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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... |
| pm2.18OLD 129 | Obsolete version of ~ pm2.... |
| pm2.18dOLD 130 | Obsolete version of ~ pm2.... |
| pm2.18i 131 | Inference associated with ... |
| notnotr 132 | Double negation eliminatio... |
| notnotri 133 | Inference associated with ... |
| notnotriALT 134 | Alternate proof of ~ notno... |
| notnotrd 135 | Deduction associated with ... |
| con2d 136 | A contraposition deduction... |
| con2 137 | Contraposition. Theorem *... |
| mt2d 138 | Modus tollens deduction. ... |
| mt2i 139 | Modus tollens inference. ... |
| nsyl3 140 | A negated syllogism infere... |
| con2i 141 | A contraposition inference... |
| nsyl 142 | A negated syllogism infere... |
| nsyl2 143 | A negated syllogism infere... |
| notnot 144 | Double negation introducti... |
| notnoti 145 | Inference associated with ... |
| notnotd 146 | Deduction associated with ... |
| con1d 147 | A contraposition deduction... |
| con1 148 | Contraposition. Theorem *... |
| con1i 149 | A contraposition inference... |
| mt3d 150 | Modus tollens deduction. ... |
| mt3i 151 | Modus tollens inference. ... |
| nsyl2OLD 152 | Obsolete version of ~ nsyl... |
| pm2.24i 153 | Inference associated with ... |
| pm2.24d 154 | Deduction form of ~ pm2.24... |
| con3d 155 | A contraposition deduction... |
| con3 156 | Contraposition. Theorem *... |
| con3i 157 | A contraposition inference... |
| con3rr3 158 | Rotate through consequent ... |
| nsyld 159 | A negated syllogism deduct... |
| nsyli 160 | A negated syllogism infere... |
| nsyl4 161 | A negated syllogism infere... |
| pm3.2im 162 | Theorem *3.2 of [Whitehead... |
| jc 163 | Deduction joining the cons... |
| jcn 164 | Theorem joining the conseq... |
| jcnd 165 | Deduction joining the cons... |
| impi 166 | An importation inference. ... |
| expi 167 | An exportation inference. ... |
| simprim 168 | Simplification. Similar t... |
| simplim 169 | Simplification. Similar t... |
| pm2.5g 170 | General instance of Theore... |
| pm2.5 171 | Theorem *2.5 of [Whitehead... |
| conax1 172 | Contrapositive of ~ ax-1 .... |
| conax1k 173 | Weakening of ~ conax1 . G... |
| pm2.51 174 | Theorem *2.51 of [Whitehea... |
| pm2.52 175 | Theorem *2.52 of [Whitehea... |
| pm2.521g 176 | A general instance of Theo... |
| pm2.521g2 177 | A general instance of Theo... |
| pm2.521 178 | Theorem *2.521 of [Whitehe... |
| expt 179 | Exportation theorem ~ pm3.... |
| impt 180 | Importation theorem ~ pm3.... |
| pm2.61d 181 | Deduction eliminating an a... |
| pm2.61d1 182 | Inference eliminating an a... |
| pm2.61d2 183 | Inference eliminating an a... |
| pm2.61i 184 | Inference eliminating an a... |
| pm2.61ii 185 | Inference eliminating two ... |
| pm2.61nii 186 | Inference eliminating two ... |
| pm2.61iii 187 | Inference eliminating thre... |
| ja 188 | Inference joining the ante... |
| jad 189 | Deduction form of ~ ja . ... |
| pm2.61iOLD 190 | Obsolete version of ~ pm2.... |
| pm2.01 191 | Weak Clavius law. If a fo... |
| pm2.01d 192 | Deduction based on reducti... |
| pm2.6 193 | Theorem *2.6 of [Whitehead... |
| pm2.61 194 | Theorem *2.61 of [Whitehea... |
| pm2.65 195 | Theorem *2.65 of [Whitehea... |
| pm2.65i 196 | Inference for proof by con... |
| pm2.21dd 197 | A contradiction implies an... |
| pm2.65d 198 | Deduction for proof by con... |
| mto 199 | The rule of modus tollens.... |
| mtod 200 | Modus tollens deduction. ... |
| mtoi 201 | Modus tollens inference. ... |
| mt2 202 | A rule similar to modus to... |
| mt3 203 | A rule similar to modus to... |
| peirce 204 | Peirce's axiom. A non-int... |
| looinv 205 | The Inversion Axiom of the... |
| bijust0 206 | A self-implication (see ~ ... |
| bijust 207 | Theorem used to justify th... |
| impbi 210 | Property of the biconditio... |
| impbii 211 | Infer an equivalence from ... |
| impbidd 212 | Deduce an equivalence from... |
| impbid21d 213 | Deduce an equivalence from... |
| impbid 214 | Deduce an equivalence from... |
| dfbi1 215 | Relate the biconditional c... |
| dfbi1ALT 216 | Alternate proof of ~ dfbi1... |
| biimp 217 | Property of the biconditio... |
| biimpi 218 | Infer an implication from ... |
| sylbi 219 | A mixed syllogism inferenc... |
| sylib 220 | A mixed syllogism inferenc... |
| sylbb 221 | A mixed syllogism inferenc... |
| biimpr 222 | Property of the biconditio... |
| bicom1 223 | Commutative law for the bi... |
| bicom 224 | Commutative law for the bi... |
| bicomd 225 | Commute two sides of a bic... |
| bicomi 226 | Inference from commutative... |
| impbid1 227 | Infer an equivalence from ... |
| impbid2 228 | Infer an equivalence from ... |
| impcon4bid 229 | A variation on ~ impbid wi... |
| biimpri 230 | Infer a converse implicati... |
| biimpd 231 | Deduce an implication from... |
| mpbi 232 | An inference from a bicond... |
| mpbir 233 | An inference from a bicond... |
| mpbid 234 | A deduction from a bicondi... |
| mpbii 235 | An inference from a nested... |
| sylibr 236 | A mixed syllogism inferenc... |
| sylbir 237 | A mixed syllogism inferenc... |
| sylbbr 238 | A mixed syllogism inferenc... |
| sylbb1 239 | A mixed syllogism inferenc... |
| sylbb2 240 | A mixed syllogism inferenc... |
| sylibd 241 | A syllogism deduction. (C... |
| sylbid 242 | A syllogism deduction. (C... |
| mpbidi 243 | A deduction from a bicondi... |
| syl5bi 244 | A mixed syllogism inferenc... |
| syl5bir 245 | A mixed syllogism inferenc... |
| syl5ib 246 | A mixed syllogism inferenc... |
| syl5ibcom 247 | A mixed syllogism inferenc... |
| syl5ibr 248 | A mixed syllogism inferenc... |
| syl5ibrcom 249 | A mixed syllogism inferenc... |
| biimprd 250 | Deduce a converse implicat... |
| biimpcd 251 | Deduce a commuted implicat... |
| biimprcd 252 | Deduce a converse commuted... |
| syl6ib 253 | A mixed syllogism inferenc... |
| syl6ibr 254 | A mixed syllogism inferenc... |
| syl6bi 255 | A mixed syllogism inferenc... |
| syl6bir 256 | A mixed syllogism inferenc... |
| syl7bi 257 | A mixed syllogism inferenc... |
| syl8ib 258 | A syllogism rule of infere... |
| mpbird 259 | A deduction from a bicondi... |
| mpbiri 260 | An inference from a nested... |
| sylibrd 261 | A syllogism deduction. (C... |
| sylbird 262 | A syllogism deduction. (C... |
| biid 263 | Principle of identity for ... |
| biidd 264 | Principle of identity with... |
| pm5.1im 265 | Two propositions are equiv... |
| 2th 266 | Two truths are equivalent.... |
| 2thd 267 | Two truths are equivalent.... |
| monothetic 268 | Two self-implications (see... |
| ibi 269 | Inference that converts a ... |
| ibir 270 | Inference that converts a ... |
| ibd 271 | Deduction that converts a ... |
| pm5.74 272 | Distribution of implicatio... |
| pm5.74i 273 | Distribution of implicatio... |
| pm5.74ri 274 | Distribution of implicatio... |
| pm5.74d 275 | Distribution of implicatio... |
| pm5.74rd 276 | Distribution of implicatio... |
| bitri 277 | An inference from transiti... |
| bitr2i 278 | An inference from transiti... |
| bitr3i 279 | An inference from transiti... |
| bitr4i 280 | An inference from transiti... |
| bitrd 281 | Deduction form of ~ bitri ... |
| bitr2d 282 | Deduction form of ~ bitr2i... |
| bitr3d 283 | Deduction form of ~ bitr3i... |
| bitr4d 284 | Deduction form of ~ bitr4i... |
| syl5bb 285 | A syllogism inference from... |
| syl5rbb 286 | A syllogism inference from... |
| syl5bbr 287 | A syllogism inference from... |
| syl5rbbr 288 | A syllogism inference from... |
| syl6bb 289 | A syllogism inference from... |
| syl6rbb 290 | A syllogism inference from... |
| syl6bbr 291 | A syllogism inference from... |
| syl6rbbr 292 | A syllogism inference from... |
| 3imtr3i 293 | A mixed syllogism inferenc... |
| 3imtr4i 294 | A mixed syllogism inferenc... |
| 3imtr3d 295 | More general version of ~ ... |
| 3imtr4d 296 | More general version of ~ ... |
| 3imtr3g 297 | More general version of ~ ... |
| 3imtr4g 298 | More general version of ~ ... |
| 3bitri 299 | A chained inference from t... |
| 3bitrri 300 | A chained inference from t... |
| 3bitr2i 301 | A chained inference from t... |
| 3bitr2ri 302 | A chained inference from t... |
| 3bitr3i 303 | A chained inference from t... |
| 3bitr3ri 304 | A chained inference from t... |
| 3bitr4i 305 | A chained inference from t... |
| 3bitr4ri 306 | A chained inference from t... |
| 3bitrd 307 | Deduction from transitivit... |
| 3bitrrd 308 | Deduction from transitivit... |
| 3bitr2d 309 | Deduction from transitivit... |
| 3bitr2rd 310 | Deduction from transitivit... |
| 3bitr3d 311 | Deduction from transitivit... |
| 3bitr3rd 312 | Deduction from transitivit... |
| 3bitr4d 313 | Deduction from transitivit... |
| 3bitr4rd 314 | Deduction from transitivit... |
| 3bitr3g 315 | More general version of ~ ... |
| 3bitr4g 316 | More general version of ~ ... |
| notnotb 317 | Double negation. Theorem ... |
| con34b 318 | A biconditional form of co... |
| con4bid 319 | A contraposition deduction... |
| notbid 320 | Deduction negating both si... |
| notbi 321 | Contraposition. Theorem *... |
| notbii 322 | Negate both sides of a log... |
| con4bii 323 | A contraposition inference... |
| mtbi 324 | An inference from a bicond... |
| mtbir 325 | An inference from a bicond... |
| mtbid 326 | A deduction from a bicondi... |
| mtbird 327 | A deduction from a bicondi... |
| mtbii 328 | An inference from a bicond... |
| mtbiri 329 | An inference from a bicond... |
| sylnib 330 | A mixed syllogism inferenc... |
| sylnibr 331 | A mixed syllogism inferenc... |
| sylnbi 332 | A mixed syllogism inferenc... |
| sylnbir 333 | A mixed syllogism inferenc... |
| xchnxbi 334 | Replacement of a subexpres... |
| xchnxbir 335 | Replacement of a subexpres... |
| xchbinx 336 | Replacement of a subexpres... |
| xchbinxr 337 | Replacement of a subexpres... |
| imbi2i 338 | Introduce an antecedent to... |
| jcndOLD 339 | Obsolete version of ~ jcnd... |
| bibi2i 340 | Inference adding a bicondi... |
| bibi1i 341 | Inference adding a bicondi... |
| bibi12i 342 | The equivalence of two equ... |
| imbi2d 343 | Deduction adding an antece... |
| imbi1d 344 | Deduction adding a consequ... |
| bibi2d 345 | Deduction adding a bicondi... |
| bibi1d 346 | Deduction adding a bicondi... |
| imbi12d 347 | Deduction joining two equi... |
| bibi12d 348 | Deduction joining two equi... |
| imbi12 349 | Closed form of ~ imbi12i .... |
| imbi1 350 | Theorem *4.84 of [Whitehea... |
| imbi2 351 | Theorem *4.85 of [Whitehea... |
| imbi1i 352 | Introduce a consequent to ... |
| imbi12i 353 | Join two logical equivalen... |
| bibi1 354 | Theorem *4.86 of [Whitehea... |
| bitr3 355 | Closed nested implication ... |
| con2bi 356 | Contraposition. Theorem *... |
| con2bid 357 | A contraposition deduction... |
| con1bid 358 | A contraposition deduction... |
| con1bii 359 | A contraposition inference... |
| con2bii 360 | A contraposition inference... |
| con1b 361 | Contraposition. Bidirecti... |
| con2b 362 | Contraposition. Bidirecti... |
| biimt 363 | A wff is equivalent to its... |
| pm5.5 364 | Theorem *5.5 of [Whitehead... |
| a1bi 365 | Inference introducing a th... |
| mt2bi 366 | A false consequent falsifi... |
| mtt 367 | Modus-tollens-like theorem... |
| imnot 368 | If a proposition is false,... |
| pm5.501 369 | Theorem *5.501 of [Whitehe... |
| ibib 370 | Implication in terms of im... |
| ibibr 371 | Implication in terms of im... |
| tbt 372 | A wff is equivalent to its... |
| nbn2 373 | The negation of a wff is e... |
| bibif 374 | Transfer negation via an e... |
| nbn 375 | The negation of a wff is e... |
| nbn3 376 | Transfer falsehood via equ... |
| pm5.21im 377 | Two propositions are equiv... |
| 2false 378 | Two falsehoods are equival... |
| 2falsed 379 | Two falsehoods are equival... |
| 2falsedOLD 380 | Obsolete version of ~ 2fal... |
| pm5.21ni 381 | Two propositions implying ... |
| pm5.21nii 382 | Eliminate an antecedent im... |
| pm5.21ndd 383 | Eliminate an antecedent im... |
| bija 384 | Combine antecedents into a... |
| pm5.18 385 | Theorem *5.18 of [Whitehea... |
| xor3 386 | Two ways to express "exclu... |
| nbbn 387 | Move negation outside of b... |
| biass 388 | Associative law for the bi... |
| biluk 389 | Lukasiewicz's shortest axi... |
| pm5.19 390 | Theorem *5.19 of [Whitehea... |
| bi2.04 391 | Logical equivalence of com... |
| pm5.4 392 | Antecedent absorption impl... |
| imdi 393 | Distributive law for impli... |
| pm5.41 394 | Theorem *5.41 of [Whitehea... |
| pm4.8 395 | Theorem *4.8 of [Whitehead... |
| pm4.81 396 | A formula is equivalent to... |
| imim21b 397 | Simplify an implication be... |
| pm4.63 400 | Theorem *4.63 of [Whitehea... |
| pm4.67 401 | Theorem *4.67 of [Whitehea... |
| imnan 402 | Express an implication in ... |
| imnani 403 | Infer an implication from ... |
| iman 404 | Implication in terms of co... |
| pm3.24 405 | Law of noncontradiction. ... |
| annim 406 | Express a conjunction in t... |
| pm4.61 407 | Theorem *4.61 of [Whitehea... |
| pm4.65 408 | Theorem *4.65 of [Whitehea... |
| imp 409 | Importation inference. (C... |
| impcom 410 | Importation inference with... |
| con3dimp 411 | Variant of ~ con3d with im... |
| mpnanrd 412 | Eliminate the right side o... |
| impd 413 | Importation deduction. (C... |
| impcomd 414 | Importation deduction with... |
| ex 415 | Exportation inference. (T... |
| expcom 416 | Exportation inference with... |
| expdcom 417 | Commuted form of ~ expd . ... |
| expd 418 | Exportation deduction. (C... |
| expcomd 419 | Deduction form of ~ expcom... |
| imp31 420 | An importation inference. ... |
| imp32 421 | An importation inference. ... |
| exp31 422 | An exportation inference. ... |
| exp32 423 | An exportation inference. ... |
| imp4b 424 | An importation inference. ... |
| imp4a 425 | An importation inference. ... |
| imp4c 426 | An importation inference. ... |
| imp4d 427 | An importation inference. ... |
| imp41 428 | An importation inference. ... |
| imp42 429 | An importation inference. ... |
| imp43 430 | An importation inference. ... |
| imp44 431 | An importation inference. ... |
| imp45 432 | An importation inference. ... |
| exp4b 433 | An exportation inference. ... |
| exp4a 434 | An exportation inference. ... |
| exp4c 435 | An exportation inference. ... |
| exp4d 436 | An exportation inference. ... |
| exp41 437 | An exportation inference. ... |
| exp42 438 | An exportation inference. ... |
| exp43 439 | An exportation inference. ... |
| exp44 440 | An exportation inference. ... |
| exp45 441 | An exportation inference. ... |
| imp5d 442 | An importation inference. ... |
| imp5a 443 | An importation inference. ... |
| imp5g 444 | An importation inference. ... |
| imp55 445 | An importation inference. ... |
| imp511 446 | An importation inference. ... |
| exp5c 447 | An exportation inference. ... |
| exp5j 448 | An exportation inference. ... |
| exp5l 449 | An exportation inference. ... |
| exp53 450 | An exportation inference. ... |
| pm3.3 451 | Theorem *3.3 (Exp) of [Whi... |
| pm3.31 452 | Theorem *3.31 (Imp) of [Wh... |
| impexp 453 | Import-export theorem. Pa... |
| impancom 454 | Mixed importation/commutat... |
| expdimp 455 | A deduction version of exp... |
| expimpd 456 | Exportation followed by a ... |
| impr 457 | Import a wff into a right ... |
| impl 458 | Export a wff from a left c... |
| expr 459 | Export a wff from a right ... |
| expl 460 | Export a wff from a left c... |
| ancoms 461 | Inference commuting conjun... |
| pm3.22 462 | Theorem *3.22 of [Whitehea... |
| ancom 463 | Commutative law for conjun... |
| ancomd 464 | Commutation of conjuncts i... |
| biancomi 465 | Commuting conjunction in a... |
| biancomd 466 | Commuting conjunction in a... |
| ancomst 467 | Closed form of ~ ancoms . ... |
| ancomsd 468 | Deduction commuting conjun... |
| anasss 469 | Associative law for conjun... |
| anassrs 470 | Associative law for conjun... |
| anass 471 | Associative law for conjun... |
| pm3.2 472 | Join antecedents with conj... |
| pm3.2i 473 | Infer conjunction of premi... |
| pm3.21 474 | Join antecedents with conj... |
| pm3.43i 475 | Nested conjunction of ante... |
| pm3.43 476 | Theorem *3.43 (Comp) of [W... |
| dfbi2 477 | A theorem similar to the s... |
| dfbi 478 | Definition ~ df-bi rewritt... |
| biimpa 479 | Importation inference from... |
| biimpar 480 | Importation inference from... |
| biimpac 481 | Importation inference from... |
| biimparc 482 | Importation inference from... |
| adantr 483 | Inference adding a conjunc... |
| adantl 484 | Inference adding a conjunc... |
| simpl 485 | Elimination of a conjunct.... |
| simpli 486 | Inference eliminating a co... |
| simpr 487 | Elimination of a conjunct.... |
| simpri 488 | Inference eliminating a co... |
| intnan 489 | Introduction of conjunct i... |
| intnanr 490 | Introduction of conjunct i... |
| intnand 491 | Introduction of conjunct i... |
| intnanrd 492 | Introduction of conjunct i... |
| adantld 493 | Deduction adding a conjunc... |
| adantrd 494 | Deduction adding a conjunc... |
| pm3.41 495 | Theorem *3.41 of [Whitehea... |
| pm3.42 496 | Theorem *3.42 of [Whitehea... |
| simpld 497 | Deduction eliminating a co... |
| simprd 498 | Deduction eliminating a co... |
| simprbi 499 | Deduction eliminating a co... |
| simplbi 500 | Deduction eliminating a co... |
| simprbda 501 | Deduction eliminating a co... |
| simplbda 502 | Deduction eliminating a co... |
| simplbi2 503 | Deduction eliminating a co... |
| simplbi2comt 504 | Closed form of ~ simplbi2c... |
| simplbi2com 505 | A deduction eliminating a ... |
| simpl2im 506 | Implication from an elimin... |
| simplbiim 507 | Implication from an elimin... |
| impel 508 | An inference for implicati... |
| mpan9 509 | Modus ponens conjoining di... |
| sylan9 510 | Nested syllogism inference... |
| sylan9r 511 | Nested syllogism inference... |
| sylan9bb 512 | Nested syllogism inference... |
| sylan9bbr 513 | Nested syllogism inference... |
| jca 514 | Deduce conjunction of the ... |
| jcad 515 | Deduction conjoining the c... |
| jca2 516 | Inference conjoining the c... |
| jca31 517 | Join three consequents. (... |
| jca32 518 | Join three consequents. (... |
| jcai 519 | Deduction replacing implic... |
| jcab 520 | Distributive law for impli... |
| pm4.76 521 | Theorem *4.76 of [Whitehea... |
| jctil 522 | Inference conjoining a the... |
| jctir 523 | Inference conjoining a the... |
| jccir 524 | Inference conjoining a con... |
| jccil 525 | Inference conjoining a con... |
| jctl 526 | Inference conjoining a the... |
| jctr 527 | Inference conjoining a the... |
| jctild 528 | Deduction conjoining a the... |
| jctird 529 | Deduction conjoining a the... |
| iba 530 | Introduction of antecedent... |
| ibar 531 | Introduction of antecedent... |
| biantru 532 | A wff is equivalent to its... |
| biantrur 533 | A wff is equivalent to its... |
| biantrud 534 | A wff is equivalent to its... |
| biantrurd 535 | A wff is equivalent to its... |
| bianfi 536 | A wff conjoined with false... |
| bianfd 537 | A wff conjoined with false... |
| baib 538 | Move conjunction outside o... |
| baibr 539 | Move conjunction outside o... |
| rbaibr 540 | Move conjunction outside o... |
| rbaib 541 | Move conjunction outside o... |
| baibd 542 | Move conjunction outside o... |
| rbaibd 543 | Move conjunction outside o... |
| bianabs 544 | Absorb a hypothesis into t... |
| pm5.44 545 | Theorem *5.44 of [Whitehea... |
| pm5.42 546 | Theorem *5.42 of [Whitehea... |
| ancl 547 | Conjoin antecedent to left... |
| anclb 548 | Conjoin antecedent to left... |
| ancr 549 | Conjoin antecedent to righ... |
| ancrb 550 | Conjoin antecedent to righ... |
| ancli 551 | Deduction conjoining antec... |
| ancri 552 | Deduction conjoining antec... |
| ancld 553 | Deduction conjoining antec... |
| ancrd 554 | Deduction conjoining antec... |
| impac 555 | Importation with conjuncti... |
| anc2l 556 | Conjoin antecedent to left... |
| anc2r 557 | Conjoin antecedent to righ... |
| anc2li 558 | Deduction conjoining antec... |
| anc2ri 559 | Deduction conjoining antec... |
| pm4.71 560 | Implication in terms of bi... |
| pm4.71r 561 | Implication in terms of bi... |
| pm4.71i 562 | Inference converting an im... |
| pm4.71ri 563 | Inference converting an im... |
| pm4.71d 564 | Deduction converting an im... |
| pm4.71rd 565 | Deduction converting an im... |
| pm4.24 566 | Theorem *4.24 of [Whitehea... |
| anidm 567 | Idempotent law for conjunc... |
| anidmdbi 568 | Conjunction idempotence wi... |
| anidms 569 | Inference from idempotent ... |
| imdistan 570 | Distribution of implicatio... |
| imdistani 571 | Distribution of implicatio... |
| imdistanri 572 | Distribution of implicatio... |
| imdistand 573 | Distribution of implicatio... |
| imdistanda 574 | Distribution of implicatio... |
| pm5.3 575 | Theorem *5.3 of [Whitehead... |
| pm5.32 576 | Distribution of implicatio... |
| pm5.32i 577 | Distribution of implicatio... |
| pm5.32ri 578 | Distribution of implicatio... |
| pm5.32d 579 | Distribution of implicatio... |
| pm5.32rd 580 | Distribution of implicatio... |
| pm5.32da 581 | Distribution of implicatio... |
| sylan 582 | A syllogism inference. (C... |
| sylanb 583 | A syllogism inference. (C... |
| sylanbr 584 | A syllogism inference. (C... |
| sylanbrc 585 | Syllogism inference. (Con... |
| syl2anc 586 | Syllogism inference combin... |
| syl2anc2 587 | Double syllogism inference... |
| sylancl 588 | Syllogism inference combin... |
| sylancr 589 | Syllogism inference combin... |
| sylancom 590 | Syllogism inference with c... |
| sylanblc 591 | Syllogism inference combin... |
| sylanblrc 592 | Syllogism inference combin... |
| syldan 593 | A syllogism deduction with... |
| sylan2 594 | A syllogism inference. (C... |
| sylan2b 595 | A syllogism inference. (C... |
| sylan2br 596 | A syllogism inference. (C... |
| syl2an 597 | A double syllogism inferen... |
| syl2anr 598 | A double syllogism inferen... |
| syl2anb 599 | A double syllogism inferen... |
| syl2anbr 600 | A double syllogism inferen... |
| sylancb 601 | A syllogism inference comb... |
| sylancbr 602 | A syllogism inference comb... |
| syldanl 603 | A syllogism deduction with... |
| syland 604 | A syllogism deduction. (C... |
| sylani 605 | A syllogism inference. (C... |
| sylan2d 606 | A syllogism deduction. (C... |
| sylan2i 607 | A syllogism inference. (C... |
| syl2ani 608 | A syllogism inference. (C... |
| syl2and 609 | A syllogism deduction. (C... |
| anim12d 610 | Conjoin antecedents and co... |
| anim12d1 611 | Variant of ~ anim12d where... |
| anim1d 612 | Add a conjunct to right of... |
| anim2d 613 | Add a conjunct to left of ... |
| anim12i 614 | Conjoin antecedents and co... |
| anim12ci 615 | Variant of ~ anim12i with ... |
| anim1i 616 | Introduce conjunct to both... |
| anim1ci 617 | Introduce conjunct to both... |
| anim2i 618 | Introduce conjunct to both... |
| anim12ii 619 | Conjoin antecedents and co... |
| anim12dan 620 | Conjoin antecedents and co... |
| im2anan9 621 | Deduction joining nested i... |
| im2anan9r 622 | Deduction joining nested i... |
| pm3.45 623 | Theorem *3.45 (Fact) of [W... |
| anbi2i 624 | Introduce a left conjunct ... |
| anbi1i 625 | Introduce a right conjunct... |
| anbi2ci 626 | Variant of ~ anbi2i with c... |
| anbi1ci 627 | Variant of ~ anbi1i with c... |
| anbi12i 628 | Conjoin both sides of two ... |
| anbi12ci 629 | Variant of ~ anbi12i with ... |
| anbi2d 630 | Deduction adding a left co... |
| anbi1d 631 | Deduction adding a right c... |
| anbi12d 632 | Deduction joining two equi... |
| anbi1 633 | Introduce a right conjunct... |
| anbi2 634 | Introduce a left conjunct ... |
| anbi1cd 635 | Introduce a proposition as... |
| pm4.38 636 | Theorem *4.38 of [Whitehea... |
| bi2anan9 637 | Deduction joining two equi... |
| bi2anan9r 638 | Deduction joining two equi... |
| bi2bian9 639 | Deduction joining two bico... |
| bianass 640 | An inference to merge two ... |
| bianassc 641 | An inference to merge two ... |
| an21 642 | Swap two conjuncts. (Cont... |
| an12 643 | Swap two conjuncts. Note ... |
| an32 644 | A rearrangement of conjunc... |
| an13 645 | A rearrangement of conjunc... |
| an31 646 | A rearrangement of conjunc... |
| an12s 647 | Swap two conjuncts in ante... |
| ancom2s 648 | Inference commuting a nest... |
| an13s 649 | Swap two conjuncts in ante... |
| an32s 650 | Swap two conjuncts in ante... |
| ancom1s 651 | Inference commuting a nest... |
| an31s 652 | Swap two conjuncts in ante... |
| anass1rs 653 | Commutative-associative la... |
| an4 654 | Rearrangement of 4 conjunc... |
| an42 655 | Rearrangement of 4 conjunc... |
| an43 656 | Rearrangement of 4 conjunc... |
| an3 657 | A rearrangement of conjunc... |
| an4s 658 | Inference rearranging 4 co... |
| an42s 659 | Inference rearranging 4 co... |
| anabs1 660 | Absorption into embedded c... |
| anabs5 661 | Absorption into embedded c... |
| anabs7 662 | Absorption into embedded c... |
| anabsan 663 | Absorption of antecedent w... |
| anabss1 664 | Absorption of antecedent i... |
| anabss4 665 | Absorption of antecedent i... |
| anabss5 666 | Absorption of antecedent i... |
| anabsi5 667 | Absorption of antecedent i... |
| anabsi6 668 | Absorption of antecedent i... |
| anabsi7 669 | Absorption of antecedent i... |
| anabsi8 670 | Absorption of antecedent i... |
| anabss7 671 | Absorption of antecedent i... |
| anabsan2 672 | Absorption of antecedent w... |
| anabss3 673 | Absorption of antecedent i... |
| anandi 674 | Distribution of conjunctio... |
| anandir 675 | Distribution of conjunctio... |
| anandis 676 | Inference that undistribut... |
| anandirs 677 | Inference that undistribut... |
| sylanl1 678 | A syllogism inference. (C... |
| sylanl2 679 | A syllogism inference. (C... |
| sylanr1 680 | A syllogism inference. (C... |
| sylanr2 681 | A syllogism inference. (C... |
| syl6an 682 | A syllogism deduction comb... |
| syl2an2r 683 | ~ syl2anr with antecedents... |
| syl2an2 684 | ~ syl2an with antecedents ... |
| mpdan 685 | An inference based on modu... |
| mpancom 686 | An inference based on modu... |
| mpidan 687 | A deduction which "stacks"... |
| mpan 688 | An inference based on modu... |
| mpan2 689 | An inference based on modu... |
| mp2an 690 | An inference based on modu... |
| mp4an 691 | An inference based on modu... |
| mpan2d 692 | A deduction based on modus... |
| mpand 693 | A deduction based on modus... |
| mpani 694 | An inference based on modu... |
| mpan2i 695 | An inference based on modu... |
| mp2ani 696 | An inference based on modu... |
| mp2and 697 | A deduction based on modus... |
| mpanl1 698 | An inference based on modu... |
| mpanl2 699 | An inference based on modu... |
| mpanl12 700 | An inference based on modu... |
| mpanr1 701 | An inference based on modu... |
| mpanr2 702 | An inference based on modu... |
| mpanr12 703 | An inference based on modu... |
| mpanlr1 704 | An inference based on modu... |
| mpbirand 705 | Detach truth from conjunct... |
| mpbiran2d 706 | Detach truth from conjunct... |
| mpbiran 707 | Detach truth from conjunct... |
| mpbiran2 708 | Detach truth from conjunct... |
| mpbir2an 709 | Detach a conjunction of tr... |
| mpbi2and 710 | Detach a conjunction of tr... |
| mpbir2and 711 | Detach a conjunction of tr... |
| adantll 712 | Deduction adding a conjunc... |
| adantlr 713 | Deduction adding a conjunc... |
| adantrl 714 | Deduction adding a conjunc... |
| adantrr 715 | Deduction adding a conjunc... |
| adantlll 716 | Deduction adding a conjunc... |
| adantllr 717 | Deduction adding a conjunc... |
| adantlrl 718 | Deduction adding a conjunc... |
| adantlrr 719 | Deduction adding a conjunc... |
| adantrll 720 | Deduction adding a conjunc... |
| adantrlr 721 | Deduction adding a conjunc... |
| adantrrl 722 | Deduction adding a conjunc... |
| adantrrr 723 | Deduction adding a conjunc... |
| ad2antrr 724 | Deduction adding two conju... |
| ad2antlr 725 | Deduction adding two conju... |
| ad2antrl 726 | Deduction adding two conju... |
| ad2antll 727 | Deduction adding conjuncts... |
| ad3antrrr 728 | Deduction adding three con... |
| ad3antlr 729 | Deduction adding three con... |
| ad4antr 730 | Deduction adding 4 conjunc... |
| ad4antlr 731 | Deduction adding 4 conjunc... |
| ad5antr 732 | Deduction adding 5 conjunc... |
| ad5antlr 733 | Deduction adding 5 conjunc... |
| ad6antr 734 | Deduction adding 6 conjunc... |
| ad6antlr 735 | Deduction adding 6 conjunc... |
| ad7antr 736 | Deduction adding 7 conjunc... |
| ad7antlr 737 | Deduction adding 7 conjunc... |
| ad8antr 738 | Deduction adding 8 conjunc... |
| ad8antlr 739 | Deduction adding 8 conjunc... |
| ad9antr 740 | Deduction adding 9 conjunc... |
| ad9antlr 741 | Deduction adding 9 conjunc... |
| ad10antr 742 | Deduction adding 10 conjun... |
| ad10antlr 743 | Deduction adding 10 conjun... |
| ad2ant2l 744 | Deduction adding two conju... |
| ad2ant2r 745 | Deduction adding two conju... |
| ad2ant2lr 746 | Deduction adding two conju... |
| ad2ant2rl 747 | Deduction adding two conju... |
| adantl3r 748 | Deduction adding 1 conjunc... |
| ad4ant13 749 | Deduction adding conjuncts... |
| ad4ant14 750 | Deduction adding conjuncts... |
| ad4ant23 751 | Deduction adding conjuncts... |
| ad4ant24 752 | Deduction adding conjuncts... |
| adantl4r 753 | Deduction adding 1 conjunc... |
| ad5ant12 754 | Deduction adding conjuncts... |
| ad5ant13 755 | Deduction adding conjuncts... |
| ad5ant14 756 | Deduction adding conjuncts... |
| ad5ant15 757 | Deduction adding conjuncts... |
| ad5ant23 758 | Deduction adding conjuncts... |
| ad5ant24 759 | Deduction adding conjuncts... |
| ad5ant25 760 | Deduction adding conjuncts... |
| adantl5r 761 | Deduction adding 1 conjunc... |
| adantl6r 762 | Deduction adding 1 conjunc... |
| pm3.33 763 | Theorem *3.33 (Syll) of [W... |
| pm3.34 764 | Theorem *3.34 (Syll) of [W... |
| simpll 765 | Simplification of a conjun... |
| simplld 766 | Deduction form of ~ simpll... |
| simplr 767 | Simplification of a conjun... |
| simplrd 768 | Deduction eliminating a do... |
| simprl 769 | Simplification of a conjun... |
| simprld 770 | Deduction eliminating a do... |
| simprr 771 | Simplification of a conjun... |
| simprrd 772 | Deduction form of ~ simprr... |
| simplll 773 | Simplification of a conjun... |
| simpllr 774 | Simplification of a conjun... |
| simplrl 775 | Simplification of a conjun... |
| simplrr 776 | Simplification of a conjun... |
| simprll 777 | Simplification of a conjun... |
| simprlr 778 | Simplification of a conjun... |
| simprrl 779 | Simplification of a conjun... |
| simprrr 780 | Simplification of a conjun... |
| simp-4l 781 | Simplification of a conjun... |
| simp-4r 782 | Simplification of a conjun... |
| simp-5l 783 | Simplification of a conjun... |
| simp-5r 784 | Simplification of a conjun... |
| simp-6l 785 | Simplification of a conjun... |
| simp-6r 786 | Simplification of a conjun... |
| simp-7l 787 | Simplification of a conjun... |
| simp-7r 788 | Simplification of a conjun... |
| simp-8l 789 | Simplification of a conjun... |
