849 | An indexed set of absolute... |
| allbutfiinf 42850 | Given a "for all but finit... |
| supxrrernmpt 42851 | The real and extended real... |
| suprleubrnmpt 42852 | The supremum of a nonempty... |
| infrnmptle 42853 | An indexed infimum of exte... |
| infxrunb3 42854 | The infimum of an unbounde... |
| uzn0d 42855 | The upper integers are all... |
| uzssd3 42856 | Subset relationship for tw... |
| rexabsle2 42857 | An indexed set of absolute... |
| infxrunb3rnmpt 42858 | The infimum of an unbounde... |
| supxrre3rnmpt 42859 | The indexed supremum of a ... |
| uzublem 42860 | A set of reals, indexed by... |
| uzub 42861 | A set of reals, indexed by... |
| ssrexr 42862 | A subset of the reals is a... |
| supxrmnf2 42863 | Removing minus infinity fr... |
| supxrcli 42864 | The supremum of an arbitra... |
| uzid3 42865 | Membership of the least me... |
| infxrlesupxr 42866 | The supremum of a nonempty... |
| xnegeqd 42867 | Equality of two extended n... |
| xnegrecl 42868 | The extended real negative... |
| xnegnegi 42869 | Extended real version of ~... |
| xnegeqi 42870 | Equality of two extended n... |
| nfxnegd 42871 | Deduction version of ~ nfx... |
| xnegnegd 42872 | Extended real version of ~... |
| uzred 42873 | An upper integer is a real... |
| xnegcli 42874 | Closure of extended real n... |
| supminfrnmpt 42875 | The indexed supremum of a ... |
| infxrpnf 42876 | Adding plus infinity to a ... |
| infxrrnmptcl 42877 | The infimum of an arbitrar... |
| leneg2d 42878 | Negative of one side of 'l... |
| supxrltinfxr 42879 | The supremum of the empty ... |
| max1d 42880 | A number is less than or e... |
| supxrleubrnmptf 42881 | The supremum of a nonempty... |
| nleltd 42882 | 'Not less than or equal to... |
| zxrd 42883 | An integer is an extended ... |
| infxrgelbrnmpt 42884 | The infimum of an indexed ... |
| rphalfltd 42885 | Half of a positive real is... |
| uzssz2 42886 | An upper set of integers i... |
| leneg3d 42887 | Negative of one side of 'l... |
| max2d 42888 | A number is less than or e... |
| uzn0bi 42889 | The upper integers functio... |
| xnegrecl2 42890 | If the extended real negat... |
| nfxneg 42891 | Bound-variable hypothesis ... |
| uzxrd 42892 | An upper integer is an ext... |
| infxrpnf2 42893 | Removing plus infinity fro... |
| supminfxr 42894 | The extended real suprema ... |
| infrpgernmpt 42895 | The infimum of a nonempty,... |
| xnegre 42896 | An extended real is real i... |
| xnegrecl2d 42897 | If the extended real negat... |
| uzxr 42898 | An upper integer is an ext... |
| supminfxr2 42899 | The extended real suprema ... |
| xnegred 42900 | An extended real is real i... |
| supminfxrrnmpt 42901 | The indexed supremum of a ... |
| min1d 42902 | The minimum of two numbers... |
| min2d 42903 | The minimum of two numbers... |
| pnfged 42904 | Plus infinity is an upper ... |
| xrnpnfmnf 42905 | An extended real that is n... |
| uzsscn 42906 | An upper set of integers i... |
| absimnre 42907 | The absolute value of the ... |
| uzsscn2 42908 | An upper set of integers i... |
| xrtgcntopre 42909 | The standard topologies on... |
| absimlere 42910 | The absolute value of the ... |
| rpssxr 42911 | The positive reals are a s... |
| monoordxrv 42912 | Ordering relation for a mo... |
| monoordxr 42913 | Ordering relation for a mo... |
| monoord2xrv 42914 | Ordering relation for a mo... |
| monoord2xr 42915 | Ordering relation for a mo... |
| xrpnf 42916 | An extended real is plus i... |
| xlenegcon1 42917 | Extended real version of ~... |
| xlenegcon2 42918 | Extended real version of ~... |
| gtnelioc 42919 | A real number larger than ... |
| ioossioc 42920 | An open interval is a subs... |
| ioondisj2 42921 | A condition for two open i... |
| ioondisj1 42922 | A condition for two open i... |
| ioogtlb 42923 | An element of a closed int... |
| evthiccabs 42924 | Extreme Value Theorem on y... |
| ltnelicc 42925 | A real number smaller than... |
| eliood 42926 | Membership in an open real... |
| iooabslt 42927 | An upper bound for the dis... |
| gtnelicc 42928 | A real number greater than... |
| iooinlbub 42929 | An open interval has empty... |
| iocgtlb 42930 | An element of a left-open ... |
| iocleub 42931 | An element of a left-open ... |
| eliccd 42932 | Membership in a closed rea... |
| eliccre 42933 | A member of a closed inter... |
| eliooshift 42934 | Element of an open interva... |
| eliocd 42935 | Membership in a left-open ... |
| icoltub 42936 | An element of a left-close... |
| eliocre 42937 | A member of a left-open ri... |
| iooltub 42938 | An element of an open inte... |
| ioontr 42939 | The interior of an interva... |
| snunioo1 42940 | The closure of one end of ... |
| lbioc 42941 | A left-open right-closed i... |
| ioomidp 42942 | The midpoint is an element... |
| iccdifioo 42943 | If the open inverval is re... |
| iccdifprioo 42944 | An open interval is the cl... |
| ioossioobi 42945 | Biconditional form of ~ io... |
| iccshift 42946 | A closed interval shifted ... |
| iccsuble 42947 | An upper bound to the dist... |
| iocopn 42948 | A left-open right-closed i... |
| eliccelioc 42949 | Membership in a closed int... |
| iooshift 42950 | An open interval shifted b... |
| iccintsng 42951 | Intersection of two adiace... |
| icoiccdif 42952 | Left-closed right-open int... |
| icoopn 42953 | A left-closed right-open i... |
| icoub 42954 | A left-closed, right-open ... |
| eliccxrd 42955 | Membership in a closed rea... |
| pnfel0pnf 42956 | ` +oo ` is a nonnegative e... |
| eliccnelico 42957 | An element of a closed int... |
| eliccelicod 42958 | A member of a closed inter... |
| ge0xrre 42959 | A nonnegative extended rea... |
| ge0lere 42960 | A nonnegative extended Rea... |
| elicores 42961 | Membership in a left-close... |
| inficc 42962 | The infimum of a nonempty ... |
| qinioo 42963 | The rational numbers are d... |
| lenelioc 42964 | A real number smaller than... |
| ioonct 42965 | A nonempty open interval i... |
| xrgtnelicc 42966 | A real number greater than... |
| iccdificc 42967 | The difference of two clos... |
| iocnct 42968 | A nonempty left-open, righ... |
| iccnct 42969 | A closed interval, with mo... |
| iooiinicc 42970 | A closed interval expresse... |
| iccgelbd 42971 | An element of a closed int... |
| iooltubd 42972 | An element of an open inte... |
| icoltubd 42973 | An element of a left-close... |
| qelioo 42974 | The rational numbers are d... |
| tgqioo2 42975 | Every open set of reals is... |
| iccleubd 42976 | An element of a closed int... |
| elioored 42977 | A member of an open interv... |
| ioogtlbd 42978 | An element of a closed int... |
| ioofun 42979 | ` (,) ` is a function. (C... |
| icomnfinre 42980 | A left-closed, right-open,... |
| sqrlearg 42981 | The square compared with i... |
| ressiocsup 42982 | If the supremum belongs to... |
| ressioosup 42983 | If the supremum does not b... |
| iooiinioc 42984 | A left-open, right-closed ... |
| ressiooinf 42985 | If the infimum does not be... |
| icogelbd 42986 | An element of a left-close... |
| iocleubd 42987 | An element of a left-open ... |
| uzinico 42988 | An upper interval of integ... |
| preimaiocmnf 42989 | Preimage of a right-closed... |
| uzinico2 42990 | An upper interval of integ... |
| uzinico3 42991 | An upper interval of integ... |
| icossico2 42992 | Condition for a closed-bel... |
| dmico 42993 | The domain of the closed-b... |
| ndmico 42994 | The closed-below, open-abo... |
| uzubioo 42995 | The upper integers are unb... |
| uzubico 42996 | The upper integers are unb... |
| uzubioo2 42997 | The upper integers are unb... |
| uzubico2 42998 | The upper integers are unb... |
| iocgtlbd 42999 | An element of a left-open ... |
| xrtgioo2 43000 | The topology on the extend... |
| tgioo4 43001 | The standard topology on t... |
| fsummulc1f 43002 | Closure of a finite sum of... |
| fsumnncl 43003 | Closure of a nonempty, fin... |
| fsumge0cl 43004 | The finite sum of nonnegat... |
| fsumf1of 43005 | Re-index a finite sum usin... |
| fsumiunss 43006 | Sum over a disjoint indexe... |
| fsumreclf 43007 | Closure of a finite sum of... |
| fsumlessf 43008 | A shorter sum of nonnegati... |
| fsumsupp0 43009 | Finite sum of function val... |
| fsumsermpt 43010 | A finite sum expressed in ... |
| fmul01 43011 | Multiplying a finite numbe... |
| fmulcl 43012 | If ' Y ' is closed under t... |
| fmuldfeqlem1 43013 | induction step for the pro... |
| fmuldfeq 43014 | X and Z are two equivalent... |
| fmul01lt1lem1 43015 | Given a finite multiplicat... |
| fmul01lt1lem2 43016 | Given a finite multiplicat... |
| fmul01lt1 43017 | Given a finite multiplicat... |
| cncfmptss 43018 | A continuous complex funct... |
| rrpsscn 43019 | The positive reals are a s... |
| mulc1cncfg 43020 | A version of ~ mulc1cncf u... |
| infrglb 43021 | The infimum of a nonempty ... |
| expcnfg 43022 | If ` F ` is a complex cont... |
| prodeq2ad 43023 | Equality deduction for pro... |
| fprodsplit1 43024 | Separate out a term in a f... |
| fprodexp 43025 | Positive integer exponenti... |
| fprodabs2 43026 | The absolute value of a fi... |
| fprod0 43027 | A finite product with a ze... |
| mccllem 43028 | * Induction step for ~ mcc... |
| mccl 43029 | A multinomial coefficient,... |
| fprodcnlem 43030 | A finite product of functi... |
| fprodcn 43031 | A finite product of functi... |
| clim1fr1 43032 | A class of sequences of fr... |
| isumneg 43033 | Negation of a converging s... |
| climrec 43034 | Limit of the reciprocal of... |
| climmulf 43035 | A version of ~ climmul usi... |
| climexp 43036 | The limit of natural power... |
| climinf 43037 | A bounded monotonic noninc... |
| climsuselem1 43038 | The subsequence index ` I ... |
| climsuse 43039 | A subsequence ` G ` of a c... |
| climrecf 43040 | A version of ~ climrec usi... |
| climneg 43041 | Complex limit of the negat... |
| climinff 43042 | A version of ~ climinf usi... |
| climdivf 43043 | Limit of the ratio of two ... |
| climreeq 43044 | If ` F ` is a real functio... |
| ellimciota 43045 | An explicit value for the ... |
| climaddf 43046 | A version of ~ climadd usi... |
| mullimc 43047 | Limit of the product of tw... |
| ellimcabssub0 43048 | An equivalent condition fo... |
| limcdm0 43049 | If a function has empty do... |
| islptre 43050 | An equivalence condition f... |
| limccog 43051 | Limit of the composition o... |
| limciccioolb 43052 | The limit of a function at... |
| climf 43053 | Express the predicate: Th... |
| mullimcf 43054 | Limit of the multiplicatio... |
| constlimc 43055 | Limit of constant function... |
| rexlim2d 43056 | Inference removing two res... |
| idlimc 43057 | Limit of the identity func... |
| divcnvg 43058 | The sequence of reciprocal... |
| limcperiod 43059 | If ` F ` is a periodic fun... |
| limcrecl 43060 | If ` F ` is a real-valued ... |
| sumnnodd 43061 | A series indexed by ` NN `... |
| lptioo2 43062 | The upper bound of an open... |
| lptioo1 43063 | The lower bound of an open... |
| elprn1 43064 | A member of an unordered p... |
| elprn2 43065 | A member of an unordered p... |
| limcmptdm 43066 | The domain of a maps-to fu... |
| clim2f 43067 | Express the predicate: Th... |
| limcicciooub 43068 | The limit of a function at... |
| ltmod 43069 | A sufficient condition for... |
| islpcn 43070 | A characterization for a l... |
| lptre2pt 43071 | If a set in the real line ... |
| limsupre 43072 | If a sequence is bounded, ... |
| limcresiooub 43073 | The left limit doesn't cha... |
| limcresioolb 43074 | The right limit doesn't ch... |
| limcleqr 43075 | If the left and the right ... |
| lptioo2cn 43076 | The upper bound of an open... |
| lptioo1cn 43077 | The lower bound of an open... |
| neglimc 43078 | Limit of the negative func... |
| addlimc 43079 | Sum of two limits. (Contr... |
| 0ellimcdiv 43080 | If the numerator converges... |
| clim2cf 43081 | Express the predicate ` F ... |
| limclner 43082 | For a limit point, both fr... |
| sublimc 43083 | Subtraction of two limits.... |
| reclimc 43084 | Limit of the reciprocal of... |
| clim0cf 43085 | Express the predicate ` F ... |
| limclr 43086 | For a limit point, both fr... |
| divlimc 43087 | Limit of the quotient of t... |
| expfac 43088 | Factorial grows faster tha... |
| climconstmpt 43089 | A constant sequence conver... |
| climresmpt 43090 | A function restricted to u... |
| climsubmpt 43091 | Limit of the difference of... |
| climsubc2mpt 43092 | Limit of the difference of... |
| climsubc1mpt 43093 | Limit of the difference of... |
| fnlimfv 43094 | The value of the limit fun... |
| climreclf 43095 | The limit of a convergent ... |
| climeldmeq 43096 | Two functions that are eve... |
| climf2 43097 | Express the predicate: Th... |
| fnlimcnv 43098 | The sequence of function v... |
| climeldmeqmpt 43099 | Two functions that are eve... |
| climfveq 43100 | Two functions that are eve... |
| clim2f2 43101 | Express the predicate: Th... |
| climfveqmpt 43102 | Two functions that are eve... |
| climd 43103 | Express the predicate: Th... |
| clim2d 43104 | The limit of complex numbe... |
| fnlimfvre 43105 | The limit function of real... |
| allbutfifvre 43106 | Given a sequence of real-v... |
| climleltrp 43107 | The limit of complex numbe... |
| fnlimfvre2 43108 | The limit function of real... |
| fnlimf 43109 | The limit function of real... |
| fnlimabslt 43110 | A sequence of function val... |
| climfveqf 43111 | Two functions that are eve... |
| climmptf 43112 | Exhibit a function ` G ` w... |
| climfveqmpt3 43113 | Two functions that are eve... |
| climeldmeqf 43114 | Two functions that are eve... |
| climreclmpt 43115 | The limit of B convergent ... |
| limsupref 43116 | If a sequence is bounded, ... |
| limsupbnd1f 43117 | If a sequence is eventuall... |
| climbddf 43118 | A converging sequence of c... |
| climeqf 43119 | Two functions that are eve... |
| climeldmeqmpt3 43120 | Two functions that are eve... |
| limsupcld 43121 | Closure of the superior li... |
| climfv 43122 | The limit of a convergent ... |
| limsupval3 43123 | The superior limit of an i... |
| climfveqmpt2 43124 | Two functions that are eve... |
| limsup0 43125 | The superior limit of the ... |
| climeldmeqmpt2 43126 | Two functions that are eve... |
| limsupresre 43127 | The supremum limit of a fu... |
| climeqmpt 43128 | Two functions that are eve... |
| climfvd 43129 | The limit of a convergent ... |
| limsuplesup 43130 | An upper bound for the sup... |
| limsupresico 43131 | The superior limit doesn't... |
| limsuppnfdlem 43132 | If the restriction of a fu... |
| limsuppnfd 43133 | If the restriction of a fu... |
| limsupresuz 43134 | If the real part of the do... |
| limsupub 43135 | If the limsup is not ` +oo... |
| limsupres 43136 | The superior limit of a re... |
| climinf2lem 43137 | A convergent, nonincreasin... |
| climinf2 43138 | A convergent, nonincreasin... |
| limsupvaluz 43139 | The superior limit, when t... |
| limsupresuz2 43140 | If the domain of a functio... |
| limsuppnflem 43141 | If the restriction of a fu... |
| limsuppnf 43142 | If the restriction of a fu... |
| limsupubuzlem 43143 | If the limsup is not ` +oo... |
| limsupubuz 43144 | For a real-valued function... |
| climinf2mpt 43145 | A bounded below, monotonic... |
| climinfmpt 43146 | A bounded below, monotonic... |
| climinf3 43147 | A convergent, nonincreasin... |
| limsupvaluzmpt 43148 | The superior limit, when t... |
| limsupequzmpt2 43149 | Two functions that are eve... |
| limsupubuzmpt 43150 | If the limsup is not ` +oo... |
| limsupmnflem 43151 | The superior limit of a fu... |
| limsupmnf 43152 | The superior limit of a fu... |
| limsupequzlem 43153 | Two functions that are eve... |
| limsupequz 43154 | Two functions that are eve... |
| limsupre2lem 43155 | Given a function on the ex... |
| limsupre2 43156 | Given a function on the ex... |
| limsupmnfuzlem 43157 | The superior limit of a fu... |
| limsupmnfuz 43158 | The superior limit of a fu... |
| limsupequzmptlem 43159 | Two functions that are eve... |
| limsupequzmpt 43160 | Two functions that are eve... |
| limsupre2mpt 43161 | Given a function on the ex... |
| limsupequzmptf 43162 | Two functions that are eve... |
| limsupre3lem 43163 | Given a function on the ex... |
| limsupre3 43164 | Given a function on the ex... |
| limsupre3mpt 43165 | Given a function on the ex... |
| limsupre3uzlem 43166 | Given a function on the ex... |
| limsupre3uz 43167 | Given a function on the ex... |
| limsupreuz 43168 | Given a function on the re... |
| limsupvaluz2 43169 | The superior limit, when t... |
| limsupreuzmpt 43170 | Given a function on the re... |
| supcnvlimsup 43171 | If a function on a set of ... |
| supcnvlimsupmpt 43172 | If a function on a set of ... |
| 0cnv 43173 | If ` (/) ` is a complex nu... |
| climuzlem 43174 | Express the predicate: Th... |
| climuz 43175 | Express the predicate: Th... |
| lmbr3v 43176 | Express the binary relatio... |
| climisp 43177 | If a sequence converges to... |
| lmbr3 43178 | Express the binary relatio... |
| climrescn 43179 | A sequence converging w.r.... |
| climxrrelem 43180 | If a seqence ranging over ... |
| climxrre 43181 | If a sequence ranging over... |
| limsuplt2 43184 | The defining property of t... |
| liminfgord 43185 | Ordering property of the i... |
| limsupvald 43186 | The superior limit of a se... |
| limsupresicompt 43187 | The superior limit doesn't... |
| limsupcli 43188 | Closure of the superior li... |
| liminfgf 43189 | Closure of the inferior li... |
| liminfval 43190 | The inferior limit of a se... |
| climlimsup 43191 | A sequence of real numbers... |
| limsupge 43192 | The defining property of t... |
| liminfgval 43193 | Value of the inferior limi... |
| liminfcl 43194 | Closure of the inferior li... |
| liminfvald 43195 | The inferior limit of a se... |
| liminfval5 43196 | The inferior limit of an i... |
| limsupresxr 43197 | The superior limit of a fu... |
| liminfresxr 43198 | The inferior limit of a fu... |
| liminfval2 43199 | The superior limit, relati... |
| climlimsupcex 43200 | Counterexample for ~ climl... |
| liminfcld 43201 | Closure of the inferior li... |
| liminfresico 43202 | The inferior limit doesn't... |
| limsup10exlem 43203 | The range of the given fun... |
| limsup10ex 43204 | The superior limit of a fu... |
| liminf10ex 43205 | The inferior limit of a fu... |
| liminflelimsuplem 43206 | The superior limit is grea... |
| liminflelimsup 43207 | The superior limit is grea... |
| limsupgtlem 43208 | For any positive real, the... |
| limsupgt 43209 | Given a sequence of real n... |
| liminfresre 43210 | The inferior limit of a fu... |
| liminfresicompt 43211 | The inferior limit doesn't... |
| liminfltlimsupex 43212 | An example where the ` lim... |
| liminfgelimsup 43213 | The inferior limit is grea... |
| liminfvalxr 43214 | Alternate definition of ` ... |
| liminfresuz 43215 | If the real part of the do... |
| liminflelimsupuz 43216 | The superior limit is grea... |
| liminfvalxrmpt 43217 | Alternate definition of ` ... |
| liminfresuz2 43218 | If the domain of a functio... |
| liminfgelimsupuz 43219 | The inferior limit is grea... |
| liminfval4 43220 | Alternate definition of ` ... |
| liminfval3 43221 | Alternate definition of ` ... |
| liminfequzmpt2 43222 | Two functions that are eve... |
| liminfvaluz 43223 | Alternate definition of ` ... |
| liminf0 43224 | The inferior limit of the ... |
| limsupval4 43225 | Alternate definition of ` ... |
| liminfvaluz2 43226 | Alternate definition of ` ... |
| liminfvaluz3 43227 | Alternate definition of ` ... |
| liminflelimsupcex 43228 | A counterexample for ~ lim... |
| limsupvaluz3 43229 | Alternate definition of ` ... |
| liminfvaluz4 43230 | Alternate definition of ` ... |
| limsupvaluz4 43231 | Alternate definition of ` ... |
| climliminflimsupd 43232 | If a sequence of real numb... |
| liminfreuzlem 43233 | Given a function on the re... |
| liminfreuz 43234 | Given a function on the re... |
| liminfltlem 43235 | Given a sequence of real n... |
| liminflt 43236 | Given a sequence of real n... |
| climliminf 43237 | A sequence of real numbers... |
| liminflimsupclim 43238 | A sequence of real numbers... |
| climliminflimsup 43239 | A sequence of real numbers... |
| climliminflimsup2 43240 | A sequence of real numbers... |
| climliminflimsup3 43241 | A sequence of real numbers... |
| climliminflimsup4 43242 | A sequence of real numbers... |
| limsupub2 43243 | A extended real valued fun... |
| limsupubuz2 43244 | A sequence with values in ... |
| xlimpnfxnegmnf 43245 | A sequence converges to ` ... |
| liminflbuz2 43246 | A sequence with values in ... |
| liminfpnfuz 43247 | The inferior limit of a fu... |
| liminflimsupxrre 43248 | A sequence with values in ... |
| xlimrel 43251 | The limit on extended real... |
| xlimres 43252 | A function converges iff i... |
| xlimcl 43253 | The limit of a sequence of... |
| rexlimddv2 43254 | Restricted existential eli... |
| xlimclim 43255 | Given a sequence of reals,... |
| xlimconst 43256 | A constant sequence conver... |
| climxlim 43257 | A converging sequence in t... |
| xlimbr 43258 | Express the binary relatio... |
| fuzxrpmcn 43259 | A function mapping from an... |
| cnrefiisplem 43260 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 43261 | A non-real, complex number... |
| xlimxrre 43262 | If a sequence ranging over... |
| xlimmnfvlem1 43263 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 43264 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 43265 | A function converges to mi... |
| xlimconst2 43266 | A sequence that eventually... |
| xlimpnfvlem1 43267 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 43268 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 43269 | A function converges to pl... |
| xlimclim2lem 43270 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 43271 | Given a sequence of extend... |
| xlimmnf 43272 | A function converges to mi... |
| xlimpnf 43273 | A function converges to pl... |
| xlimmnfmpt 43274 | A function converges to pl... |
| xlimpnfmpt 43275 | A function converges to pl... |
| climxlim2lem 43276 | In this lemma for ~ climxl... |
| climxlim2 43277 | A sequence of extended rea... |
| dfxlim2v 43278 | An alternative definition ... |
| dfxlim2 43279 | An alternative definition ... |