| simp-8r 790 | Simplification of a conjun... |
| simp-9l 791 | Simplification of a conjun... |
| simp-9r 792 | Simplification of a conjun... |
| simp-10l 793 | Simplification of a conjun... |
| simp-10r 794 | Simplification of a conjun... |
| simp-11l 795 | Simplification of a conjun... |
| simp-11r 796 | Simplification of a conjun... |
| pm2.01da 797 | Deduction based on reducti... |
| pm2.18da 798 | Deduction based on reducti... |
| impbida 799 | Deduce an equivalence from... |
| pm5.21nd 800 | Eliminate an antecedent im... |
| pm3.35 801 | Conjunctive detachment. T... |
| pm5.74da 802 | Distribution of implicatio... |
| bitr 803 | Theorem *4.22 of [Whitehea... |
| biantr 804 | A transitive law of equiva... |
| pm4.14 805 | Theorem *4.14 of [Whitehea... |
| pm3.37 806 | Theorem *3.37 (Transp) of ... |
| anim12 807 | Conjoin antecedents and co... |
| pm3.4 808 | Conjunction implies implic... |
| exbiri 809 | Inference form of ~ exbir ... |
| pm2.61ian 810 | Elimination of an antecede... |
| pm2.61dan 811 | Elimination of an antecede... |
| pm2.61ddan 812 | Elimination of two anteced... |
| pm2.61dda 813 | Elimination of two anteced... |
| mtand 814 | A modus tollens deduction.... |
| pm2.65da 815 | Deduction for proof by con... |
| condan 816 | Proof by contradiction. (... |
| biadan 817 | An implication is equivale... |
| biadani 818 | Inference associated with ... |
| biadaniALT 819 | Alternate proof of ~ biada... |
| biadanii 820 | Inference associated with ... |
| pm5.1 821 | Two propositions are equiv... |
| pm5.21 822 | Two propositions are equiv... |
| pm5.35 823 | Theorem *5.35 of [Whitehea... |
| abai 824 | Introduce one conjunct as ... |
| pm4.45im 825 | Conjunction with implicati... |
| impimprbi 826 | An implication and its rev... |
| nan 827 | Theorem to move a conjunct... |
| pm5.31 828 | Theorem *5.31 of [Whitehea... |
| pm5.31r 829 | Variant of ~ pm5.31 . (Co... |
| pm4.15 830 | Theorem *4.15 of [Whitehea... |
| pm5.36 831 | Theorem *5.36 of [Whitehea... |
| annotanannot 832 | A conjunction with a negat... |
| pm5.33 833 | Theorem *5.33 of [Whitehea... |
| syl12anc 834 | Syllogism combined with co... |
| syl21anc 835 | Syllogism combined with co... |
| syl22anc 836 | Syllogism combined with co... |
| syl1111anc 837 | Four-hypothesis eliminatio... |
| 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 wff disjoined with truth... |
| 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 ... |
| ecase3 1027 | Inference for elimination ... |
| ecase 1028 | Inference for elimination ... |
| ecase3d 1029 | Deduction for elimination ... |
| ecased 1030 | Deduction for elimination ... |
| ecase3ad 1031 | Deduction for elimination ... |
| ccase 1032 | Inference for combining ca... |
| ccased 1033 | Deduction for combining ca... |
| ccase2 1034 | Inference for combining ca... |
| 4cases 1035 | Inference eliminating two ... |
| 4casesdan 1036 | Deduction eliminating two ... |
| cases 1037 | Case disjunction according... |
| dedlem0a 1038 | Lemma for an alternate ver... |
| dedlem0b 1039 | Lemma for an alternate ver... |
| dedlema 1040 | Lemma for weak deduction t... |
| dedlemb 1041 | Lemma for weak deduction t... |
| cases2 1042 | Case disjunction according... |
| cases2ALT 1043 | Alternate proof of ~ cases... |
| dfbi3 1044 | An alternate definition of... |
| pm5.24 1045 | Theorem *5.24 of [Whitehea... |
| 4exmid 1046 | The disjunction of the fou... |
| consensus 1047 | The consensus theorem. Th... |
| pm4.42 1048 | Theorem *4.42 of [Whitehea... |
| prlem1 1049 | A specialized lemma for se... |
| prlem2 1050 | A specialized lemma for se... |
| oplem1 1051 | A specialized lemma for se... |
| dn1 1052 | A single axiom for Boolean... |
| bianir 1053 | A closed form of ~ mpbir ,... |
| jaoi2 1054 | Inference removing a negat... |
| jaoi3 1055 | Inference separating a dis... |
| ornld 1056 | Selecting one statement fr... |
| dfifp2 1059 | Alternate definition of th... |
| dfifp3 1060 | Alternate definition of th... |
| dfifp4 1061 | Alternate definition of th... |
| dfifp5 1062 | Alternate definition of th... |
| dfifp6 1063 | Alternate definition of th... |
| dfifp7 1064 | Alternate definition of th... |
| anifp 1065 | The conditional operator i... |
| ifpor 1066 | The conditional operator i... |
| ifpn 1067 | Conditional operator for t... |
| ifptru 1068 | Value of the conditional o... |
| ifpfal 1069 | Value of the conditional o... |
| ifpid 1070 | Value of the conditional o... |
| casesifp 1071 | Version of ~ cases express... |
| ifpbi123d 1072 | Equality deduction for con... |
| ifpbi123dOLD 1073 | Obsolete version of ~ ifpb... |
| ifpimpda 1074 | Separation of the values o... |
| 1fpid3 1075 | The value of the condition... |
| elimh 1076 | Hypothesis builder for the... |
| elimhOLD 1077 | Obsolete version of ~ elim... |
| dedt 1078 | The weak deduction theorem... |
| dedtOLD 1079 | Obsolete version of ~ dedt... |
| con3ALT 1080 | Proof of ~ con3 from its a... |
| con3ALTOLD 1081 | Obsolete version of ~ con3... |
| 3orass 1086 | Associative law for triple... |
| 3orel1 1087 | Partial elimination of a t... |
| 3orrot 1088 | Rotation law for triple di... |
| 3orcoma 1089 | Commutation law for triple... |
| 3orcomb 1090 | Commutation law for triple... |
| 3anass 1091 | Associative law for triple... |
| 3anan12 1092 | Convert triple conjunction... |
| 3anan32 1093 | Convert triple conjunction... |
| 3ancoma 1094 | Commutation law for triple... |
| 3ancomb 1095 | Commutation law for triple... |
| 3anrot 1096 | Rotation law for triple co... |
| 3anrev 1097 | Reversal law for triple co... |
| anandi3 1098 | Distribution of triple con... |
| anandi3r 1099 | Distribution of triple con... |
| 3anidm 1100 | Idempotent law for conjunc... |
| 3an4anass 1101 | Associative law for four c... |
| 3ioran 1102 | Negated triple disjunction... |
| 3ianor 1103 | Negated triple conjunction... |
| 3anor 1104 | Triple conjunction express... |
| 3oran 1105 | Triple disjunction in term... |
| 3impa 1106 | Importation from double to... |
| 3imp 1107 | Importation inference. (C... |
| 3imp31 1108 | The importation inference ... |
| 3imp231 1109 | Importation inference. (C... |
| 3imp21 1110 | The importation inference ... |
| 3impb 1111 | Importation from double to... |
| 3impib 1112 | Importation to triple conj... |
| 3impia 1113 | Importation to triple conj... |
| 3expa 1114 | Exportation from triple to... |
| 3exp 1115 | Exportation inference. (C... |
| 3expb 1116 | Exportation from triple to... |
| 3expia 1117 | Exportation from triple co... |
| 3expib 1118 | Exportation from triple co... |
| 3com12 1119 | Commutation in antecedent.... |
| 3com13 1120 | Commutation in antecedent.... |
| 3comr 1121 | Commutation in antecedent.... |
| 3com23 1122 | Commutation in antecedent.... |
| 3coml 1123 | Commutation in antecedent.... |
| 3jca 1124 | Join consequents with conj... |
| 3jcad 1125 | Deduction conjoining the c... |
| 3adant1 1126 | Deduction adding a conjunc... |
| 3adant2 1127 | Deduction adding a conjunc... |
| 3adant3 1128 | Deduction adding a conjunc... |
| 3ad2ant1 1129 | Deduction adding conjuncts... |
| 3ad2ant2 1130 | Deduction adding conjuncts... |
| 3ad2ant3 1131 | Deduction adding conjuncts... |
| simp1 1132 | Simplification of triple c... |
| simp2 1133 | Simplification of triple c... |
| simp3 1134 | Simplification of triple c... |
| simp1i 1135 | Infer a conjunct from a tr... |
| simp2i 1136 | Infer a conjunct from a tr... |
| simp3i 1137 | Infer a conjunct from a tr... |
| simp1d 1138 | Deduce a conjunct from a t... |
| simp2d 1139 | Deduce a conjunct from a t... |
| simp3d 1140 | Deduce a conjunct from a t... |
| simp1bi 1141 | Deduce a conjunct from a t... |
| simp2bi 1142 | Deduce a conjunct from a t... |
| simp3bi 1143 | Deduce a conjunct from a t... |
| 3simpa 1144 | Simplification of triple c... |
| 3simpb 1145 | Simplification of triple c... |
| 3simpc 1146 | Simplification of triple c... |
| 3anim123i 1147 | Join antecedents and conse... |
| 3anim1i 1148 | Add two conjuncts to antec... |
| 3anim2i 1149 | Add two conjuncts to antec... |
| 3anim3i 1150 | Add two conjuncts to antec... |
| 3anbi123i 1151 | Join 3 biconditionals with... |
| 3orbi123i 1152 | Join 3 biconditionals with... |
| 3anbi1i 1153 | Inference adding two conju... |
| 3anbi2i 1154 | Inference adding two conju... |
| 3anbi3i 1155 | Inference adding two conju... |
| syl3an 1156 | A triple syllogism inferen... |
| syl3anb 1157 | A triple syllogism inferen... |
| syl3anbr 1158 | A triple syllogism inferen... |
| syl3an1 1159 | A syllogism inference. (C... |
| syl3an2 1160 | A syllogism inference. (C... |
| syl3an3 1161 | A syllogism inference. (C... |
| 3adantl1 1162 | Deduction adding a conjunc... |
| 3adantl2 1163 | Deduction adding a conjunc... |
| 3adantl3 1164 | Deduction adding a conjunc... |
| 3adantr1 1165 | Deduction adding a conjunc... |
| 3adantr2 1166 | Deduction adding a conjunc... |
| 3adantr3 1167 | Deduction adding a conjunc... |
| ad4ant123 1168 | Deduction adding conjuncts... |
| ad4ant124 1169 | Deduction adding conjuncts... |
| ad4ant134 1170 | Deduction adding conjuncts... |
| ad4ant234 1171 | Deduction adding conjuncts... |
| 3adant1l 1172 | Deduction adding a conjunc... |
| 3adant1r 1173 | Deduction adding a conjunc... |
| 3adant2l 1174 | Deduction adding a conjunc... |
| 3adant2r 1175 | Deduction adding a conjunc... |
| 3adant3l 1176 | Deduction adding a conjunc... |
| 3adant3r 1177 | Deduction adding a conjunc... |
| 3adant3r1 1178 | Deduction adding a conjunc... |
| 3adant3r2 1179 | Deduction adding a conjunc... |
| 3adant3r3 1180 | Deduction adding a conjunc... |
| 3ad2antl1 1181 | Deduction adding conjuncts... |
| 3ad2antl2 1182 | Deduction adding conjuncts... |
| 3ad2antl3 1183 | Deduction adding conjuncts... |
| 3ad2antr1 1184 | Deduction adding conjuncts... |
| 3ad2antr2 1185 | Deduction adding conjuncts... |
| 3ad2antr3 1186 | Deduction adding conjuncts... |
| simpl1 1187 | Simplification of conjunct... |
| simpl2 1188 | Simplification of conjunct... |
| simpl3 1189 | Simplification of conjunct... |
| simpr1 1190 | Simplification of conjunct... |
| simpr2 1191 | Simplification of conjunct... |
| simpr3 1192 | Simplification of conjunct... |
| simp1l 1193 | Simplification of triple c... |
| simp1r 1194 | Simplification of triple c... |
| simp2l 1195 | Simplification of triple c... |
| simp2r 1196 | Simplification of triple c... |
| simp3l 1197 | Simplification of triple c... |
| simp3r 1198 | Simplification of triple c... |
| simp11 1199 | Simplification of doubly t... |
| simp12 1200 | Simplification of doubly t... |
| simp13 1201 | Simplification of doubly t... |
| simp21 1202 | Simplification of doubly t... |
| simp22 1203 | Simplification of doubly t... |
| simp23 1204 | Simplification of doubly t... |
| simp31 1205 | Simplification of doubly t... |
| simp32 1206 | Simplification of doubly t... |
| simp33 1207 | Simplification of doubly t... |
| simpll1 1208 | Simplification of conjunct... |
| simpll2 1209 | Simplification of conjunct... |
| simpll3 1210 | Simplification of conjunct... |
| simplr1 1211 | Simplification of conjunct... |
| simplr2 1212 | Simplification of conjunct... |
| simplr3 1213 | Simplification of conjunct... |
| simprl1 1214 | Simplification of conjunct... |
| simprl2 1215 | Simplification of conjunct... |
| simprl3 1216 | Simplification of conjunct... |
| simprr1 1217 | Simplification of conjunct... |
| simprr2 1218 | Simplification of conjunct... |
| simprr3 1219 | Simplification of conjunct... |
| simpl1l 1220 | Simplification of conjunct... |
| simpl1r 1221 | Simplification of conjunct... |
| simpl2l 1222 | Simplification of conjunct... |
| simpl2r 1223 | Simplification of conjunct... |
| simpl3l 1224 | Simplification of conjunct... |
| simpl3r 1225 | Simplification of conjunct... |
| simpr1l 1226 | Simplification of conjunct... |
| simpr1r 1227 | Simplification of conjunct... |
| simpr2l 1228 | Simplification of conjunct... |
| simpr2r 1229 | Simplification of conjunct... |
| simpr3l 1230 | Simplification of conjunct... |
| simpr3r 1231 | Simplification of conjunct... |
| simp1ll 1232 | Simplification of conjunct... |
| simp1lr 1233 | Simplification of conjunct... |
| simp1rl 1234 | Simplification of conjunct... |
| simp1rr 1235 | Simplification of conjunct... |
| simp2ll 1236 | Simplification of conjunct... |
| simp2lr 1237 | Simplification of conjunct... |
| simp2rl 1238 | Simplification of conjunct... |
| simp2rr 1239 | Simplification of conjunct... |
| simp3ll 1240 | Simplification of conjunct... |
| simp3lr 1241 | Simplification of conjunct... |
| simp3rl 1242 | Simplification of conjunct... |
| simp3rr 1243 | Simplification of conjunct... |
| simpl11 1244 | Simplification of conjunct... |
| simpl12 1245 | Simplification of conjunct... |
| simpl13 1246 | Simplification of conjunct... |
| simpl21 1247 | Simplification of conjunct... |
| simpl22 1248 | Simplification of conjunct... |
| simpl23 1249 | Simplification of conjunct... |
| simpl31 1250 | Simplification of conjunct... |
| simpl32 1251 | Simplification of conjunct... |
| simpl33 1252 | Simplification of conjunct... |
| simpr11 1253 | Simplification of conjunct... |
| simpr12 1254 | Simplification of conjunct... |
| simpr13 1255 | Simplification of conjunct... |
| simpr21 1256 | Simplification of conjunct... |
| simpr22 1257 | Simplification of conjunct... |
| simpr23 1258 | Simplification of conjunct... |
| simpr31 1259 | Simplification of conjunct... |
| simpr32 1260 | Simplification of conjunct... |
| simpr33 1261 | Simplification of conjunct... |
| simp1l1 1262 | Simplification of conjunct... |
| simp1l2 1263 | Simplification of conjunct... |
| simp1l3 1264 | Simplification of conjunct... |
| simp1r1 1265 | Simplification of conjunct... |
| simp1r2 1266 | Simplification of conjunct... |
| simp1r3 1267 | Simplification of conjunct... |
| simp2l1 1268 | Simplification of conjunct... |
| simp2l2 1269 | Simplification of conjunct... |
| simp2l3 1270 | Simplification of conjunct... |
| simp2r1 1271 | Simplification of conjunct... |
| simp2r2 1272 | Simplification of conjunct... |
| simp2r3 1273 | Simplification of conjunct... |
| simp3l1 1274 | Simplification of conjunct... |
| simp3l2 1275 | Simplification of conjunct... |
| simp3l3 1276 | Simplification of conjunct... |
| simp3r1 1277 | Simplification of conjunct... |
| simp3r2 1278 | Simplification of conjunct... |
| simp3r3 1279 | Simplification of conjunct... |
| simp11l 1280 | Simplification of conjunct... |
| simp11r 1281 | Simplification of conjunct... |
| simp12l 1282 | Simplification of conjunct... |
| simp12r 1283 | Simplification of conjunct... |
| simp13l 1284 | Simplification of conjunct... |
| simp13r 1285 | Simplification of conjunct... |
| simp21l 1286 | Simplification of conjunct... |
| simp21r 1287 | Simplification of conjunct... |
| simp22l 1288 | Simplification of conjunct... |
| simp22r 1289 | Simplification of conjunct... |
| simp23l 1290 | Simplification of conjunct... |
| simp23r 1291 | Simplification of conjunct... |
| simp31l 1292 | Simplification of conjunct... |
| simp31r 1293 | Simplification of conjunct... |
| simp32l 1294 | Simplification of conjunct... |
| simp32r 1295 | Simplification of conjunct... |
| simp33l 1296 | Simplification of conjunct... |
| simp33r 1297 | Simplification of conjunct... |
| simp111 1298 | Simplification of conjunct... |
| simp112 1299 | Simplification of conjunct... |
| simp113 1300 | Simplification of conjunct... |
| simp121 1301 | Simplification of conjunct... |
| simp122 1302 | Simplification of conjunct... |
| simp123 1303 | Simplification of conjunct... |
| simp131 1304 | Simplification of conjunct... |
| simp132 1305 | Simplification of conjunct... |
| simp133 1306 | Simplification of conjunct... |
| simp211 1307 | Simplification of conjunct... |
| simp212 1308 | Simplification of conjunct... |
| simp213 1309 | Simplification of conjunct... |
| simp221 1310 | Simplification of conjunct... |
| simp222 1311 | Simplification of conjunct... |
| simp223 1312 | Simplification of conjunct... |
| simp231 1313 | Simplification of conjunct... |
| simp232 1314 | Simplification of conjunct... |
| simp233 1315 | Simplification of conjunct... |
| simp311 1316 | Simplification of conjunct... |
| simp312 1317 | Simplification of conjunct... |
| simp313 1318 | Simplification of conjunct... |
| simp321 1319 | Simplification of conjunct... |
| simp322 1320 | Simplification of conjunct... |
| simp323 1321 | Simplification of conjunct... |
| simp331 1322 | Simplification of conjunct... |
| simp332 1323 | Simplification of conjunct... |
| simp333 1324 | Simplification of conjunct... |
| 3anibar 1325 | Remove a hypothesis from t... |
| 3mix1 1326 | Introduction in triple dis... |
| 3mix2 1327 | Introduction in triple dis... |
| 3mix3 1328 | Introduction in triple dis... |
| 3mix1i 1329 | Introduction in triple dis... |
| 3mix2i 1330 | Introduction in triple dis... |
| 3mix3i 1331 | Introduction in triple dis... |
| 3mix1d 1332 | Deduction introducing trip... |
| 3mix2d 1333 | Deduction introducing trip... |
| 3mix3d 1334 | Deduction introducing trip... |
| 3pm3.2i 1335 | Infer conjunction of premi... |
| pm3.2an3 1336 | Version of ~ pm3.2 for a t... |
| mpbir3an 1337 | Detach a conjunction of tr... |
| mpbir3and 1338 | Detach a conjunction of tr... |
| syl3anbrc 1339 | Syllogism inference. (Con... |
| syl21anbrc 1340 | Syllogism inference. (Con... |
| 3imp3i2an 1341 | An elimination deduction. ... |
| ex3 1342 | Apply ~ ex to a hypothesis... |
| 3imp1 1343 | Importation to left triple... |
| 3impd 1344 | Importation deduction for ... |
| 3imp2 1345 | Importation to right tripl... |
| 3impdi 1346 | Importation inference (und... |
| 3impdir 1347 | Importation inference (und... |
| 3exp1 1348 | Exportation from left trip... |
| 3expd 1349 | Exportation deduction for ... |
| 3exp2 1350 | Exportation from right tri... |
| exp5o 1351 | A triple exportation infer... |
| exp516 1352 | A triple exportation infer... |
| exp520 1353 | A triple exportation infer... |
| 3impexp 1354 | Version of ~ impexp for a ... |
| 3an1rs 1355 | Swap conjuncts. (Contribu... |
| 3anassrs 1356 | Associative law for conjun... |
| ad5ant245 1357 | Deduction adding conjuncts... |
| ad5ant234 1358 | Deduction adding conjuncts... |
| ad5ant235 1359 | Deduction adding conjuncts... |
| ad5ant123 1360 | Deduction adding conjuncts... |
| ad5ant124 1361 | Deduction adding conjuncts... |
| ad5ant125 1362 | Deduction adding conjuncts... |
| ad5ant134 1363 | Deduction adding conjuncts... |
| ad5ant135 1364 | Deduction adding conjuncts... |
| ad5ant145 1365 | Deduction adding conjuncts... |
| ad5ant2345 1366 | Deduction adding conjuncts... |
| syl3anc 1367 | Syllogism combined with co... |
| syl13anc 1368 | Syllogism combined with co... |
| syl31anc 1369 | Syllogism combined with co... |
| syl112anc 1370 | Syllogism combined with co... |
| syl121anc 1371 | Syllogism combined with co... |
| syl211anc 1372 | Syllogism combined with co... |
| syl23anc 1373 | Syllogism combined with co... |
| syl32anc 1374 | Syllogism combined with co... |
| syl122anc 1375 | Syllogism combined with co... |
| syl212anc 1376 | Syllogism combined with co... |
| syl221anc 1377 | Syllogism combined with co... |
| syl113anc 1378 | Syllogism combined with co... |
| syl131anc 1379 | Syllogism combined with co... |
| syl311anc 1380 | Syllogism combined with co... |
| syl33anc 1381 | Syllogism combined with co... |
| syl222anc 1382 | Syllogism combined with co... |
| syl123anc 1383 | Syllogism combined with co... |
| syl132anc 1384 | Syllogism combined with co... |
| syl213anc 1385 | Syllogism combined with co... |
| syl231anc 1386 | Syllogism combined with co... |
| syl312anc 1387 | Syllogism combined with co... |
| syl321anc 1388 | Syllogism combined with co... |
| syl133anc 1389 | Syllogism combined with co... |
| syl313anc 1390 | Syllogism combined with co... |
| syl331anc 1391 | Syllogism combined with co... |
| syl223anc 1392 | Syllogism combined with co... |
| syl232anc 1393 | Syllogism combined with co... |
| syl322anc 1394 | Syllogism combined with co... |
| syl233anc 1395 | Syllogism combined with co... |
| syl323anc 1396 | Syllogism combined with co... |
| syl332anc 1397 | Syllogism combined with co... |
| syl333anc 1398 | A syllogism inference comb... |
| syl3an1b 1399 | A syllogism inference. (C... |
| syl3an2b 1400 | A syllogism inference. (C... |
| syl3an3b 1401 | A syllogism inference. (C... |
| syl3an1br 1402 | A syllogism inference. (C... |
| syl3an2br 1403 | A syllogism inference. (C... |
| syl3an3br 1404 | A syllogism inference. (C... |
| syld3an3 1405 | A syllogism inference. (C... |
| syld3an1 1406 | A syllogism inference. (C... |
| syld3an2 1407 | A syllogism inference. (C... |
| syl3anl1 1408 | A syllogism inference. (C... |
| syl3anl2 1409 | A syllogism inference. (C... |
| syl3anl3 1410 | A syllogism inference. (C... |
| syl3anl 1411 | A triple syllogism inferen... |
| syl3anr1 1412 | A syllogism inference. (C... |
| syl3anr2 1413 | A syllogism inference. (C... |
| syl3anr3 1414 | A syllogism inference. (C... |
| 3anidm12 1415 | Inference from idempotent ... |
| 3anidm13 1416 | Inference from idempotent ... |
| 3anidm23 1417 | Inference from idempotent ... |
| syl2an3an 1418 | ~ syl3an with antecedents ... |
| syl2an23an 1419 | Deduction related to ~ syl... |
| 3ori 1420 | Infer implication from tri... |
| 3jao 1421 | Disjunction of three antec... |
| 3jaob 1422 | Disjunction of three antec... |
| 3jaoi 1423 | Disjunction of three antec... |
| 3jaod 1424 | Disjunction of three antec... |
| 3jaoian 1425 | Disjunction of three antec... |
| 3jaodan 1426 | Disjunction of three antec... |
| mpjao3dan 1427 | Eliminate a three-way disj... |
| mpjao3danOLD 1428 | Obsolete version of ~ mpja... |
| 3jaao 1429 | Inference conjoining and d... |
| syl3an9b 1430 | Nested syllogism inference... |
| 3orbi123d 1431 | Deduction joining 3 equiva... |
| 3anbi123d 1432 | Deduction joining 3 equiva... |
| 3anbi12d 1433 | Deduction conjoining and a... |
| 3anbi13d 1434 | Deduction conjoining and a... |
| 3anbi23d 1435 | Deduction conjoining and a... |
| 3anbi1d 1436 | Deduction adding conjuncts... |
| 3anbi2d 1437 | Deduction adding conjuncts... |
| 3anbi3d 1438 | Deduction adding conjuncts... |
| 3anim123d 1439 | Deduction joining 3 implic... |
| 3orim123d 1440 | Deduction joining 3 implic... |
| an6 1441 | Rearrangement of 6 conjunc... |
| 3an6 1442 | Analogue of ~ an4 for trip... |
| 3or6 1443 | Analogue of ~ or4 for trip... |
| mp3an1 1444 | An inference based on modu... |
| mp3an2 1445 | An inference based on modu... |
| mp3an3 1446 | An inference based on modu... |
| mp3an12 1447 | An inference based on modu... |
| mp3an13 1448 | An inference based on modu... |
| mp3an23 1449 | An inference based on modu... |
| mp3an1i 1450 | An inference based on modu... |
| mp3anl1 1451 | An inference based on modu... |
| mp3anl2 1452 | An inference based on modu... |
| mp3anl3 1453 | An inference based on modu... |
| mp3anr1 1454 | An inference based on modu... |
| mp3anr2 1455 | An inference based on modu... |
| mp3anr3 1456 | An inference based on modu... |
| mp3an 1457 | An inference based on modu... |
| mpd3an3 1458 | An inference based on modu... |
| mpd3an23 1459 | An inference based on modu... |
| mp3and 1460 | A deduction based on modus... |
| mp3an12i 1461 | ~ mp3an with antecedents i... |
| mp3an2i 1462 | ~ mp3an with antecedents i... |
| mp3an3an 1463 | ~ mp3an with antecedents i... |
| mp3an2ani 1464 | An elimination deduction. ... |
| biimp3a 1465 | Infer implication from a l... |
| biimp3ar 1466 | Infer implication from a l... |
| 3anandis 1467 | Inference that undistribut... |
| 3anandirs 1468 | Inference that undistribut... |
| ecase23d 1469 | Deduction for elimination ... |
| 3ecase 1470 | Inference for elimination ... |
| 3bior1fd 1471 | A disjunction is equivalen... |
| 3bior1fand 1472 | A disjunction is equivalen... |
| 3bior2fd 1473 | A wff is equivalent to its... |
| 3biant1d 1474 | A conjunction is equivalen... |
| intn3an1d 1475 | Introduction of a triple c... |
| intn3an2d 1476 | Introduction of a triple c... |
| intn3an3d 1477 | Introduction of a triple c... |
| an3andi 1478 | Distribution of conjunctio... |
| an33rean 1479 | Rearrange a 9-fold conjunc... |
| nanan 1482 | Conjunction in terms of al... |
| nanimn 1483 | Alternative denial in term... |
| nanor 1484 | Alternative denial in term... |
| nancom 1485 | Alternative denial is comm... |
| nannan 1486 | Nested alternative denials... |
| nanim 1487 | Implication in terms of al... |
| nannot 1488 | Negation in terms of alter... |
| nanbi 1489 | Biconditional in terms of ... |
| nanbi1 1490 | Introduce a right anti-con... |
| nanbi2 1491 | Introduce a left anti-conj... |
| nanbi12 1492 | Join two logical equivalen... |
| nanbi1i 1493 | Introduce a right anti-con... |
| nanbi2i 1494 | Introduce a left anti-conj... |
| nanbi12i 1495 | Join two logical equivalen... |
| nanbi1d 1496 | Introduce a right anti-con... |
| nanbi2d 1497 | Introduce a left anti-conj... |
| nanbi12d 1498 | Join two logical equivalen... |
| nanass 1499 | A characterization of when... |
| xnor 1502 | Two ways to write XNOR. (C... |
| xorcom 1503 | The connector ` \/_ ` is c... |
| xorass 1504 | The connector ` \/_ ` is a... |
| excxor 1505 | This tautology shows that ... |
| xor2 1506 | Two ways to express "exclu... |
| xoror 1507 | XOR implies OR. (Contribut... |
| xornan 1508 | XOR implies NAND. (Contrib... |
| xornan2 1509 | XOR implies NAND (written ... |
| xorneg2 1510 | The connector ` \/_ ` is n... |
| xorneg1 1511 | The connector ` \/_ ` is n... |
| xorneg 1512 | The connector ` \/_ ` is u... |
| xorbi12i 1513 | Equality property for XOR.... |
| xorbi12d 1514 | Equality property for XOR.... |
| anxordi 1515 | Conjunction distributes ov... |
| xorexmid 1516 | Exclusive-or variant of th... |
| norcom 1519 | The connector ` -\/ ` is c... |
| nornot 1520 | ` -. ` is expressible via ... |
| nornotOLD 1521 | Obsolete version of ~ norn... |
| noran 1522 | ` /\ ` is expressible via ... |
| noranOLD 1523 | Obsolete version of ~ nora... |
| noror 1524 | ` \/ ` is expressible via ... |
| nororOLD 1525 | Obsolete version of ~ noro... |
| norasslem1 1526 | This lemma shows the equiv... |
| norasslem2 1527 | This lemma specializes ~ b... |
| norasslem3 1528 | This lemma specializes ~ b... |
| norass 1529 | A characterization of when... |
| norassOLD 1530 | Obsolete version of ~ nora... |
| trujust 1535 | Soundness justification th... |
| tru 1537 | The truth value ` T. ` is ... |
| dftru2 1538 | An alternate definition of... |
| trut 1539 | A proposition is equivalen... |
| mptru 1540 | Eliminate ` T. ` as an ant... |
| tbtru 1541 | A proposition is equivalen... |
| bitru 1542 | A theorem is equivalent to... |
| trud 1543 | Anything implies ` T. ` . ... |
| truan 1544 | True can be removed from a... |
| fal 1547 | The truth value ` F. ` is ... |
| nbfal 1548 | The negation of a proposit... |
| bifal 1549 | A contradiction is equival... |
| falim 1550 | The truth value ` F. ` imp... |
| falimd 1551 | The truth value ` F. ` imp... |
| dfnot 1552 | Given falsum ` F. ` , we c... |
| inegd 1553 | Negation introduction rule... |
| efald 1554 | Deduction based on reducti... |
| pm2.21fal 1555 | If a wff and its negation ... |
| truimtru 1556 | A ` -> ` identity. (Contr... |
| truimfal 1557 | A ` -> ` identity. (Contr... |
| falimtru 1558 | A ` -> ` identity. (Contr... |
| falimfal 1559 | A ` -> ` identity. (Contr... |
| nottru 1560 | A ` -. ` identity. (Contr... |
| notfal 1561 | A ` -. ` identity. (Contr... |
| trubitru 1562 | A ` <-> ` identity. (Cont... |
| falbitru 1563 | A ` <-> ` identity. (Cont... |
| trubifal 1564 | A ` <-> ` identity. (Cont... |
| falbifal 1565 | A ` <-> ` identity. (Cont... |
| truantru 1566 | A ` /\ ` identity. (Contr... |
| truanfal 1567 | A ` /\ ` identity. (Contr... |
| falantru 1568 | A ` /\ ` identity. (Contr... |
| falanfal 1569 | A ` /\ ` identity. (Contr... |
| truortru 1570 | A ` \/ ` identity. (Contr... |
| truorfal 1571 | A ` \/ ` identity. (Contr... |
| falortru 1572 | A ` \/ ` identity. (Contr... |
| falorfal 1573 | A ` \/ ` identity. (Contr... |
| trunantru 1574 | A ` -/\ ` identity. (Cont... |
| trunanfal 1575 | A ` -/\ ` identity. (Cont... |
| falnantru 1576 | A ` -/\ ` identity. (Cont... |
| falnanfal 1577 | A ` -/\ ` identity. (Cont... |
| truxortru 1578 | A ` \/_ ` identity. (Cont... |
| truxorfal 1579 | A ` \/_ ` identity. (Cont... |
| falxortru 1580 | A ` \/_ ` identity. (Cont... |
| falxorfal 1581 | A ` \/_ ` identity. (Cont... |
| trunortru 1582 | A ` -\/ ` identity. (Cont... |
| trunortruOLD 1583 | Obsolete version of ~ trun... |
| trunorfal 1584 | A ` -\/ ` identity. (Cont... |
| trunorfalOLD 1585 | Obsolete version of ~ trun... |
| falnortru 1586 | A ` -\/ ` identity. (Cont... |
| falnorfal 1587 | A ` -\/ ` identity. (Cont... |
| falnorfalOLD 1588 | Obsolete version of ~ faln... |
| hadbi123d 1591 | Equality theorem for the a... |
| hadbi123i 1592 | Equality theorem for the a... |
| hadass 1593 | Associative law for the ad... |
| hadbi 1594 | The adder sum is the same ... |
| hadcoma 1595 | Commutative law for the ad... |
| hadcomaOLD 1596 | Commutative law for the ad... |
| hadcomb 1597 | Commutative law for the ad... |
| hadrot 1598 | Rotation law for the adder... |
| hadnot 1599 | The adder sum distributes ... |
| had1 1600 | If the first input is true... |
| had0 1601 | If the first input is fals... |
| hadifp 1602 | The value of the adder sum... |
| cador 1605 | The adder carry in disjunc... |
| cadan 1606 | The adder carry in conjunc... |
| cadbi123d 1607 | Equality theorem for the a... |
| cadbi123i 1608 | Equality theorem for the a... |
| cadcoma 1609 | Commutative law for the ad... |
| cadcomb 1610 | Commutative law for the ad... |
| cadrot 1611 | Rotation law for the adder... |
| cadnot 1612 | The adder carry distribute... |
| cad1 1613 | If one input is true, then... |
| cad0 1614 | If one input is false, the... |
| cadifp 1615 | The value of the carry is,... |
| cad11 1616 | If (at least) two inputs a... |
| cadtru 1617 | The adder carry is true as... |
| minimp 1618 | A single axiom for minimal... |
| minimp-syllsimp 1619 | Derivation of Syll-Simp ( ... |
| minimp-ax1 1620 | Derivation of ~ ax-1 from ... |
| minimp-ax2c 1621 | Derivation of a commuted f... |
| minimp-ax2 1622 | Derivation of ~ ax-2 from ... |
| minimp-pm2.43 1623 | Derivation of ~ pm2.43 (al... |
| impsingle 1624 | The shortest single axiom ... |
| impsingle-step4 1625 | Derivation of impsingle-st... |
| impsingle-step8 1626 | Derivation of impsingle-st... |
| impsingle-ax1 1627 | Derivation of impsingle-ax... |
| impsingle-step15 1628 | Derivation of impsingle-st... |
| impsingle-step18 1629 | Derivation of impsingle-st... |
| impsingle-step19 1630 | Derivation of impsingle-st... |
| impsingle-step20 1631 | Derivation of impsingle-st... |
| impsingle-step21 1632 | Derivation of impsingle-st... |
| impsingle-step22 1633 | Derivation of impsingle-st... |
| impsingle-step25 1634 | Derivation of impsingle-st... |
| impsingle-imim1 1635 | Derivation of impsingle-im... |
| impsingle-peirce 1636 | Derivation of impsingle-pe... |
| tarski-bernays-ax2 1637 | Derivation of ~ ax-2 from ... |
| meredith 1638 | Carew Meredith's sole axio... |
| merlem1 1639 | Step 3 of Meredith's proof... |
| merlem2 1640 | Step 4 of Meredith's proof... |
| merlem3 1641 | Step 7 of Meredith's proof... |
| merlem4 1642 | Step 8 of Meredith's proof... |
| merlem5 1643 | Step 11 of Meredith's proo... |
| merlem6 1644 | Step 12 of Meredith's proo... |
| merlem7 1645 | Between steps 14 and 15 of... |
| merlem8 1646 | Step 15 of Meredith's proo... |
| merlem9 1647 | Step 18 of Meredith's proo... |
| merlem10 1648 | Step 19 of Meredith's proo... |
| merlem11 1649 | Step 20 of Meredith's proo... |
| merlem12 1650 | Step 28 of Meredith's proo... |
| merlem13 1651 | Step 35 of Meredith's proo... |
| luk-1 1652 | 1 of 3 axioms for proposit... |
| luk-2 1653 | 2 of 3 axioms for proposit... |
| luk-3 1654 | 3 of 3 axioms for proposit... |
| luklem1 1655 | Used to rederive standard ... |
| luklem2 1656 | Used to rederive standard ... |
| luklem3 1657 | Used to rederive standard ... |
| luklem4 1658 | Used to rederive standard ... |
| luklem5 1659 | Used to rederive standard ... |
| luklem6 1660 | Used to rederive standard ... |
| luklem7 1661 | Used to rederive standard ... |
| luklem8 1662 | Used to rederive standard ... |
| ax1 1663 | Standard propositional axi... |
| ax2 1664 | Standard propositional axi... |
| ax3 1665 | Standard propositional axi... |
| nic-dfim 1666 | This theorem "defines" imp... |
| nic-dfneg 1667 | This theorem "defines" neg... |
| nic-mp 1668 | Derive Nicod's rule of mod... |
| nic-mpALT 1669 | A direct proof of ~ nic-mp... |
| nic-ax 1670 | Nicod's axiom derived from... |
| nic-axALT 1671 | A direct proof of ~ nic-ax... |
| nic-imp 1672 | Inference for ~ nic-mp usi... |
| nic-idlem1 1673 | Lemma for ~ nic-id . (Con... |
| nic-idlem2 1674 | Lemma for ~ nic-id . Infe... |
| nic-id 1675 | Theorem ~ id expressed wit... |
| nic-swap 1676 | The connector ` -/\ ` is s... |
| nic-isw1 1677 | Inference version of ~ nic... |
| nic-isw2 1678 | Inference for swapping nes... |
| nic-iimp1 1679 | Inference version of ~ nic... |
| nic-iimp2 1680 | Inference version of ~ nic... |
| nic-idel 1681 | Inference to remove the tr... |
| nic-ich 1682 | Chained inference. (Contr... |
| nic-idbl 1683 | Double the terms. Since d... |
| nic-bijust 1684 | Biconditional justificatio... |
| nic-bi1 1685 | Inference to extract one s... |
| nic-bi2 1686 | Inference to extract the o... |
| nic-stdmp 1687 | Derive the standard modus ... |
| nic-luk1 1688 | Proof of ~ luk-1 from ~ ni... |
| nic-luk2 1689 | Proof of ~ luk-2 from ~ ni... |
| nic-luk3 1690 | Proof of ~ luk-3 from ~ ni... |
| lukshef-ax1 1691 | This alternative axiom for... |
| lukshefth1 1692 | Lemma for ~ renicax . (Co... |
| lukshefth2 1693 | Lemma for ~ renicax . (Co... |
| renicax 1694 | A rederivation of ~ nic-ax... |
| tbw-bijust 1695 | Justification for ~ tbw-ne... |
| tbw-negdf 1696 | The definition of negation... |
| tbw-ax1 1697 | The first of four axioms i... |
| tbw-ax2 1698 | The second of four axioms ... |
| tbw-ax3 1699 | The third of four axioms i... |
| tbw-ax4 1700 | The fourth of four axioms ... |
| tbwsyl 1701 | Used to rederive the Lukas... |
| tbwlem1 1702 | Used to rederive the Lukas... |
| tbwlem2 1703 | Used to rederive the Lukas... |
| tbwlem3 1704 | Used to rederive the Lukas... |
| tbwlem4 1705 | Used to rederive the Lukas... |
| tbwlem5 1706 | Used to rederive the Lukas... |
| re1luk1 1707 | ~ luk-1 derived from the T... |
| re1luk2 1708 | ~ luk-2 derived from the T... |
| re1luk3 1709 | ~ luk-3 derived from the T... |
| merco1 1710 | A single axiom for proposi... |
| merco1lem1 1711 | Used to rederive the Tarsk... |
| retbwax4 1712 | ~ tbw-ax4 rederived from ~... |
| retbwax2 1713 | ~ tbw-ax2 rederived from ~... |
| merco1lem2 1714 | Used to rederive the Tarsk... |
| merco1lem3 1715 | Used to rederive the Tarsk... |
| merco1lem4 1716 | Used to rederive the Tarsk... |
| merco1lem5 1717 | Used to rederive the Tarsk... |
| merco1lem6 1718 | Used to rederive the Tarsk... |
| merco1lem7 1719 | Used to rederive the Tarsk... |
| retbwax3 1720 | ~ tbw-ax3 rederived from ~... |
| merco1lem8 1721 | Used to rederive the Tarsk... |
| merco1lem9 1722 | Used to rederive the Tarsk... |
| merco1lem10 1723 | Used to rederive the Tarsk... |
| merco1lem11 1724 | Used to rederive the Tarsk... |
| merco1lem12 1725 | Used to rederive the Tarsk... |
| merco1lem13 1726 | Used to rederive the Tarsk... |
| merco1lem14 1727 | Used to rederive the Tarsk... |
| merco1lem15 1728 | Used to rederive the Tarsk... |
| merco1lem16 1729 | Used to rederive the Tarsk... |
| merco1lem17 1730 | Used to rederive the Tarsk... |
| merco1lem18 1731 | Used to rederive the Tarsk... |
| retbwax1 1732 | ~ tbw-ax1 rederived from ~... |
| merco2 1733 | A single axiom for proposi... |
| mercolem1 1734 | Used to rederive the Tarsk... |
| mercolem2 1735 | Used to rederive the Tarsk... |
| mercolem3 1736 | Used to rederive the Tarsk... |
| mercolem4 1737 | Used to rederive the Tarsk... |
| mercolem5 1738 | Used to rederive the Tarsk... |
| mercolem6 1739 | Used to rederive the Tarsk... |
| mercolem7 1740 | Used to rederive the Tarsk... |
| mercolem8 1741 | Used to rederive the Tarsk... |
| re1tbw1 1742 | ~ tbw-ax1 rederived from ~... |
| re1tbw2 1743 | ~ tbw-ax2 rederived from ~... |
| re1tbw3 1744 | ~ tbw-ax3 rederived from ~... |
| re1tbw4 1745 | ~ tbw-ax4 rederived from ~... |
| rb-bijust 1746 | Justification for ~ rb-imd... |
| rb-imdf 1747 | The definition of implicat... |
| anmp 1748 | Modus ponens for ` \/ ` ` ... |
| rb-ax1 1749 | The first of four axioms i... |
| rb-ax2 1750 | The second of four axioms ... |
| rb-ax3 1751 | The third of four axioms i... |
| rb-ax4 1752 | The fourth of four axioms ... |
| rbsyl 1753 | Used to rederive the Lukas... |
| rblem1 1754 | Used to rederive the Lukas... |
| rblem2 1755 | Used to rederive the Lukas... |
| rblem3 1756 | Used to rederive the Lukas... |
| rblem4 1757 | Used to rederive the Lukas... |
| rblem5 1758 | Used to rederive the Lukas... |
| rblem6 1759 | Used to rederive the Lukas... |
| rblem7 1760 | Used to rederive the Lukas... |
| re1axmp 1761 | ~ ax-mp derived from Russe... |
| re2luk1 1762 | ~ luk-1 derived from Russe... |
| re2luk2 1763 | ~ luk-2 derived from Russe... |
| re2luk3 1764 | ~ luk-3 derived from Russe... |
| mptnan 1765 | Modus ponendo tollens 1, o... |
| mptxor 1766 | Modus ponendo tollens 2, o... |
| mtpor 1767 | Modus tollendo ponens (inc... |
| mtpxor 1768 | Modus tollendo ponens (ori... |
| stoic1a 1769 | Stoic logic Thema 1 (part ... |
| stoic1b 1770 | Stoic logic Thema 1 (part ... |
| stoic2a 1771 | Stoic logic Thema 2 versio... |
| stoic2b 1772 | Stoic logic Thema 2 versio... |
| stoic3 1773 | Stoic logic Thema 3. Stat... |
| stoic4a 1774 | Stoic logic Thema 4 versio... |
| stoic4b 1775 | Stoic logic Thema 4 versio... |
| alnex 1778 | Universal quantification o... |
| eximal 1779 | An equivalence between an ... |
| nf2 1782 | Alternate definition of no... |
| nf3 1783 | Alternate definition of no... |
| nf4 1784 | Alternate definition of no... |
| nfi 1785 | Deduce that ` x ` is not f... |
| nfri 1786 | Consequence of the definit... |
| nfd 1787 | Deduce that ` x ` is not f... |
| nfrd 1788 | Consequence of the definit... |
| nftht 1789 | Closed form of ~ nfth . (... |
| nfntht 1790 | Closed form of ~ nfnth . ... |
| nfntht2 1791 | Closed form of ~ nfnth . ... |
| gen2 1793 | Generalization applied twi... |
| mpg 1794 | Modus ponens combined with... |
| mpgbi 1795 | Modus ponens on biconditio... |
| mpgbir 1796 | Modus ponens on biconditio... |
| nex 1797 | Generalization rule for ne... |
| nfth 1798 | No variable is (effectivel... |
| nfnth 1799 | No variable is (effectivel... |
| hbth 1800 | No variable is (effectivel... |
| nftru 1801 | The true constant has no f... |
| nffal 1802 | The false constant has no ... |
| sptruw 1803 | Version of ~ sp when ` ph ... |
| altru 1804 | For all sets, ` T. ` is tr... |
| alfal 1805 | For all sets, ` -. F. ` is... |
| alim 1807 | Restatement of Axiom ~ ax-... |
| alimi 1808 | Inference quantifying both... |
| 2alimi 1809 | Inference doubly quantifyi... |
| ala1 1810 | Add an antecedent in a uni... |
| al2im 1811 | Closed form of ~ al2imi . ... |
| al2imi 1812 | Inference quantifying ante... |
| alanimi 1813 | Variant of ~ al2imi with c... |
| alimdh 1814 | Deduction form of Theorem ... |
| albi 1815 | Theorem 19.15 of [Margaris... |
| albii 1816 | Inference adding universal... |
| 2albii 1817 | Inference adding two unive... |
| sylgt 1818 | Closed form of ~ sylg . (... |
| sylg 1819 | A syllogism combined with ... |
| alrimih 1820 | Inference form of Theorem ... |
| hbxfrbi 1821 | A utility lemma to transfe... |
| alex 1822 | Universal quantifier in te... |
| exnal 1823 | Existential quantification... |
| 2nalexn 1824 | Part of theorem *11.5 in [... |
| 2exnaln 1825 | Theorem *11.22 in [Whitehe... |
| 2nexaln 1826 | Theorem *11.25 in [Whitehe... |
| alimex 1827 | An equivalence between an ... |
| aleximi 1828 | A variant of ~ al2imi : in... |
| alexbii 1829 | Biconditional form of ~ al... |
| exim 1830 | Theorem 19.22 of [Margaris... |
| eximi 1831 | Inference adding existenti... |
| 2eximi 1832 | Inference adding two exist... |
| eximii 1833 | Inference associated with ... |
| exa1 1834 | Add an antecedent in an ex... |
| 19.38 1835 | Theorem 19.38 of [Margaris... |
| 19.38a 1836 | Under a non-freeness hypot... |
| 19.38b 1837 | Under a non-freeness hypot... |
| imnang 1838 | Quantified implication in ... |
| alinexa 1839 | A transformation of quanti... |
| exnalimn 1840 | Existential quantification... |
| alexn 1841 | A relationship between two... |
| 2exnexn 1842 | Theorem *11.51 in [Whitehe... |
| exbi 1843 | Theorem 19.18 of [Margaris... |
| exbii 1844 | Inference adding existenti... |
| 2exbii 1845 | Inference adding two exist... |
| 3exbii 1846 | Inference adding three exi... |
| nfbiit 1847 | Equivalence theorem for th... |
| nfbii 1848 | Equality theorem for the n... |
| nfxfr 1849 | A utility lemma to transfe... |
| nfxfrd 1850 | A utility lemma to transfe... |
| nfnbi 1851 | A variable is non-free in ... |
| nfnt 1852 | If a variable is non-free ... |
| nfn 1853 | Inference associated with ... |
| nfnd 1854 | Deduction associated with ... |
| exanali 1855 | A transformation of quanti... |
| 2exanali 1856 | Theorem *11.521 in [Whiteh... |
| exancom 1857 | Commutation of conjunction... |
| exan 1858 | Place a conjunct in the sc... |
| exanOLD 1859 | Obsolete proof of ~ exan a... |
| alrimdh 1860 | Deduction form of Theorem ... |
| eximdh 1861 | Deduction from Theorem 19.... |
| nexdh 1862 | Deduction for generalizati... |
| albidh 1863 | Formula-building rule for ... |
| exbidh 1864 | Formula-building rule for ... |
| exsimpl 1865 | Simplification of an exist... |
| exsimpr 1866 | Simplification of an exist... |
| 19.26 1867 | Theorem 19.26 of [Margaris... |
| 19.26-2 1868 | Theorem ~ 19.26 with two q... |
| 19.26-3an 1869 | Theorem ~ 19.26 with tripl... |
| 19.29 1870 | Theorem 19.29 of [Margaris... |
| 19.29r 1871 | Variation of ~ 19.29 . (C... |
| 19.29r2 1872 | Variation of ~ 19.29r with... |
| 19.29x 1873 | Variation of ~ 19.29 with ... |
| 19.35 1874 | Theorem 19.35 of [Margaris... |
| 19.35i 1875 | Inference associated with ... |
| 19.35ri 1876 | Inference associated with ... |
| 19.25 1877 | Theorem 19.25 of [Margaris... |
| 19.30 1878 | Theorem 19.30 of [Margaris... |
| 19.43 1879 | Theorem 19.43 of [Margaris... |
| 19.43OLD 1880 | Obsolete proof of ~ 19.43 ... |
| 19.33 1881 | Theorem 19.33 of [Margaris... |
| 19.33b 1882 | The antecedent provides a ... |
| 19.40 1883 | Theorem 19.40 of [Margaris... |
| 19.40-2 1884 | Theorem *11.42 in [Whitehe... |
| 19.40b 1885 | The antecedent provides a ... |
| albiim 1886 | Split a biconditional and ... |
| 2albiim 1887 | Split a biconditional and ... |
| exintrbi 1888 | Add/remove a conjunct in t... |
| exintr 1889 | Introduce a conjunct in th... |
| alsyl 1890 | Universally quantified and... |
| nfimd 1891 | If in a context ` x ` is n... |
| nfimt 1892 | Closed form of ~ nfim and ... |
| nfim 1893 | If ` x ` is not free in ` ... |
| nfand 1894 | If in a context ` x ` is n... |
| nf3and 1895 | Deduction form of bound-va... |
| nfan 1896 | If ` x ` is not free in ` ... |
| nfnan 1897 | If ` x ` is not free in ` ... |
| nf3an 1898 | If ` x ` is not free in ` ... |
| nfbid 1899 | If in a context ` x ` is n... |
| nfbi 1900 | If ` x ` is not free in ` ... |
| nfor 1901 | If ` x ` is not free in ` ... |
| nf3or 1902 | If ` x ` is not free in ` ... |
| empty 1903 | Two characterizations of t... |
| emptyex 1904 | On the empty domain, any e... |
| emptyal 1905 | On the empty domain, any u... |
| emptynf 1906 | On the empty domain, any v... |
| ax5d 1908 | Version of ~ ax-5 with ant... |
| ax5e 1909 | A rephrasing of ~ ax-5 usi... |
| ax5ea 1910 | If a formula holds for som... |
| nfv 1911 | If ` x ` is not present in... |
| nfvd 1912 | ~ nfv with antecedent. Us... |
| alimdv 1913 | Deduction form of Theorem ... |
| eximdv 1914 | Deduction form of Theorem ... |
| 2alimdv 1915 | Deduction form of Theorem ... |
| 2eximdv 1916 | Deduction form of Theorem ... |
| albidv 1917 | Formula-building rule for ... |
| exbidv 1918 | Formula-building rule for ... |
| nfbidv 1919 | An equality theorem for no... |
| 2albidv 1920 | Formula-building rule for ... |
| 2exbidv 1921 | Formula-building rule for ... |
| 3exbidv 1922 | Formula-building rule for ... |
| 4exbidv 1923 | Formula-building rule for ... |
| alrimiv 1924 | Inference form of Theorem ... |
| alrimivv 1925 | Inference form of Theorem ... |
| alrimdv 1926 | Deduction form of Theorem ... |
| exlimiv 1927 | Inference form of Theorem ... |
| exlimiiv 1928 | Inference (Rule C) associa... |
| exlimivv 1929 | Inference form of Theorem ... |
| exlimdv 1930 | Deduction form of Theorem ... |
| exlimdvv 1931 | Deduction form of Theorem ... |
| exlimddv 1932 | Existential elimination ru... |
| nexdv 1933 | Deduction for generalizati... |
| 2ax5 1934 | Quantification of two vari... |
| stdpc5v 1935 | Version of ~ stdpc5 with a... |
| 19.21v 1936 | Version of ~ 19.21 with a ... |
| 19.32v 1937 | Version of ~ 19.32 with a ... |
| 19.31v 1938 | Version of ~ 19.31 with a ... |
| 19.23v 1939 | Version of ~ 19.23 with a ... |
| 19.23vv 1940 | Theorem ~ 19.23v extended ... |
| pm11.53v 1941 | Version of ~ pm11.53 with ... |
| 19.36imv 1942 | One direction of ~ 19.36v ... |
| 19.36iv 1943 | Inference associated with ... |
| 19.37imv 1944 | One direction of ~ 19.37v ... |
| 19.37iv 1945 | Inference associated with ... |
| 19.41v 1946 | Version of ~ 19.41 with a ... |
| 19.41vv 1947 | Version of ~ 19.41 with tw... |
| 19.41vvv 1948 | Version of ~ 19.41 with th... |
| 19.41vvvv 1949 | Version of ~ 19.41 with fo... |
| 19.42v 1950 | Version of ~ 19.42 with a ... |
| exdistr 1951 | Distribution of existentia... |
| exdistrv 1952 | Distribute a pair of exist... |
| 4exdistrv 1953 | Distribute two pairs of ex... |
| 19.42vv 1954 | Version of ~ 19.42 with tw... |
| exdistr2 1955 | Distribution of existentia... |
| 19.42vvv 1956 | Version of ~ 19.42 with th... |
| 19.42vvvOLD 1957 | Obsolete version of ~ 19.4... |
| 3exdistr 1958 | Distribution of existentia... |
| 4exdistr 1959 | Distribution of existentia... |
| weq 1960 | Extend wff definition to i... |
| equs3OLD 1961 | Obsolete as of 12-Aug-2023... |
| speimfw 1962 | Specialization, with addit... |
| speimfwALT 1963 | Alternate proof of ~ speim... |
| spimfw 1964 | Specialization, with addit... |
| ax12i 1965 | Inference that has ~ ax-12... |
| ax6v 1967 | Axiom B7 of [Tarski] p. 75... |
| ax6ev 1968 | At least one individual ex... |
| spimw 1969 | Specialization. Lemma 8 o... |
| spimew 1970 | Existential introduction, ... |
| spimehOLD 1971 | Obsolete version of ~ spim... |
| speiv 1972 | Inference from existential... |
| speivw 1973 | Version of ~ spei with a d... |
| exgen 1974 | Rule of existential genera... |
| exgenOLD 1975 | Obsolete version of ~ exge... |
| extru 1976 | There exists a variable su... |
| 19.2 1977 | Theorem 19.2 of [Margaris]... |
| 19.2d 1978 | Deduction associated with ... |
| 19.8w 1979 | Weak version of ~ 19.8a an... |
| spnfw 1980 | Weak version of ~ sp . Us... |
| spvw 1981 | Version of ~ sp when ` x `... |
| 19.3v 1982 | Version of ~ 19.3 with a d... |
| 19.8v 1983 | Version of ~ 19.8a with a ... |
| 19.9v 1984 | Version of ~ 19.9 with a d... |
| 19.3vOLD 1985 | Obsolete version of ~ 19.3... |
| spvwOLD 1986 | Obsolete version of ~ spvw... |
| 19.39 1987 | Theorem 19.39 of [Margaris... |
| 19.24 1988 | Theorem 19.24 of [Margaris... |
| 19.34 1989 | Theorem 19.34 of [Margaris... |
| 19.36v 1990 | Version of ~ 19.36 with a ... |
| 19.12vvv 1991 | Version of ~ 19.12vv with ... |
| 19.27v 1992 | Version of ~ 19.27 with a ... |
| 19.28v 1993 | Version of ~ 19.28 with a ... |
| 19.37v 1994 | Version of ~ 19.37 with a ... |
| 19.44v 1995 | Version of ~ 19.44 with a ... |
| 19.45v 1996 | Version of ~ 19.45 with a ... |
| spimevw 1997 | Existential introduction, ... |
| spimvw 1998 | A weak form of specializat... |
| spvv 1999 | Specialization, using impl... |
| spfalw 2000 | Version of ~ sp when ` ph ... |
| chvarvv 2001 | Implicit substitution of `... |
| equs4v 2002 | Version of ~ equs4 with a ... |
| alequexv 2003 | Version of ~ equs4v with i... |
| exsbim 2004 | One direction of the equiv... |
| equsv 2005 | If a formula does not cont... |
| equsalvw 2006 | Version of ~ equsalv with ... |
| equsexvw 2007 | Version of ~ equsexv with ... |
| equsexvwOLD 2008 | Obsolete version of ~ equs... |
| cbvaliw 2009 | Change bound variable. Us... |
| cbvalivw 2010 | Change bound variable. Us... |
| ax7v 2012 | Weakened version of ~ ax-7... |
| ax7v1 2013 | First of two weakened vers... |
| ax7v2 2014 | Second of two weakened ver... |
| equid 2015 | Identity law for equality.... |
| nfequid 2016 | Bound-variable hypothesis ... |
| equcomiv 2017 | Weaker form of ~ equcomi w... |
| ax6evr 2018 | A commuted form of ~ ax6ev... |
| ax7 2019 | Proof of ~ ax-7 from ~ ax7... |
| equcomi 2020 | Commutative law for equali... |
| equcom 2021 | Commutative law for equali... |
| equcomd 2022 | Deduction form of ~ equcom... |
| equcoms 2023 | An inference commuting equ... |
| equtr 2024 | A transitive law for equal... |
| equtrr 2025 | A transitive law for equal... |
| equeuclr 2026 | Commuted version of ~ eque... |
| equeucl 2027 | Equality is a left-Euclide... |
| equequ1 2028 | An equivalence law for equ... |
| equequ2 2029 | An equivalence law for equ... |
| equtr2 2030 | Equality is a left-Euclide... |
| stdpc6 2031 | One of the two equality ax... |
| equvinv 2032 | A variable introduction la... |
| equvinva 2033 | A modified version of the ... |
| equvelv 2034 | A biconditional form of ~ ... |
| ax13b 2035 | An equivalence between two... |
| spfw 2036 | Weak version of ~ sp . Us... |
| spw 2037 | Weak version of the specia... |
| cbvalw 2038 | Change bound variable. Us... |
| cbvalvw 2039 | Change bound variable. Us... |
| cbvexvw 2040 | Change bound variable. Us... |
| cbvaldvaw 2041 | Rule used to change the bo... |
| cbvexdvaw 2042 | Rule used to change the bo... |
| cbval2vw 2043 | Rule used to change bound ... |
| cbvex2vw 2044 | Rule used to change bound ... |
| cbvex4vw 2045 | Rule used to change bound ... |
| alcomiw 2046 | Weak version of ~ alcom . ... |
| alcomiwOLD 2047 | Obsolete version of ~ alco... |
| hbn1fw 2048 | Weak version of ~ ax-10 fr... |
| hbn1w 2049 | Weak version of ~ hbn1 . ... |
| hba1w 2050 | Weak version of ~ hba1 . ... |
| hbe1w 2051 | Weak version of ~ hbe1 . ... |
| hbalw 2052 | Weak version of ~ hbal . ... |
| spaev 2053 | A special instance of ~ sp... |
| cbvaev 2054 | Change bound variable in a... |
| aevlem0 2055 | Lemma for ~ aevlem . Inst... |
| aevlem 2056 | Lemma for ~ aev and ~ axc1... |
| aeveq 2057 | The antecedent ` A. x x = ... |
| aev 2058 | A "distinctor elimination"... |
| aev2 2059 | A version of ~ aev with tw... |
| hbaev 2060 | All variables are effectiv... |
| naev 2061 | If some set variables can ... |
| naev2 2062 | Generalization of ~ hbnaev... |
| hbnaev 2063 | Any variable is free in ` ... |
| sbjust 2064 | Justification theorem for ... |
| sbt 2067 | A substitution into a theo... |
| sbtru 2068 | The result of substituting... |
| stdpc4 2069 | The specialization axiom o... |
| sbtALT 2070 | Alternate proof of ~ sbt ,... |
| 2stdpc4 2071 | A double specialization us... |
| sbi1 2072 | Distribute substitution ov... |
| spsbim 2073 | Distribute substitution ov... |
| spsbbi 2074 | Biconditional property for... |
| sbimi 2075 | Distribute substitution ov... |
| sb2imi 2076 | Distribute substitution ov... |
| sbbii 2077 | Infer substitution into bo... |
| 2sbbii 2078 | Infer double substitution ... |
| sbimdv 2079 | Deduction substituting bot... |
| sbbidv 2080 | Deduction substituting bot... |
| sbbidvOLD 2081 | Obsolete version of ~ sbbi... |
| sban 2082 | Conjunction inside and out... |
| sb3an 2083 | Threefold conjunction insi... |
| spsbe 2084 | Existential generalization... |
| spsbeOLD 2085 | Obsolete version of ~ spsb... |
| sbequ 2086 | Equality property for subs... |
| sbequi 2087 | An equality theorem for su... |
| sbequOLD 2088 | Obsolete proof of ~ sbequ ... |
| sb6 2089 | Alternate definition of su... |
| 2sb6 2090 | Equivalence for double sub... |
| sb1v 2091 | One direction of ~ sb5 , p... |
| sb4vOLD 2092 | Obsolete as of 30-Jul-2023... |
| sb2vOLD 2093 | Obsolete as of 30-Jul-2023... |
| sbv 2094 | Substitution for a variabl... |
| sbcom4 2095 | Commutativity law for subs... |
| pm11.07 2096 | Axiom *11.07 in [Whitehead... |
| sbrimvlem 2097 | Common proof template for ... |
| sbrimvw 2098 | Substitution in an implica... |
| sbievw 2099 | Conversion of implicit sub... |
| sbiedvw 2100 | Conversion of implicit sub... |
| 2sbievw 2101 | Conversion of double impli... |
| sbcom3vv 2102 | Substituting ` y ` for ` x... |
| sbievw2 2103 | ~ sbievw applied twice, av... |
| sbco2vv 2104 | A composition law for subs... |
| equsb3 2105 | Substitution in an equalit... |
| equsb3r 2106 | Substitution applied to th... |
| equsb3rOLD 2107 | Obsolete version of ~ equs... |
| equsb1v 2108 | Substitution applied to an... |
| equsb1vOLD 2109 | Obsolete version of ~ equs... |
| wel 2111 | Extend wff definition to i... |
| ax8v 2113 | Weakened version of ~ ax-8... |
| ax8v1 2114 | First of two weakened vers... |
| ax8v2 2115 | Second of two weakened ver... |
| ax8 2116 | Proof of ~ ax-8 from ~ ax8... |
| elequ1 2117 | An identity law for the no... |
| elsb3 2118 | Substitution applied to an... |
| cleljust 2119 | When the class variables i... |
| ax9v 2121 | Weakened version of ~ ax-9... |
| ax9v1 2122 | First of two weakened vers... |
| ax9v2 2123 | Second of two weakened ver... |
| ax9 2124 | Proof of ~ ax-9 from ~ ax9... |
| elequ2 2125 | An identity law for the no... |
| elsb4 2126 | Substitution applied to an... |
| elequ2g 2127 | A form of ~ elequ2 with a ... |
| ax6dgen 2128 | Tarski's system uses the w... |
| ax10w 2129 | Weak version of ~ ax-10 fr... |
| ax11w 2130 | Weak version of ~ ax-11 fr... |
| ax11dgen 2131 | Degenerate instance of ~ a... |
| ax12wlem 2132 | Lemma for weak version of ... |
| ax12w 2133 | Weak version of ~ ax-12 fr... |
| ax12dgen 2134 | Degenerate instance of ~ a... |
| ax12wdemo 2135 | Example of an application ... |
| ax13w 2136 | Weak version (principal in... |
| ax13dgen1 2137 | Degenerate instance of ~ a... |
| ax13dgen2 2138 | Degenerate instance of ~ a... |
| ax13dgen3 2139 | Degenerate instance of ~ a... |
| ax13dgen4 2140 | Degenerate instance of ~ a... |
| hbn1 2142 | Alias for ~ ax-10 to be us... |
| hbe1 2143 | The setvar ` x ` is not fr... |
| hbe1a 2144 | Dual statement of ~ hbe1 .... |
| nf5-1 2145 | One direction of ~ nf5 can... |
| nf5i 2146 | Deduce that ` x ` is not f... |
| nf5dh 2147 | Deduce that ` x ` is not f... |
| nf5dv 2148 | Apply the definition of no... |
| nfnaew 2149 | All variables are effectiv... |
| nfe1 2150 | The setvar ` x ` is not fr... |
| nfa1 2151 | The setvar ` x ` is not fr... |
| nfna1 2152 | A convenience theorem part... |
| nfia1 2153 | Lemma 23 of [Monk2] p. 114... |
| nfnf1 2154 | The setvar ` x ` is not fr... |
| modal5 2155 | The analogue in our predic... |
| nfs1v 2156 | The setvar ` x ` is not fr... |
| alcoms 2158 | Swap quantifiers in an ant... |
| alcom 2159 | Theorem 19.5 of [Margaris]... |
| alrot3 2160 | Theorem *11.21 in [Whitehe... |
| alrot4 2161 | Rotate four universal quan... |
| sbal 2162 | Move universal quantifier ... |
| sbalv 2163 | Quantify with new variable... |
| sbcom2 2164 | Commutativity law for subs... |
| excom 2165 | Theorem 19.11 of [Margaris... |
| excomim 2166 | One direction of Theorem 1... |
| excom13 2167 | Swap 1st and 3rd existenti... |
| exrot3 2168 | Rotate existential quantif... |
| exrot4 2169 | Rotate existential quantif... |
| hbal 2170 | If ` x ` is not free in ` ... |
| hbald 2171 | Deduction form of bound-va... |
| 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... |
| exlimddOLD 2217 | Obsolete version of ~ exli... |
| exlimimddOLD 2218 | Obsolete version of ~ exli... |
| nexd 2219 | Deduction for generalizati... |
| albid 2220 | Formula-building rule for ... |
| exbid 2221 | Formula-building rule for ... |
| nfbidf 2222 | An equality theorem for ef... |
| 19.16 2223 | Theorem 19.16 of [Margaris... |
| 19.17 2224 | Theorem 19.17 of [Margaris... |
| 19.27 2225 | Theorem 19.27 of [Margaris... |
| 19.28 2226 | Theorem 19.28 of [Margaris... |
| 19.19 2227 | Theorem 19.19 of [Margaris... |
| 19.36 2228 | Theorem 19.36 of [Margaris... |
| 19.36i 2229 | Inference associated with ... |
| 19.37 2230 | Theorem 19.37 of [Margaris... |
| 19.32 2231 | Theorem 19.32 of [Margaris... |
| 19.31 2232 | Theorem 19.31 of [Margaris... |
| 19.41 2233 | Theorem 19.41 of [Margaris... |
| 19.42 2234 | Theorem 19.42 of [Margaris... |
| 19.44 2235 | Theorem 19.44 of [Margaris... |
| 19.45 2236 | Theorem 19.45 of [Margaris... |
| spimfv 2237 | Specialization, using impl... |
| chvarfv 2238 | Implicit substitution of `... |
| cbv3v2 2239 | Version of ~ cbv3 with two... |
| sb4av 2240 | Version of ~ sb4a with a d... |
| sbimd 2241 | Deduction substituting bot... |
| sbbid 2242 | Deduction substituting bot... |
| 2sbbid 2243 | Deduction doubly substitut... |
| sbbidOLD 2244 | Obsolete version of ~ sbbi... |
| sbequ1 2245 | An equality theorem for su... |
| sbequ2 2246 | An equality theorem for su... |
| sbequ2OLD 2247 | Obsolete version of ~ sbeq... |
| stdpc7 2248 | One of the two equality ax... |
| sbequ12 2249 | An equality theorem for su... |
| sbequ12r 2250 | An equality theorem for su... |
| sbelx 2251 | Elimination of substitutio... |
| sbequ12a 2252 | An equality theorem for su... |
| sbid 2253 | An identity theorem for su... |
| sbcov 2254 | A composition law for subs... |
| sb6a 2255 | Equivalence for substituti... |
| sbid2vw 2256 | Reverting substitution yie... |
| axc16g 2257 | Generalization of ~ axc16 ... |
| axc16 2258 | Proof of older axiom ~ ax-... |
| axc16gb 2259 | Biconditional strengthenin... |
| axc16nf 2260 | If ~ dtru is false, then t... |
| axc11v 2261 | Version of ~ axc11 with a ... |
| axc11rv 2262 | Version of ~ axc11r with a... |
| drsb2 2263 | Formula-building lemma for... |
| equsalv 2264 | An equivalence related to ... |
| equsexv 2265 | An equivalence related to ... |
| sbft 2266 | Substitution has no effect... |
| sbf 2267 | Substitution for a variabl... |
| sbf2 2268 | Substitution has no effect... |
| sbh 2269 | Substitution for a variabl... |
| hbs1 2270 | The setvar ` x ` is not fr... |
| nfs1f 2271 | If ` x ` is not free in ` ... |
| sb5 2272 | Alternate definition of su... |
| sb56 2273 | Two equivalent ways of exp... |
| sb56OLD 2274 | Obsolete version of ~ sb56... |
| equs5av 2275 | A property related to subs... |
| sb6OLD 2276 | Obsolete version of ~ sb6 ... |
| sb5OLD 2277 | Obsolete version of ~ sb5 ... |
| 2sb5 2278 | Equivalence for double sub... |
| sbco4lem 2279 | Lemma for ~ sbco4 . It re... |
| sbco4 2280 | Two ways of exchanging two... |
| dfsb7 2281 | An alternate definition of... |
| dfsb7OLD 2282 | Obsolete version of ~ dfsb... |
| sbn 2283 | Negation inside and outsid... |
| sbex 2284 | Move existential quantifie... |
| sbbibOLD 2285 | Obsolete version of ~ sbbi... |
| nf5 2286 | Alternate definition of ~ ... |
| nf6 2287 | An alternate definition of... |
| nf5d 2288 | Deduce that ` x ` is not f... |
| nf5di 2289 | Since the converse holds b... |
| 19.9h 2290 | A wff may be existentially... |
| 19.21h 2291 | Theorem 19.21 of [Margaris... |
| 19.23h 2292 | Theorem 19.23 of [Margaris... |
| exlimih 2293 | Inference associated with ... |
| exlimdh 2294 | Deduction form of Theorem ... |
| equsalhw 2295 | Version of ~ equsalh with ... |
| equsexhv 2296 | An equivalence related to ... |
| hba1 2297 | The setvar ` x ` is not fr... |
| hbnt 2298 | Closed theorem version of ... |
| hbn 2299 | If ` x ` is not free in ` ... |
| hbnd 2300 | Deduction form of bound-va... |
| hbim1 2301 | A closed form of ~ hbim . ... |
| hbimd 2302 | Deduction form of bound-va... |
| hbim 2303 | If ` x ` is not free in ` ... |
| hban 2304 | If ` x ` is not free in ` ... |
| hb3an 2305 | If ` x ` is not free in ` ... |
| sbi2 2306 | Introduction of implicatio... |
| sbim 2307 | Implication inside and out... |
| sbanOLD 2308 | Obsolete version of ~ sban... |
| sbrim 2309 | Substitution in an implica... |
| sbrimv 2310 | Substitution in an implica... |
| sblim 2311 | Substitution in an implica... |
| sbor 2312 | Disjunction inside and out... |
| sbbi 2313 | Equivalence inside and out... |
| spsbbiOLD 2314 | Obsolete version of ~ spsb... |
| sblbis 2315 | Introduce left bicondition... |
| sbrbis 2316 | Introduce right biconditio... |
| sbrbif 2317 | Introduce right biconditio... |
| sbnvOLD 2318 | Obsolete version of ~ sbn ... |
| sbi1vOLD 2319 | Obsolete version of ~ sbi1... |
| sbi2vOLD 2320 | Obsolete version of ~ sbi2... |
| sbimvOLD 2321 | Obsolete version of ~ sbim... |
| sbanvOLD 2322 | Obsolete version of ~ sban... |
| sbbivOLD 2323 | Obsolete version of ~ sbbi... |
| spsbimvOLD 2324 | Obsolete version of ~ spsb... |
| sblbisvOLD 2325 | Obsolete version of ~ sblb... |
| sbiev 2326 | Conversion of implicit sub... |
| sbievOLD 2327 | Obsolete proof of ~ sbiev ... |
| sbiedw 2328 | Conversion of implicit sub... |
| sbiedwOLD 2329 | Obsolete version of ~ sbie... |
| sbequivvOLD 2330 | Obsolete version of ~ sbeq... |
| sbequvvOLD 2331 | Obsolete version of ~ sbeq... |
| axc7 2332 | Show that the original axi... |
| axc7e 2333 | Abbreviated version of ~ a... |
| modal-b 2334 | The analogue in our predic... |
| 19.9ht 2335 | A closed version of ~ 19.9... |
| axc4 2336 | Show that the original axi... |
| axc4i 2337 | Inference version of ~ axc... |
| nfal 2338 | If ` x ` is not free in ` ... |