| climresd 43280 | A function restricted to u... |
| climresdm 43281 | A real function converges ... |
| dmclimxlim 43282 | A real valued sequence tha... |
| xlimmnflimsup2 43283 | A sequence of extended rea... |
| xlimuni 43284 | An infinite sequence conve... |
| xlimclimdm 43285 | A sequence of extended rea... |
| xlimfun 43286 | The convergence relation o... |
| xlimmnflimsup 43287 | If a sequence of extended ... |
| xlimdm 43288 | Two ways to express that a... |
| xlimpnfxnegmnf2 43289 | A sequence converges to ` ... |
| xlimresdm 43290 | A function converges in th... |
| xlimpnfliminf 43291 | If a sequence of extended ... |
| xlimpnfliminf2 43292 | A sequence of extended rea... |
| xlimliminflimsup 43293 | A sequence of extended rea... |
| xlimlimsupleliminf 43294 | A sequence of extended rea... |
| coseq0 43295 | A complex number whose cos... |
| sinmulcos 43296 | Multiplication formula for... |
| coskpi2 43297 | The cosine of an integer m... |
| cosnegpi 43298 | The cosine of negative ` _... |
| sinaover2ne0 43299 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 43300 | The cosine of an integer m... |
| mulcncff 43301 | The multiplication of two ... |
| cncfmptssg 43302 | A continuous complex funct... |
| constcncfg 43303 | A constant function is a c... |
| idcncfg 43304 | The identity function is a... |
| cncfshift 43305 | A periodic continuous func... |
| resincncf 43306 | ` sin ` restricted to real... |
| addccncf2 43307 | Adding a constant is a con... |
| 0cnf 43308 | The empty set is a continu... |
| fsumcncf 43309 | The finite sum of continuo... |
| cncfperiod 43310 | A periodic continuous func... |
| subcncff 43311 | The subtraction of two con... |
| negcncfg 43312 | The opposite of a continuo... |
| cnfdmsn 43313 | A function with a singleto... |
| cncfcompt 43314 | Composition of continuous ... |
| addcncff 43315 | The sum of two continuous ... |
| ioccncflimc 43316 | Limit at the upper bound o... |
| cncfuni 43317 | A complex function on a su... |
| icccncfext 43318 | A continuous function on a... |
| cncficcgt0 43319 | A the absolute value of a ... |
| icocncflimc 43320 | Limit at the lower bound, ... |
| cncfdmsn 43321 | A complex function with a ... |
| divcncff 43322 | The quotient of two contin... |
| cncfshiftioo 43323 | A periodic continuous func... |
| cncfiooicclem1 43324 | A continuous function ` F ... |
| cncfiooicc 43325 | A continuous function ` F ... |
| cncfiooiccre 43326 | A continuous function ` F ... |
| cncfioobdlem 43327 | ` G ` actually extends ` F... |
| cncfioobd 43328 | A continuous function ` F ... |
| jumpncnp 43329 | Jump discontinuity or disc... |
| cxpcncf2 43330 | The complex power function... |
| fprodcncf 43331 | The finite product of cont... |
| add1cncf 43332 | Addition to a constant is ... |
| add2cncf 43333 | Addition to a constant is ... |
| sub1cncfd 43334 | Subtracting a constant is ... |
| sub2cncfd 43335 | Subtraction from a constan... |
| fprodsub2cncf 43336 | ` F ` is continuous. (Con... |
| fprodadd2cncf 43337 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 43338 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 43339 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 43340 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 43341 | The sequence ` S ` of fini... |
| dvsinexp 43342 | The derivative of sin^N . ... |
| dvcosre 43343 | The real derivative of the... |
| dvsinax 43344 | Derivative exercise: the d... |
| dvsubf 43345 | The subtraction rule for e... |
| dvmptconst 43346 | Function-builder for deriv... |
| dvcnre 43347 | From compex differentiatio... |
| dvmptidg 43348 | Function-builder for deriv... |
| dvresntr 43349 | Function-builder for deriv... |
| fperdvper 43350 | The derivative of a period... |
| dvasinbx 43351 | Derivative exercise: the d... |
| dvresioo 43352 | Restriction of a derivativ... |
| dvdivf 43353 | The quotient rule for ever... |
| dvdivbd 43354 | A sufficient condition for... |
| dvsubcncf 43355 | A sufficient condition for... |
| dvmulcncf 43356 | A sufficient condition for... |
| dvcosax 43357 | Derivative exercise: the d... |
| dvdivcncf 43358 | A sufficient condition for... |
| dvbdfbdioolem1 43359 | Given a function with boun... |
| dvbdfbdioolem2 43360 | A function on an open inte... |
| dvbdfbdioo 43361 | A function on an open inte... |
| ioodvbdlimc1lem1 43362 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 43363 | Limit at the lower bound o... |
| ioodvbdlimc1 43364 | A real function with bound... |
| ioodvbdlimc2lem 43365 | Limit at the upper bound o... |
| ioodvbdlimc2 43366 | A real function with bound... |
| dvdmsscn 43367 | ` X ` is a subset of ` CC ... |
| dvmptmulf 43368 | Function-builder for deriv... |
| dvnmptdivc 43369 | Function-builder for itera... |
| dvdsn1add 43370 | If ` K ` divides ` N ` but... |
| dvxpaek 43371 | Derivative of the polynomi... |
| dvnmptconst 43372 | The ` N ` -th derivative o... |
| dvnxpaek 43373 | The ` n ` -th derivative o... |
| dvnmul 43374 | Function-builder for the `... |
| dvmptfprodlem 43375 | Induction step for ~ dvmpt... |
| dvmptfprod 43376 | Function-builder for deriv... |
| dvnprodlem1 43377 | ` D ` is bijective. (Cont... |
| dvnprodlem2 43378 | Induction step for ~ dvnpr... |
| dvnprodlem3 43379 | The multinomial formula fo... |
| dvnprod 43380 | The multinomial formula fo... |
| itgsin0pilem1 43381 | Calculation of the integra... |
| ibliccsinexp 43382 | sin^n on a closed interval... |
| itgsin0pi 43383 | Calculation of the integra... |
| iblioosinexp 43384 | sin^n on an open integral ... |
| itgsinexplem1 43385 | Integration by parts is ap... |
| itgsinexp 43386 | A recursive formula for th... |
| iblconstmpt 43387 | A constant function is int... |
| itgeq1d 43388 | Equality theorem for an in... |
| mbfres2cn 43389 | Measurability of a piecewi... |
| vol0 43390 | The measure of the empty s... |
| ditgeqiooicc 43391 | A function ` F ` on an ope... |
| volge0 43392 | The volume of a set is alw... |
| cnbdibl 43393 | A continuous bounded funct... |
| snmbl 43394 | A singleton is measurable.... |
| ditgeq3d 43395 | Equality theorem for the d... |
| iblempty 43396 | The empty function is inte... |
| iblsplit 43397 | The union of two integrabl... |
| volsn 43398 | A singleton has 0 Lebesgue... |
| itgvol0 43399 | If the domani is negligibl... |
| itgcoscmulx 43400 | Exercise: the integral of ... |
| iblsplitf 43401 | A version of ~ iblsplit us... |
| ibliooicc 43402 | If a function is integrabl... |
| volioc 43403 | The measure of a left-open... |
| iblspltprt 43404 | If a function is integrabl... |
| itgsincmulx 43405 | Exercise: the integral of ... |
| itgsubsticclem 43406 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 43407 | Integration by u-substitut... |
| itgioocnicc 43408 | The integral of a piecewis... |
| iblcncfioo 43409 | A continuous function ` F ... |
| itgspltprt 43410 | The ` S. ` integral splits... |
| itgiccshift 43411 | The integral of a function... |
| itgperiod 43412 | The integral of a periodic... |
| itgsbtaddcnst 43413 | Integral substitution, add... |
| volico 43414 | The measure of left-closed... |
| sublevolico 43415 | The Lebesgue measure of a ... |
| dmvolss 43416 | Lebesgue measurable sets a... |
| ismbl3 43417 | The predicate " ` A ` is L... |
| volioof 43418 | The function that assigns ... |
| ovolsplit 43419 | The Lebesgue outer measure... |
| fvvolioof 43420 | The function value of the ... |
| volioore 43421 | The measure of an open int... |
| fvvolicof 43422 | The function value of the ... |
| voliooico 43423 | An open interval and a lef... |
| ismbl4 43424 | The predicate " ` A ` is L... |
| volioofmpt 43425 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 43426 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 43427 | The Lebesgue measure of op... |
| volicofmpt 43428 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 43429 | The Lebesgue measure of a ... |
| voliccico 43430 | A closed interval and a le... |
| mbfdmssre 43431 | The domain of a measurable... |
| stoweidlem1 43432 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 43433 | lemma for ~ stoweid : here... |
| stoweidlem3 43434 | Lemma for ~ stoweid : if `... |
| stoweidlem4 43435 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 43436 | There exists a δ as ... |
| stoweidlem6 43437 | Lemma for ~ stoweid : two ... |
| stoweidlem7 43438 | This lemma is used to prov... |
| stoweidlem8 43439 | Lemma for ~ stoweid : two ... |
| stoweidlem9 43440 | Lemma for ~ stoweid : here... |
| stoweidlem10 43441 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 43442 | This lemma is used to prov... |
| stoweidlem12 43443 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 43444 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 43445 | There exists a ` k ` as in... |
| stoweidlem15 43446 | This lemma is used to prov... |
| stoweidlem16 43447 | Lemma for ~ stoweid . The... |
| stoweidlem17 43448 | This lemma proves that the... |
| stoweidlem18 43449 | This theorem proves Lemma ... |
| stoweidlem19 43450 | If a set of real functions... |
| stoweidlem20 43451 | If a set A of real functio... |
| stoweidlem21 43452 | Once the Stone Weierstrass... |
| stoweidlem22 43453 | If a set of real functions... |
| stoweidlem23 43454 | This lemma is used to prov... |
| stoweidlem24 43455 | This lemma proves that for... |
| stoweidlem25 43456 | This lemma proves that for... |
| stoweidlem26 43457 | This lemma is used to prov... |
| stoweidlem27 43458 | This lemma is used to prov... |
| stoweidlem28 43459 | There exists a δ as ... |
| stoweidlem29 43460 | When the hypothesis for th... |
| stoweidlem30 43461 | This lemma is used to prov... |
| stoweidlem31 43462 | This lemma is used to prov... |
| stoweidlem32 43463 | If a set A of real functio... |
| stoweidlem33 43464 | If a set of real functions... |
| stoweidlem34 43465 | This lemma proves that for... |
| stoweidlem35 43466 | This lemma is used to prov... |
| stoweidlem36 43467 | This lemma is used to prov... |
| stoweidlem37 43468 | This lemma is used to prov... |
| stoweidlem38 43469 | This lemma is used to prov... |
| stoweidlem39 43470 | This lemma is used to prov... |
| stoweidlem40 43471 | This lemma proves that q_n... |
| stoweidlem41 43472 | This lemma is used to prov... |
| stoweidlem42 43473 | This lemma is used to prov... |
| stoweidlem43 43474 | This lemma is used to prov... |
| stoweidlem44 43475 | This lemma is used to prov... |
| stoweidlem45 43476 | This lemma proves that, gi... |
| stoweidlem46 43477 | This lemma proves that set... |
| stoweidlem47 43478 | Subtracting a constant fro... |
| stoweidlem48 43479 | This lemma is used to prov... |
| stoweidlem49 43480 | There exists a function q_... |
| stoweidlem50 43481 | This lemma proves that set... |
| stoweidlem51 43482 | There exists a function x ... |
| stoweidlem52 43483 | There exists a neighborhoo... |
| stoweidlem53 43484 | This lemma is used to prov... |
| stoweidlem54 43485 | There exists a function ` ... |
| stoweidlem55 43486 | This lemma proves the exis... |
| stoweidlem56 43487 | This theorem proves Lemma ... |
| stoweidlem57 43488 | There exists a function x ... |
| stoweidlem58 43489 | This theorem proves Lemma ... |
| stoweidlem59 43490 | This lemma proves that the... |
| stoweidlem60 43491 | This lemma proves that the... |
| stoweidlem61 43492 | This lemma proves that the... |
| stoweidlem62 43493 | This theorem proves the St... |
| stoweid 43494 | This theorem proves the St... |
| stowei 43495 | This theorem proves the St... |
| wallispilem1 43496 | ` I ` is monotone: increas... |
| wallispilem2 43497 | A first set of properties ... |
| wallispilem3 43498 | I maps to real values. (C... |
| wallispilem4 43499 | ` F ` maps to explicit exp... |
| wallispilem5 43500 | The sequence ` H ` converg... |
| wallispi 43501 | Wallis' formula for π :... |
| wallispi2lem1 43502 | An intermediate step betwe... |
| wallispi2lem2 43503 | Two expressions are proven... |
| wallispi2 43504 | An alternative version of ... |
| stirlinglem1 43505 | A simple limit of fraction... |
| stirlinglem2 43506 | ` A ` maps to positive rea... |
| stirlinglem3 43507 | Long but simple algebraic ... |
| stirlinglem4 43508 | Algebraic manipulation of ... |
| stirlinglem5 43509 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 43510 | A series that converges to... |
| stirlinglem7 43511 | Algebraic manipulation of ... |
| stirlinglem8 43512 | If ` A ` converges to ` C ... |
| stirlinglem9 43513 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 43514 | A bound for any B(N)-B(N +... |
| stirlinglem11 43515 | ` B ` is decreasing. (Con... |
| stirlinglem12 43516 | The sequence ` B ` is boun... |
| stirlinglem13 43517 | ` B ` is decreasing and ha... |
| stirlinglem14 43518 | The sequence ` A ` converg... |
| stirlinglem15 43519 | The Stirling's formula is ... |
| stirling 43520 | Stirling's approximation f... |
| stirlingr 43521 | Stirling's approximation f... |
| dirkerval 43522 | The N_th Dirichlet Kernel.... |
| dirker2re 43523 | The Dirichlet Kernel value... |
| dirkerdenne0 43524 | The Dirichlet Kernel denom... |
| dirkerval2 43525 | The N_th Dirichlet Kernel ... |
| dirkerre 43526 | The Dirichlet Kernel at an... |
| dirkerper 43527 | the Dirichlet Kernel has p... |
| dirkerf 43528 | For any natural number ` N... |
| dirkertrigeqlem1 43529 | Sum of an even number of a... |
| dirkertrigeqlem2 43530 | Trigonomic equality lemma ... |
| dirkertrigeqlem3 43531 | Trigonometric equality lem... |
| dirkertrigeq 43532 | Trigonometric equality for... |
| dirkeritg 43533 | The definite integral of t... |
| dirkercncflem1 43534 | If ` Y ` is a multiple of ... |
| dirkercncflem2 43535 | Lemma used to prove that t... |
| dirkercncflem3 43536 | The Dirichlet Kernel is co... |
| dirkercncflem4 43537 | The Dirichlet Kernel is co... |
| dirkercncf 43538 | For any natural number ` N... |
| fourierdlem1 43539 | A partition interval is a ... |
| fourierdlem2 43540 | Membership in a partition.... |
| fourierdlem3 43541 | Membership in a partition.... |
| fourierdlem4 43542 | ` E ` is a function that m... |
| fourierdlem5 43543 | ` S ` is a function. (Con... |
| fourierdlem6 43544 | ` X ` is in the periodic p... |
| fourierdlem7 43545 | The difference between the... |
| fourierdlem8 43546 | A partition interval is a ... |
| fourierdlem9 43547 | ` H ` is a complex functio... |
| fourierdlem10 43548 | Condition on the bounds of... |
| fourierdlem11 43549 | If there is a partition, t... |
| fourierdlem12 43550 | A point of a partition is ... |
| fourierdlem13 43551 | Value of ` V ` in terms of... |
| fourierdlem14 43552 | Given the partition ` V ` ... |
| fourierdlem15 43553 | The range of the partition... |
| fourierdlem16 43554 | The coefficients of the fo... |
| fourierdlem17 43555 | The defined ` L ` is actua... |
| fourierdlem18 43556 | The function ` S ` is cont... |
| fourierdlem19 43557 | If two elements of ` D ` h... |
| fourierdlem20 43558 | Every interval in the part... |
| fourierdlem21 43559 | The coefficients of the fo... |
| fourierdlem22 43560 | The coefficients of the fo... |
| fourierdlem23 43561 | If ` F ` is continuous and... |
| fourierdlem24 43562 | A sufficient condition for... |
| fourierdlem25 43563 | If ` C ` is not in the ran... |
| fourierdlem26 43564 | Periodic image of a point ... |
| fourierdlem27 43565 | A partition open interval ... |
| fourierdlem28 43566 | Derivative of ` ( F `` ( X... |
| fourierdlem29 43567 | Explicit function value fo... |
| fourierdlem30 43568 | Sum of three small pieces ... |
| fourierdlem31 43569 | If ` A ` is finite and for... |
| fourierdlem32 43570 | Limit of a continuous func... |
| fourierdlem33 43571 | Limit of a continuous func... |
| fourierdlem34 43572 | A partition is one to one.... |
| fourierdlem35 43573 | There is a single point in... |
| fourierdlem36 43574 | ` F ` is an isomorphism. ... |
| fourierdlem37 43575 | ` I ` is a function that m... |
| fourierdlem38 43576 | The function ` F ` is cont... |
| fourierdlem39 43577 | Integration by parts of ... |
| fourierdlem40 43578 | ` H ` is a continuous func... |
| fourierdlem41 43579 | Lemma used to prove that e... |
| fourierdlem42 43580 | The set of points in a mov... |
| fourierdlem43 43581 | ` K ` is a real function. ... |
| fourierdlem44 43582 | A condition for having ` (... |
| fourierdlem46 43583 | The function ` F ` has a l... |
| fourierdlem47 43584 | For ` r ` large enough, th... |
| fourierdlem48 43585 | The given periodic functio... |
| fourierdlem49 43586 | The given periodic functio... |
| fourierdlem50 43587 | Continuity of ` O ` and it... |
| fourierdlem51 43588 | ` X ` is in the periodic p... |
| fourierdlem52 43589 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 43590 | The limit of ` F ( s ) ` a... |
| fourierdlem54 43591 | Given a partition ` Q ` an... |
| fourierdlem55 43592 | ` U ` is a real function. ... |
| fourierdlem56 43593 | Derivative of the ` K ` fu... |
| fourierdlem57 43594 | The derivative of ` O ` . ... |
| fourierdlem58 43595 | The derivative of ` K ` is... |
| fourierdlem59 43596 | The derivative of ` H ` is... |
| fourierdlem60 43597 | Given a differentiable fun... |
| fourierdlem61 43598 | Given a differentiable fun... |
| fourierdlem62 43599 | The function ` K ` is cont... |
| fourierdlem63 43600 | The upper bound of interva... |
| fourierdlem64 43601 | The partition ` V ` is fin... |
| fourierdlem65 43602 | The distance of two adjace... |
| fourierdlem66 43603 | Value of the ` G ` functio... |
| fourierdlem67 43604 | ` G ` is a function. (Con... |
| fourierdlem68 43605 | The derivative of ` O ` is... |
| fourierdlem69 43606 | A piecewise continuous fun... |
| fourierdlem70 43607 | A piecewise continuous fun... |
| fourierdlem71 43608 | A periodic piecewise conti... |
| fourierdlem72 43609 | The derivative of ` O ` is... |
| fourierdlem73 43610 | A version of the Riemann L... |
| fourierdlem74 43611 | Given a piecewise smooth f... |
| fourierdlem75 43612 | Given a piecewise smooth f... |
| fourierdlem76 43613 | Continuity of ` O ` and it... |
| fourierdlem77 43614 | If ` H ` is bounded, then ... |
| fourierdlem78 43615 | ` G ` is continuous when r... |
| fourierdlem79 43616 | ` E ` projects every inter... |
| fourierdlem80 43617 | The derivative of ` O ` is... |
| fourierdlem81 43618 | The integral of a piecewis... |
| fourierdlem82 43619 | Integral by substitution, ... |
| fourierdlem83 43620 | The fourier partial sum fo... |
| fourierdlem84 43621 | If ` F ` is piecewise coni... |
| fourierdlem85 43622 | Limit of the function ` G ... |
| fourierdlem86 43623 | Continuity of ` O ` and it... |
| fourierdlem87 43624 | The integral of ` G ` goes... |
| fourierdlem88 43625 | Given a piecewise continuo... |
| fourierdlem89 43626 | Given a piecewise continuo... |
| fourierdlem90 43627 | Given a piecewise continuo... |
| fourierdlem91 43628 | Given a piecewise continuo... |
| fourierdlem92 43629 | The integral of a piecewis... |
| fourierdlem93 43630 | Integral by substitution (... |
| fourierdlem94 43631 | For a piecewise smooth fun... |
| fourierdlem95 43632 | Algebraic manipulation of ... |
| fourierdlem96 43633 | limit for ` F ` at the low... |
| fourierdlem97 43634 | ` F ` is continuous on the... |
| fourierdlem98 43635 | ` F ` is continuous on the... |
| fourierdlem99 43636 | limit for ` F ` at the upp... |
| fourierdlem100 43637 | A piecewise continuous fun... |
| fourierdlem101 43638 | Integral by substitution f... |
| fourierdlem102 43639 | For a piecewise smooth fun... |
| fourierdlem103 43640 | The half lower part of the... |
| fourierdlem104 43641 | The half upper part of the... |
| fourierdlem105 43642 | A piecewise continuous fun... |
| fourierdlem106 43643 | For a piecewise smooth fun... |
| fourierdlem107 43644 | The integral of a piecewis... |
| fourierdlem108 43645 | The integral of a piecewis... |
| fourierdlem109 43646 | The integral of a piecewis... |
| fourierdlem110 43647 | The integral of a piecewis... |
| fourierdlem111 43648 | The fourier partial sum fo... |
| fourierdlem112 43649 | Here abbreviations (local ... |
| fourierdlem113 43650 | Fourier series convergence... |
| fourierdlem114 43651 | Fourier series convergence... |
| fourierdlem115 43652 | Fourier serier convergence... |
| fourierd 43653 | Fourier series convergence... |
| fourierclimd 43654 | Fourier series convergence... |
| fourierclim 43655 | Fourier series convergence... |
| fourier 43656 | Fourier series convergence... |
| fouriercnp 43657 | If ` F ` is continuous at ... |
| fourier2 43658 | Fourier series convergence... |
| sqwvfoura 43659 | Fourier coefficients for t... |
| sqwvfourb 43660 | Fourier series ` B ` coeff... |
| fourierswlem 43661 | The Fourier series for the... |
| fouriersw 43662 | Fourier series convergence... |
| fouriercn 43663 | If the derivative of ` F `... |
| elaa2lem 43664 | Elementhood in the set of ... |
| elaa2 43665 | Elementhood in the set of ... |
| etransclem1 43666 | ` H ` is a function. (Con... |
| etransclem2 43667 | Derivative of ` G ` . (Co... |
| etransclem3 43668 | The given ` if ` term is a... |
| etransclem4 43669 | ` F ` expressed as a finit... |
| etransclem5 43670 | A change of bound variable... |
| etransclem6 43671 | A change of bound variable... |
| etransclem7 43672 | The given product is an in... |
| etransclem8 43673 | ` F ` is a function. (Con... |
| etransclem9 43674 | If ` K ` divides ` N ` but... |
| etransclem10 43675 | The given ` if ` term is a... |
| etransclem11 43676 | A change of bound variable... |
| etransclem12 43677 | ` C ` applied to ` N ` . ... |
| etransclem13 43678 | ` F ` applied to ` Y ` . ... |
| etransclem14 43679 | Value of the term ` T ` , ... |
| etransclem15 43680 | Value of the term ` T ` , ... |
| etransclem16 43681 | Every element in the range... |
| etransclem17 43682 | The ` N ` -th derivative o... |
| etransclem18 43683 | The given function is inte... |
| etransclem19 43684 | The ` N ` -th derivative o... |
| etransclem20 43685 | ` H ` is smooth. (Contrib... |
| etransclem21 43686 | The ` N ` -th derivative o... |
| etransclem22 43687 | The ` N ` -th derivative o... |
| etransclem23 43688 | This is the claim proof in... |
| etransclem24 43689 | ` P ` divides the I -th de... |