| nfex 2339 | If ` x ` is not free in ` ... |
| hbex 2340 | If ` x ` is not free in ` ... |
| nfnf 2341 | If ` x ` is not free in ` ... |
| 19.12 2342 | Theorem 19.12 of [Margaris... |
| nfald 2343 | Deduction form of ~ nfal .... |
| nfexd 2344 | If ` x ` is not free in ` ... |
| nfsbv 2345 | If ` z ` is not free in ` ... |
| nfsbvOLD 2346 | Obsolete version of ~ nfsb... |
| hbsbw 2347 | If ` z ` is not free in ` ... |
| sbco2v 2348 | A composition law for subs... |
| aaan 2349 | Rearrange universal quanti... |
| eeor 2350 | Rearrange existential quan... |
| cbv3v 2351 | Rule used to change bound ... |
| cbv1v 2352 | Rule used to change bound ... |
| cbv2w 2353 | Rule used to change bound ... |
| cbvaldw 2354 | Deduction used to change b... |
| cbvexdw 2355 | Deduction used to change b... |
| cbv3hv 2356 | Rule used to change bound ... |
| cbvalv1 2357 | Rule used to change bound ... |
| cbvexv1 2358 | Rule used to change bound ... |
| cbval2v 2359 | Rule used to change bound ... |
| cbval2vOLD 2360 | Obsolete version of ~ cbva... |
| cbvex2v 2361 | Rule used to change bound ... |
| dvelimhw 2362 | Proof of ~ dvelimh without... |
| pm11.53 2363 | Theorem *11.53 in [Whitehe... |
| 19.12vv 2364 | Special case of ~ 19.12 wh... |
| eean 2365 | Rearrange existential quan... |
| eeanv 2366 | Distribute a pair of exist... |
| eeeanv 2367 | Distribute three existenti... |
| ee4anv 2368 | Distribute two pairs of ex... |
| sb8v 2369 | Substitution of variable i... |
| sb8ev 2370 | Substitution of variable i... |
| 2sb8ev 2371 | An equivalent expression f... |
| sb6rfv 2372 | Reversed substitution. Ve... |
| sbnf2 2373 | Two ways of expressing " `... |
| exsb 2374 | An equivalent expression f... |
| 2exsb 2375 | An equivalent expression f... |
| sbbib 2376 | Reversal of substitution. ... |
| sbbibvv 2377 | Reversal of substitution. ... |
| cleljustALT 2378 | Alternate proof of ~ clelj... |
| cleljustALT2 2379 | Alternate proof of ~ clelj... |
| equs5aALT 2380 | Alternate proof of ~ equs5... |
| equs5eALT 2381 | Alternate proof of ~ equs5... |
| axc11r 2382 | Same as ~ axc11 but with r... |
| dral1v 2383 | Formula-building lemma for... |
| drex1v 2384 | Formula-building lemma for... |
| drnf1v 2385 | Formula-building lemma for... |
| ax13v 2387 | A weaker version of ~ ax-1... |
| ax13lem1 2388 | A version of ~ ax13v with ... |
| ax13 2389 | Derive ~ ax-13 from ~ ax13... |
| ax13lem2 2390 | Lemma for ~ nfeqf2 . This... |
| nfeqf2 2391 | An equation between setvar... |
| dveeq2 2392 | Quantifier introduction wh... |
| nfeqf1 2393 | An equation between setvar... |
| dveeq1 2394 | Quantifier introduction wh... |
| nfeqf 2395 | A variable is effectively ... |
| axc9 2396 | Derive set.mm's original ~... |
| ax6e 2397 | At least one individual ex... |
| ax6 2398 | Theorem showing that ~ ax-... |
| axc10 2399 | Show that the original axi... |
| spimt 2400 | Closed theorem form of ~ s... |
| spim 2401 | Specialization, using impl... |
| spimed 2402 | Deduction version of ~ spi... |
| spime 2403 | Existential introduction, ... |
| spimv 2404 | A version of ~ spim with a... |
| spimvALT 2405 | Alternate proof of ~ spimv... |
| spimev 2406 | Distinct-variable version ... |
| spv 2407 | Specialization, using impl... |
| spei 2408 | Inference from existential... |
| chvar 2409 | Implicit substitution of `... |
| chvarv 2410 | Implicit substitution of `... |
| cbv3 2411 | Rule used to change bound ... |
| cbval 2412 | Rule used to change bound ... |
| cbvex 2413 | Rule used to change bound ... |
| cbvalv 2414 | Rule used to change bound ... |
| cbvexv 2415 | Rule used to change bound ... |
| cbvalvOLD 2416 | Obsolete version of ~ cbva... |
| cbvexvOLD 2417 | Obsolete version of ~ cbve... |
| cbv1 2418 | Rule used to change bound ... |
| cbv2 2419 | Rule used to change bound ... |
| cbv3h 2420 | Rule used to change bound ... |
| cbv1h 2421 | Rule used to change bound ... |
| cbv2h 2422 | Rule used to change bound ... |
| cbv2OLD 2423 | Obsolete version of ~ cbv2... |
| cbvald 2424 | Deduction used to change b... |
| cbvexd 2425 | Deduction used to change b... |
| cbvaldva 2426 | Rule used to change the bo... |
| cbvexdva 2427 | Rule used to change the bo... |
| cbval2 2428 | Rule used to change bound ... |
| cbval2OLD 2429 | Obsolete version of ~ cbva... |
| cbvex2 2430 | Rule used to change bound ... |
| cbval2vv 2431 | Rule used to change bound ... |
| cbvex2vv 2432 | Rule used to change bound ... |
| cbvex4v 2433 | Rule used to change bound ... |
| equs4 2434 | Lemma used in proofs of im... |
| equsal 2435 | An equivalence related to ... |
| equsex 2436 | An equivalence related to ... |
| equsexALT 2437 | Alternate proof of ~ equse... |
| equsalh 2438 | An equivalence related to ... |
| equsexh 2439 | An equivalence related to ... |
| axc15 2440 | Derivation of set.mm's ori... |
| ax12 2441 | Rederivation of axiom ~ ax... |
| ax12b 2442 | A bidirectional version of... |
| ax13ALT 2443 | Alternate proof of ~ ax13 ... |
| axc11n 2444 | Derive set.mm's original ~... |
| aecom 2445 | Commutation law for identi... |
| aecoms 2446 | A commutation rule for ide... |
| naecoms 2447 | A commutation rule for dis... |
| axc11 2448 | Show that ~ ax-c11 can be ... |
| hbae 2449 | All variables are effectiv... |
| hbnae 2450 | All variables are effectiv... |
| nfae 2451 | All variables are effectiv... |
| nfnae 2452 | All variables are effectiv... |
| hbnaes 2453 | Rule that applies ~ hbnae ... |
| axc16i 2454 | Inference with ~ axc16 as ... |
| axc16nfALT 2455 | Alternate proof of ~ axc16... |
| dral2 2456 | Formula-building lemma for... |
| dral1 2457 | Formula-building lemma for... |
| dral1ALT 2458 | Alternate proof of ~ dral1... |
| drex1 2459 | Formula-building lemma for... |
| drex2 2460 | Formula-building lemma for... |
| drnf1 2461 | Formula-building lemma for... |
| drnf2 2462 | Formula-building lemma for... |
| nfald2 2463 | Variation on ~ nfald which... |
| nfexd2 2464 | Variation on ~ nfexd which... |
| exdistrf 2465 | Distribution of existentia... |
| dvelimf 2466 | Version of ~ dvelimv witho... |
| dvelimdf 2467 | Deduction form of ~ dvelim... |
| dvelimh 2468 | Version of ~ dvelim withou... |
| dvelim 2469 | This theorem can be used t... |
| dvelimv 2470 | Similar to ~ dvelim with f... |
| dvelimnf 2471 | Version of ~ dvelim using ... |
| dveeq2ALT 2472 | Alternate proof of ~ dveeq... |
| equvini 2473 | A variable introduction la... |
| equviniOLD 2474 | Obsolete version of ~ equv... |
| equvel 2475 | A variable elimination law... |
| equs5a 2476 | A property related to subs... |
| equs5e 2477 | A property related to subs... |
| equs45f 2478 | Two ways of expressing sub... |
| equs5 2479 | Lemma used in proofs of su... |
| dveel1 2480 | Quantifier introduction wh... |
| dveel2 2481 | Quantifier introduction wh... |
| axc14 2482 | Axiom ~ ax-c14 is redundan... |
| sb6x 2483 | Equivalence involving subs... |
| sbequ5 2484 | Substitution does not chan... |
| sbequ6 2485 | Substitution does not chan... |
| sb5rf 2486 | Reversed substitution. Us... |
| sb6rf 2487 | Reversed substitution. Fo... |
| ax12vALT 2488 | Alternate proof of ~ ax12v... |
| 2ax6elem 2489 | We can always find values ... |
| 2ax6e 2490 | We can always find values ... |
| 2ax6eOLD 2491 | Obsolete version of ~ 2ax6... |
| 2sb5rf 2492 | Reversed double substituti... |
| 2sb6rf 2493 | Reversed double substituti... |
| sbel2x 2494 | Elimination of double subs... |
| sb4b 2495 | Simplified definition of s... |
| sb4bOLD 2496 | Obsolete version of ~ sb4b... |
| sb3b 2497 | Simplified definition of s... |
| sb3 2498 | One direction of a simplif... |
| sb1 2499 | One direction of a simplif... |
| sb2 2500 | One direction of a simplif... |
| sb3OLD 2501 | Obsolete version of ~ sb3 ... |
| sb4OLD 2502 | Obsolete as of 30-Jul-2023... |
| sb1OLD 2503 | Obsolete version of ~ sb1 ... |
| sb3bOLD 2504 | Obsolete version of ~ sb3b... |
| sb4a 2505 | A version of one implicati... |
| dfsb1 2506 | Alternate definition of su... |
| spsbeOLDOLD 2507 | Obsolete version of ~ spsb... |
| sb2vOLDOLD 2508 | Obsolete version of ~ sb2 ... |
| sb4vOLDOLD 2509 | Obsolete version of ~ sb4v... |
| sbequ2OLDOLD 2510 | Obsolete version of ~ sbeq... |
| sbimiOLD 2511 | Obsolete version of ~ sbim... |
| sbimdvOLD 2512 | Obsolete version of ~ sbim... |
| equsb1vOLDOLD 2513 | Obsolete version of ~ equs... |
| sbimdOLD 2514 | Obsolete version of sbimd ... |
| sbtvOLD 2515 | Obsolete version of ~ sbt ... |
| sbequ1OLD 2516 | Obsolete version of ~ sbeq... |
| hbsb2 2517 | Bound-variable hypothesis ... |
| nfsb2 2518 | Bound-variable hypothesis ... |
| hbsb2a 2519 | Special case of a bound-va... |
| sb4e 2520 | One direction of a simplif... |
| hbsb2e 2521 | Special case of a bound-va... |
| hbsb3 2522 | If ` y ` is not free in ` ... |
| nfs1 2523 | If ` y ` is not free in ` ... |
| axc16ALT 2524 | Alternate proof of ~ axc16... |
| axc16gALT 2525 | Alternate proof of ~ axc16... |
| equsb1 2526 | Substitution applied to an... |
| equsb2 2527 | Substitution applied to an... |
| dfsb2 2528 | An alternate definition of... |
| dfsb3 2529 | An alternate definition of... |
| sbequiOLD 2530 | Obsolete proof of ~ sbequi... |
| drsb1 2531 | Formula-building lemma for... |
| sb2ae 2532 | In the case of two success... |
| sb6f 2533 | Equivalence for substituti... |
| sb5f 2534 | Equivalence for substituti... |
| nfsb4t 2535 | A variable not free in a p... |
| nfsb4 2536 | A variable not free in a p... |
| sbnOLD 2537 | Obsolete version of ~ sbn ... |
| sbi1OLD 2538 | Obsolete version of ~ sbi1... |
| sbequ8 2539 | Elimination of equality fr... |
| sbie 2540 | Conversion of implicit sub... |
| sbied 2541 | Conversion of implicit sub... |
| sbiedv 2542 | Conversion of implicit sub... |
| 2sbiev 2543 | Conversion of double impli... |
| sbcom3 2544 | Substituting ` y ` for ` x... |
| sbco 2545 | A composition law for subs... |
| sbid2 2546 | An identity law for substi... |
| sbid2v 2547 | An identity law for substi... |
| sbidm 2548 | An idempotent law for subs... |
| sbco2 2549 | A composition law for subs... |
| sbco2d 2550 | A composition law for subs... |
| sbco3 2551 | A composition law for subs... |
| sbcom 2552 | A commutativity law for su... |
| sbtrt 2553 | Partially closed form of ~... |
| sbtr 2554 | A partial converse to ~ sb... |
| sb8 2555 | Substitution of variable i... |
| sb8e 2556 | Substitution of variable i... |
| sb9 2557 | Commutation of quantificat... |
| sb9i 2558 | Commutation of quantificat... |
| sbhb 2559 | Two ways of expressing " `... |
| nfsbd 2560 | Deduction version of ~ nfs... |
| nfsb 2561 | If ` z ` is not free in ` ... |
| nfsbOLD 2562 | Obsolete version of ~ nfsb... |
| hbsb 2563 | If ` z ` is not free in ` ... |
| sb7f 2564 | This version of ~ dfsb7 do... |
| sb7h 2565 | This version of ~ dfsb7 do... |
| dfsb7OLDOLD 2566 | Obsolete version of ~ dfsb... |
| sb10f 2567 | Hao Wang's identity axiom ... |
| sbal1 2568 | Obsolete version of ~ sbal... |
| sbal2 2569 | Move quantifier in and out... |
| sbal2OLD 2570 | Obsolete version of ~ sbal... |
| sbalOLD 2571 | Obsolete version of ~ sbal... |
| 2sb8e 2572 | An equivalent expression f... |
| sbimiALT 2573 | Alternate version of ~ sbi... |
| sbbiiALT 2574 | Alternate version of ~ sbb... |
| sbequ1ALT 2575 | Alternate version of ~ sbe... |
| sbequ2ALT 2576 | Alternate version of ~ sbe... |
| sbequ12ALT 2577 | Alternate version of ~ sbe... |
| sb1ALT 2578 | Alternate version of ~ sb1... |
| sb2vOLDALT 2579 | Alternate version of ~ sb2... |
| sb4vOLDALT 2580 | Alternate version of ~ sb4... |
| sb6ALT 2581 | Alternate version of ~ sb6... |
| sb5ALT2 2582 | Alternate version of ~ sb5... |
| sb2ALT 2583 | Alternate version of ~ sb2... |
| sb4ALT 2584 | Alternate version of one i... |
| spsbeALT 2585 | Alternate version of ~ sps... |
| stdpc4ALT 2586 | Alternate version of ~ std... |
| dfsb2ALT 2587 | Alternate version of ~ dfs... |
| dfsb3ALT 2588 | Alternate version of ~ dfs... |
| sbftALT 2589 | Alternate version of ~ sbf... |
| sbfALT 2590 | Alternate version of ~ sbf... |
| sbnALT 2591 | Alternate version of ~ sbn... |
| sbequiALT 2592 | Alternate version of ~ sbe... |
| sbequALT 2593 | Alternate version of ~ sbe... |
| sb4aALT 2594 | Alternate version of ~ sb4... |
| hbsb2ALT 2595 | Alternate version of ~ hbs... |
| nfsb2ALT 2596 | Alternate version of ~ nfs... |
| equsb1ALT 2597 | Alternate version of ~ equ... |
| sb6fALT 2598 | Alternate version of ~ sb6... |
| sb5fALT 2599 | Alternate version of ~ sb5... |
| nfsb4tALT 2600 | Alternate version of ~ nfs... |
| nfsb4ALT 2601 | Alternate version of ~ nfs... |
| sbi1ALT 2602 | Alternate version of ~ sbi... |
| sbi2ALT 2603 | Alternate version of ~ sbi... |
| sbimALT 2604 | Alternate version of ~ sbi... |
| sbrimALT 2605 | Alternate version of ~ sbr... |
| sbanALT 2606 | Alternate version of ~ sba... |
| sbbiALT 2607 | Alternate version of ~ sbb... |
| sblbisALT 2608 | Alternate version of ~ sbl... |
| sbieALT 2609 | Alternate version of ~ sbi... |
| sbiedALT 2610 | Alternate version of ~ sbi... |
| sbco2ALT 2611 | Alternate version of ~ sbc... |
| sb7fALT 2612 | Alternate version of ~ sb7... |
| dfsb7ALT 2613 | Alternate version of ~ dfs... |
| dfmoeu 2614 | An elementary proof of ~ m... |
| dfeumo 2615 | An elementary proof showin... |
| mojust 2617 | Soundness justification th... |
| nexmo 2619 | Nonexistence implies uniqu... |
| exmo 2620 | Any proposition holds for ... |
| moabs 2621 | Absorption of existence co... |
| moim 2622 | The at-most-one quantifier... |
| moimi 2623 | The at-most-one quantifier... |
| moimiOLD 2624 | Obsolete version of ~ moim... |
| moimdv 2625 | The at-most-one quantifier... |
| mobi 2626 | Equivalence theorem for th... |
| mobii 2627 | Formula-building rule for ... |
| mobiiOLD 2628 | Obsolete version of ~ mobi... |
| mobidv 2629 | Formula-building rule for ... |
| mobid 2630 | Formula-building rule for ... |
| moa1 2631 | If an implication holds fo... |
| moan 2632 | "At most one" is still the... |
| moani 2633 | "At most one" is still tru... |
| moor 2634 | "At most one" is still the... |
| mooran1 2635 | "At most one" imports disj... |
| mooran2 2636 | "At most one" exports disj... |
| nfmo1 2637 | Bound-variable hypothesis ... |
| nfmod2 2638 | Bound-variable hypothesis ... |
| nfmodv 2639 | Bound-variable hypothesis ... |
| nfmov 2640 | Bound-variable hypothesis ... |
| nfmod 2641 | Bound-variable hypothesis ... |
| nfmo 2642 | Bound-variable hypothesis ... |
| mof 2643 | Version of ~ df-mo with di... |
| mo3 2644 | Alternate definition of th... |
| mo 2645 | Equivalent definitions of ... |
| mo4 2646 | At-most-one quantifier exp... |
| mo4f 2647 | At-most-one quantifier exp... |
| mo4OLD 2648 | Obsolete version of ~ mo4 ... |
| eu3v 2651 | An alternate way to expres... |
| eujust 2652 | Soundness justification th... |
| eujustALT 2653 | Alternate proof of ~ eujus... |
| eu6lem 2654 | Lemma of ~ eu6im . A diss... |
| eu6 2655 | Alternate definition of th... |
| eu6im 2656 | One direction of ~ eu6 nee... |
| euf 2657 | Version of ~ eu6 with disj... |
| euex 2658 | Existential uniqueness imp... |
| eumo 2659 | Existential uniqueness imp... |
| eumoi 2660 | Uniqueness inferred from e... |
| exmoeub 2661 | Existence implies that uni... |
| exmoeu 2662 | Existence is equivalent to... |
| moeuex 2663 | Uniqueness implies that ex... |
| moeu 2664 | Uniqueness is equivalent t... |
| eubi 2665 | Equivalence theorem for th... |
| eubii 2666 | Introduce unique existenti... |
| eubiiOLD 2667 | Obsolete version of ~ eubi... |
| eubidv 2668 | Formula-building rule for ... |
| eubid 2669 | Formula-building rule for ... |
| nfeu1 2670 | Bound-variable hypothesis ... |
| nfeu1ALT 2671 | Alternate proof of ~ nfeu1... |
| nfeud2 2672 | Bound-variable hypothesis ... |
| nfeudw 2673 | Bound-variable hypothesis ... |
| nfeud 2674 | Bound-variable hypothesis ... |
| nfeuw 2675 | Bound-variable hypothesis ... |
| nfeu 2676 | Bound-variable hypothesis ... |
| dfeu 2677 | Rederive ~ df-eu from the ... |
| dfmo 2678 | Rederive ~ df-mo from the ... |
| euequ 2679 | There exists a unique set ... |
| sb8eulem 2680 | Lemma. Factor out the com... |
| sb8euv 2681 | Variable substitution in u... |
| sb8eu 2682 | Variable substitution in u... |
| sb8mo 2683 | Variable substitution for ... |
| cbvmow 2684 | Rule used to change bound ... |
| cbvmo 2685 | Rule used to change bound ... |
| cbveuw 2686 | Version of ~ cbveu with a ... |
| cbveu 2687 | Rule used to change bound ... |
| cbveuALT 2688 | Alternative proof of ~ cbv... |
| eu2 2689 | An alternate way of defini... |
| eu1 2690 | An alternate way to expres... |
| euor 2691 | Introduce a disjunct into ... |
| euorv 2692 | Introduce a disjunct into ... |
| euor2 2693 | Introduce or eliminate a d... |
| sbmo 2694 | Substitution into an at-mo... |
| eu4 2695 | Uniqueness using implicit ... |
| euimmo 2696 | Existential uniqueness imp... |
| euim 2697 | Add unique existential qua... |
| euimOLD 2698 | Obsolete version of ~ euim... |
| moanimlem 2699 | Factor out the common proo... |
| moanimv 2700 | Introduction of a conjunct... |
| moanim 2701 | Introduction of a conjunct... |
| euan 2702 | Introduction of a conjunct... |
| moanmo 2703 | Nested at-most-one quantif... |
| moaneu 2704 | Nested at-most-one and uni... |
| euanv 2705 | Introduction of a conjunct... |
| mopick 2706 | "At most one" picks a vari... |
| moexexlem 2707 | Factor out the proof skele... |
| 2moexv 2708 | Double quantification with... |
| moexexvw 2709 | "At most one" double quant... |
| 2moswapv 2710 | A condition allowing to sw... |
| 2euswapv 2711 | A condition allowing to sw... |
| 2euexv 2712 | Double quantification with... |
| 2exeuv 2713 | Double existential uniquen... |
| eupick 2714 | Existential uniqueness "pi... |
| eupicka 2715 | Version of ~ eupick with c... |
| eupickb 2716 | Existential uniqueness "pi... |
| eupickbi 2717 | Theorem *14.26 in [Whitehe... |
| mopick2 2718 | "At most one" can show the... |
| moexex 2719 | "At most one" double quant... |
| moexexv 2720 | "At most one" double quant... |
| 2moex 2721 | Double quantification with... |
| 2euex 2722 | Double quantification with... |
| 2eumo 2723 | Nested unique existential ... |
| 2eu2ex 2724 | Double existential uniquen... |
| 2moswap 2725 | A condition allowing to sw... |
| 2euswap 2726 | A condition allowing to sw... |
| 2exeu 2727 | Double existential uniquen... |
| 2mo2 2728 | Two ways of expressing "th... |
| 2mo 2729 | Two ways of expressing "th... |
| 2mos 2730 | Double "exists at most one... |
| 2eu1 2731 | Double existential uniquen... |
| 2eu1OLD 2732 | Obsolete version of ~ 2eu1... |
| 2eu1v 2733 | Double existential uniquen... |
| 2eu2 2734 | Double existential uniquen... |
| 2eu3 2735 | Double existential uniquen... |
| 2eu3OLD 2736 | Obsolete version of ~ 2eu3... |
| 2eu4 2737 | This theorem provides us w... |
| 2eu5 2738 | An alternate definition of... |
| 2eu5OLD 2739 | Obsolete version of ~ 2eu5... |
| 2eu6 2740 | Two equivalent expressions... |
| 2eu7 2741 | Two equivalent expressions... |
| 2eu8 2742 | Two equivalent expressions... |
| euae 2743 | Two ways to express "exact... |
| exists1 2744 | Two ways to express "exact... |
| exists2 2745 | A condition implying that ... |
| barbara 2746 | "Barbara", one of the fund... |
| celarent 2747 | "Celarent", one of the syl... |
| darii 2748 | "Darii", one of the syllog... |
| dariiALT 2749 | Alternate proof of ~ darii... |
| ferio 2750 | "Ferio" ("Ferioque"), one ... |
| barbarilem 2751 | Lemma for ~ barbari and th... |
| barbari 2752 | "Barbari", one of the syll... |
| barbariALT 2753 | Alternate proof of ~ barba... |
| celaront 2754 | "Celaront", one of the syl... |
| cesare 2755 | "Cesare", one of the syllo... |
| camestres 2756 | "Camestres", one of the sy... |
| festino 2757 | "Festino", one of the syll... |
| festinoALT 2758 | Alternate proof of ~ festi... |
| baroco 2759 | "Baroco", one of the syllo... |
| barocoALT 2760 | Alternate proof of ~ festi... |
| cesaro 2761 | "Cesaro", one of the syllo... |
| camestros 2762 | "Camestros", one of the sy... |
| datisi 2763 | "Datisi", one of the syllo... |
| disamis 2764 | "Disamis", one of the syll... |
| ferison 2765 | "Ferison", one of the syll... |
| bocardo 2766 | "Bocardo", one of the syll... |
| darapti 2767 | "Darapti", one of the syll... |
| daraptiALT 2768 | Alternate proof of ~ darap... |
| felapton 2769 | "Felapton", one of the syl... |
| calemes 2770 | "Calemes", one of the syll... |
| dimatis 2771 | "Dimatis", one of the syll... |
| fresison 2772 | "Fresison", one of the syl... |
| calemos 2773 | "Calemos", one of the syll... |
| fesapo 2774 | "Fesapo", one of the syllo... |
| bamalip 2775 | "Bamalip", one of the syll... |
| axia1 2776 | Left 'and' elimination (in... |
| axia2 2777 | Right 'and' elimination (i... |
| axia3 2778 | 'And' introduction (intuit... |
| axin1 2779 | 'Not' introduction (intuit... |
| axin2 2780 | 'Not' elimination (intuiti... |
| axio 2781 | Definition of 'or' (intuit... |
| axi4 2782 | Specialization (intuitioni... |
| axi5r 2783 | Converse of ~ axc4 (intuit... |
| axial 2784 | The setvar ` x ` is not fr... |
| axie1 2785 | The setvar ` x ` is not fr... |
| axie2 2786 | A key property of existent... |
| axi9 2787 | Axiom of existence (intuit... |
| axi10 2788 | Axiom of Quantifier Substi... |
| axi12 2789 | Axiom of Quantifier Introd... |
| axi12OLD 2790 | Obsolete version of ~ axi1... |
| axbnd 2791 | Axiom of Bundling (intuiti... |
| axbndOLD 2792 | Obsolete version of ~ axbn... |
| axexte 2794 | The axiom of extensionalit... |
| axextg 2795 | A generalization of the ax... |
| axextb 2796 | A bidirectional version of... |
| axextmo 2797 | There exists at most one s... |
| nulmo 2798 | There exists at most one e... |
| eleq1ab 2801 | Extension (in the sense of... |
| cleljustab 2802 | Extension of ~ cleljust fr... |
| abid 2803 | Simplification of class ab... |
| vexwt 2804 | A standard theorem of pred... |
| vexw 2805 | If ` ph ` is a theorem, th... |
| vextru 2806 | Every setvar is a member o... |
| hbab1 2807 | Bound-variable hypothesis ... |
| nfsab1 2808 | Bound-variable hypothesis ... |
| nfsab1OLD 2809 | Obsolete version of ~ nfsa... |
| hbab 2810 | Bound-variable hypothesis ... |
| hbabg 2811 | Bound-variable hypothesis ... |
| nfsab 2812 | Bound-variable hypothesis ... |
| nfsabg 2813 | Bound-variable hypothesis ... |
| dfcleq 2815 | The defining characterizat... |
| cvjust 2816 | Every set is a class. Pro... |
| ax9ALT 2817 | Proof of ~ ax-9 from Tarsk... |
| eqriv 2818 | Infer equality of classes ... |
| eqrdv 2819 | Deduce equality of classes... |
| eqrdav 2820 | Deduce equality of classes... |
| eqid 2821 | Law of identity (reflexivi... |
| eqidd 2822 | Class identity law with an... |
| eqeq1d 2823 | Deduction from equality to... |
| eqeq1dALT 2824 | Shorter proof of ~ eqeq1d ... |
| eqeq1 2825 | Equality implies equivalen... |
| eqeq1i 2826 | Inference from equality to... |
| eqcomd 2827 | Deduction from commutative... |
| eqcom 2828 | Commutative law for class ... |
| eqcoms 2829 | Inference applying commuta... |
| eqcomi 2830 | Inference from commutative... |
| neqcomd 2831 | Commute an inequality. (C... |
| eqeq2d 2832 | Deduction from equality to... |
| eqeq2 2833 | Equality implies equivalen... |
| eqeq2i 2834 | Inference from equality to... |
| eqeq12 2835 | Equality relationship amon... |
| eqeq12i 2836 | A useful inference for sub... |
| eqeq12d 2837 | A useful inference for sub... |
| eqeqan12d 2838 | A useful inference for sub... |
| eqeqan12dALT 2839 | Alternate proof of ~ eqeqa... |
| eqeqan12rd 2840 | A useful inference for sub... |
| eqtr 2841 | Transitive law for class e... |
| eqtr2 2842 | A transitive law for class... |
| eqtr3 2843 | A transitive law for class... |
| eqtri 2844 | An equality transitivity i... |
| eqtr2i 2845 | An equality transitivity i... |
| eqtr3i 2846 | An equality transitivity i... |
| eqtr4i 2847 | An equality transitivity i... |
| 3eqtri 2848 | An inference from three ch... |
| 3eqtrri 2849 | An inference from three ch... |
| 3eqtr2i 2850 | An inference from three ch... |
| 3eqtr2ri 2851 | An inference from three ch... |
| 3eqtr3i 2852 | An inference from three ch... |
| 3eqtr3ri 2853 | An inference from three ch... |
| 3eqtr4i 2854 | An inference from three ch... |
| 3eqtr4ri 2855 | An inference from three ch... |
| eqtrd 2856 | An equality transitivity d... |
| eqtr2d 2857 | An equality transitivity d... |
| eqtr3d 2858 | An equality transitivity e... |
| eqtr4d 2859 | An equality transitivity e... |
| 3eqtrd 2860 | A deduction from three cha... |
| 3eqtrrd 2861 | A deduction from three cha... |
| 3eqtr2d 2862 | A deduction from three cha... |
| 3eqtr2rd 2863 | A deduction from three cha... |
| 3eqtr3d 2864 | A deduction from three cha... |
| 3eqtr3rd 2865 | A deduction from three cha... |
| 3eqtr4d 2866 | A deduction from three cha... |
| 3eqtr4rd 2867 | A deduction from three cha... |
| syl5eq 2868 | An equality transitivity d... |
| syl5req 2869 | An equality transitivity d... |
| syl5eqr 2870 | An equality transitivity d... |
| syl5reqr 2871 | An equality transitivity d... |
| syl6eq 2872 | An equality transitivity d... |
| syl6req 2873 | An equality transitivity d... |
| syl6eqr 2874 | An equality transitivity d... |
| syl6reqr 2875 | An equality transitivity d... |
| sylan9eq 2876 | An equality transitivity d... |
| sylan9req 2877 | An equality transitivity d... |
| sylan9eqr 2878 | An equality transitivity d... |
| 3eqtr3g 2879 | A chained equality inferen... |
| 3eqtr3a 2880 | A chained equality inferen... |
| 3eqtr4g 2881 | A chained equality inferen... |
| 3eqtr4a 2882 | A chained equality inferen... |
| eq2tri 2883 | A compound transitive infe... |
| abbi1 2884 | Equivalent formulas yield ... |
| abbidv 2885 | Equivalent wff's yield equ... |
| abbii 2886 | Equivalent wff's yield equ... |
| abbid 2887 | Equivalent wff's yield equ... |
| abbi 2888 | Equivalent formulas define... |
| cbvabv 2889 | Rule used to change bound ... |
| cbvabw 2890 | Rule used to change bound ... |
| cbvab 2891 | Rule used to change bound ... |
| cbvabvOLD 2892 | Obsolete version of ~ cbva... |
| dfclel 2894 | Characterization of the el... |
| eleq1w 2895 | Weaker version of ~ eleq1 ... |
| eleq2w 2896 | Weaker version of ~ eleq2 ... |
| eleq1d 2897 | Deduction from equality to... |
| eleq2d 2898 | Deduction from equality to... |
| eleq2dALT 2899 | Alternate proof of ~ eleq2... |
| eleq1 2900 | Equality implies equivalen... |
| eleq2 2901 | Equality implies equivalen... |
| eleq12 2902 | Equality implies equivalen... |
| eleq1i 2903 | Inference from equality to... |
| eleq2i 2904 | Inference from equality to... |
| eleq12i 2905 | Inference from equality to... |
| eqneltri 2906 | If a class is not an eleme... |
| eleq12d 2907 | Deduction from equality to... |
| eleq1a 2908 | A transitive-type law rela... |
| eqeltri 2909 | Substitution of equal clas... |
| eqeltrri 2910 | Substitution of equal clas... |
| eleqtri 2911 | Substitution of equal clas... |
| eleqtrri 2912 | Substitution of equal clas... |
| eqeltrd 2913 | Substitution of equal clas... |
| eqeltrrd 2914 | Deduction that substitutes... |
| eleqtrd 2915 | Deduction that substitutes... |
| eleqtrrd 2916 | Deduction that substitutes... |
| eqeltrid 2917 | A membership and equality ... |
| eqeltrrid 2918 | A membership and equality ... |
| eleqtrid 2919 | A membership and equality ... |
| eleqtrrid 2920 | A membership and equality ... |
| eqeltrdi 2921 | A membership and equality ... |
| eqeltrrdi 2922 | A membership and equality ... |
| eleqtrdi 2923 | A membership and equality ... |
| eleqtrrdi 2924 | A membership and equality ... |
| 3eltr3i 2925 | Substitution of equal clas... |
| 3eltr4i 2926 | Substitution of equal clas... |
| 3eltr3d 2927 | Substitution of equal clas... |
| 3eltr4d 2928 | Substitution of equal clas... |
| 3eltr3g 2929 | Substitution of equal clas... |
| 3eltr4g 2930 | Substitution of equal clas... |
| eleq2s 2931 | Substitution of equal clas... |
| eqneltrd 2932 | If a class is not an eleme... |
| eqneltrrd 2933 | If a class is not an eleme... |
| neleqtrd 2934 | If a class is not an eleme... |
| neleqtrrd 2935 | If a class is not an eleme... |
| cleqh 2936 | Establish equality between... |
| nelneq 2937 | A way of showing two class... |
| nelneq2 2938 | A way of showing two class... |
| eqsb3 2939 | Substitution applied to an... |
| clelsb3 2940 | Substitution applied to an... |
| clelsb3vOLD 2941 | Obsolete version of ~ clel... |
| hbxfreq 2942 | A utility lemma to transfe... |
| hblem 2943 | Change the free variable o... |
| hblemg 2944 | Change the free variable o... |
| abeq2 2945 | Equality of a class variab... |
| abeq1 2946 | Equality of a class variab... |
| abeq2d 2947 | Equality of a class variab... |
| abeq2i 2948 | Equality of a class variab... |
| abeq1i 2949 | Equality of a class variab... |
| abbi2dv 2950 | Deduction from a wff to a ... |
| abbi2dvOLD 2951 | Obsolete version of ~ abbi... |
| abbi1dv 2952 | Deduction from a wff to a ... |
| abbi2i 2953 | Equality of a class variab... |
| abbi2iOLD 2954 | Obsolete version of ~ abbi... |
| abbiOLD 2955 | Obsolete proof of ~ abbi a... |
| abid1 2956 | Every class is equal to a ... |
| abid2 2957 | A simplification of class ... |
| clelab 2958 | Membership of a class vari... |
| clabel 2959 | Membership of a class abst... |
| sbab 2960 | The right-hand side of the... |
| nfcjust 2962 | Justification theorem for ... |
| nfci 2964 | Deduce that a class ` A ` ... |
| nfcii 2965 | Deduce that a class ` A ` ... |
| nfcr 2966 | Consequence of the not-fre... |
| nfcriv 2967 | Consequence of the not-fre... |
| nfcd 2968 | Deduce that a class ` A ` ... |
| nfcrd 2969 | Consequence of the not-fre... |
| nfcrii 2970 | Consequence of the not-fre... |
| nfcri 2971 | Consequence of the not-fre... |
| nfceqdf 2972 | An equality theorem for ef... |
| nfceqi 2973 | Equality theorem for class... |
| nfceqiOLD 2974 | Obsolete proof of ~ nfceqi... |
| nfcxfr 2975 | A utility lemma to transfe... |
| nfcxfrd 2976 | A utility lemma to transfe... |
| nfcv 2977 | If ` x ` is disjoint from ... |
| nfcvd 2978 | If ` x ` is disjoint from ... |
| nfab1 2979 | Bound-variable hypothesis ... |
| nfnfc1 2980 | The setvar ` x ` is bound ... |
| clelsb3fw 2981 | Substitution applied to an... |
| clelsb3f 2982 | Substitution applied to an... |
| clelsb3fOLD 2983 | Obsolete version of ~ clel... |
| nfab 2984 | Bound-variable hypothesis ... |
| nfabg 2985 | Bound-variable hypothesis ... |
| nfaba1 2986 | Bound-variable hypothesis ... |
| nfaba1g 2987 | Bound-variable hypothesis ... |
| nfeqd 2988 | Hypothesis builder for equ... |
| nfeld 2989 | Hypothesis builder for ele... |
| nfnfc 2990 | Hypothesis builder for ` F... |
| nfeq 2991 | Hypothesis builder for equ... |
| nfel 2992 | Hypothesis builder for ele... |
| nfeq1 2993 | Hypothesis builder for equ... |
| nfel1 2994 | Hypothesis builder for ele... |
| nfeq2 2995 | Hypothesis builder for equ... |
| nfel2 2996 | Hypothesis builder for ele... |
| drnfc1 2997 | Formula-building lemma for... |
| drnfc1OLD 2998 | Obsolete version of ~ drnf... |
| drnfc2 2999 | Formula-building lemma for... |
| nfabdw 3000 | Bound-variable hypothesis ... |
| nfabd 3001 | Bound-variable hypothesis ... |
| nfabd2 3002 | Bound-variable hypothesis ... |
| nfabd2OLD 3003 | Obsolete version of ~ nfab... |
| nfabdOLD 3004 | Obsolete version of ~ nfab... |
| dvelimdc 3005 | Deduction form of ~ dvelim... |
| dvelimc 3006 | Version of ~ dvelim for cl... |
| nfcvf 3007 | If ` x ` and ` y ` are dis... |
| nfcvf2 3008 | If ` x ` and ` y ` are dis... |
| nfcvfOLD 3009 | Obsolete version of ~ nfcv... |
| cleqf 3010 | Establish equality between... |
| cleqfOLD 3011 | Obsolete version of ~ cleq... |
| abid2f 3012 | A simplification of class ... |
| abeq2f 3013 | Equality of a class variab... |
| abeq2fOLD 3014 | Obsolete version of ~ abeq... |
| sbabel 3015 | Theorem to move a substitu... |
| neii 3018 | Inference associated with ... |
| neir 3019 | Inference associated with ... |
| nne 3020 | Negation of inequality. (... |
| neneqd 3021 | Deduction eliminating ineq... |
| neneq 3022 | From inequality to non-equ... |
| neqned 3023 | If it is not the case that... |
| neqne 3024 | From non-equality to inequ... |
| neirr 3025 | No class is unequal to its... |
| exmidne 3026 | Excluded middle with equal... |
| eqneqall 3027 | A contradiction concerning... |
| nonconne 3028 | Law of noncontradiction wi... |
| necon3ad 3029 | Contrapositive law deducti... |
| necon3bd 3030 | Contrapositive law deducti... |
| necon2ad 3031 | Contrapositive inference f... |
| necon2bd 3032 | Contrapositive inference f... |
| necon1ad 3033 | Contrapositive deduction f... |
| necon1bd 3034 | Contrapositive deduction f... |
| necon4ad 3035 | Contrapositive inference f... |
| necon4bd 3036 | Contrapositive inference f... |
| necon3d 3037 | Contrapositive law deducti... |
| necon1d 3038 | Contrapositive law deducti... |
| necon2d 3039 | Contrapositive inference f... |
| necon4d 3040 | Contrapositive inference f... |
| necon3ai 3041 | Contrapositive inference f... |
| necon3bi 3042 | Contrapositive inference f... |
| necon1ai 3043 | Contrapositive inference f... |
| necon1bi 3044 | Contrapositive inference f... |
| necon2ai 3045 | Contrapositive inference f... |
| necon2bi 3046 | Contrapositive inference f... |
| necon4ai 3047 | Contrapositive inference f... |
| necon3i 3048 | Contrapositive inference f... |
| necon1i 3049 | Contrapositive inference f... |
| necon2i 3050 | Contrapositive inference f... |
| necon4i 3051 | Contrapositive inference f... |
| necon3abid 3052 | Deduction from equality to... |
| necon3bbid 3053 | Deduction from equality to... |
| necon1abid 3054 | Contrapositive deduction f... |
| necon1bbid 3055 | Contrapositive inference f... |
| necon4abid 3056 | Contrapositive law deducti... |
| necon4bbid 3057 | Contrapositive law deducti... |
| necon2abid 3058 | Contrapositive deduction f... |
| necon2bbid 3059 | Contrapositive deduction f... |
| necon3bid 3060 | Deduction from equality to... |
| necon4bid 3061 | Contrapositive law deducti... |
| necon3abii 3062 | Deduction from equality to... |
| necon3bbii 3063 | Deduction from equality to... |
| necon1abii 3064 | Contrapositive inference f... |
| necon1bbii 3065 | Contrapositive inference f... |
| necon2abii 3066 | Contrapositive inference f... |
| necon2bbii 3067 | Contrapositive inference f... |
| necon3bii 3068 | Inference from equality to... |
| necom 3069 | Commutation of inequality.... |
| necomi 3070 | Inference from commutative... |
| necomd 3071 | Deduction from commutative... |
| nesym 3072 | Characterization of inequa... |
| nesymi 3073 | Inference associated with ... |
| nesymir 3074 | Inference associated with ... |
| neeq1d 3075 | Deduction for inequality. ... |
| neeq2d 3076 | Deduction for inequality. ... |
| neeq12d 3077 | Deduction for inequality. ... |
| neeq1 3078 | Equality theorem for inequ... |
| neeq2 3079 | Equality theorem for inequ... |
| neeq1i 3080 | Inference for inequality. ... |
| neeq2i 3081 | Inference for inequality. ... |
| neeq12i 3082 | Inference for inequality. ... |
| eqnetrd 3083 | Substitution of equal clas... |
| eqnetrrd 3084 | Substitution of equal clas... |
| neeqtrd 3085 | Substitution of equal clas... |
| eqnetri 3086 | Substitution of equal clas... |
| eqnetrri 3087 | Substitution of equal clas... |
| neeqtri 3088 | Substitution of equal clas... |
| neeqtrri 3089 | Substitution of equal clas... |
| neeqtrrd 3090 | Substitution of equal clas... |
| eqnetrrid 3091 | A chained equality inferen... |
| 3netr3d 3092 | Substitution of equality i... |
| 3netr4d 3093 | Substitution of equality i... |
| 3netr3g 3094 | Substitution of equality i... |
| 3netr4g 3095 | Substitution of equality i... |
| nebi 3096 | Contraposition law for ine... |
| pm13.18 3097 | Theorem *13.18 in [Whitehe... |
| pm13.18OLD 3098 | Obsolete version of ~ pm13... |
| pm13.181 3099 | Theorem *13.181 in [Whiteh... |
| pm2.61ine 3100 | Inference eliminating an i... |
| pm2.21ddne 3101 | A contradiction implies an... |
| pm2.61ne 3102 | Deduction eliminating an i... |
| pm2.61dne 3103 | Deduction eliminating an i... |
| pm2.61dane 3104 | Deduction eliminating an i... |
| pm2.61da2ne 3105 | Deduction eliminating two ... |
| pm2.61da3ne 3106 | Deduction eliminating thre... |
| pm2.61iine 3107 | Equality version of ~ pm2.... |
| neor 3108 | Logical OR with an equalit... |
| neanior 3109 | A De Morgan's law for ineq... |
| ne3anior 3110 | A De Morgan's law for ineq... |
| neorian 3111 | A De Morgan's law for ineq... |
| nemtbir 3112 | An inference from an inequ... |
| nelne1 3113 | Two classes are different ... |
| nelne1OLD 3114 | Obsolete version of ~ neln... |
| nelne2 3115 | Two classes are different ... |
| nelne2OLD 3116 | Obsolete version of ~ neln... |
| nelelne 3117 | Two classes are different ... |
| neneor 3118 | If two classes are differe... |
| nfne 3119 | Bound-variable hypothesis ... |
| nfned 3120 | Bound-variable hypothesis ... |
| nabbi 3121 | Not equivalent wff's corre... |
| mteqand 3122 | A modus tollens deduction ... |
| neli 3125 | Inference associated with ... |
| nelir 3126 | Inference associated with ... |
| neleq12d 3127 | Equality theorem for negat... |
| neleq1 3128 | Equality theorem for negat... |
| neleq2 3129 | Equality theorem for negat... |
| nfnel 3130 | Bound-variable hypothesis ... |
| nfneld 3131 | Bound-variable hypothesis ... |
| nnel 3132 | Negation of negated member... |
| elnelne1 3133 | Two classes are different ... |
| elnelne2 3134 | Two classes are different ... |
| nelcon3d 3135 | Contrapositive law deducti... |
| elnelall 3136 | A contradiction concerning... |
| pm2.61danel 3137 | Deduction eliminating an e... |
| rgen 3148 | Generalization rule for re... |
| ralel 3149 | All elements of a class ar... |
| rgenw 3150 | Generalization rule for re... |
| rgen2w 3151 | Generalization rule for re... |
| mprg 3152 | Modus ponens combined with... |
| mprgbir 3153 | Modus ponens on biconditio... |
| alral 3154 | Universal quantification i... |
| raln 3155 | Restricted universally qua... |
| ral2imi 3156 | Inference quantifying ante... |
| ralimi2 3157 | Inference quantifying both... |
| ralimia 3158 | Inference quantifying both... |
| ralimiaa 3159 | Inference quantifying both... |
| ralimi 3160 | Inference quantifying both... |
| 2ralimi 3161 | Inference quantifying both... |
| ralim 3162 | Distribution of restricted... |
| ralbii2 3163 | Inference adding different... |
| ralbiia 3164 | Inference adding restricte... |
| ralbii 3165 | Inference adding restricte... |
| 2ralbii 3166 | Inference adding two restr... |
| ralbi 3167 | Distribute a restricted un... |
| ralanid 3168 | Cancellation law for restr... |
| ralanidOLD 3169 | Obsolete version of ~ rala... |
| r19.26 3170 | Restricted quantifier vers... |
| r19.26-2 3171 | Restricted quantifier vers... |
| r19.26-3 3172 | Version of ~ r19.26 with t... |
| r19.26m 3173 | Version of ~ 19.26 and ~ r... |
| ralbiim 3174 | Split a biconditional and ... |
| r19.21v 3175 | Restricted quantifier vers... |
| ralimdv2 3176 | Inference quantifying both... |
| ralimdva 3177 | Deduction quantifying both... |
| ralimdv 3178 | Deduction quantifying both... |
| ralimdvva 3179 | Deduction doubly quantifyi... |
| hbralrimi 3180 | Inference from Theorem 19.... |
| ralrimiv 3181 | Inference from Theorem 19.... |
| ralrimiva 3182 | Inference from Theorem 19.... |
| ralrimivw 3183 | Inference from Theorem 19.... |
| r19.27v 3184 | Restricted quantitifer ver... |
| r19.27vOLD 3185 | Obsolete version of ~ r19.... |
| r19.28v 3186 | Restricted quantifier vers... |
| r19.28vOLD 3187 | Obsolete version of ~ r19.... |
| ralrimdv 3188 | Inference from Theorem 19.... |
| ralrimdva 3189 | Inference from Theorem 19.... |
| ralrimivv 3190 | Inference from Theorem 19.... |
| ralrimivva 3191 | Inference from Theorem 19.... |
| ralrimivvva 3192 | Inference from Theorem 19.... |
| ralrimdvv 3193 | Inference from Theorem 19.... |
| ralrimdvva 3194 | Inference from Theorem 19.... |
| ralbidv2 3195 | Formula-building rule for ... |
| ralbidva 3196 | Formula-building rule for ... |
| ralbidv 3197 | Formula-building rule for ... |
| 2ralbidva 3198 | Formula-building rule for ... |
| 2ralbidv 3199 | Formula-building rule for ... |
| r2allem 3200 | Lemma factoring out common... |
| r2al 3201 | Double restricted universa... |
| r3al 3202 | Triple restricted universa... |
| rgen2 3203 | Generalization rule for re... |
| rgen3 3204 | Generalization rule for re... |
| rsp 3205 | Restricted specialization.... |
| rspa 3206 | Restricted specialization.... |
| rspec 3207 | Specialization rule for re... |
| r19.21bi 3208 | Inference from Theorem 19.... |
| r19.21biOLD 3209 | Obsolete version of ~ r19.... |
| r19.21be 3210 | Inference from Theorem 19.... |
| rspec2 3211 | Specialization rule for re... |
| rspec3 3212 | Specialization rule for re... |
| rsp2 3213 | Restricted specialization,... |
| r19.21t 3214 | Restricted quantifier vers... |
| r19.21 3215 | Restricted quantifier vers... |
| ralrimi 3216 | Inference from Theorem 19.... |
| ralimdaa 3217 | Deduction quantifying both... |
| ralrimd 3218 | Inference from Theorem 19.... |
| nfra1 3219 | The setvar ` x ` is not fr... |
| hbra1 3220 | The setvar ` x ` is not fr... |
| hbral 3221 | Bound-variable hypothesis ... |
| r2alf 3222 | Double restricted universa... |
| nfraldw 3223 | Deduction version of ~ nfr... |
| nfrald 3224 | Deduction version of ~ nfr... |
| nfralw 3225 | Bound-variable hypothesis ... |
| nfral 3226 | Bound-variable hypothesis ... |
| nfra2w 3227 | Similar to Lemma 24 of [Mo... |
| nfra2 3228 | Similar to Lemma 24 of [Mo... |
| rgen2a 3229 | Generalization rule for re... |
| ralbida 3230 | Formula-building rule for ... |
| ralbid 3231 | Formula-building rule for ... |
| 2ralbida 3232 | Formula-building rule for ... |
| ralbiOLD 3233 | Obsolete version of ~ ralb... |
| raleqbii 3234 | Equality deduction for res... |
| ralcom4 3235 | Commutation of restricted ... |
| ralnex 3236 | Relationship between restr... |
| dfral2 3237 | Relationship between restr... |
| rexnal 3238 | Relationship between restr... |
| dfrex2 3239 | Relationship between restr... |
| rexex 3240 | Restricted existence impli... |
| rexim 3241 | Theorem 19.22 of [Margaris... |
| reximia 3242 | Inference quantifying both... |
| reximi 3243 | Inference quantifying both... |
| reximi2 3244 | Inference quantifying both... |
| rexbii2 3245 | Inference adding different... |
| rexbiia 3246 | Inference adding restricte... |
| rexbii 3247 | Inference adding restricte... |
| 2rexbii 3248 | Inference adding two restr... |
| rexcom4 3249 | Commutation of restricted ... |
| 2ex2rexrot 3250 | Rotate two existential qua... |
| rexcom4a 3251 | Specialized existential co... |
| rexanid 3252 | Cancellation law for restr... |
| rexanidOLD 3253 | Obsolete version of ~ rexa... |
| r19.29 3254 | Restricted quantifier vers... |
| r19.29r 3255 | Restricted quantifier vers... |
| r19.29rOLD 3256 | Obsolete version of ~ r19.... |
| r19.29imd 3257 | Theorem 19.29 of [Margaris... |
| rexnal2 3258 | Relationship between two r... |
| rexnal3 3259 | Relationship between three... |
| ralnex2 3260 | Relationship between two r... |
| ralnex2OLD 3261 | Obsolete version of ~ raln... |
| ralnex3 3262 | Relationship between three... |
| ralnex3OLD 3263 | Obsolete version of ~ raln... |
| ralinexa 3264 | A transformation of restri... |
| rexanali 3265 | A transformation of restri... |
| nrexralim 3266 | Negation of a complex pred... |
| risset 3267 | Two ways to say " ` A ` be... |
| nelb 3268 | A definition of ` -. A e. ... |
| nrex 3269 | Inference adding restricte... |
| nrexdv 3270 | Deduction adding restricte... |
| reximdv2 3271 | Deduction quantifying both... |
| reximdvai 3272 | Deduction quantifying both... |
| reximdv 3273 | Deduction from Theorem 19.... |
| reximdva 3274 | Deduction quantifying both... |
| reximddv 3275 | Deduction from Theorem 19.... |
| reximssdv 3276 | Derivation of a restricted... |
| reximdvva 3277 | Deduction doubly quantifyi... |
| reximddv2 3278 | Double deduction from Theo... |
| r19.23v 3279 | Restricted quantifier vers... |
| rexlimiv 3280 | Inference from Theorem 19.... |
| rexlimiva 3281 | Inference from Theorem 19.... |
| rexlimivw 3282 | Weaker version of ~ rexlim... |
| rexlimdv 3283 | Inference from Theorem 19.... |
| rexlimdva 3284 | Inference from Theorem 19.... |
| rexlimdvaa 3285 | Inference from Theorem 19.... |
| rexlimdv3a 3286 | Inference from Theorem 19.... |
| rexlimdva2 3287 | Inference from Theorem 19.... |
| r19.29an 3288 | A commonly used pattern in... |
| r19.29a 3289 | A commonly used pattern in... |
| rexlimdvw 3290 | Inference from Theorem 19.... |
| rexlimddv 3291 | Restricted existential eli... |
| rexlimivv 3292 | Inference from Theorem 19.... |
| rexlimdvv 3293 | Inference from Theorem 19.... |
| rexlimdvva 3294 | Inference from Theorem 19.... |
| rexbidv2 3295 | Formula-building rule for ... |
| rexbidva 3296 | Formula-building rule for ... |
| rexbidv 3297 | Formula-building rule for ... |
| 2rexbiia 3298 | Inference adding two restr... |
| 2rexbidva 3299 | Formula-building rule for ... |
| 2rexbidv 3300 | Formula-building rule for ... |
| rexralbidv 3301 | Formula-building rule for ... |
| r2exlem 3302 | Lemma factoring out common... |
| r2ex 3303 | Double restricted existent... |
| rspe 3304 | Restricted specialization.... |
| rsp2e 3305 | Restricted specialization.... |
| nfre1 3306 | The setvar ` x ` is not fr... |
| nfrexd 3307 | Deduction version of ~ nfr... |
| nfrexdg 3308 | Deduction version of ~ nfr... |
| nfrex 3309 | Bound-variable hypothesis ... |
| nfrexg 3310 | Bound-variable hypothesis ... |
| reximdai 3311 | Deduction from Theorem 19.... |
| reximd2a 3312 | Deduction quantifying both... |
| r19.23t 3313 | Closed theorem form of ~ r... |
| r19.23 3314 | Restricted quantifier vers... |
| rexlimi 3315 | Restricted quantifier vers... |
| rexlimd2 3316 | Version of ~ rexlimd with ... |
| rexlimd 3317 | Deduction form of ~ rexlim... |
| rexbida 3318 | Formula-building rule for ... |
| rexbidvaALT 3319 | Alternate proof of ~ rexbi... |
| rexbid 3320 | Formula-building rule for ... |
| rexbidvALT 3321 | Alternate proof of ~ rexbi... |
| ralrexbid 3322 | Formula-building rule for ... |
| ralrexbidOLD 3323 | Obsolete version of ~ ralr... |
| r19.12 3324 | Restricted quantifier vers... |
| r2exf 3325 | Double restricted existent... |
| rexeqbii 3326 | Equality deduction for res... |
| r19.12OLD 3327 | Obsolete version of ~ r19.... |
| reuanid 3328 | Cancellation law for restr... |
| rmoanid 3329 | Cancellation law for restr... |
| r19.29af2 3330 | A commonly used pattern ba... |
| r19.29af 3331 | A commonly used pattern ba... |
| r19.29anOLD 3332 | Obsolete version of ~ r19.... |
| r19.29aOLD 3333 | Obsolete proof of ~ r19.29... |
| 2r19.29 3334 | Theorem ~ r19.29 with two ... |
| r19.29d2r 3335 | Theorem 19.29 of [Margaris... |
| r19.29vva 3336 | A commonly used pattern ba... |
| r19.29vvaOLD 3337 | Obsolete version of ~ r19.... |
| r19.30 3338 | Restricted quantifier vers... |
| r19.30OLD 3339 | Obsolete version of ~ r19.... |
| r19.32v 3340 | Restricted quantifier vers... |
| r19.35 3341 | Restricted quantifier vers... |
| r19.36v 3342 | Restricted quantifier vers... |
| r19.37 3343 | Restricted quantifier vers... |
| r19.37v 3344 | Restricted quantifier vers... |
| r19.37vOLD 3345 | Obsolete version of ~ r19.... |
| r19.40 3346 | Restricted quantifier vers... |
| r19.41v 3347 | Restricted quantifier vers... |
| r19.41 3348 | Restricted quantifier vers... |
| r19.41vv 3349 | Version of ~ r19.41v with ... |
| r19.42v 3350 | Restricted quantifier vers... |
| r19.43 3351 | Restricted quantifier vers... |
| r19.44v 3352 | One direction of a restric... |
| r19.45v 3353 | Restricted quantifier vers... |
| ralcom 3354 | Commutation of restricted ... |
| rexcom 3355 | Commutation of restricted ... |
| rexcomOLD 3356 | Obsolete version of ~ rexc... |
| ralcomf 3357 | Commutation of restricted ... |
| rexcomf 3358 | Commutation of restricted ... |
| ralcom13 3359 | Swap first and third restr... |
| rexcom13 3360 | Swap first and third restr... |
| ralrot3 3361 | Rotate three restricted un... |
| rexrot4 3362 | Rotate four restricted exi... |
| ralcom2 3363 | Commutation of restricted ... |
| ralcom3 3364 | A commutation law for rest... |
| reeanlem 3365 | Lemma factoring out common... |
| reean 3366 | Rearrange restricted exist... |
| reeanv 3367 | Rearrange restricted exist... |
| 3reeanv 3368 | Rearrange three restricted... |
| 2ralor 3369 | Distribute restricted univ... |
| nfreu1 3370 | The setvar ` x ` is not fr... |
| nfrmo1 3371 | The setvar ` x ` is not fr... |
| nfreud 3372 | Deduction version of ~ nfr... |
| nfrmod 3373 | Deduction version of ~ nfr... |
| nfreuw 3374 | Bound-variable hypothesis ... |
| nfrmow 3375 | Bound-variable hypothesis ... |
| nfreu 3376 | Bound-variable hypothesis ... |
| nfrmo 3377 | Bound-variable hypothesis ... |
| rabid 3378 | An "identity" law of concr... |
| rabrab 3379 | Abstract builder restricte... |
| rabidim1 3380 | Membership in a restricted... |
| rabid2 3381 | An "identity" law for rest... |
| rabid2f 3382 | An "identity" law for rest... |
| rabbi 3383 | Equivalent wff's correspon... |
| nfrab1 3384 | The abstraction variable i... |
| nfrabw 3385 | A variable not free in a w... |
| nfrab 3386 | A variable not free in a w... |
| reubida 3387 | Formula-building rule for ... |
| reubidva 3388 | Formula-building rule for ... |
| reubidv 3389 | Formula-building rule for ... |
| reubiia 3390 | Formula-building rule for ... |
| reubii 3391 | Formula-building rule for ... |
| rmobida 3392 | Formula-building rule for ... |
| rmobidva 3393 | Formula-building rule for ... |
| rmobidv 3394 | Formula-building rule for ... |
| rmobiia 3395 | Formula-building rule for ... |
| rmobii 3396 | Formula-building rule for ... |
| raleqf 3397 | Equality theorem for restr... |
| rexeqf 3398 | Equality theorem for restr... |
| reueq1f 3399 | Equality theorem for restr... |
| rmoeq1f 3400 | Equality theorem for restr... |
| raleqbidv 3401 | Equality deduction for res... |
| rexeqbidv 3402 | Equality deduction for res... |
| raleqbi1dv 3403 | Equality deduction for res... |
| rexeqbi1dv 3404 | Equality deduction for res... |
| raleq 3405 | Equality theorem for restr... |
| rexeq 3406 | Equality theorem for restr... |
| reueq1 3407 | Equality theorem for restr... |
| rmoeq1 3408 | Equality theorem for restr... |
| raleqOLD 3409 | Obsolete version of ~ rale... |
| rexeqOLD 3410 | Obsolete version of ~ rexe... |
| reueq1OLD 3411 | Obsolete version of ~ reue... |
| rmoeq1OLD 3412 | Obsolete version of ~ rmoe... |
| raleqi 3413 | Equality inference for res... |
| rexeqi 3414 | Equality inference for res... |
| raleqdv 3415 | Equality deduction for res... |
| rexeqdv 3416 | Equality deduction for res... |
| raleqbi1dvOLD 3417 | Obsolete version of ~ rale... |
| rexeqbi1dvOLD 3418 | Obsolete version of ~ rexe... |
| reueqd 3419 | Equality deduction for res... |
| rmoeqd 3420 | Equality deduction for res... |
| raleqbid 3421 | Equality deduction for res... |
| rexeqbid 3422 | Equality deduction for res... |
| raleqbidvOLD 3423 | Obsolete version of ~ rale... |
| rexeqbidvOLD 3424 | Obsolete version of ~ rexe... |
| raleqbidva 3425 | Equality deduction for res... |
| rexeqbidva 3426 | Equality deduction for res... |
| raleleq 3427 | All elements of a class ar... |
| raleleqALT 3428 | Alternate proof of ~ ralel... |
| mormo 3429 | Unrestricted "at most one"... |
| reu5 3430 | Restricted uniqueness in t... |
| reurex 3431 | Restricted unique existenc... |
| 2reu2rex 3432 | Double restricted existent... |
| reurmo 3433 | Restricted existential uni... |
| rmo5 3434 | Restricted "at most one" i... |
| nrexrmo 3435 | Nonexistence implies restr... |
| reueubd 3436 | Restricted existential uni... |
| cbvralfw 3437 | Rule used to change bound ... |
| cbvrexfw 3438 | Rule used to change bound ... |
| cbvralf 3439 | Rule used to change bound ... |
| cbvrexf 3440 | Rule used to change bound ... |
| cbvralw 3441 | Rule used to change bound ... |
| cbvrexw 3442 | Rule used to change bound ... |
| cbvreuw 3443 | Change the bound variable ... |
| cbvrmow 3444 | Change the bound variable ... |
| cbvral 3445 | Rule used to change bound ... |
| cbvrex 3446 | Rule used to change bound ... |
| cbvreu 3447 | Change the bound variable ... |
| cbvrmo 3448 | Change the bound variable ... |
| cbvralvw 3449 | Change the bound variable ... |
| cbvrexvw 3450 | Change the bound variable ... |
| cbvreuvw 3451 | Change the bound variable ... |
| cbvralv 3452 | Change the bound variable ... |
| cbvrexv 3453 | Change the bound variable ... |
| cbvreuv 3454 | Change the bound variable ... |
| cbvrmov 3455 | Change the bound variable ... |
| cbvraldva2 3456 | Rule used to change the bo... |
| cbvrexdva2 3457 | Rule used to change the bo... |
| cbvrexdva2OLD 3458 | Obsolete version of ~ cbvr... |
| cbvraldva 3459 | Rule used to change the bo... |
| cbvrexdva 3460 | Rule used to change the bo... |
| cbvral2vw 3461 | Change bound variables of ... |
| cbvrex2vw 3462 | Change bound variables of ... |
| cbvral3vw 3463 | Change bound variables of ... |
| cbvral2v 3464 | Change bound variables of ... |
| cbvrex2v 3465 | Change bound variables of ... |
| cbvral3v 3466 | Change bound variables of ... |
| cbvralsvw 3467 | Change bound variable by u... |
| cbvrexsvw 3468 | Change bound variable by u... |
| cbvralsv 3469 | Change bound variable by u... |
| cbvrexsv 3470 | Change bound variable by u... |
| sbralie 3471 | Implicit to explicit subst... |
| rabbiia 3472 | Equivalent wff's yield equ... |
| rabbii 3473 | Equivalent wff's correspon... |
| rabbida 3474 | Equivalent wff's yield equ... |
| rabbid 3475 | Version of ~ rabbidv with ... |
| rabbidva2 3476 | Equivalent wff's yield equ... |
| rabbia2 3477 | Equivalent wff's yield equ... |
| rabbidva 3478 | Equivalent wff's yield equ... |
| rabbidvaOLD 3479 | Obsolete proof of ~ rabbid... |
| rabbidv 3480 | Equivalent wff's yield equ... |
| rabeqf 3481 | Equality theorem for restr... |
| rabeqi 3482 | Equality theorem for restr... |
| rabeq 3483 | Equality theorem for restr... |
| rabeqdv 3484 | Equality of restricted cla... |
| rabeqbidv 3485 | Equality of restricted cla... |
| rabeqbidva 3486 | Equality of restricted cla... |
| rabeq2i 3487 | Inference from equality of... |
| rabswap 3488 | Swap with a membership rel... |
| cbvrabw 3489 | Rule to change the bound v... |
| cbvrab 3490 | Rule to change the bound v... |
| cbvrabv 3491 | Rule to change the bound v... |
| cbvrabvOLD 3492 | Obsolete version of ~ cbvr... |
| rabrabi 3493 | Abstract builder restricte... |
| vjust 3495 | Soundness justification th... |
| vex 3497 | All setvar variables are s... |
| vexOLD 3498 | Obsolete version of ~ vex ... |
| elv 3499 | If a proposition is implie... |
| elvd 3500 | If a proposition is implie... |
| el2v 3501 | If a proposition is implie... |
| eqv 3502 | The universe contains ever... |
| eqvf 3503 | The universe contains ever... |
| abv 3504 | The class of sets verifyin... |
| elisset 3505 | An element of a class exis... |
| isset 3506 | Two ways to say " ` A ` is... |
| issetf 3507 | A version of ~ isset that ... |
| isseti 3508 | A way to say " ` A ` is a ... |
| issetiOLD 3509 | Obsolete version of ~ isse... |
| issetri 3510 | A way to say " ` A ` is a ... |
| eqvisset 3511 | A class equal to a variabl... |
| elex 3512 | If a class is a member of ... |
| elexi 3513 | If a class is a member of ... |
| elexd 3514 | If a class is a member of ... |
| elissetOLD 3515 | Obsolete version of ~ elis... |
| elex2 3516 | If a class contains anothe... |
| elex22 3517 | If two classes each contai... |
| prcnel 3518 | A proper class doesn't bel... |
| ralv 3519 | A universal quantifier res... |
| rexv 3520 | An existential quantifier ... |
| reuv 3521 | A unique existential quant... |
| rmov 3522 | An at-most-one quantifier ... |
| rabab 3523 | A class abstraction restri... |
| rexcom4b 3524 | Specialized existential co... |
| ralcom4OLD 3525 | Obsolete version of ~ ralc... |
| rexcom4OLD 3526 | Obsolete version of ~ rexc... |
| ceqsalt 3527 | Closed theorem version of ... |
| ceqsralt 3528 | Restricted quantifier vers... |
| ceqsalg 3529 | A representation of explic... |
| ceqsalgALT 3530 | Alternate proof of ~ ceqsa... |
| ceqsal 3531 | A representation of explic... |
| ceqsalv 3532 | A representation of explic... |
| ceqsralv 3533 | Restricted quantifier vers... |
| gencl 3534 | Implicit substitution for ... |
| 2gencl 3535 | Implicit substitution for ... |
| 3gencl 3536 | Implicit substitution for ... |
| cgsexg 3537 | Implicit substitution infe... |
| cgsex2g 3538 | Implicit substitution infe... |
| cgsex4g 3539 | An implicit substitution i... |
| ceqsex 3540 | Elimination of an existent... |
| ceqsexv 3541 | Elimination of an existent... |
| ceqsexv2d 3542 | Elimination of an existent... |
| ceqsex2 3543 | Elimination of two existen... |
| ceqsex2v 3544 | Elimination of two existen... |
| ceqsex3v 3545 | Elimination of three exist... |
| ceqsex4v 3546 | Elimination of four existe... |
| ceqsex6v 3547 | Elimination of six existen... |
| ceqsex8v 3548 | Elimination of eight exist... |
| gencbvex 3549 | Change of bound variable u... |
| gencbvex2 3550 | Restatement of ~ gencbvex ... |
| gencbval 3551 | Change of bound variable u... |
| sbhypf 3552 | Introduce an explicit subs... |
| vtoclgft 3553 | Closed theorem form of ~ v... |
| vtoclgftOLD 3554 | Obsolete version of ~ vtoc... |
| vtocldf 3555 | Implicit substitution of a... |
| vtocld 3556 | Implicit substitution of a... |
| vtocl2d 3557 | Implicit substitution of t... |
| vtoclf 3558 | Implicit substitution of a... |
| vtocl 3559 | Implicit substitution of a... |
| vtoclALT 3560 | Alternate proof of ~ vtocl... |
| vtocl2 3561 | Implicit substitution of c... |
| vtocl2OLD 3562 | Obsolete proof of ~ vtocl2... |
| vtocl3 3563 | Implicit substitution of c... |
| vtoclb 3564 | Implicit substitution of a... |
| vtoclgf 3565 | Implicit substitution of a... |
| vtoclg1f 3566 | Version of ~ vtoclgf with ... |
| vtoclg 3567 | Implicit substitution of a... |
| vtoclbg 3568 | Implicit substitution of a... |
| vtocl2gf 3569 | Implicit substitution of a... |
| vtocl3gf 3570 | Implicit substitution of a... |
| vtocl2g 3571 | Implicit substitution of 2... |
| vtoclgaf 3572 | Implicit substitution of a... |
| vtoclga 3573 | Implicit substitution of a... |
| vtocl2ga 3574 | Implicit substitution of 2... |
| vtocl2gaf 3575 | Implicit substitution of 2... |
| vtocl3gaf 3576 | Implicit substitution of 3... |
| vtocl3ga 3577 | Implicit substitution of 3... |
| vtocl4g 3578 | Implicit substitution of 4... |
| vtocl4ga 3579 | Implicit substitution of 4... |
| vtocleg 3580 | Implicit substitution of a... |
| vtoclegft 3581 | Implicit substitution of a... |
| vtoclef 3582 | Implicit substitution of a... |
| vtocle 3583 | Implicit substitution of a... |
| vtoclri 3584 | Implicit substitution of a... |
| spcimgft 3585 | A closed version of ~ spci... |
| spcgft 3586 | A closed version of ~ spcg... |
| spcimgf 3587 | Rule of specialization, us... |
| spcimegf 3588 | Existential specialization... |
| spcgf 3589 | Rule of specialization, us... |
| spcegf 3590 | Existential specialization... |
| spcimdv 3591 | Restricted specialization,... |
| spcdv 3592 | Rule of specialization, us... |
| spcimedv 3593 | Restricted existential spe... |
| spcgv 3594 | Rule of specialization, us... |
| spcgvOLD 3595 | Obsolete version of ~ spcg... |
| spcegv 3596 | Existential specialization... |
| spcegvOLD 3597 | Obsolete version of ~ spce... |
| spcedv 3598 | Existential specialization... |
| spc2egv 3599 | Existential specialization... |
| spc2gv 3600 | Specialization with two qu... |
| spc2ed 3601 | Existential specialization... |
| spc2d 3602 | Specialization with 2 quan... |
| spc3egv 3603 | Existential specialization... |
| spc3gv 3604 | Specialization with three ... |
| spcv 3605 | Rule of specialization, us... |
| spcev 3606 | Existential specialization... |
| spc2ev 3607 | Existential specialization... |
| rspct 3608 | A closed version of ~ rspc... |
| rspcdf 3609 | Restricted specialization,... |
| rspc 3610 | Restricted specialization,... |
| rspce 3611 | Restricted existential spe... |
| rspcimdv 3612 | Restricted specialization,... |
| rspcimedv 3613 | Restricted existential spe... |
| rspcdv 3614 | Restricted specialization,... |
| rspcedv 3615 | Restricted existential spe... |
| rspcebdv 3616 | Restricted existential spe... |
| rspcv 3617 | Restricted specialization,... |
| rspcvOLD 3618 | Obsolete version of ~ rspc... |
| rspccv 3619 | Restricted specialization,... |
| rspcva 3620 | Restricted specialization,... |
| rspccva 3621 | Restricted specialization,... |
| rspcev 3622 | Restricted existential spe... |
| rspcevOLD 3623 | Obsolete version of ~ rspc... |
| rspcdva 3624 | Restricted specialization,... |
| rspcedvd 3625 | Restricted existential spe... |
| rspcime 3626 | Prove a restricted existen... |
| rspceaimv 3627 | Restricted existential spe... |
| rspcedeq1vd 3628 | Restricted existential spe... |
| rspcedeq2vd 3629 | Restricted existential spe... |
| rspc2 3630 | Restricted specialization ... |
| rspc2gv 3631 | Restricted specialization ... |
| rspc2v 3632 | 2-variable restricted spec... |
| rspc2va 3633 | 2-variable restricted spec... |
| rspc2ev 3634 | 2-variable restricted exis... |
| rspc3v 3635 | 3-variable restricted spec... |
| rspc3ev 3636 | 3-variable restricted exis... |
| rspceeqv 3637 | Restricted existential spe... |
| ralxpxfr2d 3638 | Transfer a universal quant... |
| rexraleqim 3639 | Statement following from e... |
| eqvincg 3640 | A variable introduction la... |
| eqvinc 3641 | A variable introduction la... |
| eqvincf 3642 | A variable introduction la... |
| alexeqg 3643 | Two ways to express substi... |
| ceqex 3644 | Equality implies equivalen... |
| ceqsexg 3645 | A representation of explic... |
| ceqsexgv 3646 | Elimination of an existent... |