| etransclem25 43690 | ` P ` factorial divides th... |
| etransclem26 43691 | Every term in the sum of t... |
| etransclem27 43692 | The ` N ` -th derivative o... |
| etransclem28 43693 | ` ( P - 1 ) ` factorial di... |
| etransclem29 43694 | The ` N ` -th derivative o... |
| etransclem30 43695 | The ` N ` -th derivative o... |
| etransclem31 43696 | The ` N ` -th derivative o... |
| etransclem32 43697 | This is the proof for the ... |
| etransclem33 43698 | ` F ` is smooth. (Contrib... |
| etransclem34 43699 | The ` N ` -th derivative o... |
| etransclem35 43700 | ` P ` does not divide the ... |
| etransclem36 43701 | The ` N ` -th derivative o... |
| etransclem37 43702 | ` ( P - 1 ) ` factorial di... |
| etransclem38 43703 | ` P ` divides the I -th de... |
| etransclem39 43704 | ` G ` is a function. (Con... |
| etransclem40 43705 | The ` N ` -th derivative o... |
| etransclem41 43706 | ` P ` does not divide the ... |
| etransclem42 43707 | The ` N ` -th derivative o... |
| etransclem43 43708 | ` G ` is a continuous func... |
| etransclem44 43709 | The given finite sum is no... |
| etransclem45 43710 | ` K ` is an integer. (Con... |
| etransclem46 43711 | This is the proof for equa... |
| etransclem47 43712 | ` _e ` is transcendental. ... |
| etransclem48 43713 | ` _e ` is transcendental. ... |
| etransc 43714 | ` _e ` is transcendental. ... |
| rrxtopn 43715 | The topology of the genera... |
| rrxngp 43716 | Generalized Euclidean real... |
| rrxtps 43717 | Generalized Euclidean real... |
| rrxtopnfi 43718 | The topology of the n-dime... |
| rrxtopon 43719 | The topology on generalize... |
| rrxtop 43720 | The topology on generalize... |
| rrndistlt 43721 | Given two points in the sp... |
| rrxtoponfi 43722 | The topology on n-dimensio... |
| rrxunitopnfi 43723 | The base set of the standa... |
| rrxtopn0 43724 | The topology of the zero-d... |
| qndenserrnbllem 43725 | n-dimensional rational num... |
| qndenserrnbl 43726 | n-dimensional rational num... |
| rrxtopn0b 43727 | The topology of the zero-d... |
| qndenserrnopnlem 43728 | n-dimensional rational num... |
| qndenserrnopn 43729 | n-dimensional rational num... |
| qndenserrn 43730 | n-dimensional rational num... |
| rrxsnicc 43731 | A multidimensional singlet... |
| rrnprjdstle 43732 | The distance between two p... |
| rrndsmet 43733 | ` D ` is a metric for the ... |
| rrndsxmet 43734 | ` D ` is an extended metri... |
| ioorrnopnlem 43735 | The a point in an indexed ... |
| ioorrnopn 43736 | The indexed product of ope... |
| ioorrnopnxrlem 43737 | Given a point ` F ` that b... |
| ioorrnopnxr 43738 | The indexed product of ope... |
| issal 43745 | Express the predicate " ` ... |
| pwsal 43746 | The power set of a given s... |
| salunicl 43747 | SAlg sigma-algebra is clos... |
| saluncl 43748 | The union of two sets in a... |
| prsal 43749 | The pair of the empty set ... |
| saldifcl 43750 | The complement of an eleme... |
| 0sal 43751 | The empty set belongs to e... |
| salgenval 43752 | The sigma-algebra generate... |
| saliuncl 43753 | SAlg sigma-algebra is clos... |
| salincl 43754 | The intersection of two se... |
| saluni 43755 | A set is an element of any... |
| saliincl 43756 | SAlg sigma-algebra is clos... |
| saldifcl2 43757 | The difference of two elem... |
| intsaluni 43758 | The union of an arbitrary ... |
| intsal 43759 | The arbitrary intersection... |
| salgenn0 43760 | The set used in the defini... |
| salgencl 43761 | ` SalGen ` actually genera... |
| issald 43762 | Sufficient condition to pr... |
| salexct 43763 | An example of nontrivial s... |
| sssalgen 43764 | A set is a subset of the s... |
| salgenss 43765 | The sigma-algebra generate... |
| salgenuni 43766 | The base set of the sigma-... |
| issalgend 43767 | One side of ~ dfsalgen2 . ... |
| salexct2 43768 | An example of a subset tha... |
| unisalgen 43769 | The union of a set belongs... |
| dfsalgen2 43770 | Alternate characterization... |
| salexct3 43771 | An example of a sigma-alge... |
| salgencntex 43772 | This counterexample shows ... |
| salgensscntex 43773 | This counterexample shows ... |
| issalnnd 43774 | Sufficient condition to pr... |
| dmvolsal 43775 | Lebesgue measurable sets f... |
| saldifcld 43776 | The complement of an eleme... |
| saluncld 43777 | The union of two sets in a... |
| salgencld 43778 | ` SalGen ` actually genera... |
| 0sald 43779 | The empty set belongs to e... |
| iooborel 43780 | An open interval is a Bore... |
| salincld 43781 | The intersection of two se... |
| salunid 43782 | A set is an element of any... |
| unisalgen2 43783 | The union of a set belongs... |
| bor1sal 43784 | The Borel sigma-algebra on... |
| iocborel 43785 | A left-open, right-closed ... |
| subsaliuncllem 43786 | A subspace sigma-algebra i... |
| subsaliuncl 43787 | A subspace sigma-algebra i... |
| subsalsal 43788 | A subspace sigma-algebra i... |
| subsaluni 43789 | A set belongs to the subsp... |
| sge0rnre 43792 | When ` sum^ ` is applied t... |
| fge0icoicc 43793 | If ` F ` maps to nonnegati... |
| sge0val 43794 | The value of the sum of no... |
| fge0npnf 43795 | If ` F ` maps to nonnegati... |
| sge0rnn0 43796 | The range used in the defi... |
| sge0vald 43797 | The value of the sum of no... |
| fge0iccico 43798 | A range of nonnegative ext... |
| gsumge0cl 43799 | Closure of group sum, for ... |
| sge0reval 43800 | Value of the sum of nonneg... |
| sge0pnfval 43801 | If a term in the sum of no... |
| fge0iccre 43802 | A range of nonnegative ext... |
| sge0z 43803 | Any nonnegative extended s... |
| sge00 43804 | The sum of nonnegative ext... |
| fsumlesge0 43805 | Every finite subsum of non... |
| sge0revalmpt 43806 | Value of the sum of nonneg... |
| sge0sn 43807 | A sum of a nonnegative ext... |
| sge0tsms 43808 | ` sum^ ` applied to a nonn... |
| sge0cl 43809 | The arbitrary sum of nonne... |
| sge0f1o 43810 | Re-index a nonnegative ext... |
| sge0snmpt 43811 | A sum of a nonnegative ext... |
| sge0ge0 43812 | The sum of nonnegative ext... |
| sge0xrcl 43813 | The arbitrary sum of nonne... |
| sge0repnf 43814 | The of nonnegative extende... |
| sge0fsum 43815 | The arbitrary sum of a fin... |
| sge0rern 43816 | If the sum of nonnegative ... |
| sge0supre 43817 | If the arbitrary sum of no... |
| sge0fsummpt 43818 | The arbitrary sum of a fin... |
| sge0sup 43819 | The arbitrary sum of nonne... |
| sge0less 43820 | A shorter sum of nonnegati... |
| sge0rnbnd 43821 | The range used in the defi... |
| sge0pr 43822 | Sum of a pair of nonnegati... |
| sge0gerp 43823 | The arbitrary sum of nonne... |
| sge0pnffigt 43824 | If the sum of nonnegative ... |
| sge0ssre 43825 | If a sum of nonnegative ex... |
| sge0lefi 43826 | A sum of nonnegative exten... |
| sge0lessmpt 43827 | A shorter sum of nonnegati... |
| sge0ltfirp 43828 | If the sum of nonnegative ... |
| sge0prle 43829 | The sum of a pair of nonne... |
| sge0gerpmpt 43830 | The arbitrary sum of nonne... |
| sge0resrnlem 43831 | The sum of nonnegative ext... |
| sge0resrn 43832 | The sum of nonnegative ext... |
| sge0ssrempt 43833 | If a sum of nonnegative ex... |
| sge0resplit 43834 | ` sum^ ` splits into two p... |
| sge0le 43835 | If all of the terms of sum... |
| sge0ltfirpmpt 43836 | If the extended sum of non... |
| sge0split 43837 | Split a sum of nonnegative... |
| sge0lempt 43838 | If all of the terms of sum... |
| sge0splitmpt 43839 | Split a sum of nonnegative... |
| sge0ss 43840 | Change the index set to a ... |
| sge0iunmptlemfi 43841 | Sum of nonnegative extende... |
| sge0p1 43842 | The addition of the next t... |
| sge0iunmptlemre 43843 | Sum of nonnegative extende... |
| sge0fodjrnlem 43844 | Re-index a nonnegative ext... |
| sge0fodjrn 43845 | Re-index a nonnegative ext... |
| sge0iunmpt 43846 | Sum of nonnegative extende... |
| sge0iun 43847 | Sum of nonnegative extende... |
| sge0nemnf 43848 | The generalized sum of non... |
| sge0rpcpnf 43849 | The sum of an infinite num... |
| sge0rernmpt 43850 | If the sum of nonnegative ... |
| sge0lefimpt 43851 | A sum of nonnegative exten... |
| nn0ssge0 43852 | Nonnegative integers are n... |
| sge0clmpt 43853 | The generalized sum of non... |
| sge0ltfirpmpt2 43854 | If the extended sum of non... |
| sge0isum 43855 | If a series of nonnegative... |
| sge0xrclmpt 43856 | The generalized sum of non... |
| sge0xp 43857 | Combine two generalized su... |
| sge0isummpt 43858 | If a series of nonnegative... |
| sge0ad2en 43859 | The value of the infinite ... |
| sge0isummpt2 43860 | If a series of nonnegative... |
| sge0xaddlem1 43861 | The extended addition of t... |
| sge0xaddlem2 43862 | The extended addition of t... |
| sge0xadd 43863 | The extended addition of t... |
| sge0fsummptf 43864 | The generalized sum of a f... |
| sge0snmptf 43865 | A sum of a nonnegative ext... |
| sge0ge0mpt 43866 | The sum of nonnegative ext... |
| sge0repnfmpt 43867 | The of nonnegative extende... |
| sge0pnffigtmpt 43868 | If the generalized sum of ... |
| sge0splitsn 43869 | Separate out a term in a g... |
| sge0pnffsumgt 43870 | If the sum of nonnegative ... |
| sge0gtfsumgt 43871 | If the generalized sum of ... |
| sge0uzfsumgt 43872 | If a real number is smalle... |
| sge0pnfmpt 43873 | If a term in the sum of no... |
| sge0seq 43874 | A series of nonnegative re... |
| sge0reuz 43875 | Value of the generalized s... |
| sge0reuzb 43876 | Value of the generalized s... |
| ismea 43879 | Express the predicate " ` ... |
| dmmeasal 43880 | The domain of a measure is... |
| meaf 43881 | A measure is a function th... |
| mea0 43882 | The measure of the empty s... |
| nnfoctbdjlem 43883 | There exists a mapping fro... |
| nnfoctbdj 43884 | There exists a mapping fro... |
| meadjuni 43885 | The measure of the disjoin... |
| meacl 43886 | The measure of a set is a ... |
| iundjiunlem 43887 | The sets in the sequence `... |
| iundjiun 43888 | Given a sequence ` E ` of ... |
| meaxrcl 43889 | The measure of a set is an... |
| meadjun 43890 | The measure of the union o... |
| meassle 43891 | The measure of a set is gr... |
| meaunle 43892 | The measure of the union o... |
| meadjiunlem 43893 | The sum of nonnegative ext... |
| meadjiun 43894 | The measure of the disjoin... |
| ismeannd 43895 | Sufficient condition to pr... |
| meaiunlelem 43896 | The measure of the union o... |
| meaiunle 43897 | The measure of the union o... |
| psmeasurelem 43898 | ` M ` applied to a disjoin... |
| psmeasure 43899 | Point supported measure, R... |
| voliunsge0lem 43900 | The Lebesgue measure funct... |
| voliunsge0 43901 | The Lebesgue measure funct... |
| volmea 43902 | The Lebeasgue measure on t... |
| meage0 43903 | If the measure of a measur... |
| meadjunre 43904 | The measure of the union o... |
| meassre 43905 | If the measure of a measur... |
| meale0eq0 43906 | A measure that is less tha... |
| meadif 43907 | The measure of the differe... |
| meaiuninclem 43908 | Measures are continuous fr... |
| meaiuninc 43909 | Measures are continuous fr... |
| meaiuninc2 43910 | Measures are continuous fr... |
| meaiunincf 43911 | Measures are continuous fr... |
| meaiuninc3v 43912 | Measures are continuous fr... |
| meaiuninc3 43913 | Measures are continuous fr... |
| meaiininclem 43914 | Measures are continuous fr... |
| meaiininc 43915 | Measures are continuous fr... |
| meaiininc2 43916 | Measures are continuous fr... |
| caragenval 43921 | The sigma-algebra generate... |
| isome 43922 | Express the predicate " ` ... |
| caragenel 43923 | Membership in the Caratheo... |
| omef 43924 | An outer measure is a func... |
| ome0 43925 | The outer measure of the e... |
| omessle 43926 | The outer measure of a set... |
| omedm 43927 | The domain of an outer mea... |
| caragensplit 43928 | If ` E ` is in the set gen... |
| caragenelss 43929 | An element of the Caratheo... |
| carageneld 43930 | Membership in the Caratheo... |
| omecl 43931 | The outer measure of a set... |
| caragenss 43932 | The sigma-algebra generate... |
| omeunile 43933 | The outer measure of the u... |
| caragen0 43934 | The empty set belongs to a... |
| omexrcl 43935 | The outer measure of a set... |
| caragenunidm 43936 | The base set of an outer m... |
| caragensspw 43937 | The sigma-algebra generate... |
| omessre 43938 | If the outer measure of a ... |
| caragenuni 43939 | The base set of the sigma-... |
| caragenuncllem 43940 | The Caratheodory's constru... |
| caragenuncl 43941 | The Caratheodory's constru... |
| caragendifcl 43942 | The Caratheodory's constru... |
| caragenfiiuncl 43943 | The Caratheodory's constru... |
| omeunle 43944 | The outer measure of the u... |
| omeiunle 43945 | The outer measure of the i... |
| omelesplit 43946 | The outer measure of a set... |
| omeiunltfirp 43947 | If the outer measure of a ... |
| omeiunlempt 43948 | The outer measure of the i... |
| carageniuncllem1 43949 | The outer measure of ` A i... |
| carageniuncllem2 43950 | The Caratheodory's constru... |
| carageniuncl 43951 | The Caratheodory's constru... |
| caragenunicl 43952 | The Caratheodory's constru... |
| caragensal 43953 | Caratheodory's method gene... |
| caratheodorylem1 43954 | Lemma used to prove that C... |
| caratheodorylem2 43955 | Caratheodory's constructio... |
| caratheodory 43956 | Caratheodory's constructio... |
| 0ome 43957 | The map that assigns 0 to ... |
| isomenndlem 43958 | ` O ` is sub-additive w.r.... |
| isomennd 43959 | Sufficient condition to pr... |
| caragenel2d 43960 | Membership in the Caratheo... |
| omege0 43961 | If the outer measure of a ... |
| omess0 43962 | If the outer measure of a ... |
| caragencmpl 43963 | A measure built with the C... |
| vonval 43968 | Value of the Lebesgue meas... |
| ovnval 43969 | Value of the Lebesgue oute... |
| elhoi 43970 | Membership in a multidimen... |
| icoresmbl 43971 | A closed-below, open-above... |
| hoissre 43972 | The projection of a half-o... |
| ovnval2 43973 | Value of the Lebesgue oute... |
| volicorecl 43974 | The Lebesgue measure of a ... |
| hoiprodcl 43975 | The pre-measure of half-op... |
| hoicvr 43976 | ` I ` is a countable set o... |
| hoissrrn 43977 | A half-open interval is a ... |
| ovn0val 43978 | The Lebesgue outer measure... |
| ovnn0val 43979 | The value of a (multidimen... |
| ovnval2b 43980 | Value of the Lebesgue oute... |
| volicorescl 43981 | The Lebesgue measure of a ... |
| ovnprodcl 43982 | The product used in the de... |
| hoiprodcl2 43983 | The pre-measure of half-op... |
| hoicvrrex 43984 | Any subset of the multidim... |
| ovnsupge0 43985 | The set used in the defini... |
| ovnlecvr 43986 | Given a subset of multidim... |
| ovnpnfelsup 43987 | ` +oo ` is an element of t... |
| ovnsslelem 43988 | The (multidimensional, non... |
| ovnssle 43989 | The (multidimensional) Leb... |
| ovnlerp 43990 | The Lebesgue outer measure... |
| ovnf 43991 | The Lebesgue outer measure... |
| ovncvrrp 43992 | The Lebesgue outer measure... |
| ovn0lem 43993 | For any finite dimension, ... |
| ovn0 43994 | For any finite dimension, ... |
| ovncl 43995 | The Lebesgue outer measure... |
| ovn02 43996 | For the zero-dimensional s... |
| ovnxrcl 43997 | The Lebesgue outer measure... |
| ovnsubaddlem1 43998 | The Lebesgue outer measure... |
| ovnsubaddlem2 43999 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 44000 | ` ( voln* `` X ) ` is suba... |
| ovnome 44001 | ` ( voln* `` X ) ` is an o... |
| vonmea 44002 | ` ( voln `` X ) ` is a mea... |
| volicon0 44003 | The measure of a nonempty ... |
| hsphoif 44004 | ` H ` is a function (that ... |
| hoidmvval 44005 | The dimensional volume of ... |
| hoissrrn2 44006 | A half-open interval is a ... |
| hsphoival 44007 | ` H ` is a function (that ... |
| hoiprodcl3 44008 | The pre-measure of half-op... |
| volicore 44009 | The Lebesgue measure of a ... |
| hoidmvcl 44010 | The dimensional volume of ... |
| hoidmv0val 44011 | The dimensional volume of ... |
| hoidmvn0val 44012 | The dimensional volume of ... |
| hsphoidmvle2 44013 | The dimensional volume of ... |
| hsphoidmvle 44014 | The dimensional volume of ... |
| hoidmvval0 44015 | The dimensional volume of ... |
| hoiprodp1 44016 | The dimensional volume of ... |
| sge0hsphoire 44017 | If the generalized sum of ... |
| hoidmvval0b 44018 | The dimensional volume of ... |
| hoidmv1lelem1 44019 | The supremum of ` U ` belo... |
| hoidmv1lelem2 44020 | This is the contradiction ... |
| hoidmv1lelem3 44021 | The dimensional volume of ... |
| hoidmv1le 44022 | The dimensional volume of ... |
| hoidmvlelem1 44023 | The supremum of ` U ` belo... |
| hoidmvlelem2 44024 | This is the contradiction ... |
| hoidmvlelem3 44025 | This is the contradiction ... |
| hoidmvlelem4 44026 | The dimensional volume of ... |
| hoidmvlelem5 44027 | The dimensional volume of ... |
| hoidmvle 44028 | The dimensional volume of ... |
| ovnhoilem1 44029 | The Lebesgue outer measure... |
| ovnhoilem2 44030 | The Lebesgue outer measure... |
| ovnhoi 44031 | The Lebesgue outer measure... |
| dmovn 44032 | The domain of the Lebesgue... |
| hoicoto2 44033 | The half-open interval exp... |
| dmvon 44034 | Lebesgue measurable n-dime... |
| hoi2toco 44035 | The half-open interval exp... |
| hoidifhspval 44036 | ` D ` is a function that r... |
| hspval 44037 | The value of the half-spac... |
| ovnlecvr2 44038 | Given a subset of multidim... |
| ovncvr2 44039 | ` B ` and ` T ` are the le... |
| dmovnsal 44040 | The domain of the Lebesgue... |
| unidmovn 44041 | Base set of the n-dimensio... |
| rrnmbl 44042 | The set of n-dimensional R... |
| hoidifhspval2 44043 | ` D ` is a function that r... |
| hspdifhsp 44044 | A n-dimensional half-open ... |
| unidmvon 44045 | Base set of the n-dimensio... |
| hoidifhspf 44046 | ` D ` is a function that r... |
| hoidifhspval3 44047 | ` D ` is a function that r... |
| hoidifhspdmvle 44048 | The dimensional volume of ... |
| voncmpl 44049 | The Lebesgue measure is co... |
| hoiqssbllem1 44050 | The center of the n-dimens... |
| hoiqssbllem2 44051 | The center of the n-dimens... |
| hoiqssbllem3 44052 | A n-dimensional ball conta... |
| hoiqssbl 44053 | A n-dimensional ball conta... |
| hspmbllem1 44054 | Any half-space of the n-di... |
| hspmbllem2 44055 | Any half-space of the n-di... |
| hspmbllem3 44056 | Any half-space of the n-di... |
| hspmbl 44057 | Any half-space of the n-di... |
| hoimbllem 44058 | Any n-dimensional half-ope... |
| hoimbl 44059 | Any n-dimensional half-ope... |
| opnvonmbllem1 44060 | The half-open interval exp... |
| opnvonmbllem2 44061 | An open subset of the n-di... |
| opnvonmbl 44062 | An open subset of the n-di... |
| opnssborel 44063 | Open sets of a generalized... |
| borelmbl 44064 | All Borel subsets of the n... |
| volicorege0 44065 | The Lebesgue measure of a ... |
| isvonmbl 44066 | The predicate " ` A ` is m... |
| mblvon 44067 | The n-dimensional Lebesgue... |
| vonmblss 44068 | n-dimensional Lebesgue mea... |
| volico2 44069 | The measure of left-closed... |
| vonmblss2 44070 | n-dimensional Lebesgue mea... |
| ovolval2lem 44071 | The value of the Lebesgue ... |
| ovolval2 44072 | The value of the Lebesgue ... |
| ovnsubadd2lem 44073 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 44074 | ` ( voln* `` X ) ` is suba... |
| ovolval3 44075 | The value of the Lebesgue ... |
| ovnsplit 44076 | The n-dimensional Lebesgue... |
| ovolval4lem1 44077 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 44078 | The value of the Lebesgue ... |
| ovolval4 44079 | The value of the Lebesgue ... |
| ovolval5lem1 44080 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 44081 | ` |- ( ( ph /\ n e. NN ) -... |
| ovolval5lem3 44082 | The value of the Lebesgue ... |
| ovolval5 44083 | The value of the Lebesgue ... |
| ovnovollem1 44084 | if ` F ` is a cover of ` B... |
| ovnovollem2 44085 | if ` I ` is a cover of ` (... |
| ovnovollem3 44086 | The 1-dimensional Lebesgue... |
| ovnovol 44087 | The 1-dimensional Lebesgue... |
| vonvolmbllem 44088 | If a subset ` B ` of real ... |
| vonvolmbl 44089 | A subset of Real numbers i... |
| vonvol 44090 | The 1-dimensional Lebesgue... |
| vonvolmbl2 44091 | A subset ` X ` of the spac... |
| vonvol2 44092 | The 1-dimensional Lebesgue... |
| hoimbl2 44093 | Any n-dimensional half-ope... |
| voncl 44094 | The Lebesgue measure of a ... |
| vonhoi 44095 | The Lebesgue outer measure... |
| vonxrcl 44096 | The Lebesgue measure of a ... |
| ioosshoi 44097 | A n-dimensional open inter... |
| vonn0hoi 44098 | The Lebesgue outer measure... |
| von0val 44099 | The Lebesgue measure (for ... |
| vonhoire 44100 | The Lebesgue measure of a ... |
| iinhoiicclem 44101 | A n-dimensional closed int... |
| iinhoiicc 44102 | A n-dimensional closed int... |
| iunhoiioolem 44103 | A n-dimensional open inter... |
| iunhoiioo 44104 | A n-dimensional open inter... |
| ioovonmbl 44105 | Any n-dimensional open int... |
| iccvonmbllem 44106 | Any n-dimensional closed i... |
| iccvonmbl 44107 | Any n-dimensional closed i... |
| vonioolem1 44108 | The sequence of the measur... |
| vonioolem2 44109 | The n-dimensional Lebesgue... |
| vonioo 44110 | The n-dimensional Lebesgue... |
| vonicclem1 44111 | The sequence of the measur... |
| vonicclem2 44112 | The n-dimensional Lebesgue... |
| vonicc 44113 | The n-dimensional Lebesgue... |
| snvonmbl 44114 | A n-dimensional singleton ... |
| vonn0ioo 44115 | The n-dimensional Lebesgue... |
| vonn0icc 44116 | The n-dimensional Lebesgue... |
| ctvonmbl 44117 | Any n-dimensional countabl... |
| vonn0ioo2 44118 | The n-dimensional Lebesgue... |
| vonsn 44119 | The n-dimensional Lebesgue... |
| vonn0icc2 44120 | The n-dimensional Lebesgue... |
| vonct 44121 | The n-dimensional Lebesgue... |
| vitali2 44122 | There are non-measurable s... |
| pimltmnf2 44125 | Given a real-valued functi... |
| preimagelt 44126 | The preimage of a right-op... |
| preimalegt 44127 | The preimage of a left-ope... |
| pimconstlt0 44128 | Given a constant function,... |
| pimconstlt1 44129 | Given a constant function,... |
| pimltpnf 44130 | Given a real-valued functi... |
| pimgtpnf2 44131 | Given a real-valued functi... |
| salpreimagelt 44132 | If all the preimages of le... |
| pimrecltpos 44133 | The preimage of an unbound... |
| salpreimalegt 44134 | If all the preimages of ri... |
| pimiooltgt 44135 | The preimage of an open in... |
| preimaicomnf 44136 | Preimage of an open interv... |