| ceqsexgvOLD 3647 | Obsolete version of ~ ceqs... |
| ceqsrexv 3648 | Elimination of a restricte... |
| ceqsrexbv 3649 | Elimination of a restricte... |
| ceqsrex2v 3650 | Elimination of a restricte... |
| clel2g 3651 | An alternate definition of... |
| clel2 3652 | An alternate definition of... |
| clel3g 3653 | An alternate definition of... |
| clel3 3654 | An alternate definition of... |
| clel4 3655 | An alternate definition of... |
| clel5 3656 | Alternate definition of cl... |
| clel5OLD 3657 | Obsolete version of ~ clel... |
| pm13.183 3658 | Compare theorem *13.183 in... |
| pm13.183OLD 3659 | Obsolete version of ~ pm13... |
| rr19.3v 3660 | Restricted quantifier vers... |
| rr19.28v 3661 | Restricted quantifier vers... |
| elabgt 3662 | Membership in a class abst... |
| elabgf 3663 | Membership in a class abst... |
| elabf 3664 | Membership in a class abst... |
| elabg 3665 | Membership in a class abst... |
| elab 3666 | Membership in a class abst... |
| elab2g 3667 | Membership in a class abst... |
| elabd 3668 | Explicit demonstration the... |
| elab2 3669 | Membership in a class abst... |
| elab4g 3670 | Membership in a class abst... |
| elab3gf 3671 | Membership in a class abst... |
| elab3g 3672 | Membership in a class abst... |
| elab3 3673 | Membership in a class abst... |
| elrabi 3674 | Implication for the member... |
| elrabf 3675 | Membership in a restricted... |
| rabtru 3676 | Abstract builder using the... |
| rabeqc 3677 | A restricted class abstrac... |
| elrab3t 3678 | Membership in a restricted... |
| elrab 3679 | Membership in a restricted... |
| elrab3 3680 | Membership in a restricted... |
| elrabd 3681 | Membership in a restricted... |
| elrab2 3682 | Membership in a class abst... |
| ralab 3683 | Universal quantification o... |
| ralrab 3684 | Universal quantification o... |
| rexab 3685 | Existential quantification... |
| rexrab 3686 | Existential quantification... |
| ralab2 3687 | Universal quantification o... |
| ralab2OLD 3688 | Obsolete version of ~ rala... |
| ralrab2 3689 | Universal quantification o... |
| rexab2 3690 | Existential quantification... |
| rexab2OLD 3691 | Obsolete version of ~ rexa... |
| rexrab2 3692 | Existential quantification... |
| abidnf 3693 | Identity used to create cl... |
| dedhb 3694 | A deduction theorem for co... |
| nelrdva 3695 | Deduce negative membership... |
| eqeu 3696 | A condition which implies ... |
| moeq 3697 | There exists at most one s... |
| eueq 3698 | A class is a set if and on... |
| eueqi 3699 | There exists a unique set ... |
| eueq2 3700 | Equality has existential u... |
| eueq3 3701 | Equality has existential u... |
| moeq3 3702 | "At most one" property of ... |
| mosub 3703 | "At most one" remains true... |
| mo2icl 3704 | Theorem for inferring "at ... |
| mob2 3705 | Consequence of "at most on... |
| moi2 3706 | Consequence of "at most on... |
| mob 3707 | Equality implied by "at mo... |
| moi 3708 | Equality implied by "at mo... |
| morex 3709 | Derive membership from uni... |
| euxfr2w 3710 | Transfer existential uniqu... |
| euxfrw 3711 | Transfer existential uniqu... |
| euxfr2 3712 | Transfer existential uniqu... |
| euxfr 3713 | Transfer existential uniqu... |
| euind 3714 | Existential uniqueness via... |
| reu2 3715 | A way to express restricte... |
| reu6 3716 | A way to express restricte... |
| reu3 3717 | A way to express restricte... |
| reu6i 3718 | A condition which implies ... |
| eqreu 3719 | A condition which implies ... |
| rmo4 3720 | Restricted "at most one" u... |
| reu4 3721 | Restricted uniqueness usin... |
| reu7 3722 | Restricted uniqueness usin... |
| reu8 3723 | Restricted uniqueness usin... |
| rmo3f 3724 | Restricted "at most one" u... |
| rmo4f 3725 | Restricted "at most one" u... |
| reu2eqd 3726 | Deduce equality from restr... |
| reueq 3727 | Equality has existential u... |
| rmoeq 3728 | Equality's restricted exis... |
| rmoan 3729 | Restricted "at most one" s... |
| rmoim 3730 | Restricted "at most one" i... |
| rmoimia 3731 | Restricted "at most one" i... |
| rmoimi 3732 | Restricted "at most one" i... |
| rmoimi2 3733 | Restricted "at most one" i... |
| 2reu5a 3734 | Double restricted existent... |
| reuimrmo 3735 | Restricted uniqueness impl... |
| 2reuswap 3736 | A condition allowing swap ... |
| 2reuswap2 3737 | A condition allowing swap ... |
| reuxfrd 3738 | Transfer existential uniqu... |
| reuxfr 3739 | Transfer existential uniqu... |
| reuxfr1d 3740 | Transfer existential uniqu... |
| reuxfr1ds 3741 | Transfer existential uniqu... |
| reuxfr1 3742 | Transfer existential uniqu... |
| reuind 3743 | Existential uniqueness via... |
| 2rmorex 3744 | Double restricted quantifi... |
| 2reu5lem1 3745 | Lemma for ~ 2reu5 . Note ... |
| 2reu5lem2 3746 | Lemma for ~ 2reu5 . (Cont... |
| 2reu5lem3 3747 | Lemma for ~ 2reu5 . This ... |
| 2reu5 3748 | Double restricted existent... |
| 2reurex 3749 | Double restricted quantifi... |
| 2reurmo 3750 | Double restricted quantifi... |
| 2rmoswap 3751 | A condition allowing to sw... |
| 2rexreu 3752 | Double restricted existent... |
| cdeqi 3755 | Deduce conditional equalit... |
| cdeqri 3756 | Property of conditional eq... |
| cdeqth 3757 | Deduce conditional equalit... |
| cdeqnot 3758 | Distribute conditional equ... |
| cdeqal 3759 | Distribute conditional equ... |
| cdeqab 3760 | Distribute conditional equ... |
| cdeqal1 3761 | Distribute conditional equ... |
| cdeqab1 3762 | Distribute conditional equ... |
| cdeqim 3763 | Distribute conditional equ... |
| cdeqcv 3764 | Conditional equality for s... |
| cdeqeq 3765 | Distribute conditional equ... |
| cdeqel 3766 | Distribute conditional equ... |
| nfcdeq 3767 | If we have a conditional e... |
| nfccdeq 3768 | Variation of ~ nfcdeq for ... |
| rru 3769 | Relative version of Russel... |
| ru 3770 | Russell's Paradox. Propos... |
| dfsbcq 3773 | Proper substitution of a c... |
| dfsbcq2 3774 | This theorem, which is sim... |
| sbsbc 3775 | Show that ~ df-sb and ~ df... |
| sbceq1d 3776 | Equality theorem for class... |
| sbceq1dd 3777 | Equality theorem for class... |
| sbceqbid 3778 | Equality theorem for class... |
| sbc8g 3779 | This is the closest we can... |
| sbc2or 3780 | The disjunction of two equ... |
| sbcex 3781 | By our definition of prope... |
| sbceq1a 3782 | Equality theorem for class... |
| sbceq2a 3783 | Equality theorem for class... |
| spsbc 3784 | Specialization: if a formu... |
| spsbcd 3785 | Specialization: if a formu... |
| sbcth 3786 | A substitution into a theo... |
| sbcthdv 3787 | Deduction version of ~ sbc... |
| sbcid 3788 | An identity theorem for su... |
| nfsbc1d 3789 | Deduction version of ~ nfs... |
| nfsbc1 3790 | Bound-variable hypothesis ... |
| nfsbc1v 3791 | Bound-variable hypothesis ... |
| nfsbcdw 3792 | Deduction version of ~ nfs... |
| nfsbcw 3793 | Bound-variable hypothesis ... |
| sbccow 3794 | A composition law for clas... |
| nfsbcd 3795 | Deduction version of ~ nfs... |
| nfsbc 3796 | Bound-variable hypothesis ... |
| sbcco 3797 | A composition law for clas... |
| sbcco2 3798 | A composition law for clas... |
| sbc5 3799 | An equivalence for class s... |
| sbc6g 3800 | An equivalence for class s... |
| sbc6 3801 | An equivalence for class s... |
| sbc7 3802 | An equivalence for class s... |
| cbvsbcw 3803 | Change bound variables in ... |
| cbvsbcvw 3804 | Change the bound variable ... |
| cbvsbc 3805 | Change bound variables in ... |
| cbvsbcv 3806 | Change the bound variable ... |
| sbciegft 3807 | Conversion of implicit sub... |
| sbciegf 3808 | Conversion of implicit sub... |
| sbcieg 3809 | Conversion of implicit sub... |
| sbcie2g 3810 | Conversion of implicit sub... |
| sbcie 3811 | Conversion of implicit sub... |
| sbciedf 3812 | Conversion of implicit sub... |
| sbcied 3813 | Conversion of implicit sub... |
| sbcied2 3814 | Conversion of implicit sub... |
| elrabsf 3815 | Membership in a restricted... |
| eqsbc3 3816 | Substitution applied to an... |
| eqsbc3OLD 3817 | Obsolete version of ~ eqsb... |
| sbcng 3818 | Move negation in and out o... |
| sbcimg 3819 | Distribution of class subs... |
| sbcan 3820 | Distribution of class subs... |
| sbcor 3821 | Distribution of class subs... |
| sbcbig 3822 | Distribution of class subs... |
| sbcn1 3823 | Move negation in and out o... |
| sbcim1 3824 | Distribution of class subs... |
| sbcbid 3825 | Formula-building deduction... |
| sbcbidv 3826 | Formula-building deduction... |
| sbcbidvOLD 3827 | Obsolete version of ~ sbcb... |
| sbcbii 3828 | Formula-building inference... |
| sbcbi1 3829 | Distribution of class subs... |
| sbcbi2 3830 | Substituting into equivale... |
| sbcbi2OLD 3831 | Obsolete proof of ~ sbcbi2... |
| sbcal 3832 | Move universal quantifier ... |
| sbcex2 3833 | Move existential quantifie... |
| sbceqal 3834 | Class version of one impli... |
| sbeqalb 3835 | Theorem *14.121 in [Whiteh... |
| eqsbc3r 3836 | ~ eqsbc3 with setvar varia... |
| sbc3an 3837 | Distribution of class subs... |
| sbcel1v 3838 | Class substitution into a ... |
| sbcel1vOLD 3839 | Obsolete version of ~ sbce... |
| sbcel2gv 3840 | Class substitution into a ... |
| sbcel21v 3841 | Class substitution into a ... |
| sbcimdv 3842 | Substitution analogue of T... |
| sbctt 3843 | Substitution for a variabl... |
| sbcgf 3844 | Substitution for a variabl... |
| sbc19.21g 3845 | Substitution for a variabl... |
| sbcg 3846 | Substitution for a variabl... |
| sbcgfi 3847 | Substitution for a variabl... |
| sbc2iegf 3848 | Conversion of implicit sub... |
| sbc2ie 3849 | Conversion of implicit sub... |
| sbc2iedv 3850 | Conversion of implicit sub... |
| sbc3ie 3851 | Conversion of implicit sub... |
| sbccomlem 3852 | Lemma for ~ sbccom . (Con... |
| sbccom 3853 | Commutative law for double... |
| sbcralt 3854 | Interchange class substitu... |
| sbcrext 3855 | Interchange class substitu... |
| sbcralg 3856 | Interchange class substitu... |
| sbcrex 3857 | Interchange class substitu... |
| sbcreu 3858 | Interchange class substitu... |
| reu8nf 3859 | Restricted uniqueness usin... |
| sbcabel 3860 | Interchange class substitu... |
| rspsbc 3861 | Restricted quantifier vers... |
| rspsbca 3862 | Restricted quantifier vers... |
| rspesbca 3863 | Existence form of ~ rspsbc... |
| spesbc 3864 | Existence form of ~ spsbc ... |
| spesbcd 3865 | form of ~ spsbc . (Contri... |
| sbcth2 3866 | A substitution into a theo... |
| ra4v 3867 | Version of ~ ra4 with a di... |
| ra4 3868 | Restricted quantifier vers... |
| rmo2 3869 | Alternate definition of re... |
| rmo2i 3870 | Condition implying restric... |
| rmo3 3871 | Restricted "at most one" u... |
| rmo3OLD 3872 | Obsolete version of ~ rmo3... |
| rmob 3873 | Consequence of "at most on... |
| rmoi 3874 | Consequence of "at most on... |
| rmob2 3875 | Consequence of "restricted... |
| rmoi2 3876 | Consequence of "restricted... |
| rmoanim 3877 | Introduction of a conjunct... |
| rmoanimALT 3878 | Alternate proof of ~ rmoan... |
| reuan 3879 | Introduction of a conjunct... |
| 2reu1 3880 | Double restricted existent... |
| 2reu2 3881 | Double restricted existent... |
| csb2 3884 | Alternate expression for t... |
| csbeq1 3885 | Analogue of ~ dfsbcq for p... |
| csbeq1d 3886 | Equality deduction for pro... |
| csbeq2 3887 | Substituting into equivale... |
| csbeq2d 3888 | Formula-building deduction... |
| csbeq2dv 3889 | Formula-building deduction... |
| csbeq2i 3890 | Formula-building inference... |
| csbeq12dv 3891 | Formula-building inference... |
| cbvcsbw 3892 | Change bound variables in ... |
| cbvcsb 3893 | Change bound variables in ... |
| cbvcsbv 3894 | Change the bound variable ... |
| csbid 3895 | Analogue of ~ sbid for pro... |
| csbeq1a 3896 | Equality theorem for prope... |
| csbcow 3897 | Composition law for chaine... |
| csbco 3898 | Composition law for chaine... |
| csbtt 3899 | Substitution doesn't affec... |
| csbconstgf 3900 | Substitution doesn't affec... |
| csbconstg 3901 | Substitution doesn't affec... |
| csbgfi 3902 | Substitution for a variabl... |
| csbconstgi 3903 | The proper substitution of... |
| nfcsb1d 3904 | Bound-variable hypothesis ... |
| nfcsb1 3905 | Bound-variable hypothesis ... |
| nfcsb1v 3906 | Bound-variable hypothesis ... |
| nfcsbd 3907 | Deduction version of ~ nfc... |
| nfcsbw 3908 | Bound-variable hypothesis ... |
| nfcsb 3909 | Bound-variable hypothesis ... |
| csbhypf 3910 | Introduce an explicit subs... |
| csbiebt 3911 | Conversion of implicit sub... |
| csbiedf 3912 | Conversion of implicit sub... |
| csbieb 3913 | Bidirectional conversion b... |
| csbiebg 3914 | Bidirectional conversion b... |
| csbiegf 3915 | Conversion of implicit sub... |
| csbief 3916 | Conversion of implicit sub... |
| csbie 3917 | Conversion of implicit sub... |
| csbied 3918 | Conversion of implicit sub... |
| csbied2 3919 | Conversion of implicit sub... |
| csbie2t 3920 | Conversion of implicit sub... |
| csbie2 3921 | Conversion of implicit sub... |
| csbie2g 3922 | Conversion of implicit sub... |
| cbvrabcsfw 3923 | Version of ~ cbvrabcsf wit... |
| cbvralcsf 3924 | A more general version of ... |
| cbvrexcsf 3925 | A more general version of ... |
| cbvreucsf 3926 | A more general version of ... |
| cbvrabcsf 3927 | A more general version of ... |
| cbvralv2 3928 | Rule used to change the bo... |
| cbvrexv2 3929 | Rule used to change the bo... |
| vtocl2dOLD 3930 | Obsolete version of ~ vtoc... |
| rspc2vd 3931 | Deduction version of 2-var... |
| difjust 3937 | Soundness justification th... |
| unjust 3939 | Soundness justification th... |
| injust 3941 | Soundness justification th... |
| dfin5 3943 | Alternate definition for t... |
| dfdif2 3944 | Alternate definition of cl... |
| eldif 3945 | Expansion of membership in... |
| eldifd 3946 | If a class is in one class... |
| eldifad 3947 | If a class is in the diffe... |
| eldifbd 3948 | If a class is in the diffe... |
| elneeldif 3949 | The elements of a set diff... |
| velcomp 3950 | Characterization of setvar... |
| dfss 3952 | Variant of subclass defini... |
| dfss2 3954 | Alternate definition of th... |
| dfss3 3955 | Alternate definition of su... |
| dfss6 3956 | Alternate definition of su... |
| dfss2f 3957 | Equivalence for subclass r... |
| dfss3f 3958 | Equivalence for subclass r... |
| nfss 3959 | If ` x ` is not free in ` ... |
| ssel 3960 | Membership relationships f... |
| ssel2 3961 | Membership relationships f... |
| sseli 3962 | Membership implication fro... |
| sselii 3963 | Membership inference from ... |
| sseldi 3964 | Membership inference from ... |
| sseld 3965 | Membership deduction from ... |
| sselda 3966 | Membership deduction from ... |
| sseldd 3967 | Membership inference from ... |
| ssneld 3968 | If a class is not in anoth... |
| ssneldd 3969 | If an element is not in a ... |
| ssriv 3970 | Inference based on subclas... |
| ssrd 3971 | Deduction based on subclas... |
| ssrdv 3972 | Deduction based on subclas... |
| sstr2 3973 | Transitivity of subclass r... |
| sstr 3974 | Transitivity of subclass r... |
| sstri 3975 | Subclass transitivity infe... |
| sstrd 3976 | Subclass transitivity dedu... |
| sstrid 3977 | Subclass transitivity dedu... |
| sstrdi 3978 | Subclass transitivity dedu... |
| sylan9ss 3979 | A subclass transitivity de... |
| sylan9ssr 3980 | A subclass transitivity de... |
| eqss 3981 | The subclass relationship ... |
| eqssi 3982 | Infer equality from two su... |
| eqssd 3983 | Equality deduction from tw... |
| sssseq 3984 | If a class is a subclass o... |
| eqrd 3985 | Deduce equality of classes... |
| eqri 3986 | Infer equality of classes ... |
| eqelssd 3987 | Equality deduction from su... |
| ssid 3988 | Any class is a subclass of... |
| ssidd 3989 | Weakening of ~ ssid . (Co... |
| ssv 3990 | Any class is a subclass of... |
| sseq1 3991 | Equality theorem for subcl... |
| sseq2 3992 | Equality theorem for the s... |
| sseq12 3993 | Equality theorem for the s... |
| sseq1i 3994 | An equality inference for ... |
| sseq2i 3995 | An equality inference for ... |
| sseq12i 3996 | An equality inference for ... |
| sseq1d 3997 | An equality deduction for ... |
| sseq2d 3998 | An equality deduction for ... |
| sseq12d 3999 | An equality deduction for ... |
| eqsstri 4000 | Substitution of equality i... |
| eqsstrri 4001 | Substitution of equality i... |
| sseqtri 4002 | Substitution of equality i... |
| sseqtrri 4003 | Substitution of equality i... |
| eqsstrd 4004 | Substitution of equality i... |
| eqsstrrd 4005 | Substitution of equality i... |
| sseqtrd 4006 | Substitution of equality i... |
| sseqtrrd 4007 | Substitution of equality i... |
| 3sstr3i 4008 | Substitution of equality i... |
| 3sstr4i 4009 | Substitution of equality i... |
| 3sstr3g 4010 | Substitution of equality i... |
| 3sstr4g 4011 | Substitution of equality i... |
| 3sstr3d 4012 | Substitution of equality i... |
| 3sstr4d 4013 | Substitution of equality i... |
| eqsstrid 4014 | A chained subclass and equ... |
| eqsstrrid 4015 | A chained subclass and equ... |
| sseqtrdi 4016 | A chained subclass and equ... |
| sseqtrrdi 4017 | A chained subclass and equ... |
| sseqtrid 4018 | Subclass transitivity dedu... |
| sseqtrrid 4019 | Subclass transitivity dedu... |
| eqsstrdi 4020 | A chained subclass and equ... |
| eqsstrrdi 4021 | A chained subclass and equ... |
| eqimss 4022 | Equality implies the subcl... |
| eqimss2 4023 | Equality implies the subcl... |
| eqimssi 4024 | Infer subclass relationshi... |
| eqimss2i 4025 | Infer subclass relationshi... |
| nssne1 4026 | Two classes are different ... |
| nssne2 4027 | Two classes are different ... |
| nss 4028 | Negation of subclass relat... |
| nelss 4029 | Demonstrate by witnesses t... |
| ssrexf 4030 | Restricted existential qua... |
| ssrmof 4031 | "At most one" existential ... |
| ssralv 4032 | Quantification restricted ... |
| ssrexv 4033 | Existential quantification... |
| ss2ralv 4034 | Two quantifications restri... |
| ss2rexv 4035 | Two existential quantifica... |
| ralss 4036 | Restricted universal quant... |
| rexss 4037 | Restricted existential qua... |
| ss2ab 4038 | Class abstractions in a su... |
| abss 4039 | Class abstraction in a sub... |
| ssab 4040 | Subclass of a class abstra... |
| ssabral 4041 | The relation for a subclas... |
| ss2abi 4042 | Inference of abstraction s... |
| ss2abdv 4043 | Deduction of abstraction s... |
| abssdv 4044 | Deduction of abstraction s... |
| abssi 4045 | Inference of abstraction s... |
| ss2rab 4046 | Restricted abstraction cla... |
| rabss 4047 | Restricted class abstracti... |
| ssrab 4048 | Subclass of a restricted c... |
| ssrabdv 4049 | Subclass of a restricted c... |
| rabssdv 4050 | Subclass of a restricted c... |
| ss2rabdv 4051 | Deduction of restricted ab... |
| ss2rabi 4052 | Inference of restricted ab... |
| rabss2 4053 | Subclass law for restricte... |
| ssab2 4054 | Subclass relation for the ... |
| ssrab2 4055 | Subclass relation for a re... |
| ssrab3 4056 | Subclass relation for a re... |
| rabssrabd 4057 | Subclass of a restricted c... |
| ssrabeq 4058 | If the restricting class o... |
| rabssab 4059 | A restricted class is a su... |
| uniiunlem 4060 | A subset relationship usef... |
| dfpss2 4061 | Alternate definition of pr... |
| dfpss3 4062 | Alternate definition of pr... |
| psseq1 4063 | Equality theorem for prope... |
| psseq2 4064 | Equality theorem for prope... |
| psseq1i 4065 | An equality inference for ... |
| psseq2i 4066 | An equality inference for ... |
| psseq12i 4067 | An equality inference for ... |
| psseq1d 4068 | An equality deduction for ... |
| psseq2d 4069 | An equality deduction for ... |
| psseq12d 4070 | An equality deduction for ... |
| pssss 4071 | A proper subclass is a sub... |
| pssne 4072 | Two classes in a proper su... |
| pssssd 4073 | Deduce subclass from prope... |
| pssned 4074 | Proper subclasses are uneq... |
| sspss 4075 | Subclass in terms of prope... |
| pssirr 4076 | Proper subclass is irrefle... |
| pssn2lp 4077 | Proper subclass has no 2-c... |
| sspsstri 4078 | Two ways of stating tricho... |
| ssnpss 4079 | Partial trichotomy law for... |
| psstr 4080 | Transitive law for proper ... |
| sspsstr 4081 | Transitive law for subclas... |
| psssstr 4082 | Transitive law for subclas... |
| psstrd 4083 | Proper subclass inclusion ... |
| sspsstrd 4084 | Transitivity involving sub... |
| psssstrd 4085 | Transitivity involving sub... |
| npss 4086 | A class is not a proper su... |
| ssnelpss 4087 | A subclass missing a membe... |
| ssnelpssd 4088 | Subclass inclusion with on... |
| ssexnelpss 4089 | If there is an element of ... |
| dfdif3 4090 | Alternate definition of cl... |
| difeq1 4091 | Equality theorem for class... |
| difeq2 4092 | Equality theorem for class... |
| difeq12 4093 | Equality theorem for class... |
| difeq1i 4094 | Inference adding differenc... |
| difeq2i 4095 | Inference adding differenc... |
| difeq12i 4096 | Equality inference for cla... |
| difeq1d 4097 | Deduction adding differenc... |
| difeq2d 4098 | Deduction adding differenc... |
| difeq12d 4099 | Equality deduction for cla... |
| difeqri 4100 | Inference from membership ... |
| nfdif 4101 | Bound-variable hypothesis ... |
| eldifi 4102 | Implication of membership ... |
| eldifn 4103 | Implication of membership ... |
| elndif 4104 | A set does not belong to a... |
| neldif 4105 | Implication of membership ... |
| difdif 4106 | Double class difference. ... |
| difss 4107 | Subclass relationship for ... |
| difssd 4108 | A difference of two classe... |
| difss2 4109 | If a class is contained in... |
| difss2d 4110 | If a class is contained in... |
| ssdifss 4111 | Preservation of a subclass... |
| ddif 4112 | Double complement under un... |
| ssconb 4113 | Contraposition law for sub... |
| sscon 4114 | Contraposition law for sub... |
| ssdif 4115 | Difference law for subsets... |
| ssdifd 4116 | If ` A ` is contained in `... |
| sscond 4117 | If ` A ` is contained in `... |
| ssdifssd 4118 | If ` A ` is contained in `... |
| ssdif2d 4119 | If ` A ` is contained in `... |
| raldifb 4120 | Restricted universal quant... |
| rexdifi 4121 | Restricted existential qua... |
| complss 4122 | Complementation reverses i... |
| compleq 4123 | Two classes are equal if a... |
| elun 4124 | Expansion of membership in... |
| elunnel1 4125 | A member of a union that i... |
| uneqri 4126 | Inference from membership ... |
| unidm 4127 | Idempotent law for union o... |
| uncom 4128 | Commutative law for union ... |
| equncom 4129 | If a class equals the unio... |
| equncomi 4130 | Inference form of ~ equnco... |
| uneq1 4131 | Equality theorem for the u... |
| uneq2 4132 | Equality theorem for the u... |
| uneq12 4133 | Equality theorem for the u... |
| uneq1i 4134 | Inference adding union to ... |
| uneq2i 4135 | Inference adding union to ... |
| uneq12i 4136 | Equality inference for the... |
| uneq1d 4137 | Deduction adding union to ... |
| uneq2d 4138 | Deduction adding union to ... |
| uneq12d 4139 | Equality deduction for the... |
| nfun 4140 | Bound-variable hypothesis ... |
| unass 4141 | Associative law for union ... |
| un12 4142 | A rearrangement of union. ... |
| un23 4143 | A rearrangement of union. ... |
| un4 4144 | A rearrangement of the uni... |
| unundi 4145 | Union distributes over its... |
| unundir 4146 | Union distributes over its... |
| ssun1 4147 | Subclass relationship for ... |
| ssun2 4148 | Subclass relationship for ... |
| ssun3 4149 | Subclass law for union of ... |
| ssun4 4150 | Subclass law for union of ... |
| elun1 4151 | Membership law for union o... |
| elun2 4152 | Membership law for union o... |
| elunant 4153 | A statement is true for ev... |
| unss1 4154 | Subclass law for union of ... |
| ssequn1 4155 | A relationship between sub... |
| unss2 4156 | Subclass law for union of ... |
| unss12 4157 | Subclass law for union of ... |
| ssequn2 4158 | A relationship between sub... |
| unss 4159 | The union of two subclasse... |
| unssi 4160 | An inference showing the u... |
| unssd 4161 | A deduction showing the un... |
| unssad 4162 | If ` ( A u. B ) ` is conta... |
| unssbd 4163 | If ` ( A u. B ) ` is conta... |
| ssun 4164 | A condition that implies i... |
| rexun 4165 | Restricted existential qua... |
| ralunb 4166 | Restricted quantification ... |
| ralun 4167 | Restricted quantification ... |
| elin 4168 | Expansion of membership in... |
| elini 4169 | Membership in an intersect... |
| elind 4170 | Deduce membership in an in... |
| elinel1 4171 | Membership in an intersect... |
| elinel2 4172 | Membership in an intersect... |
| elin2 4173 | Membership in a class defi... |
| elin1d 4174 | Elementhood in the first s... |
| elin2d 4175 | Elementhood in the first s... |
| elin3 4176 | Membership in a class defi... |
| incom 4177 | Commutative law for inters... |
| incomOLD 4178 | Obsolete version of ~ inco... |
| ineqri 4179 | Inference from membership ... |
| ineq1 4180 | Equality theorem for inter... |
| ineq1OLD 4181 | Obsolete version of ~ ineq... |
| ineq2 4182 | Equality theorem for inter... |
| ineq12 4183 | Equality theorem for inter... |
| ineq1i 4184 | Equality inference for int... |
| ineq2i 4185 | Equality inference for int... |
| ineq12i 4186 | Equality inference for int... |
| ineq1d 4187 | Equality deduction for int... |
| ineq2d 4188 | Equality deduction for int... |
| ineq12d 4189 | Equality deduction for int... |
| ineqan12d 4190 | Equality deduction for int... |
| sseqin2 4191 | A relationship between sub... |
| nfin 4192 | Bound-variable hypothesis ... |
| rabbi2dva 4193 | Deduction from a wff to a ... |
| inidm 4194 | Idempotent law for interse... |
| inass 4195 | Associative law for inters... |
| in12 4196 | A rearrangement of interse... |
| in32 4197 | A rearrangement of interse... |
| in13 4198 | A rearrangement of interse... |
| in31 4199 | A rearrangement of interse... |
| inrot 4200 | Rotate the intersection of... |
| in4 4201 | Rearrangement of intersect... |
| inindi 4202 | Intersection distributes o... |
| inindir 4203 | Intersection distributes o... |
| inss1 4204 | The intersection of two cl... |
| inss2 4205 | The intersection of two cl... |
| ssin 4206 | Subclass of intersection. ... |
| ssini 4207 | An inference showing that ... |
| ssind 4208 | A deduction showing that a... |
| ssrin 4209 | Add right intersection to ... |
| sslin 4210 | Add left intersection to s... |
| ssrind 4211 | Add right intersection to ... |
| ss2in 4212 | Intersection of subclasses... |
| ssinss1 4213 | Intersection preserves sub... |
| inss 4214 | Inclusion of an intersecti... |
| rexin 4215 | Restricted existential qua... |
| dfss7 4216 | Alternate definition of su... |
| symdifcom 4219 | Symmetric difference commu... |
| symdifeq1 4220 | Equality theorem for symme... |
| symdifeq2 4221 | Equality theorem for symme... |
| nfsymdif 4222 | Hypothesis builder for sym... |
| elsymdif 4223 | Membership in a symmetric ... |
| dfsymdif4 4224 | Alternate definition of th... |
| elsymdifxor 4225 | Membership in a symmetric ... |
| dfsymdif2 4226 | Alternate definition of th... |
| symdifass 4227 | Symmetric difference is as... |
| difsssymdif 4228 | The symmetric difference c... |
| difsymssdifssd 4229 | If the symmetric differenc... |
| unabs 4230 | Absorption law for union. ... |
| inabs 4231 | Absorption law for interse... |
| nssinpss 4232 | Negation of subclass expre... |
| nsspssun 4233 | Negation of subclass expre... |
| dfss4 4234 | Subclass defined in terms ... |
| dfun2 4235 | An alternate definition of... |
| dfin2 4236 | An alternate definition of... |
| difin 4237 | Difference with intersecti... |
| ssdifim 4238 | Implication of a class dif... |
| ssdifsym 4239 | Symmetric class difference... |
| dfss5 4240 | Alternate definition of su... |
| dfun3 4241 | Union defined in terms of ... |
| dfin3 4242 | Intersection defined in te... |
| dfin4 4243 | Alternate definition of th... |
| invdif 4244 | Intersection with universa... |
| indif 4245 | Intersection with class di... |
| indif2 4246 | Bring an intersection in a... |
| indif1 4247 | Bring an intersection in a... |
| indifcom 4248 | Commutation law for inters... |
| indi 4249 | Distributive law for inter... |
| undi 4250 | Distributive law for union... |
| indir 4251 | Distributive law for inter... |
| undir 4252 | Distributive law for union... |
| unineq 4253 | Infer equality from equali... |
| uneqin 4254 | Equality of union and inte... |
| difundi 4255 | Distributive law for class... |
| difundir 4256 | Distributive law for class... |
| difindi 4257 | Distributive law for class... |
| difindir 4258 | Distributive law for class... |
| indifdir 4259 | Distribute intersection ov... |
| difdif2 4260 | Class difference by a clas... |
| undm 4261 | De Morgan's law for union.... |
| indm 4262 | De Morgan's law for inters... |
| difun1 4263 | A relationship involving d... |
| undif3 4264 | An equality involving clas... |
| difin2 4265 | Represent a class differen... |
| dif32 4266 | Swap second and third argu... |
| difabs 4267 | Absorption-like law for cl... |
| dfsymdif3 4268 | Alternate definition of th... |
| unab 4269 | Union of two class abstrac... |
| inab 4270 | Intersection of two class ... |
| difab 4271 | Difference of two class ab... |
| notab 4272 | A class builder defined by... |
| unrab 4273 | Union of two restricted cl... |
| inrab 4274 | Intersection of two restri... |
| inrab2 4275 | Intersection with a restri... |
| difrab 4276 | Difference of two restrict... |
| dfrab3 4277 | Alternate definition of re... |
| dfrab2 4278 | Alternate definition of re... |
| notrab 4279 | Complementation of restric... |
| dfrab3ss 4280 | Restricted class abstracti... |
| rabun2 4281 | Abstraction restricted to ... |
| reuss2 4282 | Transfer uniqueness to a s... |
| reuss 4283 | Transfer uniqueness to a s... |
| reuun1 4284 | Transfer uniqueness to a s... |
| reuun2 4285 | Transfer uniqueness to a s... |
| reupick 4286 | Restricted uniqueness "pic... |
| reupick3 4287 | Restricted uniqueness "pic... |
| reupick2 4288 | Restricted uniqueness "pic... |
| euelss 4289 | Transfer uniqueness of an ... |
| dfnul2 4292 | Alternate definition of th... |
| dfnul2OLD 4293 | Obsolete version of ~ dfnu... |
| dfnul3 4294 | Alternate definition of th... |
| noel 4295 | The empty set has no eleme... |
| noelOLD 4296 | Obsolete version of ~ noel... |
| nel02 4297 | The empty set has no eleme... |
| n0i 4298 | If a class has elements, t... |
| ne0i 4299 | If a class has elements, t... |
| ne0d 4300 | Deduction form of ~ ne0i .... |
| n0ii 4301 | If a class has elements, t... |
| ne0ii 4302 | If a class has elements, t... |
| vn0 4303 | The universal class is not... |
| eq0f 4304 | A class is equal to the em... |
| neq0f 4305 | A class is not empty if an... |
| n0f 4306 | A class is nonempty if and... |
| eq0 4307 | A class is equal to the em... |
| neq0 4308 | A class is not empty if an... |
| n0 4309 | A class is nonempty if and... |
| nel0 4310 | From the general negation ... |
| reximdva0 4311 | Restricted existence deduc... |
| rspn0 4312 | Specialization for restric... |
| n0rex 4313 | There is an element in a n... |
| ssn0rex 4314 | There is an element in a c... |
| n0moeu 4315 | A case of equivalence of "... |
| rex0 4316 | Vacuous restricted existen... |
| reu0 4317 | Vacuous restricted uniquen... |
| rmo0 4318 | Vacuous restricted at-most... |
| 0el 4319 | Membership of the empty se... |
| n0el 4320 | Negated membership of the ... |
| eqeuel 4321 | A condition which implies ... |
| ssdif0 4322 | Subclass expressed in term... |
| difn0 4323 | If the difference of two s... |
| pssdifn0 4324 | A proper subclass has a no... |
| pssdif 4325 | A proper subclass has a no... |
| ndisj 4326 | Express that an intersecti... |
| difin0ss 4327 | Difference, intersection, ... |
| inssdif0 4328 | Intersection, subclass, an... |
| difid 4329 | The difference between a c... |
| difidALT 4330 | Alternate proof of ~ difid... |
| dif0 4331 | The difference between a c... |
| ab0 4332 | The class of sets verifyin... |
| dfnf5 4333 | Characterization of non-fr... |
| ab0orv 4334 | The class builder of a wff... |
| abn0 4335 | Nonempty class abstraction... |
| rab0 4336 | Any restricted class abstr... |
| rabeq0 4337 | Condition for a restricted... |
| rabn0 4338 | Nonempty restricted class ... |
| rabxm 4339 | Law of excluded middle, in... |
| rabnc 4340 | Law of noncontradiction, i... |
| elneldisj 4341 | The set of elements ` s ` ... |
| elnelun 4342 | The union of the set of el... |
| un0 4343 | The union of a class with ... |
| in0 4344 | The intersection of a clas... |
| 0un 4345 | The union of the empty set... |
| 0in 4346 | The intersection of the em... |
| inv1 4347 | The intersection of a clas... |
| unv 4348 | The union of a class with ... |
| 0ss 4349 | The null set is a subset o... |
| ss0b 4350 | Any subset of the empty se... |
| ss0 4351 | Any subset of the empty se... |
| sseq0 4352 | A subclass of an empty cla... |
| ssn0 4353 | A class with a nonempty su... |
| 0dif 4354 | The difference between the... |
| abf 4355 | A class builder with a fal... |
| eq0rdv 4356 | Deduction for equality to ... |
| csbprc 4357 | The proper substitution of... |
| csb0 4358 | The proper substitution of... |
| sbcel12 4359 | Distribute proper substitu... |
| sbceqg 4360 | Distribute proper substitu... |
| sbceqi 4361 | Distribution of class subs... |
| sbcnel12g 4362 | Distribute proper substitu... |
| sbcne12 4363 | Distribute proper substitu... |
| sbcel1g 4364 | Move proper substitution i... |
| sbceq1g 4365 | Move proper substitution t... |
| sbcel2 4366 | Move proper substitution i... |
| sbceq2g 4367 | Move proper substitution t... |
| csbcom 4368 | Commutative law for double... |
| sbcnestgfw 4369 | Nest the composition of tw... |
| csbnestgfw 4370 | Nest the composition of tw... |
| sbcnestgw 4371 | Nest the composition of tw... |
| csbnestgw 4372 | Nest the composition of tw... |
| sbcco3gw 4373 | Composition of two substit... |
| sbcnestgf 4374 | Nest the composition of tw... |
| csbnestgf 4375 | Nest the composition of tw... |
| sbcnestg 4376 | Nest the composition of tw... |
| csbnestg 4377 | Nest the composition of tw... |
| sbcco3g 4378 | Composition of two substit... |
| csbco3g 4379 | Composition of two class s... |
| csbnest1g 4380 | Nest the composition of tw... |
| csbidm 4381 | Idempotent law for class s... |
| csbvarg 4382 | The proper substitution of... |
| csbvargi 4383 | The proper substitution of... |
| sbccsb 4384 | Substitution into a wff ex... |
| sbccsb2 4385 | Substitution into a wff ex... |
| rspcsbela 4386 | Special case related to ~ ... |
| sbnfc2 4387 | Two ways of expressing " `... |
| csbab 4388 | Move substitution into a c... |
| csbun 4389 | Distribution of class subs... |
| csbin 4390 | Distribute proper substitu... |
| csbie2df 4391 | Conversion of implicit sub... |
| 2nreu 4392 | If there are two different... |
| un00 4393 | Two classes are empty iff ... |
| vss 4394 | Only the universal class h... |
| 0pss 4395 | The null set is a proper s... |
| npss0 4396 | No set is a proper subset ... |
| pssv 4397 | Any non-universal class is... |
| disj 4398 | Two ways of saying that tw... |
| disjr 4399 | Two ways of saying that tw... |
| disj1 4400 | Two ways of saying that tw... |
| reldisj 4401 | Two ways of saying that tw... |
| disj3 4402 | Two ways of saying that tw... |
| disjne 4403 | Members of disjoint sets a... |
| disjeq0 4404 | Two disjoint sets are equa... |
| disjel 4405 | A set can't belong to both... |
| disj2 4406 | Two ways of saying that tw... |
| disj4 4407 | Two ways of saying that tw... |
| ssdisj 4408 | Intersection with a subcla... |
| disjpss 4409 | A class is a proper subset... |
| undisj1 4410 | The union of disjoint clas... |
| undisj2 4411 | The union of disjoint clas... |
| ssindif0 4412 | Subclass expressed in term... |
| inelcm 4413 | The intersection of classe... |
| minel 4414 | A minimum element of a cla... |
| undif4 4415 | Distribute union over diff... |
| disjssun 4416 | Subset relation for disjoi... |
| vdif0 4417 | Universal class equality i... |
| difrab0eq 4418 | If the difference between ... |
| pssnel 4419 | A proper subclass has a me... |
| disjdif 4420 | A class and its relative c... |
| difin0 4421 | The difference of a class ... |
| unvdif 4422 | The union of a class and i... |
| undif1 4423 | Absorption of difference b... |
| undif2 4424 | Absorption of difference b... |
| undifabs 4425 | Absorption of difference b... |
| inundif 4426 | The intersection and class... |
| disjdif2 4427 | The difference of a class ... |
| difun2 4428 | Absorption of union by dif... |
| undif 4429 | Union of complementary par... |
| ssdifin0 4430 | A subset of a difference d... |
| ssdifeq0 4431 | A class is a subclass of i... |
| ssundif 4432 | A condition equivalent to ... |
| difcom 4433 | Swap the arguments of a cl... |
| pssdifcom1 4434 | Two ways to express overla... |
| pssdifcom2 4435 | Two ways to express non-co... |
| difdifdir 4436 | Distributive law for class... |
| uneqdifeq 4437 | Two ways to say that ` A `... |
| raldifeq 4438 | Equality theorem for restr... |
| r19.2z 4439 | Theorem 19.2 of [Margaris]... |
| r19.2zb 4440 | A response to the notion t... |
| r19.3rz 4441 | Restricted quantification ... |
| r19.28z 4442 | Restricted quantifier vers... |
| r19.3rzv 4443 | Restricted quantification ... |
| r19.9rzv 4444 | Restricted quantification ... |
| r19.28zv 4445 | Restricted quantifier vers... |
| r19.37zv 4446 | Restricted quantifier vers... |
| r19.45zv 4447 | Restricted version of Theo... |
| r19.44zv 4448 | Restricted version of Theo... |
| r19.27z 4449 | Restricted quantifier vers... |
| r19.27zv 4450 | Restricted quantifier vers... |
| r19.36zv 4451 | Restricted quantifier vers... |
| rzal 4452 | Vacuous quantification is ... |
| rexn0 4453 | Restricted existential qua... |
| ralidm 4454 | Idempotent law for restric... |
| ral0 4455 | Vacuous universal quantifi... |
| ralf0 4456 | The quantification of a fa... |
| ralnralall 4457 | A contradiction concerning... |
| falseral0 4458 | A false statement can only... |
| raaan 4459 | Rearrange restricted quant... |
| raaanv 4460 | Rearrange restricted quant... |
| sbss 4461 | Set substitution into the ... |
| sbcssg 4462 | Distribute proper substitu... |
| raaan2 4463 | Rearrange restricted quant... |
| 2reu4lem 4464 | Lemma for ~ 2reu4 . (Cont... |
| 2reu4 4465 | Definition of double restr... |
| dfif2 4468 | An alternate definition of... |
| dfif6 4469 | An alternate definition of... |
| ifeq1 4470 | Equality theorem for condi... |
| ifeq2 4471 | Equality theorem for condi... |
| iftrue 4472 | Value of the conditional o... |
| iftruei 4473 | Inference associated with ... |
| iftrued 4474 | Value of the conditional o... |
| iffalse 4475 | Value of the conditional o... |
| iffalsei 4476 | Inference associated with ... |
| iffalsed 4477 | Value of the conditional o... |
| ifnefalse 4478 | When values are unequal, b... |
| ifsb 4479 | Distribute a function over... |
| dfif3 4480 | Alternate definition of th... |
| dfif4 4481 | Alternate definition of th... |
| dfif5 4482 | Alternate definition of th... |
| ifeq12 4483 | Equality theorem for condi... |
| ifeq1d 4484 | Equality deduction for con... |
| ifeq2d 4485 | Equality deduction for con... |
| ifeq12d 4486 | Equality deduction for con... |
| ifbi 4487 | Equivalence theorem for co... |
| ifbid 4488 | Equivalence deduction for ... |
| ifbieq1d 4489 | Equivalence/equality deduc... |
| ifbieq2i 4490 | Equivalence/equality infer... |
| ifbieq2d 4491 | Equivalence/equality deduc... |
| ifbieq12i 4492 | Equivalence deduction for ... |
| ifbieq12d 4493 | Equivalence deduction for ... |
| nfifd 4494 | Deduction form of ~ nfif .... |
| nfif 4495 | Bound-variable hypothesis ... |
| ifeq1da 4496 | Conditional equality. (Co... |
| ifeq2da 4497 | Conditional equality. (Co... |
| ifeq12da 4498 | Equivalence deduction for ... |
| ifbieq12d2 4499 | Equivalence deduction for ... |
| ifclda 4500 | Conditional closure. (Con... |
| ifeqda 4501 | Separation of the values o... |
| elimif 4502 | Elimination of a condition... |
| ifbothda 4503 | A wff ` th ` containing a ... |
| ifboth 4504 | A wff ` th ` containing a ... |
| ifid 4505 | Identical true and false a... |
| eqif 4506 | Expansion of an equality w... |
| ifval 4507 | Another expression of the ... |
| elif 4508 | Membership in a conditiona... |
| ifel 4509 | Membership of a conditiona... |
| ifcl 4510 | Membership (closure) of a ... |
| ifcld 4511 | Membership (closure) of a ... |
| ifcli 4512 | Inference associated with ... |
| ifexg 4513 | Conditional operator exist... |
| ifex 4514 | Conditional operator exist... |
| ifeqor 4515 | The possible values of a c... |
| ifnot 4516 | Negating the first argumen... |
| ifan 4517 | Rewrite a conjunction in a... |
| ifor 4518 | Rewrite a disjunction in a... |
| 2if2 4519 | Resolve two nested conditi... |
| ifcomnan 4520 | Commute the conditions in ... |
| csbif 4521 | Distribute proper substitu... |
| dedth 4522 | Weak deduction theorem tha... |
| dedth2h 4523 | Weak deduction theorem eli... |
| dedth3h 4524 | Weak deduction theorem eli... |
| dedth4h 4525 | Weak deduction theorem eli... |
| dedth2v 4526 | Weak deduction theorem for... |
| dedth3v 4527 | Weak deduction theorem for... |
| dedth4v 4528 | Weak deduction theorem for... |
| elimhyp 4529 | Eliminate a hypothesis con... |
| elimhyp2v 4530 | Eliminate a hypothesis con... |
| elimhyp3v 4531 | Eliminate a hypothesis con... |
| elimhyp4v 4532 | Eliminate a hypothesis con... |
| elimel 4533 | Eliminate a membership hyp... |
| elimdhyp 4534 | Version of ~ elimhyp where... |
| keephyp 4535 | Transform a hypothesis ` p... |
| keephyp2v 4536 | Keep a hypothesis containi... |
| keephyp3v 4537 | Keep a hypothesis containi... |
| pwjust 4539 | Soundness justification th... |
| pweq 4541 | Equality theorem for power... |
| pweqi 4542 | Equality inference for pow... |
| pweqd 4543 | Equality deduction for pow... |
| elpwg 4544 | Membership in a power clas... |
| elpw 4545 | Membership in a power clas... |
| velpw 4546 | Setvar variable membership... |
| elpwOLD 4547 | Obsolete proof of ~ elpw a... |
| elpwgOLD 4548 | Obsolete proof of ~ elpwg ... |
| elpwd 4549 | Membership in a power clas... |
| elpwi 4550 | Subset relation implied by... |
| elpwb 4551 | Characterization of the el... |
| elpwid 4552 | An element of a power clas... |
| elelpwi 4553 | If ` A ` belongs to a part... |
| nfpw 4554 | Bound-variable hypothesis ... |
| pwidg 4555 | A set is an element of its... |
| pwidb 4556 | A class is an element of i... |
| pwid 4557 | A set is a member of its p... |
| pwss 4558 | Subclass relationship for ... |
| snjust 4559 | Soundness justification th... |
| sneq 4570 | Equality theorem for singl... |
| sneqi 4571 | Equality inference for sin... |
| sneqd 4572 | Equality deduction for sin... |
| dfsn2 4573 | Alternate definition of si... |
| elsng 4574 | There is exactly one eleme... |
| elsn 4575 | There is exactly one eleme... |
| velsn 4576 | There is only one element ... |
| elsni 4577 | There is only one element ... |
| absn 4578 | Condition for a class abst... |
| dfpr2 4579 | Alternate definition of un... |
| dfsn2ALT 4580 | Alternate definition of si... |
| elprg 4581 | A member of an unordered p... |
| elpri 4582 | If a class is an element o... |
| elpr 4583 | A member of an unordered p... |
| elpr2 4584 | A member of an unordered p... |
| 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 ... |
| rexreusng 4610 | Restricted existential uni... |
| exsnrex 4611 | There is a set being the e... |
| ralsn 4612 | Convert a quantification o... |
| rexsn 4613 | Restricted existential qua... |
| elpwunsn 4614 | Membership in an extension... |
| eqoreldif 4615 | An element of a set is eit... |
| eltpg 4616 | Members of an unordered tr... |
| eldiftp 4617 | Membership in a set with t... |
| eltpi 4618 | A member of an unordered t... |
| eltp 4619 | A member of an unordered t... |
| dftp2 4620 | Alternate definition of un... |
| nfpr 4621 | Bound-variable hypothesis ... |
| ifpr 4622 | Membership of a conditiona... |
| ralprgf 4623 | Convert a restricted unive... |
| rexprgf 4624 | Convert a restricted exist... |
| ralprg 4625 | Convert a restricted unive... |
| rexprg 4626 | Convert a restricted exist... |
| raltpg 4627 | Convert a restricted unive... |
| rextpg 4628 | Convert a restricted exist... |
| ralpr 4629 | Convert a restricted unive... |
| rexpr 4630 | Convert a restricted exist... |
| reuprg0 4631 | Convert a restricted exist... |
| reuprg 4632 | Convert a restricted exist... |
| reurexprg 4633 | Convert a restricted exist... |
| raltp 4634 | Convert a quantification o... |
| rextp 4635 | Convert a quantification o... |
| nfsn 4636 | Bound-variable hypothesis ... |
| csbsng 4637 | Distribute proper substitu... |
| csbprg 4638 | Distribute proper substitu... |
| elinsn 4639 | If the intersection of two... |
| disjsn 4640 | Intersection with the sing... |
| disjsn2 4641 | Two distinct singletons ar... |
| disjpr2 4642 | Two completely distinct un... |
| disjprsn 4643 | The disjoint intersection ... |
| disjtpsn 4644 | The disjoint intersection ... |
| disjtp2 4645 | Two completely distinct un... |
| snprc 4646 | The singleton of a proper ... |
| snnzb 4647 | A singleton is nonempty if... |
| rmosn 4648 | A restricted at-most-one q... |
| r19.12sn 4649 | Special case of ~ r19.12 w... |
| rabsn 4650 | Condition where a restrict... |
| rabsnifsb 4651 | A restricted class abstrac... |
| rabsnif 4652 | A restricted class abstrac... |
| rabrsn 4653 | A restricted class abstrac... |
| euabsn2 4654 | Another way to express exi... |
| euabsn 4655 | Another way to express exi... |
| reusn 4656 | A way to express restricte... |
| absneu 4657 | Restricted existential uni... |
| rabsneu 4658 | Restricted existential uni... |
| eusn 4659 | Two ways to express " ` A ... |
| rabsnt 4660 | Truth implied by equality ... |
| prcom 4661 | Commutative law for unorde... |
| preq1 4662 | Equality theorem for unord... |
| preq2 4663 | Equality theorem for unord... |
| preq12 4664 | Equality theorem for unord... |
| preq1i 4665 | Equality inference for uno... |
| preq2i 4666 | Equality inference for uno... |
| preq12i 4667 | Equality inference for uno... |
| preq1d 4668 | Equality deduction for uno... |
| preq2d 4669 | Equality deduction for uno... |
| preq12d 4670 | Equality deduction for uno... |
| tpeq1 4671 | Equality theorem for unord... |
| tpeq2 4672 | Equality theorem for unord... |
| tpeq3 4673 | Equality theorem for unord... |
| tpeq1d 4674 | Equality theorem for unord... |
| tpeq2d 4675 | Equality theorem for unord... |
| tpeq3d 4676 | Equality theorem for unord... |
| tpeq123d 4677 | Equality theorem for unord... |
| tprot 4678 | Rotation of the elements o... |
| tpcoma 4679 | Swap 1st and 2nd members o... |
| tpcomb 4680 | Swap 2nd and 3rd members o... |
| tpass 4681 | Split off the first elemen... |
| qdass 4682 | Two ways to write an unord... |
| qdassr 4683 | Two ways to write an unord... |
| tpidm12 4684 | Unordered triple ` { A , A... |
| tpidm13 4685 | Unordered triple ` { A , B... |
| tpidm23 4686 | Unordered triple ` { A , B... |
| tpidm 4687 | Unordered triple ` { A , A... |
| tppreq3 4688 | An unordered triple is an ... |
| prid1g 4689 | An unordered pair contains... |
| prid2g 4690 | An unordered pair contains... |
| prid1 4691 | An unordered pair contains... |
| prid2 4692 | An unordered pair contains... |
| ifpprsnss 4693 | An unordered pair is a sin... |
| prprc1 4694 | A proper class vanishes in... |
| prprc2 4695 | A proper class vanishes in... |
| prprc 4696 | An unordered pair containi... |
| tpid1 4697 | One of the three elements ... |
| tpid1g 4698 | Closed theorem form of ~ t... |
| tpid2 4699 | One of the three elements ... |
| tpid2g 4700 | Closed theorem form of ~ t... |
| tpid3g 4701 | Closed theorem form of ~ t... |
| tpid3 4702 | One of the three elements ... |
| snnzg 4703 | The singleton of a set is ... |
| snnz 4704 | The singleton of a set is ... |
| prnz 4705 | A pair containing a set is... |
| prnzg 4706 | A pair containing a set is... |
| tpnz 4707 | A triplet containing a set... |
| tpnzd 4708 | A triplet containing a set... |
| raltpd 4709 | Convert a quantification o... |
| snssg 4710 | The singleton of an elemen... |
| snss 4711 | The singleton of an elemen... |
| eldifsn 4712 | Membership in a set with a... |
| ssdifsn 4713 | Subset of a set with an el... |
| elpwdifsn 4714 | A subset of a set is an el... |
| eldifsni 4715 | Membership in a set with a... |
| eldifsnneq 4716 | An element of a difference... |
| eldifsnneqOLD 4717 | Obsolete version of ~ eldi... |
| neldifsn 4718 | The class ` A ` is not in ... |
| neldifsnd 4719 | The class ` A ` is not in ... |
| rexdifsn 4720 | Restricted existential qua... |
| raldifsni 4721 | Rearrangement of a propert... |
| raldifsnb 4722 | Restricted universal quant... |
| eldifvsn 4723 | A set is an element of the... |
| difsn 4724 | An element not in a set ca... |
| difprsnss 4725 | Removal of a singleton fro... |
| difprsn1 4726 | Removal of a singleton fro... |
| difprsn2 4727 | Removal of a singleton fro... |
| diftpsn3 4728 | Removal of a singleton fro... |
| difpr 4729 | Removing two elements as p... |
| tpprceq3 4730 | An unordered triple is an ... |
| tppreqb 4731 | An unordered triple is an ... |
| difsnb 4732 | ` ( B \ { A } ) ` equals `... |
| difsnpss 4733 | ` ( B \ { A } ) ` is a pro... |
| snssi 4734 | The singleton of an elemen... |
| snssd 4735 | The singleton of an elemen... |
| difsnid 4736 | If we remove a single elem... |
| eldifeldifsn 4737 | An element of a difference... |
| pw0 4738 | Compute the power set of t... |
| pwpw0 4739 | Compute the power set of t... |
| snsspr1 4740 | A singleton is a subset of... |
| snsspr2 4741 | A singleton is a subset of... |
| snsstp1 4742 | A singleton is a subset of... |
| snsstp2 4743 | A singleton is a subset of... |
| snsstp3 4744 | A singleton is a subset of... |
| prssg 4745 | A pair of elements of a cl... |
| prss 4746 | A pair of elements of a cl... |
| prssi 4747 | A pair of elements of a cl... |
| prssd 4748 | Deduction version of ~ prs... |
| prsspwg 4749 | An unordered pair belongs ... |
| ssprss 4750 | A pair as subset of a pair... |
| ssprsseq 4751 | A proper pair is a subset ... |
| sssn 4752 | The subsets of a singleton... |
| ssunsn2 4753 | The property of being sand... |
| ssunsn 4754 | Possible values for a set ... |
| eqsn 4755 | Two ways to express that a... |
| issn 4756 | A sufficient condition for... |
| n0snor2el 4757 | A nonempty set is either a... |
| ssunpr 4758 | Possible values for a set ... |
| sspr 4759 | The subsets of a pair. (C... |
| sstp 4760 | The subsets of a triple. ... |
| tpss 4761 | A triplet of elements of a... |
| tpssi 4762 | A triple of elements of a ... |
| sneqrg 4763 | Closed form of ~ sneqr . ... |
| sneqr 4764 | If the singletons of two s... |
| snsssn 4765 | If a singleton is a subset... |
| mosneq 4766 | There exists at most one s... |
| sneqbg 4767 | Two singletons of sets are... |
| snsspw 4768 | The singleton of a class i... |
| prsspw 4769 | An unordered pair belongs ... |
| preq1b 4770 | Biconditional equality lem... |
| preq2b 4771 | Biconditional equality lem... |
| preqr1 4772 | Reverse equality lemma for... |
| preqr2 4773 | Reverse equality lemma for... |
| preq12b 4774 | Equality relationship for ... |
| opthpr 4775 | An unordered pair has the ... |
| preqr1g 4776 | Reverse equality lemma for... |
| preq12bg 4777 | Closed form of ~ preq12b .... |
| prneimg 4778 | Two pairs are not equal if... |
| prnebg 4779 | A (proper) pair is not equ... |
| pr1eqbg 4780 | A (proper) pair is equal t... |
| pr1nebg 4781 | A (proper) pair is not equ... |
| preqsnd 4782 | Equivalence for a pair equ... |
| prnesn 4783 | A proper unordered pair is... |
| prneprprc 4784 | A proper unordered pair is... |
| preqsn 4785 | Equivalence for a pair equ... |
| preq12nebg 4786 | Equality relationship for ... |
| prel12g 4787 | Equality of two unordered ... |
| opthprneg 4788 | An unordered pair has the ... |
| elpreqprlem 4789 | Lemma for ~ elpreqpr . (C... |
| elpreqpr 4790 | Equality and membership ru... |
| elpreqprb 4791 | A set is an element of an ... |
| elpr2elpr 4792 | For an element ` A ` of an... |
| dfopif 4793 | Rewrite ~ df-op using ` if... |
| dfopg 4794 | Value of the ordered pair ... |
| dfop 4795 | Value of an ordered pair w... |
| opeq1 4796 | Equality theorem for order... |
| opeq2 4797 | Equality theorem for order... |
| opeq12 4798 | Equality theorem for order... |
| opeq1i 4799 | Equality inference for ord... |
| opeq2i 4800 | Equality inference for ord... |
| opeq12i 4801 | Equality inference for ord... |
| opeq1d 4802 | Equality deduction for ord... |
| opeq2d 4803 | Equality deduction for ord... |
| opeq12d 4804 | Equality deduction for ord... |
| oteq1 4805 | Equality theorem for order... |
| oteq2 4806 | Equality theorem for order... |
| oteq3 4807 | Equality theorem for order... |
| oteq1d 4808 | Equality deduction for ord... |
| oteq2d 4809 | Equality deduction for ord... |
| oteq3d 4810 | Equality deduction for ord... |
| oteq123d 4811 | Equality deduction for ord... |
| nfop 4812 | Bound-variable hypothesis ... |
| nfopd 4813 | Deduction version of bound... |
| csbopg 4814 | Distribution of class subs... |
| opidg 4815 | The ordered pair ` <. A , ... |
| opid 4816 | The ordered pair ` <. A , ... |
| ralunsn 4817 | Restricted quantification ... |
| 2ralunsn 4818 | Double restricted quantifi... |
| opprc 4819 | Expansion of an ordered pa... |
| opprc1 4820 | Expansion of an ordered pa... |
| opprc2 4821 | Expansion of an ordered pa... |
| oprcl 4822 | If an ordered pair has an ... |
| pwsn 4823 | The power set of a singlet... |
| pwsnALT 4824 | Alternate proof of ~ pwsn ... |
| pwpr 4825 | The power set of an unorde... |
| pwtp 4826 | The power set of an unorde... |
| pwpwpw0 4827 | Compute the power set of t... |
| pwv 4828 | The power class of the uni... |
| prproe 4829 | For an element of a proper... |
| 3elpr2eq 4830 | If there are three element... |
| dfuni2 4833 | Alternate definition of cl... |
| eluni 4834 | Membership in class union.... |
| eluni2 4835 | Membership in class union.... |
| elunii 4836 | Membership in class union.... |
| nfunid 4837 | Deduction version of ~ nfu... |
| nfuni 4838 | Bound-variable hypothesis ... |
| unieq 4839 | Equality theorem for class... |
| unieqi 4840 | Inference of equality of t... |
| unieqd 4841 | Deduction of equality of t... |
| eluniab 4842 | Membership in union of a c... |
| elunirab 4843 | Membership in union of a c... |
| unipr 4844 | The union of a pair is the... |
| uniprg 4845 | The union of a pair is the... |
| unisng 4846 | A set equals the union of ... |
| unisn 4847 | A set equals the union of ... |
| unisn3 4848 | Union of a singleton in th... |
| dfnfc2 4849 | An alternative statement o... |
| uniun 4850 | The class union of the uni... |
| uniin 4851 | The class union of the int... |
| uniss 4852 | Subclass relationship for ... |
| ssuni 4853 | Subclass relationship for ... |
| unissi 4854 | Subclass relationship for ... |
| unissd 4855 | Subclass relationship for ... |
| uni0b 4856 | The union of a set is empt... |
| uni0c 4857 | The union of a set is empt... |
| uni0 4858 | The union of the empty set... |
| csbuni 4859 | Distribute proper substitu... |
| elssuni 4860 | An element of a class is a... |
| unissel 4861 | Condition turning a subcla... |
| unissb 4862 | Relationship involving mem... |
| uniss2 4863 | A subclass condition on th... |
| unidif 4864 | If the difference ` A \ B ... |
| ssunieq 4865 | Relationship implying unio... |
| unimax 4866 | Any member of a class is t... |
| pwuni 4867 | A class is a subclass of t... |
| dfint2 4870 | Alternate definition of cl... |
| inteq 4871 | Equality law for intersect... |
| inteqi 4872 | Equality inference for cla... |
| inteqd 4873 | Equality deduction for cla... |
| elint 4874 | Membership in class inters... |
| elint2 4875 | Membership in class inters... |
| elintg 4876 | Membership in class inters... |
| elinti 4877 | Membership in class inters... |
| nfint 4878 | Bound-variable hypothesis ... |
| elintab 4879 | Membership in the intersec... |
| elintrab 4880 | Membership in the intersec... |
| elintrabg 4881 | Membership in the intersec... |
| int0 4882 | The intersection of the em... |
| intss1 4883 | An element of a class incl... |
| ssint 4884 | Subclass of a class inters... |
| ssintab 4885 | Subclass of the intersecti... |
| ssintub 4886 | Subclass of the least uppe... |
| ssmin 4887 | Subclass of the minimum va... |
| intmin 4888 | Any member of a class is t... |
| intss 4889 | Intersection of subclasses... |
| intssuni 4890 | The intersection of a none... |
| ssintrab 4891 | Subclass of the intersecti... |
| unissint 4892 | If the union of a class is... |
| intssuni2 4893 | Subclass relationship for ... |
| intminss 4894 | Under subset ordering, the... |
| intmin2 4895 | Any set is the smallest of... |
| intmin3 4896 | Under subset ordering, the... |
| intmin4 4897 | Elimination of a conjunct ... |
| intab 4898 | The intersection of a spec... |
| int0el 4899 | The intersection of a clas... |
| intun 4900 | The class intersection of ... |
| intpr 4901 | The intersection of a pair... |
| intprg 4902 | The intersection of a pair... |
| intsng 4903 | Intersection of a singleto... |
| intsn 4904 | The intersection of a sing... |
| uniintsn 4905 | Two ways to express " ` A ... |
| uniintab 4906 | The union and the intersec... |
| intunsn 4907 | Theorem joining a singleto... |
| rint0 4908 | Relative intersection of a... |
| elrint 4909 | Membership in a restricted... |
| elrint2 4910 | Membership in a restricted... |
| eliun 4915 | Membership in indexed unio... |
| eliin 4916 | Membership in indexed inte... |
| eliuni 4917 | Membership in an indexed u... |
| iuncom 4918 | Commutation of indexed uni... |
| iuncom4 4919 | Commutation of union with ... |
| iunconst 4920 | Indexed union of a constan... |
| iinconst 4921 | Indexed intersection of a ... |
| iuneqconst 4922 | Indexed union of identical... |
| iuniin 4923 | Law combining indexed unio... |
| iinssiun 4924 | An indexed intersection is... |
| iunss1 4925 | Subclass theorem for index... |
| iinss1 4926 | Subclass theorem for index... |
| iuneq1 4927 | Equality theorem for index... |
| iineq1 4928 | Equality theorem for index... |
| ss2iun 4929 | Subclass theorem for index... |
| iuneq2 4930 | Equality theorem for index... |
| iineq2 4931 | Equality theorem for index... |
| iuneq2i 4932 | Equality inference for ind... |
| iineq2i 4933 | Equality inference for ind... |
| iineq2d 4934 | Equality deduction for ind... |
| iuneq2dv 4935 | Equality deduction for ind... |
| iineq2dv 4936 | Equality deduction for ind... |
| iuneq12df 4937 | Equality deduction for ind... |
| iuneq1d 4938 | Equality theorem for index... |
| iuneq12d 4939 | Equality deduction for ind... |
| iuneq2d 4940 | Equality deduction for ind... |
| nfiun 4941 | Bound-variable hypothesis ... |
| nfiin 4942 | Bound-variable hypothesis ... |
| nfiung 4943 | Bound-variable hypothesis ... |
| nfiing 4944 | Bound-variable hypothesis ... |
| nfiu1 4945 | Bound-variable hypothesis ... |
| nfii1 4946 | Bound-variable hypothesis ... |
| dfiun2g 4947 | Alternate definition of in... |
| dfiun2gOLD 4948 | Obsolete proof of ~ dfiun2... |
| dfiin2g 4949 | Alternate definition of in... |
| dfiun2 4950 | Alternate definition of in... |
| dfiin2 4951 | Alternate definition of in... |
| dfiunv2 4952 | Define double indexed unio... |
| cbviun 4953 | Rule used to change the bo... |
| cbviin 4954 | Change bound variables in ... |
| cbviung 4955 | Rule used to change the bo... |
| cbviing 4956 | Change bound variables in ... |
| cbviunv 4957 | Rule used to change the bo... |
| cbviinv 4958 | Change bound variables in ... |
| cbviunvg 4959 | Rule used to change the bo... |
| cbviinvg 4960 | Change bound variables in ... |
| iunss 4961 | Subset theorem for an inde... |
| ssiun 4962 | Subset implication for an ... |
| ssiun2 4963 | Identity law for subset of... |
| ssiun2s 4964 | Subset relationship for an... |
| iunss2 4965 | A subclass condition on th... |
| iunssd 4966 | Subset theorem for an inde... |
| iunab 4967 | The indexed union of a cla... |
| iunrab 4968 | The indexed union of a res... |
| iunxdif2 4969 | Indexed union with a class... |
| ssiinf 4970 | Subset theorem for an inde... |
| ssiin 4971 | Subset theorem for an inde... |
| iinss 4972 | Subset implication for an ... |
| iinss2 4973 | An indexed intersection is... |
| uniiun 4974 | Class union in terms of in... |
| intiin 4975 | Class intersection in term... |
| iunid 4976 | An indexed union of single... |
| iun0 4977 | An indexed union of the em... |
| 0iun 4978 | An empty indexed union is ... |
| 0iin 4979 | An empty indexed intersect... |
| viin 4980 | Indexed intersection with ... |
| iunn0 4981 | There is a nonempty class ... |
| iinab 4982 | Indexed intersection of a ... |
| iinrab 4983 | Indexed intersection of a ... |
| iinrab2 4984 | Indexed intersection of a ... |
| iunin2 4985 | Indexed union of intersect... |
| iunin1 4986 | Indexed union of intersect... |
| iinun2 4987 | Indexed intersection of un... |
| iundif2 4988 | Indexed union of class dif... |
| iindif1 4989 | Indexed intersection of cl... |
| 2iunin 4990 | Rearrange indexed unions o... |
| iindif2 4991 | Indexed intersection of cl... |
| iinin2 4992 | Indexed intersection of in... |
| iinin1 4993 | Indexed intersection of in... |
| iinvdif 4994 | The indexed intersection o... |
| elriin 4995 | Elementhood in a relative ... |
| riin0 4996 | Relative intersection of a... |
| riinn0 4997 | Relative intersection of a... |
| riinrab 4998 | Relative intersection of a... |
| symdif0 4999 | Symmetric difference with ... |
| symdifv 5000 | The symmetric difference w... |
| symdifid 5001 | The symmetric difference o... |
| iinxsng 5002 | A singleton index picks ou... |
| iinxprg 5003 | Indexed intersection with ... |
| iunxsng 5004 | A singleton index picks ou... |
| iunxsn 5005 | A singleton index picks ou... |
| iunxsngf 5006 | A singleton index picks ou... |