| pimltpnf2 44137 | Given a real-valued functi... |
| pimgtmnf2 44138 | Given a real-valued functi... |
| pimdecfgtioc 44139 | Given a nonincreasing func... |
| pimincfltioc 44140 | Given a nondecreasing func... |
| pimdecfgtioo 44141 | Given a nondecreasing func... |
| pimincfltioo 44142 | Given a nondecreasing func... |
| preimaioomnf 44143 | Preimage of an open interv... |
| preimageiingt 44144 | A preimage of a left-close... |
| preimaleiinlt 44145 | A preimage of a left-open,... |
| pimgtmnf 44146 | Given a real-valued functi... |
| pimrecltneg 44147 | The preimage of an unbound... |
| salpreimagtge 44148 | If all the preimages of le... |
| salpreimaltle 44149 | If all the preimages of ri... |
| issmflem 44150 | The predicate " ` F ` is a... |
| issmf 44151 | The predicate " ` F ` is a... |
| salpreimalelt 44152 | If all the preimages of ri... |
| salpreimagtlt 44153 | If all the preimages of le... |
| smfpreimalt 44154 | Given a function measurabl... |
| smff 44155 | A function measurable w.r.... |
| smfdmss 44156 | The domain of a function m... |
| issmff 44157 | The predicate " ` F ` is a... |
| issmfd 44158 | A sufficient condition for... |
| smfpreimaltf 44159 | Given a function measurabl... |
| issmfdf 44160 | A sufficient condition for... |
| sssmf 44161 | The restriction of a sigma... |
| mbfresmf 44162 | A real-valued measurable f... |
| cnfsmf 44163 | A continuous function is m... |
| incsmflem 44164 | A nondecreasing function i... |
| incsmf 44165 | A real-valued, nondecreasi... |
| smfsssmf 44166 | If a function is measurabl... |
| issmflelem 44167 | The predicate " ` F ` is a... |
| issmfle 44168 | The predicate " ` F ` is a... |
| smfpimltmpt 44169 | Given a function measurabl... |
| smfpimltxr 44170 | Given a function measurabl... |
| issmfdmpt 44171 | A sufficient condition for... |
| smfconst 44172 | Given a sigma-algebra over... |
| sssmfmpt 44173 | The restriction of a sigma... |
| cnfrrnsmf 44174 | A function, continuous fro... |
| smfid 44175 | The identity function is B... |
| bormflebmf 44176 | A Borel measurable functio... |
| smfpreimale 44177 | Given a function measurabl... |
| issmfgtlem 44178 | The predicate " ` F ` is a... |
| issmfgt 44179 | The predicate " ` F ` is a... |
| issmfled 44180 | A sufficient condition for... |
| smfpimltxrmpt 44181 | Given a function measurabl... |
| smfmbfcex 44182 | A constant function, with ... |
| issmfgtd 44183 | A sufficient condition for... |
| smfpreimagt 44184 | Given a function measurabl... |
| smfaddlem1 44185 | Given the sum of two funct... |
| smfaddlem2 44186 | The sum of two sigma-measu... |
| smfadd 44187 | The sum of two sigma-measu... |
| decsmflem 44188 | A nonincreasing function i... |
| decsmf 44189 | A real-valued, nonincreasi... |
| smfpreimagtf 44190 | Given a function measurabl... |
| issmfgelem 44191 | The predicate " ` F ` is a... |
| issmfge 44192 | The predicate " ` F ` is a... |
| smflimlem1 44193 | Lemma for the proof that t... |
| smflimlem2 44194 | Lemma for the proof that t... |
| smflimlem3 44195 | The limit of sigma-measura... |
| smflimlem4 44196 | Lemma for the proof that t... |
| smflimlem5 44197 | Lemma for the proof that t... |
| smflimlem6 44198 | Lemma for the proof that t... |
| smflim 44199 | The limit of sigma-measura... |
| nsssmfmbflem 44200 | The sigma-measurable funct... |
| nsssmfmbf 44201 | The sigma-measurable funct... |
| smfpimgtxr 44202 | Given a function measurabl... |
| smfpimgtmpt 44203 | Given a function measurabl... |
| smfpreimage 44204 | Given a function measurabl... |
| mbfpsssmf 44205 | Real-valued measurable fun... |
| smfpimgtxrmpt 44206 | Given a function measurabl... |
| smfpimioompt 44207 | Given a function measurabl... |
| smfpimioo 44208 | Given a function measurabl... |
| smfresal 44209 | Given a sigma-measurable f... |
| smfrec 44210 | The reciprocal of a sigma-... |
| smfres 44211 | The restriction of sigma-m... |
| smfmullem1 44212 | The multiplication of two ... |
| smfmullem2 44213 | The multiplication of two ... |
| smfmullem3 44214 | The multiplication of two ... |
| smfmullem4 44215 | The multiplication of two ... |
| smfmul 44216 | The multiplication of two ... |
| smfmulc1 44217 | A sigma-measurable functio... |
| smfdiv 44218 | The fraction of two sigma-... |
| smfpimbor1lem1 44219 | Every open set belongs to ... |
| smfpimbor1lem2 44220 | Given a sigma-measurable f... |
| smfpimbor1 44221 | Given a sigma-measurable f... |
| smf2id 44222 | Twice the identity functio... |
| smfco 44223 | The composition of a Borel... |
| smfneg 44224 | The negative of a sigma-me... |
| smffmpt 44225 | A function measurable w.r.... |
| smflim2 44226 | The limit of a sequence of... |
| smfpimcclem 44227 | Lemma for ~ smfpimcc given... |
| smfpimcc 44228 | Given a countable set of s... |
| issmfle2d 44229 | A sufficient condition for... |
| smflimmpt 44230 | The limit of a sequence of... |
| smfsuplem1 44231 | The supremum of a countabl... |
| smfsuplem2 44232 | The supremum of a countabl... |
| smfsuplem3 44233 | The supremum of a countabl... |
| smfsup 44234 | The supremum of a countabl... |
| smfsupmpt 44235 | The supremum of a countabl... |
| smfsupxr 44236 | The supremum of a countabl... |
| smfinflem 44237 | The infimum of a countable... |
| smfinf 44238 | The infimum of a countable... |
| smfinfmpt 44239 | The infimum of a countable... |
| smflimsuplem1 44240 | If ` H ` converges, the ` ... |
| smflimsuplem2 44241 | The superior limit of a se... |
| smflimsuplem3 44242 | The limit of the ` ( H `` ... |
| smflimsuplem4 44243 | If ` H ` converges, the ` ... |
| smflimsuplem5 44244 | ` H ` converges to the sup... |
| smflimsuplem6 44245 | The superior limit of a se... |
| smflimsuplem7 44246 | The superior limit of a se... |
| smflimsuplem8 44247 | The superior limit of a se... |
| smflimsup 44248 | The superior limit of a se... |
| smflimsupmpt 44249 | The superior limit of a se... |
| smfliminflem 44250 | The inferior limit of a co... |
| smfliminf 44251 | The inferior limit of a co... |
| smfliminfmpt 44252 | The inferior limit of a co... |
| sigarval 44253 | Define the signed area by ... |
| sigarim 44254 | Signed area takes value in... |
| sigarac 44255 | Signed area is anticommuta... |
| sigaraf 44256 | Signed area is additive by... |
| sigarmf 44257 | Signed area is additive (w... |
| sigaras 44258 | Signed area is additive by... |
| sigarms 44259 | Signed area is additive (w... |
| sigarls 44260 | Signed area is linear by t... |
| sigarid 44261 | Signed area of a flat para... |
| sigarexp 44262 | Expand the signed area for... |
| sigarperm 44263 | Signed area ` ( A - C ) G ... |
| sigardiv 44264 | If signed area between vec... |
| sigarimcd 44265 | Signed area takes value in... |
| sigariz 44266 | If signed area is zero, th... |
| sigarcol 44267 | Given three points ` A ` ,... |
| sharhght 44268 | Let ` A B C ` be a triangl... |
| sigaradd 44269 | Subtracting (double) area ... |
| cevathlem1 44270 | Ceva's theorem first lemma... |
| cevathlem2 44271 | Ceva's theorem second lemm... |
| cevath 44272 | Ceva's theorem. Let ` A B... |
| simpcntrab 44273 | The center of a simple gro... |
| hirstL-ax3 44274 | The third axiom of a syste... |
| ax3h 44275 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 44276 | A closed form showing (a i... |
| aibandbiaiaiffb 44277 | A closed form showing (a i... |
| notatnand 44278 | Do not use. Use intnanr i... |
| aistia 44279 | Given a is equivalent to `... |
| aisfina 44280 | Given a is equivalent to `... |
| bothtbothsame 44281 | Given both a, b are equiva... |
| bothfbothsame 44282 | Given both a, b are equiva... |
| aiffbbtat 44283 | Given a is equivalent to b... |
| aisbbisfaisf 44284 | Given a is equivalent to b... |
| axorbtnotaiffb 44285 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 44286 | Given a is equivalent to (... |
| axorbciffatcxorb 44287 | Given a is equivalent to (... |
| aibnbna 44288 | Given a implies b, (not b)... |
| aibnbaif 44289 | Given a implies b, not b, ... |
| aiffbtbat 44290 | Given a is equivalent to b... |
| astbstanbst 44291 | Given a is equivalent to T... |
| aistbistaandb 44292 | Given a is equivalent to T... |
| aisbnaxb 44293 | Given a is equivalent to b... |
| atbiffatnnb 44294 | If a implies b, then a imp... |
| bisaiaisb 44295 | Application of bicom1 with... |
| atbiffatnnbalt 44296 | If a implies b, then a imp... |
| abnotbtaxb 44297 | Assuming a, not b, there e... |
| abnotataxb 44298 | Assuming not a, b, there e... |
| conimpf 44299 | Assuming a, not b, and a i... |
| conimpfalt 44300 | Assuming a, not b, and a i... |
| aistbisfiaxb 44301 | Given a is equivalent to T... |
| aisfbistiaxb 44302 | Given a is equivalent to F... |
| aifftbifffaibif 44303 | Given a is equivalent to T... |
| aifftbifffaibifff 44304 | Given a is equivalent to T... |
| atnaiana 44305 | Given a, it is not the cas... |
| ainaiaandna 44306 | Given a, a implies it is n... |
| abcdta 44307 | Given (((a and b) and c) a... |
| abcdtb 44308 | Given (((a and b) and c) a... |
| abcdtc 44309 | Given (((a and b) and c) a... |
| abcdtd 44310 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 44311 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 44312 | Operands in a biconditiona... |
| nabctnabc 44313 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 44314 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 44315 | From one negated implicati... |
| twonotinotbothi 44316 | From these two negated imp... |
| clifte 44317 | show d is the same as an i... |
| cliftet 44318 | show d is the same as an i... |
| clifteta 44319 | show d is the same as an i... |
| cliftetb 44320 | show d is the same as an i... |
| confun 44321 | Given the hypotheses there... |
| confun2 44322 | Confun simplified to two p... |
| confun3 44323 | Confun's more complex form... |
| confun4 44324 | An attempt at derivative. ... |
| confun5 44325 | An attempt at derivative. ... |
| plcofph 44326 | Given, a,b and a "definiti... |
| pldofph 44327 | Given, a,b c, d, "definiti... |
| plvcofph 44328 | Given, a,b,d, and "definit... |
| plvcofphax 44329 | Given, a,b,d, and "definit... |
| plvofpos 44330 | rh is derivable because ON... |
| mdandyv0 44331 | Given the equivalences set... |
| mdandyv1 44332 | Given the equivalences set... |
| mdandyv2 44333 | Given the equivalences set... |
| mdandyv3 44334 | Given the equivalences set... |
| mdandyv4 44335 | Given the equivalences set... |
| mdandyv5 44336 | Given the equivalences set... |
| mdandyv6 44337 | Given the equivalences set... |
| mdandyv7 44338 | Given the equivalences set... |
| mdandyv8 44339 | Given the equivalences set... |
| mdandyv9 44340 | Given the equivalences set... |
| mdandyv10 44341 | Given the equivalences set... |
| mdandyv11 44342 | Given the equivalences set... |
| mdandyv12 44343 | Given the equivalences set... |
| mdandyv13 44344 | Given the equivalences set... |
| mdandyv14 44345 | Given the equivalences set... |
| mdandyv15 44346 | Given the equivalences set... |
| mdandyvr0 44347 | Given the equivalences set... |
| mdandyvr1 44348 | Given the equivalences set... |
| mdandyvr2 44349 | Given the equivalences set... |
| mdandyvr3 44350 | Given the equivalences set... |
| mdandyvr4 44351 | Given the equivalences set... |
| mdandyvr5 44352 | Given the equivalences set... |
| mdandyvr6 44353 | Given the equivalences set... |
| mdandyvr7 44354 | Given the equivalences set... |
| mdandyvr8 44355 | Given the equivalences set... |
| mdandyvr9 44356 | Given the equivalences set... |
| mdandyvr10 44357 | Given the equivalences set... |
| mdandyvr11 44358 | Given the equivalences set... |
| mdandyvr12 44359 | Given the equivalences set... |
| mdandyvr13 44360 | Given the equivalences set... |
| mdandyvr14 44361 | Given the equivalences set... |
| mdandyvr15 44362 | Given the equivalences set... |
| mdandyvrx0 44363 | Given the exclusivities se... |
| mdandyvrx1 44364 | Given the exclusivities se... |
| mdandyvrx2 44365 | Given the exclusivities se... |
| mdandyvrx3 44366 | Given the exclusivities se... |
| mdandyvrx4 44367 | Given the exclusivities se... |
| mdandyvrx5 44368 | Given the exclusivities se... |
| mdandyvrx6 44369 | Given the exclusivities se... |
| mdandyvrx7 44370 | Given the exclusivities se... |
| mdandyvrx8 44371 | Given the exclusivities se... |
| mdandyvrx9 44372 | Given the exclusivities se... |
| mdandyvrx10 44373 | Given the exclusivities se... |
| mdandyvrx11 44374 | Given the exclusivities se... |
| mdandyvrx12 44375 | Given the exclusivities se... |
| mdandyvrx13 44376 | Given the exclusivities se... |
| mdandyvrx14 44377 | Given the exclusivities se... |
| mdandyvrx15 44378 | Given the exclusivities se... |
| H15NH16TH15IH16 44379 | Given 15 hypotheses and a ... |
| dandysum2p2e4 44380 | CONTRADICTION PROVED AT 1 ... |
| mdandysum2p2e4 44381 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 44382 | Replacement of a nested an... |
| adh-minim 44383 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 44384 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 44385 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 44386 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 44387 | Fourth lemma for the deriv... |
| adh-minim-ax1 44388 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 44389 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 44390 | Sixth lemma for the deriva... |
| adh-minim-ax2c 44391 | Derivation of a commuted f... |
| adh-minim-ax2 44392 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 44393 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 44394 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 44395 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 44396 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 44397 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 44398 | Third lemma for the deriva... |
| adh-minimp-sylsimp 44399 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 44400 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 44401 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 44402 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 44403 | Fourth lemma for the deriv... |
| adh-minimp-ax2 44404 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 44405 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 44406 | Derivation of ~ pm2.43 Whi... |
| eusnsn 44407 | There is a unique element ... |
| absnsb 44408 | If the class abstraction `... |
| euabsneu 44409 | Another way to express exi... |
| elprneb 44410 | An element of a proper uno... |
| oppr 44411 | Equality for ordered pairs... |
| opprb 44412 | Equality for unordered pai... |
| or2expropbilem1 44413 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 44414 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 44415 | If two classes are strictl... |
| eubrv 44416 | If there is a unique set w... |
| eubrdm 44417 | If there is a unique set w... |
| eldmressn 44418 | Element of the domain of a... |
| iota0def 44419 | Example for a defined iota... |
| iota0ndef 44420 | Example for an undefined i... |
| fveqvfvv 44421 | If a function's value at a... |
| fnresfnco 44422 | Composition of two functio... |
| funcoressn 44423 | A composition restricted t... |
| funressnfv 44424 | A restriction to a singlet... |
| funressndmfvrn 44425 | The value of a function ` ... |
| funressnvmo 44426 | A function restricted to a... |
| funressnmo 44427 | A function restricted to a... |
| funressneu 44428 | There is exactly one value... |
| fresfo 44429 | Conditions for a restricti... |
| fsetsniunop 44430 | The class of all functions... |
| fsetabsnop 44431 | The class of all functions... |
| fsetsnf 44432 | The mapping of an element ... |
| fsetsnf1 44433 | The mapping of an element ... |
| fsetsnfo 44434 | The mapping of an element ... |
| fsetsnf1o 44435 | The mapping of an element ... |
| fsetsnprcnex 44436 | The class of all functions... |
| cfsetssfset 44437 | The class of constant func... |
| cfsetsnfsetfv 44438 | The function value of the ... |
| cfsetsnfsetf 44439 | The mapping of the class o... |
| cfsetsnfsetf1 44440 | The mapping of the class o... |
| cfsetsnfsetfo 44441 | The mapping of the class o... |
| cfsetsnfsetf1o 44442 | The mapping of the class o... |
| fsetprcnexALT 44443 | First version of proof for... |
| fcoreslem1 44444 | Lemma 1 for ~ fcores . (C... |
| fcoreslem2 44445 | Lemma 2 for ~ fcores . (C... |
| fcoreslem3 44446 | Lemma 3 for ~ fcores . (C... |
| fcoreslem4 44447 | Lemma 4 for ~ fcores . (C... |
| fcores 44448 | Every composite function `... |
| fcoresf1lem 44449 | Lemma for ~ fcoresf1 . (C... |
| fcoresf1 44450 | If a composition is inject... |
| fcoresf1b 44451 | A composition is injective... |
| fcoresfo 44452 | If a composition is surjec... |
| fcoresfob 44453 | A composition is surjectiv... |
| fcoresf1ob 44454 | A composition is bijective... |
| f1cof1blem 44455 | Lemma for ~ f1cof1b and ~ ... |
| f1cof1b 44456 | If the range of ` F ` equa... |
| funfocofob 44457 | If the domain of a functio... |
| fnfocofob 44458 | If the domain of a functio... |
| focofob 44459 | If the domain of a functio... |
| f1ocof1ob 44460 | If the range of ` F ` equa... |
| f1ocof1ob2 44461 | If the range of ` F ` equa... |
| aiotajust 44463 | Soundness justification th... |
| dfaiota2 44465 | Alternate definition of th... |
| reuabaiotaiota 44466 | The iota and the alternate... |
| reuaiotaiota 44467 | The iota and the alternate... |
| aiotaexb 44468 | The alternate iota over a ... |
| aiotavb 44469 | The alternate iota over a ... |
| aiotaint 44470 | This is to ~ df-aiota what... |
| dfaiota3 44471 | Alternate definition of ` ... |
| iotan0aiotaex 44472 | If the iota over a wff ` p... |
| aiotaexaiotaiota 44473 | The alternate iota over a ... |
| aiotaval 44474 | Theorem 8.19 in [Quine] p.... |
| aiota0def 44475 | Example for a defined alte... |
| aiota0ndef 44476 | Example for an undefined a... |
| r19.32 44477 | Theorem 19.32 of [Margaris... |
| rexsb 44478 | An equivalent expression f... |
| rexrsb 44479 | An equivalent expression f... |
| 2rexsb 44480 | An equivalent expression f... |
| 2rexrsb 44481 | An equivalent expression f... |
| cbvral2 44482 | Change bound variables of ... |
| cbvrex2 44483 | Change bound variables of ... |
| ralndv1 44484 | Example for a theorem abou... |
| ralndv2 44485 | Second example for a theor... |
| reuf1odnf 44486 | There is exactly one eleme... |
| reuf1od 44487 | There is exactly one eleme... |
| euoreqb 44488 | There is a set which is eq... |
| 2reu3 44489 | Double restricted existent... |
| 2reu7 44490 | Two equivalent expressions... |
| 2reu8 44491 | Two equivalent expressions... |
| 2reu8i 44492 | Implication of a double re... |
| 2reuimp0 44493 | Implication of a double re... |
| 2reuimp 44494 | Implication of a double re... |
| ralbinrald 44501 | Elemination of a restricte... |
| nvelim 44502 | If a class is the universa... |
| alneu 44503 | If a statement holds for a... |
| eu2ndop1stv 44504 | If there is a unique secon... |
| dfateq12d 44505 | Equality deduction for "de... |
| nfdfat 44506 | Bound-variable hypothesis ... |
| dfdfat2 44507 | Alternate definition of th... |
| fundmdfat 44508 | A function is defined at a... |
| dfatprc 44509 | A function is not defined ... |
| dfatelrn 44510 | The value of a function ` ... |
| dfafv2 44511 | Alternative definition of ... |
| afveq12d 44512 | Equality deduction for fun... |
| afveq1 44513 | Equality theorem for funct... |
| afveq2 44514 | Equality theorem for funct... |
| nfafv 44515 | Bound-variable hypothesis ... |
| csbafv12g 44516 | Move class substitution in... |
| afvfundmfveq 44517 | If a class is a function r... |
| afvnfundmuv 44518 | If a set is not in the dom... |
| ndmafv 44519 | The value of a class outsi... |
| afvvdm 44520 | If the function value of a... |
| nfunsnafv 44521 | If the restriction of a cl... |
| afvvfunressn 44522 | If the function value of a... |
| afvprc 44523 | A function's value at a pr... |
| afvvv 44524 | If a function's value at a... |
| afvpcfv0 44525 | If the value of the altern... |
| afvnufveq 44526 | The value of the alternati... |
| afvvfveq 44527 | The value of the alternati... |
| afv0fv0 44528 | If the value of the altern... |
| afvfvn0fveq 44529 | If the function's value at... |
| afv0nbfvbi 44530 | The function's value at an... |
| afvfv0bi 44531 | The function's value at an... |
| afveu 44532 | The value of a function at... |
| fnbrafvb 44533 | Equivalence of function va... |
| fnopafvb 44534 | Equivalence of function va... |
| funbrafvb 44535 | Equivalence of function va... |
| funopafvb 44536 | Equivalence of function va... |
| funbrafv 44537 | The second argument of a b... |
| funbrafv2b 44538 | Function value in terms of... |
| dfafn5a 44539 | Representation of a functi... |
| dfafn5b 44540 | Representation of a functi... |
| fnrnafv 44541 | The range of a function ex... |
| afvelrnb 44542 | A member of a function's r... |
| afvelrnb0 44543 | A member of a function's r... |
| dfaimafn 44544 | Alternate definition of th... |
| dfaimafn2 44545 | Alternate definition of th... |
| afvelima 44546 | Function value in an image... |
| afvelrn 44547 | A function's value belongs... |
| fnafvelrn 44548 | A function's value belongs... |
| fafvelrn 44549 | A function's value belongs... |
| ffnafv 44550 | A function maps to a class... |
| afvres 44551 | The value of a restricted ... |
| tz6.12-afv 44552 | Function value. Theorem 6... |
| tz6.12-1-afv 44553 | Function value (Theorem 6.... |
| dmfcoafv 44554 | Domains of a function comp... |
| afvco2 44555 | Value of a function compos... |
| rlimdmafv 44556 | Two ways to express that a... |
| aoveq123d 44557 | Equality deduction for ope... |
| nfaov 44558 | Bound-variable hypothesis ... |
| csbaovg 44559 | Move class substitution in... |
| aovfundmoveq 44560 | If a class is a function r... |
| aovnfundmuv 44561 | If an ordered pair is not ... |
| ndmaov 44562 | The value of an operation ... |
| ndmaovg 44563 | The value of an operation ... |
| aovvdm 44564 | If the operation value of ... |
| nfunsnaov 44565 | If the restriction of a cl... |
| aovvfunressn 44566 | If the operation value of ... |
| aovprc 44567 | The value of an operation ... |
| aovrcl 44568 | Reverse closure for an ope... |
| aovpcov0 44569 | If the alternative value o... |
| aovnuoveq 44570 | The alternative value of t... |
| aovvoveq 44571 | The alternative value of t... |
| aov0ov0 44572 | If the alternative value o... |
| aovovn0oveq 44573 | If the operation's value a... |
| aov0nbovbi 44574 | The operation's value on a... |
| aovov0bi 44575 | The operation's value on a... |
| rspceaov 44576 | A frequently used special ... |
| fnotaovb 44577 | Equivalence of operation v... |
| ffnaov 44578 | An operation maps to a cla... |
| faovcl 44579 | Closure law for an operati... |
| aovmpt4g 44580 | Value of a function given ... |
| aoprssdm 44581 | Domain of closure of an op... |
| ndmaovcl 44582 | The "closure" of an operat... |
| ndmaovrcl 44583 | Reverse closure law, in co... |
| ndmaovcom 44584 | Any operation is commutati... |
| ndmaovass 44585 | Any operation is associati... |
| ndmaovdistr 44586 | Any operation is distribut... |
| dfatafv2iota 44589 | If a function is defined a... |
| ndfatafv2 44590 | The alternate function val... |
| ndfatafv2undef 44591 | The alternate function val... |
| dfatafv2ex 44592 | The alternate function val... |
| afv2ex 44593 | The alternate function val... |
| afv2eq12d 44594 | Equality deduction for fun... |
| afv2eq1 44595 | Equality theorem for funct... |
| afv2eq2 44596 | Equality theorem for funct... |
| nfafv2 44597 | Bound-variable hypothesis ... |
| csbafv212g 44598 | Move class substitution in... |
| fexafv2ex 44599 | The alternate function val... |
| ndfatafv2nrn 44600 | The alternate function val... |
| ndmafv2nrn 44601 | The value of a class outsi... |
| funressndmafv2rn 44602 | The alternate function val... |
| afv2ndefb 44603 | Two ways to say that an al... |
| nfunsnafv2 44604 | If the restriction of a cl... |
| afv2prc 44605 | A function's value at a pr... |
| dfatafv2rnb 44606 | The alternate function val... |
| afv2orxorb 44607 | If a set is in the range o... |
| dmafv2rnb 44608 | The alternate function val... |
| fundmafv2rnb 44609 | The alternate function val... |
| afv2elrn 44610 | An alternate function valu... |
| afv20defat 44611 | If the alternate function ... |
| fnafv2elrn 44612 | An alternate function valu... |
| fafv2elrn 44613 | An alternate function valu... |
| fafv2elrnb 44614 | An alternate function valu... |
| frnvafv2v 44615 | If the codomain of a funct... |
| tz6.12-2-afv2 44616 | Function value when ` F ` ... |
| afv2eu 44617 | The value of a function at... |
| afv2res 44618 | The value of a restricted ... |
| tz6.12-afv2 44619 | Function value (Theorem 6.... |
| tz6.12-1-afv2 44620 | Function value (Theorem 6.... |
| tz6.12c-afv2 44621 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 44622 | Corollary of Theorem 6.12(... |
| funressnbrafv2 44623 | The second argument of a b... |
| dfatbrafv2b 44624 | Equivalence of function va... |
| dfatopafv2b 44625 | Equivalence of function va... |
| funbrafv2 44626 | The second argument of a b... |
| fnbrafv2b 44627 | Equivalence of function va... |
| fnopafv2b 44628 | Equivalence of function va... |
| funbrafv22b 44629 | Equivalence of function va... |
| funopafv2b 44630 | Equivalence of function va... |
| dfatsnafv2 44631 | Singleton of function valu... |
| dfafv23 44632 | A definition of function v... |
| dfatdmfcoafv2 44633 | Domain of a function compo... |
| dfatcolem 44634 | Lemma for ~ dfatco . (Con... |
| dfatco 44635 | The predicate "defined at"... |
| afv2co2 44636 | Value of a function compos... |
| rlimdmafv2 44637 | Two ways to express that a... |
| dfafv22 44638 | Alternate definition of ` ... |
| afv2ndeffv0 44639 | If the alternate function ... |
| dfatafv2eqfv 44640 | If a function is defined a... |
| afv2rnfveq 44641 | If the alternate function ... |
| afv20fv0 44642 | If the alternate function ... |
| afv2fvn0fveq 44643 | If the function's value at... |
| afv2fv0 44644 | If the function's value at... |
| afv2fv0b 44645 | The function's value at an... |
| afv2fv0xorb 44646 | If a set is in the range o... |
| an4com24 44647 | Rearrangement of 4 conjunc... |
| 3an4ancom24 44648 | Commutative law for a conj... |
| 4an21 44649 | Rearrangement of 4 conjunc... |
| dfnelbr2 44652 | Alternate definition of th... |
| nelbr 44653 | The binary relation of a s... |
| nelbrim 44654 | If a set is related to ano... |
| nelbrnel 44655 | A set is related to anothe... |
| nelbrnelim 44656 | If a set is related to ano... |
| ralralimp 44657 | Selecting one of two alter... |
| otiunsndisjX 44658 | The union of singletons co... |
| fvifeq 44659 | Equality of function value... |
| rnfdmpr 44660 | The range of a one-to-one ... |
| imarnf1pr 44661 | The image of the range of ... |
| funop1 44662 | A function is an ordered p... |
| fun2dmnopgexmpl 44663 | A function with a domain c... |
| opabresex0d 44664 | A collection of ordered pa... |
| opabbrfex0d 44665 | A collection of ordered pa... |
| opabresexd 44666 | A collection of ordered pa... |
| opabbrfexd 44667 | A collection of ordered pa... |
| f1oresf1orab 44668 | Build a bijection by restr... |
| f1oresf1o 44669 | Build a bijection by restr... |
| f1oresf1o2 44670 | Build a bijection by restr... |
| fvmptrab 44671 | Value of a function mappin... |
| fvmptrabdm 44672 | Value of a function mappin... |
| leltletr 44673 | Transitive law, weaker for... |
| cnambpcma 44674 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 44675 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 44676 | The sum of two complex num... |
| leaddsuble 44677 | Addition and subtraction o... |
| 2leaddle2 44678 | If two real numbers are le... |
| ltnltne 44679 | Variant of trichotomy law ... |
| p1lep2 44680 | A real number increasd by ... |
| ltsubsubaddltsub 44681 | If the result of subtracti... |
| zm1nn 44682 | An integer minus 1 is posi... |
| readdcnnred 44683 | The sum of a real number a... |
| resubcnnred 44684 | The difference of a real n... |
| recnmulnred 44685 | The product of a real numb... |
| cndivrenred 44686 | The quotient of an imagina... |
| sqrtnegnre 44687 | The square root of a negat... |
| nn0resubcl 44688 | Closure law for subtractio... |
| zgeltp1eq 44689 | If an integer is between a... |
| 1t10e1p1e11 44690 | 11 is 1 times 10 to the po... |
| deccarry 44691 | Add 1 to a 2 digit number ... |
| eluzge0nn0 44692 | If an integer is greater t... |
| nltle2tri 44693 | Negated extended trichotom... |
| ssfz12 44694 | Subset relationship for fi... |
| elfz2z 44695 | Membership of an integer i... |
| 2elfz3nn0 44696 | If there are two elements ... |
| fz0addcom 44697 | The addition of two member... |
| 2elfz2melfz 44698 | If the sum of two integers... |
| fz0addge0 44699 | The sum of two integers in... |
| elfzlble 44700 | Membership of an integer i... |
| elfzelfzlble 44701 | Membership of an element o... |
| fzopred 44702 | Join a predecessor to the ... |
| fzopredsuc 44703 | Join a predecessor and a s... |
| 1fzopredsuc 44704 | Join 0 and a successor to ... |
| el1fzopredsuc 44705 | An element of an open inte... |
| subsubelfzo0 44706 | Subtracting a difference f... |
| fzoopth 44707 | A half-open integer range ... |
| 2ffzoeq 44708 | Two functions over a half-... |
| m1mod0mod1 44709 | An integer decreased by 1 ... |
| elmod2 44710 | An integer modulo 2 is eit... |
| smonoord 44711 | Ordering relation for a st... |
| fsummsndifre 44712 | A finite sum with one of i... |
| fsumsplitsndif 44713 | Separate out a term in a f... |
| fsummmodsndifre 44714 | A finite sum of summands m... |
| fsummmodsnunz 44715 | A finite sum of summands m... |
| setsidel 44716 | The injected slot is an el... |
| setsnidel 44717 | The injected slot is an el... |
| setsv 44718 | The value of the structure... |
| preimafvsnel 44719 | The preimage of a function... |
| preimafvn0 44720 | The preimage of a function... |
| uniimafveqt 44721 | The union of the image of ... |
| uniimaprimaeqfv 44722 | The union of the image of ... |
| setpreimafvex 44723 | The class ` P ` of all pre... |
| elsetpreimafvb 44724 | The characterization of an... |
| elsetpreimafv 44725 | An element of the class ` ... |
| elsetpreimafvssdm 44726 | An element of the class ` ... |
| fvelsetpreimafv 44727 | There is an element in a p... |
| preimafvelsetpreimafv 44728 | The preimage of a function... |
| preimafvsspwdm 44729 | The class ` P ` of all pre... |
| 0nelsetpreimafv 44730 | The empty set is not an el... |
| elsetpreimafvbi 44731 | An element of the preimage... |
| elsetpreimafveqfv 44732 | The elements of the preima... |
| eqfvelsetpreimafv 44733 | If an element of the domai... |
| elsetpreimafvrab 44734 | An element of the preimage... |
| imaelsetpreimafv 44735 | The image of an element of... |
| uniimaelsetpreimafv 44736 | The union of the image of ... |
| elsetpreimafveq 44737 | If two preimages of functi... |
| fundcmpsurinjlem1 44738 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 44739 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 44740 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 44741 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 44742 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 44743 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 44744 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 44745 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 44746 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 44747 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 44748 | Every function ` F : A -->... |
| fundcmpsurinj 44749 | Every function ` F : A -->... |
| fundcmpsurbijinj 44750 | Every function ` F : A -->... |
| fundcmpsurinjimaid 44751 | Every function ` F : A -->... |
| fundcmpsurinjALT 44752 | Alternate proof of ~ fundc... |
| iccpval 44755 | Partition consisting of a ... |
| iccpart 44756 | A special partition. Corr... |
| iccpartimp 44757 | Implications for a class b... |
| iccpartres 44758 | The restriction of a parti... |
| iccpartxr 44759 | If there is a partition, t... |
| iccpartgtprec 44760 | If there is a partition, t... |
| iccpartipre 44761 | If there is a partition, t... |
| iccpartiltu 44762 | If there is a partition, t... |
| iccpartigtl 44763 | If there is a partition, t... |
| iccpartlt 44764 | If there is a partition, t... |
| iccpartltu 44765 | If there is a partition, t... |
| iccpartgtl 44766 | If there is a partition, t... |
| iccpartgt 44767 | If there is a partition, t... |
| iccpartleu 44768 | If there is a partition, t... |
| iccpartgel 44769 | If there is a partition, t... |
| iccpartrn 44770 | If there is a partition, t... |
| iccpartf 44771 | The range of the partition... |
| iccpartel 44772 | If there is a partition, t... |
| iccelpart 44773 | An element of any partitio... |
| iccpartiun 44774 | A half-open interval of ex... |
| icceuelpartlem 44775 | Lemma for ~ icceuelpart . ... |
| icceuelpart 44776 | An element of a partitione... |
| iccpartdisj 44777 | The segments of a partitio... |
| iccpartnel 44778 | A point of a partition is ... |
| fargshiftfv 44779 | If a class is a function, ... |
| fargshiftf 44780 | If a class is a function, ... |
| fargshiftf1 44781 | If a function is 1-1, then... |
| fargshiftfo 44782 | If a function is onto, the... |
| fargshiftfva 44783 | The values of a shifted fu... |
| lswn0 44784 | The last symbol of a not e... |
| nfich1 44787 | The first interchangeable ... |
| nfich2 44788 | The second interchangeable... |
| ichv 44789 | Setvar variables are inter... |
| ichf 44790 | Setvar variables are inter... |
| ichid 44791 | A setvar variable is alway... |
| icht 44792 | A theorem is interchangeab... |
| ichbidv 44793 | Formula building rule for ... |
| ichcircshi 44794 | The setvar variables are i... |
| ichan 44795 | If two setvar variables ar... |
| ichn 44796 | Negation does not affect i... |
| ichim 44797 | Formula building rule for ... |
| dfich2 44798 | Alternate definition of th... |
| ichcom 44799 | The interchangeability of ... |
| ichbi12i 44800 | Equivalence for interchang... |
| icheqid 44801 | In an equality for the sam... |
| icheq 44802 | In an equality of setvar v... |
| ichnfimlem 44803 | Lemma for ~ ichnfim : A s... |
| ichnfim 44804 | If in an interchangeabilit... |
| ichnfb 44805 | If ` x ` and ` y ` are int... |
| ichal 44806 | Move a universal quantifie... |
| ich2al 44807 | Two setvar variables are a... |
| ich2ex 44808 | Two setvar variables are a... |
| ichexmpl1 44809 | Example for interchangeabl... |
| ichexmpl2 44810 | Example for interchangeabl... |
| ich2exprop 44811 | If the setvar variables ar... |
| ichnreuop 44812 | If the setvar variables ar... |
| ichreuopeq 44813 | If the setvar variables ar... |
| sprid 44814 | Two identical representati... |
| elsprel 44815 | An unordered pair is an el... |
| spr0nelg 44816 | The empty set is not an el... |
| sprval 44819 | The set of all unordered p... |
| sprvalpw 44820 | The set of all unordered p... |
| sprssspr 44821 | The set of all unordered p... |
| spr0el 44822 | The empty set is not an un... |
| sprvalpwn0 44823 | The set of all unordered p... |
| sprel 44824 | An element of the set of a... |
| prssspr 44825 | An element of a subset of ... |
| prelspr 44826 | An unordered pair of eleme... |
| prsprel 44827 | The elements of a pair fro... |
| prsssprel 44828 | The elements of a pair fro... |
| sprvalpwle2 44829 | The set of all unordered p... |
| sprsymrelfvlem 44830 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 44831 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 44832 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 44833 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 44834 | The value of the function ... |
| sprsymrelf 44835 | The mapping ` F ` is a fun... |
| sprsymrelf1 44836 | The mapping ` F ` is a one... |
| sprsymrelfo 44837 | The mapping ` F ` is a fun... |
| sprsymrelf1o 44838 | The mapping ` F ` is a bij... |
| sprbisymrel 44839 | There is a bijection betwe... |
| sprsymrelen 44840 | The class ` P ` of subsets... |
| prpair 44841 | Characterization of a prop... |
| prproropf1olem0 44842 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 44843 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 44844 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 44845 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 44846 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 44847 | There is a bijection betwe... |
| prproropen 44848 | The set of proper pairs an... |
| prproropreud 44849 | There is exactly one order... |
| pairreueq 44850 | Two equivalent representat... |
| paireqne 44851 | Two sets are not equal iff... |
| prprval 44854 | The set of all proper unor... |
| prprvalpw 44855 | The set of all proper unor... |
| prprelb 44856 | An element of the set of a... |
| prprelprb 44857 | A set is an element of the... |
| prprspr2 44858 | The set of all proper unor... |
| prprsprreu 44859 | There is a unique proper u... |
| prprreueq 44860 | There is a unique proper u... |
| sbcpr 44861 | The proper substitution of... |
| reupr 44862 | There is a unique unordere... |
| reuprpr 44863 | There is a unique proper u... |
| poprelb 44864 | Equality for unordered pai... |
| 2exopprim 44865 | The existence of an ordere... |
| reuopreuprim 44866 | There is a unique unordere... |
| fmtno 44869 | The ` N ` th Fermat number... |
| fmtnoge3 44870 | Each Fermat number is grea... |
| fmtnonn 44871 | Each Fermat number is a po... |
| fmtnom1nn 44872 | A Fermat number minus one ... |
| fmtnoodd 44873 | Each Fermat number is odd.... |
| fmtnorn 44874 | A Fermat number is a funct... |
| fmtnof1 44875 | The enumeration of the Fer... |
| fmtnoinf 44876 | The set of Fermat numbers ... |
| fmtnorec1 44877 | The first recurrence relat... |
| sqrtpwpw2p 44878 | The floor of the square ro... |
| fmtnosqrt 44879 | The floor of the square ro... |
| fmtno0 44880 | The ` 0 ` th Fermat number... |
| fmtno1 44881 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 44882 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 44883 | The second recurrence rela... |
| fmtnodvds 44884 | Any Fermat number divides ... |
| goldbachthlem1 44885 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 44886 | Lemma 2 for ~ goldbachth .... |
| goldbachth 44887 | Goldbach's theorem: Two d... |
| fmtnorec3 44888 | The third recurrence relat... |
| fmtnorec4 44889 | The fourth recurrence rela... |
| fmtno2 44890 | The ` 2 ` nd Fermat number... |
| fmtno3 44891 | The ` 3 ` rd Fermat number... |
| fmtno4 44892 | The ` 4 ` th Fermat number... |
| fmtno5lem1 44893 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 44894 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 44895 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 44896 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 44897 | The ` 5 ` th Fermat number... |
| fmtno0prm 44898 | The ` 0 ` th Fermat number... |
| fmtno1prm 44899 | The ` 1 ` st Fermat number... |
| fmtno2prm 44900 | The ` 2 ` nd Fermat number... |
| 257prm 44901 | 257 is a prime number (the... |
| fmtno3prm 44902 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 44903 | Any power of two is coprim... |
| fmtnoprmfac1lem 44904 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 44905 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 44906 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 44907 | Divisor of Fermat number (... |
| fmtnofac2lem 44908 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 44909 | Divisor of Fermat number (... |
| fmtnofac1 44910 | Divisor of Fermat number (... |
| fmtno4sqrt 44911 | The floor of the square ro... |
| fmtno4prmfac 44912 | If P was a (prime) factor ... |
| fmtno4prmfac193 44913 | If P was a (prime) factor ... |
| fmtno4nprmfac193 44914 | 193 is not a (prime) facto... |
| fmtno4prm 44915 | The ` 4 `-th Fermat number... |
| 65537prm 44916 | 65537 is a prime number (t... |
| fmtnofz04prm 44917 | The first five Fermat numb... |
| fmtnole4prm 44918 | The first five Fermat numb... |
| fmtno5faclem1 44919 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 44920 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 44921 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 44922 | The factorisation of the `... |
| fmtno5nprm 44923 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 44924 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 44925 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 44926 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 44927 | The mapping of a Fermat nu... |
| prminf2 44928 | The set of prime numbers i... |
| 2pwp1prm 44929 | For ` ( ( 2 ^ k ) + 1 ) ` ... |
| 2pwp1prmfmtno 44930 | Every prime number of the ... |
| m2prm 44931 | The second Mersenne number... |
| m3prm 44932 | The third Mersenne number ... |
| flsqrt 44933 | A condition equivalent to ... |
| flsqrt5 44934 | The floor of the square ro... |
| 3ndvds4 44935 | 3 does not divide 4. (Con... |
| 139prmALT 44936 | 139 is a prime number. In... |
| 31prm 44937 | 31 is a prime number. In ... |
| m5prm 44938 | The fifth Mersenne number ... |
| 127prm 44939 | 127 is a prime number. (C... |
| m7prm 44940 | The seventh Mersenne numbe... |
| m11nprm 44941 | The eleventh Mersenne numb... |
| mod42tp1mod8 44942 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 44943 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 44944 | If ` P ` is a Sophie Germa... |
| lighneallem1 44945 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 44946 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 44947 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 44948 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 44949 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 44950 | Lemma 3 for ~ lighneal . ... |
| lighneal 44951 | If a power of a prime ` P ... |
| modexp2m1d 44952 | The square of an integer w... |
| proththdlem 44953 | Lemma for ~ proththd . (C... |
| proththd 44954 | Proth's theorem (1878). I... |
| 5tcu2e40 44955 | 5 times the cube of 2 is 4... |
| 3exp4mod41 44956 | 3 to the fourth power is -... |
| 41prothprmlem1 44957 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 44958 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 44959 | 41 is a _Proth prime_. (C... |
| quad1 44960 | A condition for a quadrati... |
| requad01 44961 | A condition for a quadrati... |
| requad1 44962 | A condition for a quadrati... |
| requad2 44963 | A condition for a quadrati... |
| iseven 44968 | The predicate "is an even ... |
| isodd 44969 | The predicate "is an odd n... |
| evenz 44970 | An even number is an integ... |
| oddz 44971 | An odd number is an intege... |
| evendiv2z 44972 | The result of dividing an ... |
| oddp1div2z 44973 | The result of dividing an ... |
| oddm1div2z 44974 | The result of dividing an ... |
| isodd2 44975 | The predicate "is an odd n... |
| dfodd2 44976 | Alternate definition for o... |
| dfodd6 44977 | Alternate definition for o... |
| dfeven4 44978 | Alternate definition for e... |
| evenm1odd 44979 | The predecessor of an even... |
| evenp1odd 44980 | The successor of an even n... |
| oddp1eveni 44981 | The successor of an odd nu... |
| oddm1eveni 44982 | The predecessor of an odd ... |
| evennodd 44983 | An even number is not an o... |
| oddneven 44984 | An odd number is not an ev... |
| enege 44985 | The negative of an even nu... |
| onego 44986 | The negative of an odd num... |
| m1expevenALTV 44987 | Exponentiation of -1 by an... |
| m1expoddALTV 44988 | Exponentiation of -1 by an... |
| dfeven2 44989 | Alternate definition for e... |
| dfodd3 44990 | Alternate definition for o... |
| iseven2 44991 | The predicate "is an even ... |
| isodd3 44992 | The predicate "is an odd n... |
| 2dvdseven 44993 | 2 divides an even number. ... |
| m2even 44994 | A multiple of 2 is an even... |
| 2ndvdsodd 44995 | 2 does not divide an odd n... |
| 2dvdsoddp1 44996 | 2 divides an odd number in... |
| 2dvdsoddm1 44997 | 2 divides an odd number de... |
| dfeven3 44998 | Alternate definition for e... |
| dfodd4 44999 | Alternate definition for o... |
| dfodd5 45000 | Alternate definition for o... |
| zefldiv2ALTV 45001 | The floor of an even numbe... |
| zofldiv2ALTV 45002 | The floor of an odd numer ... |
| oddflALTV 45003 | Odd number representation ... |
| iseven5 45004 | The predicate "is an even ... |
| isodd7 45005 | The predicate "is an odd n... |
| dfeven5 45006 | Alternate definition for e... |
| dfodd7 45007 | Alternate definition for o... |
| gcd2odd1 45008 | The greatest common diviso... |
| zneoALTV 45009 | No even integer equals an ... |
| zeoALTV 45010 | An integer is even or odd.... |
| zeo2ALTV 45011 | An integer is even or odd ... |
| nneoALTV 45012 | A positive integer is even... |
| nneoiALTV 45013 | A positive integer is even... |
| odd2np1ALTV 45014 | An integer is odd iff it i... |
| oddm1evenALTV 45015 | An integer is odd iff its ... |
| oddp1evenALTV 45016 | An integer is odd iff its ... |
| oexpnegALTV 45017 | The exponential of the neg... |
| oexpnegnz 45018 | The exponential of the neg... |
| bits0ALTV 45019 | Value of the zeroth bit. ... |
| bits0eALTV 45020 | The zeroth bit of an even ... |
| bits0oALTV 45021 | The zeroth bit of an odd n... |
| divgcdoddALTV 45022 | Either ` A / ( A gcd B ) `... |
| opoeALTV 45023 | The sum of two odds is eve... |
| opeoALTV 45024 | The sum of an odd and an e... |
| omoeALTV 45025 | The difference of two odds... |
| omeoALTV 45026 | The difference of an odd a... |
| oddprmALTV 45027 | A prime not equal to ` 2 `... |
| 0evenALTV 45028 | 0 is an even number. (Con... |
| 0noddALTV 45029 | 0 is not an odd number. (... |
| 1oddALTV 45030 | 1 is an odd number. (Cont... |
| 1nevenALTV 45031 | 1 is not an even number. ... |
| 2evenALTV 45032 | 2 is an even number. (Con... |
| 2noddALTV 45033 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 45034 | An odd nonnegative integer... |
| nnoALTV 45035 | An alternate characterizat... |
| nn0oALTV 45036 | An alternate characterizat... |
| nn0e 45037 | An alternate characterizat... |
| nneven 45038 | An alternate characterizat... |
| nn0onn0exALTV 45039 | For each odd nonnegative i... |
| nn0enn0exALTV 45040 | For each even nonnegative ... |
| nnennexALTV 45041 | For each even positive int... |
| nnpw2evenALTV 45042 | 2 to the power of a positi... |
| epoo 45043 | The sum of an even and an ... |
| emoo 45044 | The difference of an even ... |
| epee 45045 | The sum of two even number... |
| emee 45046 | The difference of two even... |
| evensumeven 45047 | If a summand is even, the ... |
| 3odd 45048 | 3 is an odd number. (Cont... |
| 4even 45049 | 4 is an even number. (Con... |
| 5odd 45050 | 5 is an odd number. (Cont... |
| 6even 45051 | 6 is an even number. (Con... |
| 7odd 45052 | 7 is an odd number. (Cont... |
| 8even 45053 | 8 is an even number. (Con... |
| evenprm2 45054 | A prime number is even iff... |
| oddprmne2 45055 | Every prime number not bei... |
| oddprmuzge3 45056 | A prime number which is od... |
| evenltle 45057 | If an even number is great... |
| odd2prm2 45058 | If an odd number is the su... |
| even3prm2 45059 | If an even number is the s... |
| mogoldbblem 45060 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 45061 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 45062 | Lemma for ~ perfectALTV . ... |
| perfectALTV 45063 | The Euclid-Euler theorem, ... |
| fppr 45066 | The set of Fermat pseudopr... |
| fpprmod 45067 | The set of Fermat pseudopr... |
| fpprel 45068 | A Fermat pseudoprime to th... |
| fpprbasnn 45069 | The base of a Fermat pseud... |
| fpprnn 45070 | A Fermat pseudoprime to th... |
| fppr2odd 45071 | A Fermat pseudoprime to th... |
| 11t31e341 45072 | 341 is the product of 11 a... |
| 2exp340mod341 45073 | Eight to the eighth power ... |
| 341fppr2 45074 | 341 is the (smallest) _Pou... |
| 4fppr1 45075 | 4 is the (smallest) Fermat... |
| 8exp8mod9 45076 | Eight to the eighth power ... |
| 9fppr8 45077 | 9 is the (smallest) Fermat... |
| dfwppr 45078 | Alternate definition of a ... |
| fpprwppr 45079 | A Fermat pseudoprime to th... |
| fpprwpprb 45080 | An integer ` X ` which is ... |
| fpprel2 45081 | An alternate definition fo... |
| nfermltl8rev 45082 | Fermat's little theorem wi... |
| nfermltl2rev 45083 | Fermat's little theorem wi... |
| nfermltlrev 45084 | Fermat's little theorem re... |
| isgbe 45091 | The predicate "is an even ... |
| isgbow 45092 | The predicate "is a weak o... |
| isgbo 45093 | The predicate "is an odd G... |
| gbeeven 45094 | An even Goldbach number is... |
| gbowodd 45095 | A weak odd Goldbach number... |
| gbogbow 45096 | A (strong) odd Goldbach nu... |
| gboodd 45097 | An odd Goldbach number is ... |
| gbepos 45098 | Any even Goldbach number i... |
| gbowpos 45099 | Any weak odd Goldbach numb... |
| gbopos 45100 | Any odd Goldbach number is... |
| gbegt5 45101 | Any even Goldbach number i... |
| gbowgt5 45102 | Any weak odd Goldbach numb... |
| gbowge7 45103 | Any weak odd Goldbach numb... |
| gboge9 45104 | Any odd Goldbach number is... |
| gbege6 45105 | Any even Goldbach number i... |
| gbpart6 45106 | The Goldbach partition of ... |
| gbpart7 45107 | The (weak) Goldbach partit... |
| gbpart8 45108 | The Goldbach partition of ... |
| gbpart9 45109 | The (strong) Goldbach part... |
| gbpart11 45110 | The (strong) Goldbach part... |
| 6gbe 45111 | 6 is an even Goldbach numb... |
| 7gbow 45112 | 7 is a weak odd Goldbach n... |
| 8gbe 45113 | 8 is an even Goldbach numb... |
| 9gbo 45114 | 9 is an odd Goldbach numbe... |
| 11gbo 45115 | 11 is an odd Goldbach numb... |
| stgoldbwt 45116 | If the strong ternary Gold... |
| sbgoldbwt 45117 | If the strong binary Goldb... |
| sbgoldbst 45118 | If the strong binary Goldb... |
| sbgoldbaltlem1 45119 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 45120 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 45121 | An alternate (related to t... |
| sbgoldbb 45122 | If the strong binary Goldb... |
| sgoldbeven3prm 45123 | If the binary Goldbach con... |
| sbgoldbm 45124 | If the strong binary Goldb... |
| mogoldbb 45125 | If the modern version of t... |
| sbgoldbmb 45126 | The strong binary Goldbach... |
| sbgoldbo 45127 | If the strong binary Goldb... |
| nnsum3primes4 45128 | 4 is the sum of at most 3 ... |
| nnsum4primes4 45129 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 45130 | Every prime is "the sum of... |
| nnsum4primesprm 45131 | Every prime is "the sum of... |
| nnsum3primesgbe 45132 | Any even Goldbach number i... |
| nnsum4primesgbe 45133 | Any even Goldbach number i... |
| nnsum3primesle9 45134 | Every integer greater than... |
| nnsum4primesle9 45135 | Every integer greater than... |
| nnsum4primesodd 45136 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 45137 | If the (strong) ternary Go... |
| evengpop3 45138 | If the (weak) ternary Gold... |
| evengpoap3 45139 | If the (strong) ternary Go... |
| nnsum4primeseven 45140 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 45141 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 45142 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 45143 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 45144 | If the binary Goldbach con... |
| bgoldbtbndlem1 45145 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 45146 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 45147 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 45148 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 45149 | If the binary Goldbach con... |
| tgoldbachgtALTV 45152 | Variant of Thierry Arnoux'... |
| bgoldbachlt 45153 | The binary Goldbach conjec... |
| tgblthelfgott 45155 | The ternary Goldbach conje... |
| tgoldbachlt 45156 | The ternary Goldbach conje... |
| tgoldbach 45157 | The ternary Goldbach conje... |
| isomgrrel 45162 | The isomorphy relation for... |
| isomgr 45163 | The isomorphy relation for... |
| isisomgr 45164 | Implications of two graphs... |
| isomgreqve 45165 | A set is isomorphic to a h... |
| isomushgr 45166 | The isomorphy relation for... |
| isomuspgrlem1 45167 | Lemma 1 for ~ isomuspgr . ... |
| isomuspgrlem2a 45168 | Lemma 1 for ~ isomuspgrlem... |
| isomuspgrlem2b 45169 | Lemma 2 for ~ isomuspgrlem... |
| isomuspgrlem2c 45170 | Lemma 3 for ~ isomuspgrlem... |
| isomuspgrlem2d 45171 | Lemma 4 for ~ isomuspgrlem... |
| isomuspgrlem2e 45172 | Lemma 5 for ~ isomuspgrlem... |
| isomuspgrlem2 45173 | Lemma 2 for ~ isomuspgr . ... |
| isomuspgr 45174 | The isomorphy relation for... |
| isomgrref 45175 | The isomorphy relation is ... |
| isomgrsym 45176 | The isomorphy relation is ... |
| isomgrsymb 45177 | The isomorphy relation is ... |
| isomgrtrlem 45178 | Lemma for ~ isomgrtr . (C... |
| isomgrtr 45179 | The isomorphy relation is ... |
| strisomgrop 45180 | A graph represented as an ... |
| ushrisomgr 45181 | A simple hypergraph (with ... |
| 1hegrlfgr 45182 | A graph ` G ` with one hyp... |
| upwlksfval 45185 | The set of simple walks (i... |
| isupwlk 45186 | Properties of a pair of fu... |
| isupwlkg 45187 | Generalization of ~ isupwl... |
| upwlkbprop 45188 | Basic properties of a simp... |
| upwlkwlk 45189 | A simple walk is a walk. ... |
| upgrwlkupwlk 45190 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 45191 | In a pseudograph, the defi... |
| upgrisupwlkALT 45192 | Alternate proof of ~ upgri... |
| upgredgssspr 45193 | The set of edges of a pseu... |
| uspgropssxp 45194 | The set ` G ` of "simple p... |
| uspgrsprfv 45195 | The value of the function ... |
| uspgrsprf 45196 | The mapping ` F ` is a fun... |
| uspgrsprf1 45197 | The mapping ` F ` is a one... |
| uspgrsprfo 45198 | The mapping ` F ` is a fun... |
| uspgrsprf1o 45199 | The mapping ` F ` is a bij... |
| uspgrex 45200 | The class ` G ` of all "si... |
| uspgrbispr 45201 | There is a bijection betwe... |
| uspgrspren 45202 | The set ` G ` of the "simp... |
| uspgrymrelen 45203 | The set ` G ` of the "simp... |
| uspgrbisymrel 45204 | There is a bijection betwe... |
| uspgrbisymrelALT 45205 | Alternate proof of ~ uspgr... |
| ovn0dmfun 45206 | If a class operation value... |
| xpsnopab 45207 | A Cartesian product with a... |
| xpiun 45208 | A Cartesian product expres... |
| ovn0ssdmfun 45209 | If a class' operation valu... |
| fnxpdmdm 45210 | The domain of the domain o... |
| cnfldsrngbas 45211 | The base set of a subring ... |
| cnfldsrngadd 45212 | The group addition operati... |
| cnfldsrngmul 45213 | The ring multiplication op... |
| plusfreseq 45214 | If the empty set is not co... |
| mgmplusfreseq 45215 | If the empty set is not co... |
| 0mgm 45216 | A set with an empty base s... |
| mgmpropd 45217 | If two structures have the... |
| ismgmd 45218 | Deduce a magma from its pr... |
| mgmhmrcl 45223 | Reverse closure of a magma... |
| submgmrcl 45224 | Reverse closure for submag... |
| ismgmhm 45225 | Property of a magma homomo... |
| mgmhmf 45226 | A magma homomorphism is a ... |
| mgmhmpropd 45227 | Magma homomorphism depends... |
| mgmhmlin 45228 | A magma homomorphism prese... |
| mgmhmf1o 45229 | A magma homomorphism is bi... |
| idmgmhm 45230 | The identity homomorphism ... |
| issubmgm 45231 | Expand definition of a sub... |
| issubmgm2 45232 | Submagmas are subsets that... |
| rabsubmgmd 45233 | Deduction for proving that... |
| submgmss 45234 | Submagmas are subsets of t... |
| submgmid 45235 | Every magma is trivially a... |
| submgmcl 45236 | Submagmas are closed under... |
| submgmmgm 45237 | Submagmas are themselves m... |
| submgmbas 45238 | The base set of a submagma... |
| subsubmgm 45239 | A submagma of a submagma i... |
| resmgmhm 45240 | Restriction of a magma hom... |
| resmgmhm2 45241 | One direction of ~ resmgmh... |
| resmgmhm2b 45242 | Restriction of the codomai... |
| mgmhmco 45243 | The composition of magma h... |
| mgmhmima 45244 | The homomorphic image of a... |
| mgmhmeql 45245 | The equalizer of two magma... |
| submgmacs 45246 | Submagmas are an algebraic... |
| ismhm0 45247 | Property of a monoid homom... |
| mhmismgmhm 45248 | Each monoid homomorphism i... |
| opmpoismgm 45249 | A structure with a group a... |
| copissgrp 45250 | A structure with a constan... |
| copisnmnd 45251 | A structure with a constan... |
| 0nodd 45252 | 0 is not an odd integer. ... |
| 1odd 45253 | 1 is an odd integer. (Con... |
| 2nodd 45254 | 2 is not an odd integer. ... |
| oddibas 45255 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 45256 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 45257 | The structure of all odd i... |
| nnsgrpmgm 45258 | The structure of positive ... |
| nnsgrp 45259 | The structure of positive ... |
| nnsgrpnmnd 45260 | The structure of positive ... |
| nn0mnd 45261 | The set of nonnegative int... |
| gsumsplit2f 45262 | Split a group sum into two... |
| gsumdifsndf 45263 | Extract a summand from a f... |
| gsumfsupp 45264 | A group sum of a family ca... |
| iscllaw 45271 | The predicate "is a closed... |
| iscomlaw 45272 | The predicate "is a commut... |
| clcllaw 45273 | Closure of a closed operat... |
| isasslaw 45274 | The predicate "is an assoc... |
| asslawass 45275 | Associativity of an associ... |
| mgmplusgiopALT 45276 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 45277 | Slot 2 (group operation) o... |
| intopval 45284 | The internal (binary) oper... |
| intop 45285 | An internal (binary) opera... |
| clintopval 45286 | The closed (internal binar... |
| assintopval 45287 | The associative (closed in... |
| assintopmap 45288 | The associative (closed in... |
| isclintop 45289 | The predicate "is a closed... |
| clintop 45290 | A closed (internal binary)... |
| assintop 45291 | An associative (closed int... |
| isassintop 45292 | The predicate "is an assoc... |
| clintopcllaw 45293 | The closure law holds for ... |
| assintopcllaw 45294 | The closure low holds for ... |
| assintopasslaw 45295 | The associative low holds ... |
| assintopass 45296 | An associative (closed int... |
| ismgmALT 45305 | The predicate "is a magma"... |
| iscmgmALT 45306 | The predicate "is a commut... |
| issgrpALT 45307 | The predicate "is a semigr... |
| iscsgrpALT 45308 | The predicate "is a commut... |
| mgm2mgm 45309 | Equivalence of the two def... |
| sgrp2sgrp 45310 | Equivalence of the two def... |
| idfusubc0 45311 | The identity functor for a... |
| idfusubc 45312 | The identity functor for a... |
| inclfusubc 45313 | The "inclusion functor" fr... |
| lmod0rng 45314 | If the scalar ring of a mo... |
| nzrneg1ne0 45315 | The additive inverse of th... |
| 0ringdif 45316 | A zero ring is a ring whic... |
| 0ringbas 45317 | The base set of a zero rin... |
| 0ring1eq0 45318 | In a zero ring, a ring whi... |
| nrhmzr 45319 | There is no ring homomorph... |
| isrng 45322 | The predicate "is a non-un... |
| rngabl 45323 | A non-unital ring is an (a... |
| rngmgp 45324 | A non-unital ring is a sem... |
| ringrng 45325 | A unital ring is a (non-un... |
| ringssrng 45326 | The unital rings are (non-... |
| isringrng 45327 | The predicate "is a unital... |
| rngdir 45328 | Distributive law for the m... |
| rngcl 45329 | Closure of the multiplicat... |
| rnglz 45330 | The zero of a nonunital ri... |
| rnghmrcl 45335 | Reverse closure of a non-u... |
| rnghmfn 45336 | The mapping of two non-uni... |
| rnghmval 45337 | The set of the non-unital ... |
| isrnghm 45338 | A function is a non-unital... |
| isrnghmmul 45339 | A function is a non-unital... |
| rnghmmgmhm 45340 | A non-unital ring homomorp... |
| rnghmval2 45341 | The non-unital ring homomo... |
| isrngisom 45342 | An isomorphism of non-unit... |
| rngimrcl 45343 | Reverse closure for an iso... |
| rnghmghm 45344 | A non-unital ring homomorp... |
| rnghmf 45345 | A ring homomorphism is a f... |
| rnghmmul 45346 | A homomorphism of non-unit... |
| isrnghm2d 45347 | Demonstration of non-unita... |
| isrnghmd 45348 | Demonstration of non-unita... |
| rnghmf1o 45349 | A non-unital ring homomorp... |
| isrngim 45350 | An isomorphism of non-unit... |
| rngimf1o 45351 | An isomorphism of non-unit... |
| rngimrnghm 45352 | An isomorphism of non-unit... |
| rnghmco 45353 | The composition of non-uni... |
| idrnghm 45354 | The identity homomorphism ... |
| c0mgm 45355 | The constant mapping to ze... |
| c0mhm 45356 | The constant mapping to ze... |
| c0ghm 45357 | The constant mapping to ze... |
| c0rhm 45358 | The constant mapping to ze... |
| c0rnghm 45359 | The constant mapping to ze... |
| c0snmgmhm 45360 | The constant mapping to ze... |
| c0snmhm 45361 | The constant mapping to ze... |
| c0snghm 45362 | The constant mapping to ze... |
| zrrnghm 45363 | The constant mapping to ze... |
| rhmfn 45364 | The mapping of two rings t... |
| rhmval 45365 | The ring homomorphisms bet... |
| rhmisrnghm 45366 | Each unital ring homomorph... |
| lidldomn1 45367 | If a (left) ideal (which i... |
| lidlssbas 45368 | The base set of the restri... |
| lidlbas 45369 | A (left) ideal of a ring i... |
| lidlabl 45370 | A (left) ideal of a ring i... |
| lidlmmgm 45371 | The multiplicative group o... |
| lidlmsgrp 45372 | The multiplicative group o... |
| lidlrng 45373 | A (left) ideal of a ring i... |
| zlidlring 45374 | The zero (left) ideal of a... |
| uzlidlring 45375 | Only the zero (left) ideal... |
| lidldomnnring 45376 | A (left) ideal of a domain... |
| 0even 45377 | 0 is an even integer. (Co... |
| 1neven 45378 | 1 is not an even integer. ... |
| 2even 45379 | 2 is an even integer. (Co... |
| 2zlidl 45380 | The even integers are a (l... |
| 2zrng 45381 | The ring of integers restr... |
| 2zrngbas 45382 | The base set of R is the s... |
| 2zrngadd 45383 | The group addition operati... |
| 2zrng0 45384 | The additive identity of R... |
| 2zrngamgm 45385 | R is an (additive) magma. ... |
| 2zrngasgrp 45386 | R is an (additive) semigro... |
| 2zrngamnd 45387 | R is an (additive) monoid.... |
| 2zrngacmnd 45388 | R is a commutative (additi... |
| 2zrngagrp 45389 | R is an (additive) group. ... |
| 2zrngaabl 45390 | R is an (additive) abelian... |
| 2zrngmul 45391 | The ring multiplication op... |
| 2zrngmmgm 45392 | R is a (multiplicative) ma... |
| 2zrngmsgrp 45393 | R is a (multiplicative) se... |
| 2zrngALT 45394 | The ring of integers restr... |
| 2zrngnmlid 45395 | R has no multiplicative (l... |
| 2zrngnmrid 45396 | R has no multiplicative (r... |
| 2zrngnmlid2 45397 | R has no multiplicative (l... |
| 2zrngnring 45398 | R is not a unital ring. (... |
| cznrnglem 45399 | Lemma for ~ cznrng : The ... |
| cznabel 45400 | The ring constructed from ... |
| cznrng 45401 | The ring constructed from ... |
| cznnring 45402 | The ring constructed from ... |
| rngcvalALTV 45407 | Value of the category of n... |
| rngcval 45408 | Value of the category of n... |
| rnghmresfn 45409 | The class of non-unital ri... |
| rnghmresel 45410 | An element of the non-unit... |
| rngcbas 45411 | Set of objects of the cate... |
| rngchomfval 45412 | Set of arrows of the categ... |
| rngchom 45413 | Set of arrows of the categ... |
| elrngchom 45414 | A morphism of non-unital r... |
| rngchomfeqhom 45415 | The functionalized Hom-set... |
| rngccofval 45416 | Composition in the categor... |
| rngcco 45417 | Composition in the categor... |
| dfrngc2 45418 | Alternate definition of th... |
| rnghmsscmap2 45419 | The non-unital ring homomo... |
| rnghmsscmap 45420 | The non-unital ring homomo... |
| rnghmsubcsetclem1 45421 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 45422 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 45423 | The non-unital ring homomo... |
| rngccat 45424 | The category of non-unital... |
| rngcid 45425 | The identity arrow in the ... |
| rngcsect 45426 | A section in the category ... |
| rngcinv 45427 | An inverse in the category... |
| rngciso 45428 | An isomorphism in the cate... |
| rngcbasALTV 45429 | Set of objects of the cate... |
| rngchomfvalALTV 45430 | Set of arrows of the categ... |
| rngchomALTV 45431 | Set of arrows of the categ... |
| elrngchomALTV 45432 | A morphism of non-unital r... |
| rngccofvalALTV 45433 | Composition in the categor... |
| rngccoALTV 45434 | Composition in the categor... |
| rngccatidALTV 45435 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 45436 | The category of non-unital... |
| rngcidALTV 45437 | The identity arrow in the ... |
| rngcsectALTV 45438 | A section in the category ... |
| rngcinvALTV 45439 | An inverse in the category... |
| rngcisoALTV 45440 | An isomorphism in the cate... |
| rngchomffvalALTV 45441 | The value of the functiona... |
| rngchomrnghmresALTV 45442 | The value of the functiona... |
| rngcifuestrc 45443 | The "inclusion functor" fr... |
| funcrngcsetc 45444 | The "natural forgetful fun... |
| funcrngcsetcALT 45445 | Alternate proof of ~ funcr... |
| zrinitorngc 45446 | The zero ring is an initia... |
| zrtermorngc 45447 | The zero ring is a termina... |
| zrzeroorngc 45448 | The zero ring is a zero ob... |
| ringcvalALTV 45453 | Value of the category of r... |
| ringcval 45454 | Value of the category of u... |
| rhmresfn 45455 | The class of unital ring h... |
| rhmresel 45456 | An element of the unital r... |
| ringcbas 45457 | Set of objects of the cate... |
| ringchomfval 45458 | Set of arrows of the categ... |
| ringchom 45459 | Set of arrows of the categ... |
| elringchom 45460 | A morphism of unital rings... |
| ringchomfeqhom 45461 | The functionalized Hom-set... |
| ringccofval 45462 | Composition in the categor... |
| ringcco 45463 | Composition in the categor... |
| dfringc2 45464 | Alternate definition of th... |
| rhmsscmap2 45465 | The unital ring homomorphi... |
| rhmsscmap 45466 | The unital ring homomorphi... |
| rhmsubcsetclem1 45467 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 45468 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 45469 | The unital ring homomorphi... |
| ringccat 45470 | The category of unital rin... |
| ringcid 45471 | The identity arrow in the ... |
| rhmsscrnghm 45472 | The unital ring homomorphi... |
| rhmsubcrngclem1 45473 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 45474 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 45475 | The unital ring homomorphi... |
| rngcresringcat 45476 | The restriction of the cat... |
| ringcsect 45477 | A section in the category ... |
| ringcinv 45478 | An inverse in the category... |
| ringciso 45479 | An isomorphism in the cate... |
| ringcbasbas 45480 | An element of the base set... |
| funcringcsetc 45481 | The "natural forgetful fun... |
| funcringcsetcALTV2lem1 45482 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 45483 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 45484 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 45485 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 45486 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 45487 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 45488 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 45489 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 45490 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 45491 | The "natural forgetful fun... |
| ringcbasALTV 45492 | Set of objects of the cate... |
| ringchomfvalALTV 45493 | Set of arrows of the categ... |
| ringchomALTV 45494 | Set of arrows of the categ... |
| elringchomALTV 45495 | A morphism of rings is a f... |
| ringccofvalALTV 45496 | Composition in the categor... |
| ringccoALTV 45497 | Composition in the categor... |
| ringccatidALTV 45498 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 45499 | The category of rings is a... |
| ringcidALTV 45500 | The identity arrow in the ... |
| ringcsectALTV 45501 | A section in the category ... |
| ringcinvALTV 45502 | An inverse in the category... |
| ringcisoALTV 45503 | An isomorphism in the cate... |
| ringcbasbasALTV 45504 | An element of the base set... |
| funcringcsetclem1ALTV 45505 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 45506 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 45507 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 45508 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 45509 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 45510 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 45511 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 45512 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 45513 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 45514 | The "natural forgetful fun... |
| irinitoringc 45515 | The ring of integers is an... |
| zrtermoringc 45516 | The zero ring is a termina... |
| zrninitoringc 45517 | The zero ring is not an in... |
| nzerooringczr 45518 | There is no zero object in... |
| srhmsubclem1 45519 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 45520 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 45521 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 45522 | According to ~ df-subc , t... |
| sringcat 45523 | The restriction of the cat... |
| crhmsubc 45524 | According to ~ df-subc , t... |
| cringcat 45525 | The restriction of the cat... |
| drhmsubc 45526 | According to ~ df-subc , t... |
| drngcat 45527 | The restriction of the cat... |
| fldcat 45528 | The restriction of the cat... |
| fldc 45529 | The restriction of the cat... |
| fldhmsubc 45530 | According to ~ df-subc , t... |
| rngcrescrhm 45531 | The category of non-unital... |
| rhmsubclem1 45532 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 45533 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 45534 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 45535 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 45536 | According to ~ df-subc , t... |
| rhmsubccat 45537 | The restriction of the cat... |
| srhmsubcALTVlem1 45538 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 45539 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 45540 | According to ~ df-subc , t... |
| sringcatALTV 45541 | The restriction of the cat... |
| crhmsubcALTV 45542 | According to ~ df-subc , t... |
| cringcatALTV 45543 | The restriction of the cat... |
| drhmsubcALTV 45544 | According to ~ df-subc , t... |
| drngcatALTV 45545 | The restriction of the cat... |
| fldcatALTV 45546 | The restriction of the cat... |
| fldcALTV 45547 | The restriction of the cat... |
| fldhmsubcALTV 45548 | According to ~ df-subc , t... |
| rngcrescrhmALTV 45549 | The category of non-unital... |
| rhmsubcALTVlem1 45550 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 45551 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 45552 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 45553 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 45554 | According to ~ df-subc , t... |
| rhmsubcALTVcat 45555 | The restriction of the cat... |
| opeliun2xp 45556 | Membership of an ordered p... |
| eliunxp2 45557 | Membership in a union of C... |
| mpomptx2 45558 | Express a two-argument fun... |
| cbvmpox2 45559 | Rule to change the bound v... |
| dmmpossx2 45560 | The domain of a mapping is... |
| mpoexxg2 45561 | Existence of an operation ... |
| ovmpordxf 45562 | Value of an operation give... |
| ovmpordx 45563 | Value of an operation give... |
| ovmpox2 45564 | The value of an operation ... |
| fdmdifeqresdif 45565 | The restriction of a condi... |
| offvalfv 45566 | The function operation exp... |
| ofaddmndmap 45567 | The function operation app... |
| mapsnop 45568 | A singleton of an ordered ... |
| fprmappr 45569 | A function with a domain o... |
| mapprop 45570 | An unordered pair containi... |
| ztprmneprm 45571 | A prime is not an integer ... |
| 2t6m3t4e0 45572 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 45573 | For any finite subset of `... |
| nn0sumltlt 45574 | If the sum of two nonnegat... |
| bcpascm1 45575 | Pascal's rule for the bino... |
| altgsumbc 45576 | The sum of binomial coeffi... |
| altgsumbcALT 45577 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 45578 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 45579 | An element of the (base se... |
| zlmodzxz0 45580 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 45581 | The scalar multiplication ... |
| zlmodzxzadd 45582 | The addition of the ` ZZ `... |
| zlmodzxzsubm 45583 | The subtraction of the ` Z... |
| zlmodzxzsub 45584 | The subtraction of the ` Z... |
| mgpsumunsn 45585 | Extract a summand/factor f... |
| mgpsumz 45586 | If the group sum for the m... |
| mgpsumn 45587 | If the group sum for the m... |
| exple2lt6 45588 | A nonnegative integer to t... |
| pgrple2abl 45589 | Every symmetric group on a... |
| pgrpgt2nabl 45590 | Every symmetric group on a... |
| invginvrid 45591 | Identity for a multiplicat... |
| rmsupp0 45592 | The support of a mapping o... |
| domnmsuppn0 45593 | The support of a mapping o... |
| rmsuppss 45594 | The support of a mapping o... |
| mndpsuppss 45595 | The support of a mapping o... |
| scmsuppss 45596 | The support of a mapping o... |
| rmsuppfi 45597 | The support of a mapping o... |
| rmfsupp 45598 | A mapping of a multiplicat... |
| mndpsuppfi 45599 | The support of a mapping o... |
| mndpfsupp 45600 | A mapping of a scalar mult... |
| scmsuppfi 45601 | The support of a mapping o... |
| scmfsupp 45602 | A mapping of a scalar mult... |
| suppmptcfin 45603 | The support of a mapping w... |
| mptcfsupp 45604 | A mapping with value 0 exc... |
| fsuppmptdmf 45605 | A mapping with a finite do... |
| lmodvsmdi 45606 | Multiple distributive law ... |
| gsumlsscl 45607 | Closure of a group sum in ... |
| assaascl0 45608 | The scalar 0 embedded into... |
| assaascl1 45609 | The scalar 1 embedded into... |
| ply1vr1smo 45610 | The variable in a polynomi... |
| ply1ass23l 45611 | Associative identity with ... |
| ply1sclrmsm 45612 | The ring multiplication of... |
| coe1id 45613 | Coefficient vector of the ... |
| coe1sclmulval 45614 | The value of the coefficie... |
| ply1mulgsumlem1 45615 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 45616 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 45617 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 45618 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 45619 | The product of two polynom... |
| evl1at0 45620 | Polynomial evaluation for ... |
| evl1at1 45621 | Polynomial evaluation for ... |
| linply1 45622 | A term of the form ` x - C... |
| lineval 45623 | A term of the form ` x - C... |
| linevalexample 45624 | The polynomial ` x - 3 ` o... |
| dmatALTval 45629 | The algebra of ` N ` x ` N... |
| dmatALTbas 45630 | The base set of the algebr... |
| dmatALTbasel 45631 | An element of the base set... |
| dmatbas 45632 | The set of all ` N ` x ` N... |
| lincop 45637 | A linear combination as op... |
| lincval 45638 | The value of a linear comb... |
| dflinc2 45639 | Alternative definition of ... |
| lcoop 45640 | A linear combination as op... |
| lcoval 45641 | The value of a linear comb... |
| lincfsuppcl 45642 | A linear combination of ve... |
| linccl 45643 | A linear combination of ve... |
| lincval0 45644 | The value of an empty line... |
| lincvalsng 45645 | The linear combination ove... |
| lincvalsn 45646 | The linear combination ove... |
| lincvalpr 45647 | The linear combination ove... |
| lincval1 45648 | The linear combination ove... |
| lcosn0 45649 | Properties of a linear com... |
| lincvalsc0 45650 | The linear combination whe... |
| lcoc0 45651 | Properties of a linear com... |
| linc0scn0 45652 | If a set contains the zero... |
| lincdifsn 45653 | A vector is a linear combi... |
| linc1 45654 | A vector is a linear combi... |
| lincellss 45655 | A linear combination of a ... |
| lco0 45656 | The set of empty linear co... |
| lcoel0 45657 | The zero vector is always ... |
| lincsum 45658 | The sum of two linear comb... |
| lincscm 45659 | A linear combinations mult... |
| lincsumcl 45660 | The sum of two linear comb... |
| lincscmcl 45661 | The multiplication of a li... |
| lincsumscmcl 45662 | The sum of a linear combin... |
| lincolss 45663 | According to the statement... |
| ellcoellss 45664 | Every linear combination o... |
| lcoss 45665 | A set of vectors of a modu... |
| lspsslco 45666 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 45667 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 45668 | Equivalence of a _span_ of... |
| rellininds 45672 | The class defining the rel... |
| linindsv 45674 | The classes of the module ... |
| islininds 45675 | The property of being a li... |
| linindsi 45676 | The implications of being ... |
| linindslinci 45677 | The implications of being ... |
| islinindfis 45678 | The property of being a li... |
| islinindfiss 45679 | The property of being a li... |
| linindscl 45680 | A linearly independent set... |
| lindepsnlininds 45681 | A linearly dependent subse... |
| islindeps 45682 | The property of being a li... |
| lincext1 45683 | Property 1 of an extension... |
| lincext2 45684 | Property 2 of an extension... |
| lincext3 45685 | Property 3 of an extension... |
| lindslinindsimp1 45686 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 45687 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 45688 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 45689 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 45690 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 45691 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 45692 | Implication 2 for ~ lindsl... |
| lindslininds 45693 | Equivalence of definitions... |
| linds0 45694 | The empty set is always a ... |
| el0ldep 45695 | A set containing the zero ... |
| el0ldepsnzr 45696 | A set containing the zero ... |
| lindsrng01 45697 | Any subset of a module is ... |
| lindszr 45698 | Any subset of a module ove... |
| snlindsntorlem 45699 | Lemma for ~ snlindsntor . ... |
| snlindsntor 45700 | A singleton is linearly in... |
| ldepsprlem 45701 | Lemma for ~ ldepspr . (Co... |
| ldepspr 45702 | If a vector is a scalar mu... |
| lincresunit3lem3 45703 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 45704 | Lemma 1 for properties of ... |
| lincresunitlem2 45705 | Lemma for properties of a ... |
| lincresunit1 45706 | Property 1 of a specially ... |
| lincresunit2 45707 | Property 2 of a specially ... |
| lincresunit3lem1 45708 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 45709 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 45710 | Property 3 of a specially ... |
| lincreslvec3 45711 | Property 3 of a specially ... |
| islindeps2 45712 | Conditions for being a lin... |
| islininds2 45713 | Implication of being a lin... |
| isldepslvec2 45714 | Alternative definition of ... |
| lindssnlvec 45715 | A singleton not containing... |
| lmod1lem1 45716 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 45717 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 45718 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 45719 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 45720 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 45721 | The (smallest) structure r... |
| lmod1zr 45722 | The (smallest) structure r... |
| lmod1zrnlvec 45723 | There is a (left) module (... |
| lmodn0 45724 | Left modules exist. (Cont... |
| zlmodzxzequa 45725 | Example of an equation wit... |
| zlmodzxznm 45726 | Example of a linearly depe... |
| zlmodzxzldeplem 45727 | A and B are not equal. (C... |
| zlmodzxzequap 45728 | Example of an equation wit... |
| zlmodzxzldeplem1 45729 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 45730 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 45731 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 45732 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 45733 | { A , B } is a linearly de... |
| ldepsnlinclem1 45734 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 45735 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 45736 | The class of all (left) ve... |
| ldepsnlinc 45737 | The reverse implication of... |
| ldepslinc 45738 | For (left) vector spaces, ... |
| suppdm 45739 | If the range of a function... |
| eluz2cnn0n1 45740 | An integer greater than 1 ... |
| divge1b 45741 | The ratio of a real number... |
| divgt1b 45742 | The ratio of a real number... |
| ltsubaddb 45743 | Equivalence for the "less ... |
| ltsubsubb 45744 | Equivalence for the "less ... |
| ltsubadd2b 45745 | Equivalence for the "less ... |
| divsub1dir 45746 | Distribution of division o... |
| expnegico01 45747 | An integer greater than 1 ... |
| elfzolborelfzop1 45748 | An element of a half-open ... |
| pw2m1lepw2m1 45749 | 2 to the power of a positi... |
| zgtp1leeq 45750 | If an integer is between a... |
| flsubz 45751 | An integer can be moved in... |
| fldivmod 45752 | Expressing the floor of a ... |
| mod0mul 45753 | If an integer is 0 modulo ... |
| modn0mul 45754 | If an integer is not 0 mod... |
| m1modmmod 45755 | An integer decreased by 1 ... |
| difmodm1lt 45756 | The difference between an ... |
| nn0onn0ex 45757 | For each odd nonnegative i... |
| nn0enn0ex 45758 | For each even nonnegative ... |
| nnennex 45759 | For each even positive int... |
| nneop 45760 | A positive integer is even... |
| nneom 45761 | A positive integer is even... |
| nn0eo 45762 | A nonnegative integer is e... |
| nnpw2even 45763 | 2 to the power of a positi... |
| zefldiv2 45764 | The floor of an even integ... |
| zofldiv2 45765 | The floor of an odd intege... |
| nn0ofldiv2 45766 | The floor of an odd nonneg... |
| flnn0div2ge 45767 | The floor of a positive in... |
| flnn0ohalf 45768 | The floor of the half of a... |
| logcxp0 45769 | Logarithm of a complex pow... |
| regt1loggt0 45770 | The natural logarithm for ... |
| fdivval 45773 | The quotient of two functi... |
| fdivmpt 45774 | The quotient of two functi... |
| fdivmptf 45775 | The quotient of two functi... |
| refdivmptf 45776 | The quotient of two functi... |
| fdivpm 45777 | The quotient of two functi... |
| refdivpm 45778 | The quotient of two functi... |
| fdivmptfv 45779 | The function value of a qu... |
| refdivmptfv 45780 | The function value of a qu... |
| bigoval 45783 | Set of functions of order ... |
| elbigofrcl 45784 | Reverse closure of the "bi... |
| elbigo 45785 | Properties of a function o... |
| elbigo2 45786 | Properties of a function o... |
| elbigo2r 45787 | Sufficient condition for a... |
| elbigof 45788 | A function of order G(x) i... |
| elbigodm 45789 | The domain of a function o... |
| elbigoimp 45790 | The defining property of a... |
| elbigolo1 45791 | A function (into the posit... |
| rege1logbrege0 45792 | The general logarithm, wit... |
| rege1logbzge0 45793 | The general logarithm, wit... |
| fllogbd 45794 | A real number is between t... |
| relogbmulbexp 45795 | The logarithm of the produ... |
| relogbdivb 45796 | The logarithm of the quoti... |
| logbge0b 45797 | The logarithm of a number ... |
| logblt1b 45798 | The logarithm of a number ... |
| fldivexpfllog2 45799 | The floor of a positive re... |
| nnlog2ge0lt1 45800 | A positive integer is 1 if... |
| logbpw2m1 45801 | The floor of the binary lo... |
| fllog2 45802 | The floor of the binary lo... |
| blenval 45805 | The binary length of an in... |
| blen0 45806 | The binary length of 0. (... |
| blenn0 45807 | The binary length of a "nu... |
| blenre 45808 | The binary length of a pos... |
| blennn 45809 | The binary length of a pos... |
| blennnelnn 45810 | The binary length of a pos... |
| blennn0elnn 45811 | The binary length of a non... |
| blenpw2 45812 | The binary length of a pow... |
| blenpw2m1 45813 | The binary length of a pow... |
| nnpw2blen 45814 | A positive integer is betw... |
| nnpw2blenfzo 45815 | A positive integer is betw... |
| nnpw2blenfzo2 45816 | A positive integer is eith... |
| nnpw2pmod 45817 | Every positive integer can... |
| blen1 45818 | The binary length of 1. (... |
| blen2 45819 | The binary length of 2. (... |
| nnpw2p 45820 | Every positive integer can... |
| nnpw2pb 45821 | A number is a positive int... |
| blen1b 45822 | The binary length of a non... |
| blennnt2 45823 | The binary length of a pos... |
| nnolog2flm1 45824 | The floor of the binary lo... |
| blennn0em1 45825 | The binary length of the h... |
| blennngt2o2 45826 | The binary length of an od... |
| blengt1fldiv2p1 45827 | The binary length of an in... |
| blennn0e2 45828 | The binary length of an ev... |
| digfval 45831 | Operation to obtain the ` ... |
| digval 45832 | The ` K ` th digit of a no... |
| digvalnn0 45833 | The ` K ` th digit of a no... |
| nn0digval 45834 | The ` K ` th digit of a no... |
| dignn0fr 45835 | The digits of the fraction... |
| dignn0ldlem 45836 | Lemma for ~ dignnld . (Co... |
| dignnld 45837 | The leading digits of a po... |
| dig2nn0ld 45838 | The leading digits of a po... |
| dig2nn1st 45839 | The first (relevant) digit... |
| dig0 45840 | All digits of 0 are 0. (C... |
| digexp 45841 | The ` K ` th digit of a po... |
| dig1 45842 | All but one digits of 1 ar... |
| 0dig1 45843 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 45844 | The integers 0 and 1 corre... |
| dig2nn0 45845 | A digit of a nonnegative i... |
| 0dig2nn0e 45846 | The last bit of an even in... |
| 0dig2nn0o 45847 | The last bit of an odd int... |
| dig2bits 45848 | The ` K ` th digit of a no... |
| dignn0flhalflem1 45849 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 45850 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 45851 | The digits of the half of ... |
| dignn0flhalf 45852 | The digits of the rounded ... |
| nn0sumshdiglemA 45853 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 45854 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 45855 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 45856 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 45857 | A nonnegative integer can ... |
| nn0mulfsum 45858 | Trivial algorithm to calcu... |
| nn0mullong 45859 | Standard algorithm (also k... |
| naryfval 45862 | The set of the n-ary (endo... |
| naryfvalixp 45863 | The set of the n-ary (endo... |
| naryfvalel 45864 | An n-ary (endo)function on... |
| naryrcl 45865 | Reverse closure for n-ary ... |
| naryfvalelfv 45866 | The value of an n-ary (end... |
| naryfvalelwrdf 45867 | An n-ary (endo)function on... |
| 0aryfvalel 45868 | A nullary (endo)function o... |
| 0aryfvalelfv 45869 | The value of a nullary (en... |
| 1aryfvalel 45870 | A unary (endo)function on ... |
| fv1arycl 45871 | Closure of a unary (endo)f... |
| 1arympt1 45872 | A unary (endo)function in ... |
| 1arympt1fv 45873 | The value of a unary (endo... |
| 1arymaptfv 45874 | The value of the mapping o... |
| 1arymaptf 45875 | The mapping of unary (endo... |
| 1arymaptf1 45876 | The mapping of unary (endo... |
| 1arymaptfo 45877 | The mapping of unary (endo... |
| 1arymaptf1o 45878 | The mapping of unary (endo... |
| 1aryenef 45879 | The set of unary (endo)fun... |
| 1aryenefmnd 45880 | The set of unary (endo)fun... |
| 2aryfvalel 45881 | A binary (endo)function on... |
| fv2arycl 45882 | Closure of a binary (endo)... |
| 2arympt 45883 | A binary (endo)function in... |
| 2arymptfv 45884 | The value of a binary (end... |
| 2arymaptfv 45885 | The value of the mapping o... |
| 2arymaptf 45886 | The mapping of binary (end... |
| 2arymaptf1 45887 | The mapping of binary (end... |
| 2arymaptfo 45888 | The mapping of binary (end... |
| 2arymaptf1o 45889 | The mapping of binary (end... |
| 2aryenef 45890 | The set of binary (endo)fu... |
| itcoval 45895 | The value of the function ... |
| itcoval0 45896 | A function iterated zero t... |
| itcoval1 45897 | A function iterated once. ... |
| itcoval2 45898 | A function iterated twice.... |
| itcoval3 45899 | A function iterated three ... |
| itcoval0mpt 45900 | A mapping iterated zero ti... |
| itcovalsuc 45901 | The value of the function ... |
| itcovalsucov 45902 | The value of the function ... |
| itcovalendof 45903 | The n-th iterate of an end... |
| itcovalpclem1 45904 | Lemma 1 for ~ itcovalpc : ... |
| itcovalpclem2 45905 | Lemma 2 for ~ itcovalpc : ... |
| itcovalpc 45906 | The value of the function ... |
| itcovalt2lem2lem1 45907 | Lemma 1 for ~ itcovalt2lem... |
| itcovalt2lem2lem2 45908 | Lemma 2 for ~ itcovalt2lem... |
| itcovalt2lem1 45909 | Lemma 1 for ~ itcovalt2 : ... |
| itcovalt2lem2 45910 | Lemma 2 for ~ itcovalt2 : ... |
| itcovalt2 45911 | The value of the function ... |
| ackvalsuc1mpt 45912 | The Ackermann function at ... |
| ackvalsuc1 45913 | The Ackermann function at ... |
| ackval0 45914 | The Ackermann function at ... |
| ackval1 45915 | The Ackermann function at ... |
| ackval2 45916 | The Ackermann function at ... |
| ackval3 45917 | The Ackermann function at ... |
| ackendofnn0 45918 | The Ackermann function at ... |
| ackfnnn0 45919 | The Ackermann function at ... |
| ackval0val 45920 | The Ackermann function at ... |
| ackvalsuc0val 45921 | The Ackermann function at ... |
| ackvalsucsucval 45922 | The Ackermann function at ... |
| ackval0012 45923 | The Ackermann function at ... |
| ackval1012 45924 | The Ackermann function at ... |
| ackval2012 45925 | The Ackermann function at ... |
| ackval3012 45926 | The Ackermann function at ... |
| ackval40 45927 | The Ackermann function at ... |
| ackval41a 45928 | The Ackermann function at ... |
| ackval41 45929 | The Ackermann function at ... |
| ackval42 45930 | The Ackermann function at ... |
| ackval42a 45931 | The Ackermann function at ... |
| ackval50 45932 | The Ackermann function at ... |
| fv1prop 45933 | The function value of unor... |
| fv2prop 45934 | The function value of unor... |
| submuladdmuld 45935 | Transformation of a sum of... |
| affinecomb1 45936 | Combination of two real af... |
| affinecomb2 45937 | Combination of two real af... |
| affineid 45938 | Identity of an affine comb... |
| 1subrec1sub 45939 | Subtract the reciprocal of... |
| resum2sqcl 45940 | The sum of two squares of ... |
| resum2sqgt0 45941 | The sum of the square of a... |
| resum2sqrp 45942 | The sum of the square of a... |
| resum2sqorgt0 45943 | The sum of the square of t... |
| reorelicc 45944 | Membership in and outside ... |
| rrx2pxel 45945 | The x-coordinate of a poin... |
| rrx2pyel 45946 | The y-coordinate of a poin... |
| prelrrx2 45947 | An unordered pair of order... |
| prelrrx2b 45948 | An unordered pair of order... |
| rrx2pnecoorneor 45949 | If two different points ` ... |
| rrx2pnedifcoorneor 45950 | If two different points ` ... |
| rrx2pnedifcoorneorr 45951 | If two different points ` ... |
| rrx2xpref1o 45952 | There is a bijection betwe... |
| rrx2xpreen 45953 | The set of points in the t... |
| rrx2plord 45954 | The lexicographical orderi... |
| rrx2plord1 45955 | The lexicographical orderi... |
| rrx2plord2 45956 | The lexicographical orderi... |
| rrx2plordisom 45957 | The set of points in the t... |
| rrx2plordso 45958 | The lexicographical orderi... |
| ehl2eudisval0 45959 | The Euclidean distance of ... |
| ehl2eudis0lt 45960 | An upper bound of the Eucl... |
| lines 45965 | The lines passing through ... |
| line 45966 | The line passing through t... |
| rrxlines 45967 | Definition of lines passin... |
| rrxline 45968 | The line passing through t... |
| rrxlinesc 45969 | Definition of lines passin... |
| rrxlinec 45970 | The line passing through t... |
| eenglngeehlnmlem1 45971 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 45972 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 45973 | The line definition in the... |
| rrx2line 45974 | The line passing through t... |
| rrx2vlinest 45975 | The vertical line passing ... |
| rrx2linest 45976 | The line passing through t... |
| rrx2linesl 45977 | The line passing through t... |
| rrx2linest2 45978 | The line passing through t... |
| elrrx2linest2 45979 | The line passing through t... |
| spheres 45980 | The spheres for given cent... |
| sphere 45981 | A sphere with center ` X `... |
| rrxsphere 45982 | The sphere with center ` M... |
| 2sphere 45983 | The sphere with center ` M... |
| 2sphere0 45984 | The sphere around the orig... |
| line2ylem 45985 | Lemma for ~ line2y . This... |
| line2 45986 | Example for a line ` G ` p... |
| line2xlem 45987 | Lemma for ~ line2x . This... |
| line2x 45988 | Example for a horizontal l... |
| line2y 45989 | Example for a vertical lin... |
| itsclc0lem1 45990 | Lemma for theorems about i... |
| itsclc0lem2 45991 | Lemma for theorems about i... |
| itsclc0lem3 45992 | Lemma for theorems about i... |
| itscnhlc0yqe 45993 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 45994 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 45995 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 45996 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 45997 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 45998 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 45999 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 46000 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 46001 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 46002 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 46003 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 46004 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 46005 | The intersection points of... |
| itsclc0b 46006 | The intersection points of... |
| itsclinecirc0 46007 | The intersection points of... |
| itsclinecirc0b 46008 | The intersection points of... |
| itsclinecirc0in 46009 | The intersection points of... |
| itsclquadb 46010 | Quadratic equation for the... |
| itsclquadeu 46011 | Quadratic equation for the... |
| 2itscplem1 46012 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 46013 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 46014 | Lemma D for ~ 2itscp . (C... |
| 2itscp 46015 | A condition for a quadrati... |
| itscnhlinecirc02plem1 46016 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 46017 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 46018 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 46019 | Intersection of a nonhoriz... |
| inlinecirc02plem 46020 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 46021 | Intersection of a line wit... |
| inlinecirc02preu 46022 | Intersection of a line wit... |
| pm4.71da 46023 | Deduction converting a bic... |
| logic1 46024 | Distribution of implicatio... |
| logic1a 46025 | Variant of ~ logic1 . (Co... |
| logic2 46026 | Variant of ~ logic1 . (Co... |
| pm5.32dav 46027 | Distribution of implicatio... |
| pm5.32dra 46028 | Reverse distribution of im... |
| exp12bd 46029 | The import-export theorem ... |
| mpbiran3d 46030 | Equivalence with a conjunc... |
| mpbiran4d 46031 | Equivalence with a conjunc... |
| dtrucor3 46032 | An example of how ~ ax-5 w... |
| ralbidb 46033 | Formula-building rule for ... |
| ralbidc 46034 | Formula-building rule for ... |
| r19.41dv 46035 | A complex deduction form o... |
| rspceb2dv 46036 | Restricted existential spe... |
| rextru 46037 | Two ways of expressing "at... |
| rmotru 46038 | Two ways of expressing "at... |
| reutru 46039 | Two ways of expressing "ex... |
| reutruALT 46040 | Alternate proof for ~ reut... |
| ssdisjd 46041 | Subset preserves disjointn... |
| ssdisjdr 46042 | Subset preserves disjointn... |
| disjdifb 46043 | Relative complement is ant... |
| predisj 46044 | Preimages of disjoint sets... |
| vsn 46045 | The singleton of the unive... |
| mosn 46046 | "At most one" element in a... |
| mo0 46047 | "At most one" element in a... |
| mosssn 46048 | "At most one" element in a... |
| mo0sn 46049 | Two ways of expressing "at... |
| mosssn2 46050 | Two ways of expressing "at... |
| unilbss 46051 | Superclass of the greatest... |
| inpw 46052 | Two ways of expressing a c... |
| mof0 46053 | There is at most one funct... |
| mof02 46054 | A variant of ~ mof0 . (Co... |
| mof0ALT 46055 | Alternate proof for ~ mof0... |
| eufsnlem 46056 | There is exactly one funct... |
| eufsn 46057 | There is exactly one funct... |
| eufsn2 46058 | There is exactly one funct... |
| mofsn 46059 | There is at most one funct... |
| mofsn2 46060 | There is at most one funct... |
| mofsssn 46061 | There is at most one funct... |
| mofmo 46062 | There is at most one funct... |
| mofeu 46063 | The uniqueness of a functi... |
| elfvne0 46064 | If a function value has a ... |
| fdomne0 46065 | A function with non-empty ... |
| f1sn2g 46066 | A function that maps a sin... |
| f102g 46067 | A function that maps the e... |
| f1mo 46068 | A function that maps a set... |
| f002 46069 | A function with an empty c... |
| map0cor 46070 | A function exists iff an e... |
| fvconstr 46071 | Two ways of expressing ` A... |
| fvconstrn0 46072 | Two ways of expressing ` A... |
| fvconstr2 46073 | Two ways of expressing ` A... |
| fvconst0ci 46074 | A constant function's valu... |
| fvconstdomi 46075 | A constant function's valu... |
| f1omo 46076 | There is at most one eleme... |
| f1omoALT 46077 | There is at most one eleme... |
| iccin 46078 | Intersection of two closed... |
| iccdisj2 46079 | If the upper bound of one ... |
| iccdisj 46080 | If the upper bound of one ... |
| mreuniss 46081 | The union of a collection ... |
| clduni 46082 | The union of closed sets i... |
| opncldeqv 46083 | Conditions on open sets ar... |
| opndisj 46084 | Two ways of saying that tw... |
| clddisj 46085 | Two ways of saying that tw... |
| neircl 46086 | Reverse closure of the nei... |
| opnneilem 46087 | Lemma factoring out common... |
| opnneir 46088 | If something is true for a... |
| opnneirv 46089 | A variant of ~ opnneir wit... |
| opnneilv 46090 | The converse of ~ opnneir ... |
| opnneil 46091 | A variant of ~ opnneilv . ... |
| opnneieqv 46092 | The equivalence between ne... |
| opnneieqvv 46093 | The equivalence between ne... |
| restcls2lem 46094 | A closed set in a subspace... |
| restcls2 46095 | A closed set in a subspace... |
| restclsseplem 46096 | Lemma for ~ restclssep . ... |
| restclssep 46097 | Two disjoint closed sets i... |
| cnneiima 46098 | Given a continuous functio... |
| iooii 46099 | Open intervals are open se... |
| icccldii 46100 | Closed intervals are close... |
| i0oii 46101 | ` ( 0 [,) A ) ` is open in... |
| io1ii 46102 | ` ( A (,] 1 ) ` is open in... |
| sepnsepolem1 46103 | Lemma for ~ sepnsepo . (C... |
| sepnsepolem2 46104 | Open neighborhood and neig... |
| sepnsepo 46105 | Open neighborhood and neig... |
| sepdisj 46106 | Separated sets are disjoin... |
| seposep 46107 | If two sets are separated ... |
| sepcsepo 46108 | If two sets are separated ... |
| sepfsepc 46109 | If two sets are separated ... |
| seppsepf 46110 | If two sets are precisely ... |
| seppcld 46111 | If two sets are precisely ... |
| isnrm4 46112 | A topological space is nor... |
| dfnrm2 46113 | A topological space is nor... |
| dfnrm3 46114 | A topological space is nor... |
| iscnrm3lem1 46115 | Lemma for ~ iscnrm3 . Sub... |
| iscnrm3lem2 46116 | Lemma for ~ iscnrm3 provin... |
| iscnrm3lem3 46117 | Lemma for ~ iscnrm3lem4 . ... |
| iscnrm3lem4 46118 | Lemma for ~ iscnrm3lem5 an... |
| iscnrm3lem5 46119 | Lemma for ~ iscnrm3l . (C... |
| iscnrm3lem6 46120 | Lemma for ~ iscnrm3lem7 . ... |
| iscnrm3lem7 46121 | Lemma for ~ iscnrm3rlem8 a... |
| iscnrm3rlem1 46122 | Lemma for ~ iscnrm3rlem2 .... |
| iscnrm3rlem2 46123 | Lemma for ~ iscnrm3rlem3 .... |
| iscnrm3rlem3 46124 | Lemma for ~ iscnrm3r . Th... |
| iscnrm3rlem4 46125 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem5 46126 | Lemma for ~ iscnrm3rlem6 .... |
| iscnrm3rlem6 46127 | Lemma for ~ iscnrm3rlem7 .... |
| iscnrm3rlem7 46128 | Lemma for ~ iscnrm3rlem8 .... |
| iscnrm3rlem8 46129 | Lemma for ~ iscnrm3r . Di... |
| iscnrm3r 46130 | Lemma for ~ iscnrm3 . If ... |
| iscnrm3llem1 46131 | Lemma for ~ iscnrm3l . Cl... |
| iscnrm3llem2 46132 | Lemma for ~ iscnrm3l . If... |
| iscnrm3l 46133 | Lemma for ~ iscnrm3 . Giv... |
| iscnrm3 46134 | A completely normal topolo... |
| iscnrm3v 46135 | A topology is completely n... |
| iscnrm4 46136 | A completely normal topolo... |
| isprsd 46137 | Property of being a preord... |
| lubeldm2 46138 | Member of the domain of th... |
| glbeldm2 46139 | Member of the domain of th... |
| lubeldm2d 46140 | Member of the domain of th... |
| glbeldm2d 46141 | Member of the domain of th... |
| lubsscl 46142 | If a subset of ` S ` conta... |
| glbsscl 46143 | If a subset of ` S ` conta... |
| lubprlem 46144 | Lemma for ~ lubprdm and ~ ... |
| lubprdm 46145 | The set of two comparable ... |
| lubpr 46146 | The LUB of the set of two ... |
| glbprlem 46147 | Lemma for ~ glbprdm and ~ ... |
| glbprdm 46148 | The set of two comparable ... |
| glbpr 46149 | The GLB of the set of two ... |
| joindm2 46150 | The join of any two elemen... |
| joindm3 46151 | The join of any two elemen... |
| meetdm2 46152 | The meet of any two elemen... |
| meetdm3 46153 | The meet of any two elemen... |
| posjidm 46154 | Poset join is idempotent. ... |
| posmidm 46155 | Poset meet is idempotent. ... |
| toslat 46156 | A toset is a lattice. (Co... |
| isclatd 46157 | The predicate "is a comple... |
| intubeu 46158 | Existential uniqueness of ... |
| unilbeu 46159 | Existential uniqueness of ... |
| ipolublem 46160 | Lemma for ~ ipolubdm and ~... |
| ipolubdm 46161 | The domain of the LUB of t... |
| ipolub 46162 | The LUB of the inclusion p... |
| ipoglblem 46163 | Lemma for ~ ipoglbdm and ~... |
| ipoglbdm 46164 | The domain of the GLB of t... |
| ipoglb 46165 | The GLB of the inclusion p... |
| ipolub0 46166 | The LUB of the empty set i... |
| ipolub00 46167 | The LUB of the empty set i... |
| ipoglb0 46168 | The GLB of the empty set i... |
| mrelatlubALT 46169 | Least upper bounds in a Mo... |
| mrelatglbALT 46170 | Greatest lower bounds in a... |
| mreclat 46171 | A Moore space is a complet... |
| topclat 46172 | A topology is a complete l... |
| toplatglb0 46173 | The empty intersection in ... |
| toplatlub 46174 | Least upper bounds in a to... |
| toplatglb 46175 | Greatest lower bounds in a... |
| toplatjoin 46176 | Joins in a topology are re... |
| toplatmeet 46177 | Meets in a topology are re... |
| topdlat 46178 | A topology is a distributi... |
| catprslem 46179 | Lemma for ~ catprs . (Con... |
| catprs 46180 | A preorder can be extracte... |
| catprs2 46181 | A category equipped with t... |
| catprsc 46182 | A construction of the preo... |
| catprsc2 46183 | An alternate construction ... |
| endmndlem 46184 | A diagonal hom-set in a ca... |
| idmon 46185 | An identity arrow, or an i... |
| idepi 46186 | An identity arrow, or an i... |
| funcf2lem 46187 | A utility theorem for prov... |
| isthinc 46190 | The predicate "is a thin c... |
| isthinc2 46191 | A thin category is a categ... |
| isthinc3 46192 | A thin category is a categ... |
| thincc 46193 | A thin category is a categ... |
| thinccd 46194 | A thin category is a categ... |
| thincssc 46195 | A thin category is a categ... |
| isthincd2lem1 46196 | Lemma for ~ isthincd2 and ... |
| thincmo2 46197 | Morphisms in the same hom-... |
| thincmo 46198 | There is at most one morph... |
| thincmoALT 46199 | Alternate proof for ~ thin... |
| thincmod 46200 | At most one morphism in ea... |
| thincn0eu 46201 | In a thin category, a hom-... |
| thincid 46202 | In a thin category, a morp... |
| thincmon 46203 | In a thin category, all mo... |
| thincepi 46204 | In a thin category, all mo... |
| isthincd2lem2 46205 | Lemma for ~ isthincd2 . (... |
| isthincd 46206 | The predicate "is a thin c... |
| isthincd2 46207 | The predicate " ` C ` is a... |
| oppcthin 46208 | The opposite category of a... |
| subthinc 46209 | A subcategory of a thin ca... |
| functhinclem1 46210 | Lemma for ~ functhinc . G... |
| functhinclem2 46211 | Lemma for ~ functhinc . (... |
| functhinclem3 46212 | Lemma for ~ functhinc . T... |
| functhinclem4 46213 | Lemma for ~ functhinc . O... |
| functhinc 46214 | A functor to a thin catego... |
| fullthinc 46215 | A functor to a thin catego... |
| fullthinc2 46216 | A full functor to a thin c... |
| thincfth 46217 | A functor from a thin cate... |
| thincciso 46218 | Two thin categories are is... |
| 0thincg 46219 | Any structure with an empt... |
| 0thinc 46220 | The empty category (see ~ ... |
| indthinc 46221 | An indiscrete category in ... |
| indthincALT 46222 | An alternate proof for ~ i... |
| prsthinc 46223 | Preordered sets as categor... |
| setcthin 46224 | A category of sets all of ... |
| setc2othin 46225 | The category ` ( SetCat ``... |
| thincsect 46226 | In a thin category, one mo... |
| thincsect2 46227 | In a thin category, ` F ` ... |
| thincinv 46228 | In a thin category, ` F ` ... |
| thinciso 46229 | In a thin category, ` F : ... |
| thinccic 46230 | In a thin category, two ob... |
| prstcval 46233 | Lemma for ~ prstcnidlem an... |
| prstcnidlem 46234 | Lemma for ~ prstcnid and ~... |
| prstcnid 46235 | Components other than ` Ho... |
| prstcbas 46236 | The base set is unchanged.... |
| prstcleval 46237 | Value of the less-than-or-... |
| prstcle 46238 | Value of the less-than-or-... |
| prstcocval 46239 | Orthocomplementation is un... |
| prstcoc 46240 | Orthocomplementation is un... |
| prstchomval 46241 | Hom-sets of the constructe... |
| prstcprs 46242 | The category is a preorder... |
| prstcthin 46243 | The preordered set is equi... |
| prstchom 46244 | Hom-sets of the constructe... |
| prstchom2 46245 | Hom-sets of the constructe... |
| prstchom2ALT 46246 | Hom-sets of the constructe... |
| postcpos 46247 | The converted category is ... |
| postcposALT 46248 | Alternate proof for ~ post... |
| postc 46249 | The converted category is ... |
| mndtcval 46252 | Value of the category buil... |
| mndtcbasval 46253 | The base set of the catego... |
| mndtcbas 46254 | The category built from a ... |
| mndtcob 46255 | Lemma for ~ mndtchom and ~... |
| mndtcbas2 46256 | Two objects in a category ... |
| mndtchom 46257 | The only hom-set of the ca... |
| mndtcco 46258 | The composition of the cat... |
| mndtcco2 46259 | The composition of the cat... |
| mndtccatid 46260 | Lemma for ~ mndtccat and ~... |
| mndtccat 46261 | The function value is a ca... |
| mndtcid 46262 | The identity morphism, or ... |
| grptcmon 46263 | All morphisms in a categor... |
| grptcepi 46264 | All morphisms in a categor... |
| nfintd 46265 | Bound-variable hypothesis ... |
| nfiund 46266 | Bound-variable hypothesis ... |
| nfiundg 46267 | Bound-variable hypothesis ... |
| iunord 46268 | The indexed union of a col... |
| iunordi 46269 | The indexed union of a col... |
| spd 46270 | Specialization deduction, ... |
| spcdvw 46271 | A version of ~ spcdv where... |
| tfis2d 46272 | Transfinite Induction Sche... |
| bnd2d 46273 | Deduction form of ~ bnd2 .... |
| dffun3f 46274 | Alternate definition of fu... |
| setrecseq 46277 | Equality theorem for set r... |
| nfsetrecs 46278 | Bound-variable hypothesis ... |
| setrec1lem1 46279 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 46280 | Lemma for ~ setrec1 . If ... |
| setrec1lem3 46281 | Lemma for ~ setrec1 . If ... |
| setrec1lem4 46282 | Lemma for ~ setrec1 . If ... |
| setrec1 46283 | This is the first of two f... |
| setrec2fun 46284 | This is the second of two ... |
| setrec2lem1 46285 | Lemma for ~ setrec2 . The... |
| setrec2lem2 46286 | Lemma for ~ setrec2 . The... |
| setrec2 46287 | This is the second of two ... |
| setrec2v 46288 | Version of ~ setrec2 with ... |
| setis 46289 | Version of ~ setrec2 expre... |
| elsetrecslem 46290 | Lemma for ~ elsetrecs . A... |
| elsetrecs 46291 | A set ` A ` is an element ... |
| setrecsss 46292 | The ` setrecs ` operator r... |
| setrecsres 46293 | A recursively generated cl... |
| vsetrec 46294 | Construct ` _V ` using set... |
| 0setrec 46295 | If a function sends the em... |
| onsetreclem1 46296 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 46297 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 46298 | Lemma for ~ onsetrec . (C... |
| onsetrec 46299 | Construct ` On ` using set... |
| elpglem1 46302 | Lemma for ~ elpg . (Contr... |
| elpglem2 46303 | Lemma for ~ elpg . (Contr... |
| elpglem3 46304 | Lemma for ~ elpg . (Contr... |
| elpg 46305 | Membership in the class of... |
| sbidd 46306 | An identity theorem for su... |
| sbidd-misc 46307 | An identity theorem for su... |
| gte-lte 46312 | Simple relationship betwee... |
| gt-lt 46313 | Simple relationship betwee... |
| gte-lteh 46314 | Relationship between ` <_ ... |
| gt-lth 46315 | Relationship between ` < `... |
| ex-gt 46316 | Simple example of ` > ` , ... |
| ex-gte 46317 | Simple example of ` >_ ` ,... |
| sinhval-named 46324 | Value of the named sinh fu... |
| coshval-named 46325 | Value of the named cosh fu... |
| tanhval-named 46326 | Value of the named tanh fu... |
| sinh-conventional 46327 | Conventional definition of... |
| sinhpcosh 46328 | Prove that ` ( sinh `` A )... |
| secval 46335 | Value of the secant functi... |
| cscval 46336 | Value of the cosecant func... |
| cotval 46337 | Value of the cotangent fun... |
| seccl 46338 | The closure of the secant ... |
| csccl 46339 | The closure of the cosecan... |
| cotcl 46340 | The closure of the cotange... |
| reseccl 46341 | The closure of the secant ... |
| recsccl 46342 | The closure of the cosecan... |
| recotcl 46343 | The closure of the cotange... |
| recsec 46344 | The reciprocal of secant i... |
| reccsc 46345 | The reciprocal of cosecant... |
| reccot 46346 | The reciprocal of cotangen... |
| rectan 46347 | The reciprocal of tangent ... |
| sec0 46348 | The value of the secant fu... |
| onetansqsecsq 46349 | Prove the tangent squared ... |
| cotsqcscsq 46350 | Prove the tangent squared ... |
| ifnmfalse 46351 | If A is not a member of B,... |
| logb2aval 46352 | Define the value of the ` ... |
| comraddi 46359 | Commute RHS addition. See... |
| mvlraddi 46360 | Move the right term in a s... |
| mvrladdi 46361 | Move the left term in a su... |
| assraddsubi 46362 | Associate RHS addition-sub... |
| joinlmuladdmuli 46363 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 46364 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 46365 | Join AB-CB into (A-C) on L... |
| mvlrmuld 46366 | Move the right term in a p... |
| mvlrmuli 46367 | Move the right term in a p... |
| i2linesi 46368 | Solve for the intersection... |
| i2linesd 46369 | Solve for the intersection... |
| alimp-surprise 46370 | Demonstrate that when usin... |
| alimp-no-surprise 46371 | There is no "surprise" in ... |
| empty-surprise 46372 | Demonstrate that when usin... |
| empty-surprise2 46373 | "Prove" that false is true... |
| eximp-surprise 46374 | Show what implication insi... |
| eximp-surprise2 46375 | Show that "there exists" w... |
| alsconv 46380 | There is an equivalence be... |
| alsi1d 46381 | Deduction rule: Given "al... |
| alsi2d 46382 | Deduction rule: Given "al... |
| alsc1d 46383 | Deduction rule: Given "al... |
| alsc2d 46384 | Deduction rule: Given "al... |
| alscn0d 46385 | Deduction rule: Given "al... |
| alsi-no-surprise 46386 | Demonstrate that there is ... |
| 5m4e1 46387 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 46388 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 46389 | Resolution rule. This is ... |
| testable 46390 | In classical logic all wff... |
| aacllem 46391 | Lemma for other theorems a... |
| amgmwlem 46392 | Weighted version of ~ amgm... |
| amgmlemALT 46393 | Alternate proof of ~ amgml... |
| amgmw2d 46394 | Weighted arithmetic-geomet... |
| young2d 46395 | Young's inequality for ` n... |
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