| iunun 5007 | Separate a union in an ind... |
| iunxun 5008 | Separate a union in the in... |
| iunxdif3 5009 | An indexed union where som... |
| iunxprg 5010 | A pair index picks out two... |
| iunxiun 5011 | Separate an indexed union ... |
| iinuni 5012 | A relationship involving u... |
| iununi 5013 | A relationship involving u... |
| sspwuni 5014 | Subclass relationship for ... |
| pwssb 5015 | Two ways to express a coll... |
| elpwpw 5016 | Characterization of the el... |
| pwpwab 5017 | The double power class wri... |
| pwpwssunieq 5018 | The class of sets whose un... |
| elpwuni 5019 | Relationship for power cla... |
| iinpw 5020 | The power class of an inte... |
| iunpwss 5021 | Inclusion of an indexed un... |
| rintn0 5022 | Relative intersection of a... |
| dfdisj2 5025 | Alternate definition for d... |
| disjss2 5026 | If each element of a colle... |
| disjeq2 5027 | Equality theorem for disjo... |
| disjeq2dv 5028 | Equality deduction for dis... |
| disjss1 5029 | A subset of a disjoint col... |
| disjeq1 5030 | Equality theorem for disjo... |
| disjeq1d 5031 | Equality theorem for disjo... |
| disjeq12d 5032 | Equality theorem for disjo... |
| cbvdisj 5033 | Change bound variables in ... |
| cbvdisjv 5034 | Change bound variables in ... |
| nfdisjw 5035 | Bound-variable hypothesis ... |
| nfdisj 5036 | Bound-variable hypothesis ... |
| nfdisj1 5037 | Bound-variable hypothesis ... |
| disjor 5038 | Two ways to say that a col... |
| disjors 5039 | Two ways to say that a col... |
| disji2 5040 | Property of a disjoint col... |
| disji 5041 | Property of a disjoint col... |
| invdisj 5042 | If there is a function ` C... |
| invdisjrabw 5043 | Version of ~ invdisjrab wi... |
| invdisjrab 5044 | The restricted class abstr... |
| disjiun 5045 | A disjoint collection yiel... |
| disjord 5046 | Conditions for a collectio... |
| disjiunb 5047 | Two ways to say that a col... |
| disjiund 5048 | Conditions for a collectio... |
| sndisj 5049 | Any collection of singleto... |
| 0disj 5050 | Any collection of empty se... |
| disjxsn 5051 | A singleton collection is ... |
| disjx0 5052 | An empty collection is dis... |
| disjprgw 5053 | Version of ~ disjprg with ... |
| disjprg 5054 | A pair collection is disjo... |
| disjxiun 5055 | An indexed union of a disj... |
| disjxun 5056 | The union of two disjoint ... |
| disjss3 5057 | Expand a disjoint collecti... |
| breq 5060 | Equality theorem for binar... |
| breq1 5061 | Equality theorem for a bin... |
| breq2 5062 | Equality theorem for a bin... |
| breq12 5063 | Equality theorem for a bin... |
| breqi 5064 | Equality inference for bin... |
| breq1i 5065 | Equality inference for a b... |
| breq2i 5066 | Equality inference for a b... |
| breq12i 5067 | Equality inference for a b... |
| breq1d 5068 | Equality deduction for a b... |
| breqd 5069 | Equality deduction for a b... |
| breq2d 5070 | Equality deduction for a b... |
| breq12d 5071 | Equality deduction for a b... |
| breq123d 5072 | Equality deduction for a b... |
| breqdi 5073 | Equality deduction for a b... |
| breqan12d 5074 | Equality deduction for a b... |
| breqan12rd 5075 | Equality deduction for a b... |
| eqnbrtrd 5076 | Substitution of equal clas... |
| nbrne1 5077 | Two classes are different ... |
| nbrne2 5078 | Two classes are different ... |
| eqbrtri 5079 | Substitution of equal clas... |
| eqbrtrd 5080 | Substitution of equal clas... |
| eqbrtrri 5081 | Substitution of equal clas... |
| eqbrtrrd 5082 | Substitution of equal clas... |
| breqtri 5083 | Substitution of equal clas... |
| breqtrd 5084 | Substitution of equal clas... |
| breqtrri 5085 | Substitution of equal clas... |
| breqtrrd 5086 | Substitution of equal clas... |
| 3brtr3i 5087 | Substitution of equality i... |
| 3brtr4i 5088 | Substitution of equality i... |
| 3brtr3d 5089 | Substitution of equality i... |
| 3brtr4d 5090 | Substitution of equality i... |
| 3brtr3g 5091 | Substitution of equality i... |
| 3brtr4g 5092 | Substitution of equality i... |
| eqbrtrid 5093 | A chained equality inferen... |
| eqbrtrrid 5094 | A chained equality inferen... |
| breqtrid 5095 | A chained equality inferen... |
| breqtrrid 5096 | A chained equality inferen... |
| eqbrtrdi 5097 | A chained equality inferen... |
| eqbrtrrdi 5098 | A chained equality inferen... |
| breqtrdi 5099 | A chained equality inferen... |
| breqtrrdi 5100 | A chained equality inferen... |
| ssbrd 5101 | Deduction from a subclass ... |
| ssbr 5102 | Implication from a subclas... |
| ssbri 5103 | Inference from a subclass ... |
| nfbrd 5104 | Deduction version of bound... |
| nfbr 5105 | Bound-variable hypothesis ... |
| brab1 5106 | Relationship between a bin... |
| br0 5107 | The empty binary relation ... |
| brne0 5108 | If two sets are in a binar... |
| brun 5109 | The union of two binary re... |
| brin 5110 | The intersection of two re... |
| brdif 5111 | The difference of two bina... |
| sbcbr123 5112 | Move substitution in and o... |
| sbcbr 5113 | Move substitution in and o... |
| sbcbr12g 5114 | Move substitution in and o... |
| sbcbr1g 5115 | Move substitution in and o... |
| sbcbr2g 5116 | Move substitution in and o... |
| brsymdif 5117 | Characterization of the sy... |
| brralrspcev 5118 | Restricted existential spe... |
| brimralrspcev 5119 | Restricted existential spe... |
| opabss 5122 | The collection of ordered ... |
| opabbid 5123 | Equivalent wff's yield equ... |
| opabbidv 5124 | Equivalent wff's yield equ... |
| opabbii 5125 | Equivalent wff's yield equ... |
| nfopab 5126 | Bound-variable hypothesis ... |
| nfopab1 5127 | The first abstraction vari... |
| nfopab2 5128 | The second abstraction var... |
| cbvopab 5129 | Rule used to change bound ... |
| cbvopabv 5130 | Rule used to change bound ... |
| cbvopab1 5131 | Change first bound variabl... |
| cbvopab1g 5132 | Change first bound variabl... |
| cbvopab2 5133 | Change second bound variab... |
| cbvopab1s 5134 | Change first bound variabl... |
| cbvopab1v 5135 | Rule used to change the fi... |
| cbvopab2v 5136 | Rule used to change the se... |
| unopab 5137 | Union of two ordered pair ... |
| mpteq12df 5140 | An equality inference for ... |
| mpteq12f 5141 | An equality theorem for th... |
| mpteq12dva 5142 | An equality inference for ... |
| mpteq12dv 5143 | An equality inference for ... |
| mpteq12dvOLD 5144 | Obsolete version of ~ mpte... |
| mpteq12 5145 | An equality theorem for th... |
| mpteq1 5146 | An equality theorem for th... |
| mpteq1d 5147 | An equality theorem for th... |
| mpteq1i 5148 | An equality theorem for th... |
| mpteq2ia 5149 | An equality inference for ... |
| mpteq2i 5150 | An equality inference for ... |
| mpteq12i 5151 | An equality inference for ... |
| mpteq2da 5152 | Slightly more general equa... |
| mpteq2dva 5153 | Slightly more general equa... |
| mpteq2dv 5154 | An equality inference for ... |
| nfmpt 5155 | Bound-variable hypothesis ... |
| nfmpt1 5156 | Bound-variable hypothesis ... |
| cbvmptf 5157 | Rule to change the bound v... |
| cbvmptfg 5158 | Rule to change the bound v... |
| cbvmpt 5159 | Rule to change the bound v... |
| cbvmptg 5160 | Rule to change the bound v... |
| cbvmptv 5161 | Rule to change the bound v... |
| cbvmptvg 5162 | Rule to change the bound v... |
| mptv 5163 | Function with universal do... |
| dftr2 5166 | An alternate way of defini... |
| dftr5 5167 | An alternate way of defini... |
| dftr3 5168 | An alternate way of defini... |
| dftr4 5169 | An alternate way of defini... |
| treq 5170 | Equality theorem for the t... |
| trel 5171 | In a transitive class, the... |
| trel3 5172 | In a transitive class, the... |
| trss 5173 | An element of a transitive... |
| trin 5174 | The intersection of transi... |
| tr0 5175 | The empty set is transitiv... |
| trv 5176 | The universe is transitive... |
| triun 5177 | An indexed union of a clas... |
| truni 5178 | The union of a class of tr... |
| triin 5179 | An indexed intersection of... |
| trint 5180 | The intersection of a clas... |
| trintss 5181 | Any nonempty transitive cl... |
| axrep1 5183 | The version of the Axiom o... |
| axreplem 5184 | Lemma for ~ axrep2 and ~ a... |
| axrep2 5185 | Axiom of Replacement expre... |
| axrep3 5186 | Axiom of Replacement sligh... |
| axrep4 5187 | A more traditional version... |
| axrep5 5188 | Axiom of Replacement (simi... |
| axrep6 5189 | A condensed form of ~ ax-r... |
| zfrepclf 5190 | An inference based on the ... |
| zfrep3cl 5191 | An inference based on the ... |
| zfrep4 5192 | A version of Replacement u... |
| axsepgfromrep 5193 | A more general version ~ a... |
| axsep 5194 | Axiom scheme of separation... |
| axsepg 5196 | A more general version of ... |
| zfauscl 5197 | Separation Scheme (Aussond... |
| bm1.3ii 5198 | Convert implication to equ... |
| ax6vsep 5199 | Derive ~ ax6v (a weakened ... |
| axnulALT 5200 | Alternate proof of ~ axnul... |
| axnul 5201 | The Null Set Axiom of ZF s... |
| 0ex 5203 | The Null Set Axiom of ZF s... |
| al0ssb 5204 | The empty set is the uniqu... |
| sseliALT 5205 | Alternate proof of ~ sseli... |
| csbexg 5206 | The existence of proper su... |
| csbex 5207 | The existence of proper su... |
| unisn2 5208 | A version of ~ unisn witho... |
| nalset 5209 | No set contains all sets. ... |
| vnex 5210 | The universal class does n... |
| vprc 5211 | The universal class is not... |
| nvel 5212 | The universal class does n... |
| inex1 5213 | Separation Scheme (Aussond... |
| inex2 5214 | Separation Scheme (Aussond... |
| inex1g 5215 | Closed-form, generalized S... |
| inex2g 5216 | Sufficient condition for a... |
| ssex 5217 | The subset of a set is als... |
| ssexi 5218 | The subset of a set is als... |
| ssexg 5219 | The subset of a set is als... |
| ssexd 5220 | A subclass of a set is a s... |
| prcssprc 5221 | The superclass of a proper... |
| sselpwd 5222 | Elementhood to a power set... |
| difexg 5223 | Existence of a difference.... |
| difexi 5224 | Existence of a difference,... |
| zfausab 5225 | Separation Scheme (Aussond... |
| rabexg 5226 | Separation Scheme in terms... |
| rabex 5227 | Separation Scheme in terms... |
| rabexd 5228 | Separation Scheme in terms... |
| rabex2 5229 | Separation Scheme in terms... |
| rab2ex 5230 | A class abstraction based ... |
| elssabg 5231 | Membership in a class abst... |
| intex 5232 | The intersection of a none... |
| intnex 5233 | If a class intersection is... |
| intexab 5234 | The intersection of a none... |
| intexrab 5235 | The intersection of a none... |
| iinexg 5236 | The existence of a class i... |
| intabs 5237 | Absorption of a redundant ... |
| inuni 5238 | The intersection of a unio... |
| elpw2g 5239 | Membership in a power clas... |
| elpw2 5240 | Membership in a power clas... |
| elpwi2 5241 | Membership in a power clas... |
| pwnss 5242 | The power set of a set is ... |
| pwne 5243 | No set equals its power se... |
| difelpw 5244 | A difference is an element... |
| rabelpw 5245 | A restricted class abstrac... |
| class2set 5246 | Construct, from any class ... |
| class2seteq 5247 | Equality theorem based on ... |
| 0elpw 5248 | Every power class contains... |
| pwne0 5249 | A power class is never emp... |
| 0nep0 5250 | The empty set and its powe... |
| 0inp0 5251 | Something cannot be equal ... |
| unidif0 5252 | The removal of the empty s... |
| iin0 5253 | An indexed intersection of... |
| notzfaus 5254 | In the Separation Scheme ~... |
| notzfausOLD 5255 | Obsolete proof of ~ notzfa... |
| intv 5256 | The intersection of the un... |
| axpweq 5257 | Two equivalent ways to exp... |
| zfpow 5259 | Axiom of Power Sets expres... |
| axpow2 5260 | A variant of the Axiom of ... |
| axpow3 5261 | A variant of the Axiom of ... |
| el 5262 | Every set is an element of... |
| dtru 5263 | At least two sets exist (o... |
| dtrucor 5264 | Corollary of ~ dtru . Thi... |
| dtrucor2 5265 | The theorem form of the de... |
| dvdemo1 5266 | Demonstration of a theorem... |
| dvdemo2 5267 | Demonstration of a theorem... |
| nfnid 5268 | A setvar variable is not f... |
| nfcvb 5269 | The "distinctor" expressio... |
| vpwex 5270 | Power set axiom: the power... |
| pwexg 5271 | Power set axiom expressed ... |
| pwexd 5272 | Deduction version of the p... |
| pwex 5273 | Power set axiom expressed ... |
| abssexg 5274 | Existence of a class of su... |
| snexALT 5275 | Alternate proof of ~ snex ... |
| p0ex 5276 | The power set of the empty... |
| p0exALT 5277 | Alternate proof of ~ p0ex ... |
| pp0ex 5278 | The power set of the power... |
| ord3ex 5279 | The ordinal number 3 is a ... |
| dtruALT 5280 | Alternate proof of ~ dtru ... |
| axc16b 5281 | This theorem shows that ax... |
| eunex 5282 | Existential uniqueness imp... |
| eusv1 5283 | Two ways to express single... |
| eusvnf 5284 | Even if ` x ` is free in `... |
| eusvnfb 5285 | Two ways to say that ` A (... |
| eusv2i 5286 | Two ways to express single... |
| eusv2nf 5287 | Two ways to express single... |
| eusv2 5288 | Two ways to express single... |
| reusv1 5289 | Two ways to express single... |
| reusv2lem1 5290 | Lemma for ~ reusv2 . (Con... |
| reusv2lem2 5291 | Lemma for ~ reusv2 . (Con... |
| reusv2lem3 5292 | Lemma for ~ reusv2 . (Con... |
| reusv2lem4 5293 | Lemma for ~ reusv2 . (Con... |
| reusv2lem5 5294 | Lemma for ~ reusv2 . (Con... |
| reusv2 5295 | Two ways to express single... |
| reusv3i 5296 | Two ways of expressing exi... |
| reusv3 5297 | Two ways to express single... |
| eusv4 5298 | Two ways to express single... |
| alxfr 5299 | Transfer universal quantif... |
| ralxfrd 5300 | Transfer universal quantif... |
| rexxfrd 5301 | Transfer universal quantif... |
| ralxfr2d 5302 | Transfer universal quantif... |
| rexxfr2d 5303 | Transfer universal quantif... |
| ralxfrd2 5304 | Transfer universal quantif... |
| rexxfrd2 5305 | Transfer existence from a ... |
| ralxfr 5306 | Transfer universal quantif... |
| ralxfrALT 5307 | Alternate proof of ~ ralxf... |
| rexxfr 5308 | Transfer existence from a ... |
| rabxfrd 5309 | Class builder membership a... |
| rabxfr 5310 | Class builder membership a... |
| reuhypd 5311 | A theorem useful for elimi... |
| reuhyp 5312 | A theorem useful for elimi... |
| zfpair 5313 | The Axiom of Pairing of Ze... |
| axprALT 5314 | Alternate proof of ~ axpr ... |
| axprlem1 5315 | Lemma for ~ axpr . There ... |
| axprlem2 5316 | Lemma for ~ axpr . There ... |
| axprlem3 5317 | Lemma for ~ axpr . Elimin... |
| axprlem4 5318 | Lemma for ~ axpr . The fi... |
| axprlem5 5319 | Lemma for ~ axpr . The se... |
| axpr 5320 | Unabbreviated version of t... |
| zfpair2 5322 | Derive the abbreviated ver... |
| snex 5323 | A singleton is a set. The... |
| prex 5324 | The Axiom of Pairing using... |
| sels 5325 | If a class is a set, then ... |
| elALT 5326 | Alternate proof of ~ el , ... |
| dtruALT2 5327 | Alternate proof of ~ dtru ... |
| snelpwi 5328 | A singleton of a set belon... |
| snelpw 5329 | A singleton of a set belon... |
| prelpw 5330 | A pair of two sets belongs... |
| prelpwi 5331 | A pair of two sets belongs... |
| rext 5332 | A theorem similar to exten... |
| sspwb 5333 | The powerclass constructio... |
| unipw 5334 | A class equals the union o... |
| univ 5335 | The union of the universe ... |
| pwel 5336 | Quantitative version of ~ ... |
| pwtr 5337 | A class is transitive iff ... |
| ssextss 5338 | An extensionality-like pri... |
| ssext 5339 | An extensionality-like pri... |
| nssss 5340 | Negation of subclass relat... |
| pweqb 5341 | Classes are equal if and o... |
| intid 5342 | The intersection of all se... |
| moabex 5343 | "At most one" existence im... |
| rmorabex 5344 | Restricted "at most one" e... |
| euabex 5345 | The abstraction of a wff w... |
| nnullss 5346 | A nonempty class (even if ... |
| exss 5347 | Restricted existence in a ... |
| opex 5348 | An ordered pair of classes... |
| otex 5349 | An ordered triple of class... |
| elopg 5350 | Characterization of the el... |
| elop 5351 | Characterization of the el... |
| opi1 5352 | One of the two elements in... |
| opi2 5353 | One of the two elements of... |
| opeluu 5354 | Each member of an ordered ... |
| op1stb 5355 | Extract the first member o... |
| brv 5356 | Two classes are always in ... |
| opnz 5357 | An ordered pair is nonempt... |
| opnzi 5358 | An ordered pair is nonempt... |
| opth1 5359 | Equality of the first memb... |
| opth 5360 | The ordered pair theorem. ... |
| opthg 5361 | Ordered pair theorem. ` C ... |
| opth1g 5362 | Equality of the first memb... |
| opthg2 5363 | Ordered pair theorem. (Co... |
| opth2 5364 | Ordered pair theorem. (Co... |
| opthneg 5365 | Two ordered pairs are not ... |
| opthne 5366 | Two ordered pairs are not ... |
| otth2 5367 | Ordered triple theorem, wi... |
| otth 5368 | Ordered triple theorem. (... |
| otthg 5369 | Ordered triple theorem, cl... |
| eqvinop 5370 | A variable introduction la... |
| sbcop1 5371 | The proper substitution of... |
| sbcop 5372 | The proper substitution of... |
| copsexgw 5373 | Version of ~ copsexg with ... |
| copsexg 5374 | Substitution of class ` A ... |
| copsex2t 5375 | Closed theorem form of ~ c... |
| copsex2g 5376 | Implicit substitution infe... |
| copsex4g 5377 | An implicit substitution i... |
| 0nelop 5378 | A property of ordered pair... |
| opwo0id 5379 | An ordered pair is equal t... |
| opeqex 5380 | Equivalence of existence i... |
| oteqex2 5381 | Equivalence of existence i... |
| oteqex 5382 | Equivalence of existence i... |
| opcom 5383 | An ordered pair commutes i... |
| moop2 5384 | "At most one" property of ... |
| opeqsng 5385 | Equivalence for an ordered... |
| opeqsn 5386 | Equivalence for an ordered... |
| opeqpr 5387 | Equivalence for an ordered... |
| snopeqop 5388 | Equivalence for an ordered... |
| propeqop 5389 | Equivalence for an ordered... |
| propssopi 5390 | If a pair of ordered pairs... |
| snopeqopsnid 5391 | Equivalence for an ordered... |
| mosubopt 5392 | "At most one" remains true... |
| mosubop 5393 | "At most one" remains true... |
| euop2 5394 | Transfer existential uniqu... |
| euotd 5395 | Prove existential uniquene... |
| opthwiener 5396 | Justification theorem for ... |
| uniop 5397 | The union of an ordered pa... |
| uniopel 5398 | Ordered pair membership is... |
| opthhausdorff 5399 | Justification theorem for ... |
| opthhausdorff0 5400 | Justification theorem for ... |
| otsndisj 5401 | The singletons consisting ... |
| otiunsndisj 5402 | The union of singletons co... |
| iunopeqop 5403 | Implication of an ordered ... |
| opabidw 5404 | The law of concretion. Sp... |
| opabid 5405 | The law of concretion. Sp... |
| elopab 5406 | Membership in a class abst... |
| rexopabb 5407 | Restricted existential qua... |
| opelopabsbALT 5408 | The law of concretion in t... |
| opelopabsb 5409 | The law of concretion in t... |
| brabsb 5410 | The law of concretion in t... |
| opelopabt 5411 | Closed theorem form of ~ o... |
| opelopabga 5412 | The law of concretion. Th... |
| brabga 5413 | The law of concretion for ... |
| opelopab2a 5414 | Ordered pair membership in... |
| opelopaba 5415 | The law of concretion. Th... |
| braba 5416 | The law of concretion for ... |
| opelopabg 5417 | The law of concretion. Th... |
| brabg 5418 | The law of concretion for ... |
| opelopabgf 5419 | The law of concretion. Th... |
| opelopab2 5420 | Ordered pair membership in... |
| opelopab 5421 | The law of concretion. Th... |
| brab 5422 | The law of concretion for ... |
| opelopabaf 5423 | The law of concretion. Th... |
| opelopabf 5424 | The law of concretion. Th... |
| ssopab2 5425 | Equivalence of ordered pai... |
| ssopab2bw 5426 | Equivalence of ordered pai... |
| eqopab2bw 5427 | Equivalence of ordered pai... |
| ssopab2b 5428 | Equivalence of ordered pai... |
| ssopab2i 5429 | Inference of ordered pair ... |
| ssopab2dv 5430 | Inference of ordered pair ... |
| eqopab2b 5431 | Equivalence of ordered pai... |
| opabn0 5432 | Nonempty ordered pair clas... |
| opab0 5433 | Empty ordered pair class a... |
| csbopab 5434 | Move substitution into a c... |
| csbopabgALT 5435 | Move substitution into a c... |
| csbmpt12 5436 | Move substitution into a m... |
| csbmpt2 5437 | Move substitution into the... |
| iunopab 5438 | Move indexed union inside ... |
| elopabr 5439 | Membership in a class abst... |
| elopabran 5440 | Membership in a class abst... |
| rbropapd 5441 | Properties of a pair in an... |
| rbropap 5442 | Properties of a pair in a ... |
| 2rbropap 5443 | Properties of a pair in a ... |
| 0nelopab 5444 | The empty set is never an ... |
| brabv 5445 | If two classes are in a re... |
| pwin 5446 | The power class of the int... |
| pwunss 5447 | The power class of the uni... |
| pwunssOLD 5448 | The power class of the uni... |
| pwssun 5449 | The power class of the uni... |
| pwundif 5450 | Break up the power class o... |
| pwundifOLD 5451 | Obsolete proof of ~ pwundi... |
| pwun 5452 | The power class of the uni... |
| dfid4 5455 | The identity function expr... |
| dfid3 5456 | A stronger version of ~ df... |
| dfid2 5457 | Alternate definition of th... |
| epelg 5460 | The membership relation an... |
| epelgOLD 5461 | Obsolete version of ~ epel... |
| epeli 5462 | The membership relation an... |
| epel 5463 | The membership relation an... |
| 0sn0ep 5464 | An example for the members... |
| epn0 5465 | The membership relation is... |
| poss 5470 | Subset theorem for the par... |
| poeq1 5471 | Equality theorem for parti... |
| poeq2 5472 | Equality theorem for parti... |
| nfpo 5473 | Bound-variable hypothesis ... |
| nfso 5474 | Bound-variable hypothesis ... |
| pocl 5475 | Properties of partial orde... |
| ispod 5476 | Sufficient conditions for ... |
| swopolem 5477 | Perform the substitutions ... |
| swopo 5478 | A strict weak order is a p... |
| poirr 5479 | A partial order relation i... |
| potr 5480 | A partial order relation i... |
| po2nr 5481 | A partial order relation h... |
| po3nr 5482 | A partial order relation h... |
| po2ne 5483 | Two classes which are in a... |
| po0 5484 | Any relation is a partial ... |
| pofun 5485 | A function preserves a par... |
| sopo 5486 | A strict linear order is a... |
| soss 5487 | Subset theorem for the str... |
| soeq1 5488 | Equality theorem for the s... |
| soeq2 5489 | Equality theorem for the s... |
| sonr 5490 | A strict order relation is... |
| sotr 5491 | A strict order relation is... |
| solin 5492 | A strict order relation is... |
| so2nr 5493 | A strict order relation ha... |
| so3nr 5494 | A strict order relation ha... |
| sotric 5495 | A strict order relation sa... |
| sotrieq 5496 | Trichotomy law for strict ... |
| sotrieq2 5497 | Trichotomy law for strict ... |
| soasym 5498 | Asymmetry law for strict o... |
| sotr2 5499 | A transitivity relation. ... |
| issod 5500 | An irreflexive, transitive... |
| issoi 5501 | An irreflexive, transitive... |
| isso2i 5502 | Deduce strict ordering fro... |
| so0 5503 | Any relation is a strict o... |
| somo 5504 | A totally ordered set has ... |
| fri 5511 | Property of well-founded r... |
| seex 5512 | The ` R ` -preimage of an ... |
| exse 5513 | Any relation on a set is s... |
| dffr2 5514 | Alternate definition of we... |
| frc 5515 | Property of well-founded r... |
| frss 5516 | Subset theorem for the wel... |
| sess1 5517 | Subset theorem for the set... |
| sess2 5518 | Subset theorem for the set... |
| freq1 5519 | Equality theorem for the w... |
| freq2 5520 | Equality theorem for the w... |
| seeq1 5521 | Equality theorem for the s... |
| seeq2 5522 | Equality theorem for the s... |
| nffr 5523 | Bound-variable hypothesis ... |
| nfse 5524 | Bound-variable hypothesis ... |
| nfwe 5525 | Bound-variable hypothesis ... |
| frirr 5526 | A well-founded relation is... |
| fr2nr 5527 | A well-founded relation ha... |
| fr0 5528 | Any relation is well-found... |
| frminex 5529 | If an element of a well-fo... |
| efrirr 5530 | A well-founded class does ... |
| efrn2lp 5531 | A well-founded class conta... |
| epse 5532 | The membership relation is... |
| tz7.2 5533 | Similar to Theorem 7.2 of ... |
| dfepfr 5534 | An alternate way of saying... |
| epfrc 5535 | A subset of a well-founded... |
| wess 5536 | Subset theorem for the wel... |
| weeq1 5537 | Equality theorem for the w... |
| weeq2 5538 | Equality theorem for the w... |
| wefr 5539 | A well-ordering is well-fo... |
| weso 5540 | A well-ordering is a stric... |
| wecmpep 5541 | The elements of a class we... |
| wetrep 5542 | On a class well-ordered by... |
| wefrc 5543 | A nonempty subclass of a c... |
| we0 5544 | Any relation is a well-ord... |
| wereu 5545 | A subset of a well-ordered... |
| wereu2 5546 | All nonempty subclasses of... |
| xpeq1 5563 | Equality theorem for Carte... |
| xpss12 5564 | Subset theorem for Cartesi... |
| xpss 5565 | A Cartesian product is inc... |
| inxpssres 5566 | Intersection with a Cartes... |
| relxp 5567 | A Cartesian product is a r... |
| xpss1 5568 | Subset relation for Cartes... |
| xpss2 5569 | Subset relation for Cartes... |
| xpeq2 5570 | Equality theorem for Carte... |
| elxpi 5571 | Membership in a Cartesian ... |
| elxp 5572 | Membership in a Cartesian ... |
| elxp2 5573 | Membership in a Cartesian ... |
| xpeq12 5574 | Equality theorem for Carte... |
| xpeq1i 5575 | Equality inference for Car... |
| xpeq2i 5576 | Equality inference for Car... |
| xpeq12i 5577 | Equality inference for Car... |
| xpeq1d 5578 | Equality deduction for Car... |
| xpeq2d 5579 | Equality deduction for Car... |
| xpeq12d 5580 | Equality deduction for Car... |
| sqxpeqd 5581 | Equality deduction for a C... |
| nfxp 5582 | Bound-variable hypothesis ... |
| 0nelxp 5583 | The empty set is not a mem... |
| 0nelelxp 5584 | A member of a Cartesian pr... |
| opelxp 5585 | Ordered pair membership in... |
| opelxpi 5586 | Ordered pair membership in... |
| opelxpd 5587 | Ordered pair membership in... |
| opelvv 5588 | Ordered pair membership in... |
| opelvvg 5589 | Ordered pair membership in... |
| opelxp1 5590 | The first member of an ord... |
| opelxp2 5591 | The second member of an or... |
| otelxp1 5592 | The first member of an ord... |
| otel3xp 5593 | An ordered triple is an el... |
| rabxp 5594 | Membership in a class buil... |
| brxp 5595 | Binary relation on a Carte... |
| pwvrel 5596 | A set is a binary relation... |
| pwvabrel 5597 | The powerclass of the cart... |
| brrelex12 5598 | Two classes related by a b... |
| brrelex1 5599 | If two classes are related... |
| brrelex2 5600 | If two classes are related... |
| brrelex12i 5601 | Two classes that are relat... |
| brrelex1i 5602 | The first argument of a bi... |
| brrelex2i 5603 | The second argument of a b... |
| nprrel12 5604 | Proper classes are not rel... |
| nprrel 5605 | No proper class is related... |
| 0nelrel0 5606 | A binary relation does not... |
| 0nelrel 5607 | A binary relation does not... |
| fconstmpt 5608 | Representation of a consta... |
| vtoclr 5609 | Variable to class conversi... |
| opthprc 5610 | Justification theorem for ... |
| brel 5611 | Two things in a binary rel... |
| elxp3 5612 | Membership in a Cartesian ... |
| opeliunxp 5613 | Membership in a union of C... |
| xpundi 5614 | Distributive law for Carte... |
| xpundir 5615 | Distributive law for Carte... |
| xpiundi 5616 | Distributive law for Carte... |
| xpiundir 5617 | Distributive law for Carte... |
| iunxpconst 5618 | Membership in a union of C... |
| xpun 5619 | The Cartesian product of t... |
| elvv 5620 | Membership in universal cl... |
| elvvv 5621 | Membership in universal cl... |
| elvvuni 5622 | An ordered pair contains i... |
| brinxp2 5623 | Intersection with cross pr... |
| brinxp 5624 | Intersection of binary rel... |
| opelinxp 5625 | Ordered pair element in an... |
| poinxp 5626 | Intersection of partial or... |
| soinxp 5627 | Intersection of total orde... |
| frinxp 5628 | Intersection of well-found... |
| seinxp 5629 | Intersection of set-like r... |
| weinxp 5630 | Intersection of well-order... |
| posn 5631 | Partial ordering of a sing... |
| sosn 5632 | Strict ordering on a singl... |
| frsn 5633 | Founded relation on a sing... |
| wesn 5634 | Well-ordering of a singlet... |
| elopaelxp 5635 | Membership in an ordered p... |
| bropaex12 5636 | Two classes related by an ... |
| opabssxp 5637 | An abstraction relation is... |
| brab2a 5638 | The law of concretion for ... |
| optocl 5639 | Implicit substitution of c... |
| 2optocl 5640 | Implicit substitution of c... |
| 3optocl 5641 | Implicit substitution of c... |
| opbrop 5642 | Ordered pair membership in... |
| 0xp 5643 | The Cartesian product with... |
| csbxp 5644 | Distribute proper substitu... |
| releq 5645 | Equality theorem for the r... |
| releqi 5646 | Equality inference for the... |
| releqd 5647 | Equality deduction for the... |
| nfrel 5648 | Bound-variable hypothesis ... |
| sbcrel 5649 | Distribute proper substitu... |
| relss 5650 | Subclass theorem for relat... |
| ssrel 5651 | A subclass relationship de... |
| eqrel 5652 | Extensionality principle f... |
| ssrel2 5653 | A subclass relationship de... |
| relssi 5654 | Inference from subclass pr... |
| relssdv 5655 | Deduction from subclass pr... |
| eqrelriv 5656 | Inference from extensional... |
| eqrelriiv 5657 | Inference from extensional... |
| eqbrriv 5658 | Inference from extensional... |
| eqrelrdv 5659 | Deduce equality of relatio... |
| eqbrrdv 5660 | Deduction from extensional... |
| eqbrrdiv 5661 | Deduction from extensional... |
| eqrelrdv2 5662 | A version of ~ eqrelrdv . ... |
| ssrelrel 5663 | A subclass relationship de... |
| eqrelrel 5664 | Extensionality principle f... |
| elrel 5665 | A member of a relation is ... |
| rel0 5666 | The empty set is a relatio... |
| nrelv 5667 | The universal class is not... |
| relsng 5668 | A singleton is a relation ... |
| relsnb 5669 | An at-most-singleton is a ... |
| relsnopg 5670 | A singleton of an ordered ... |
| relsn 5671 | A singleton is a relation ... |
| relsnop 5672 | A singleton of an ordered ... |
| copsex2gb 5673 | Implicit substitution infe... |
| copsex2ga 5674 | Implicit substitution infe... |
| elopaba 5675 | Membership in an ordered p... |
| xpsspw 5676 | A Cartesian product is inc... |
| unixpss 5677 | The double class union of ... |
| relun 5678 | The union of two relations... |
| relin1 5679 | The intersection with a re... |
| relin2 5680 | The intersection with a re... |
| relinxp 5681 | Intersection with a Cartes... |
| reldif 5682 | A difference cutting down ... |
| reliun 5683 | An indexed union is a rela... |
| reliin 5684 | An indexed intersection is... |
| reluni 5685 | The union of a class is a ... |
| relint 5686 | The intersection of a clas... |
| relopabiv 5687 | A class of ordered pairs i... |
| relopabi 5688 | A class of ordered pairs i... |
| relopabiALT 5689 | Alternate proof of ~ relop... |
| relopab 5690 | A class of ordered pairs i... |
| mptrel 5691 | The maps-to notation alway... |
| reli 5692 | The identity relation is a... |
| rele 5693 | The membership relation is... |
| opabid2 5694 | A relation expressed as an... |
| inopab 5695 | Intersection of two ordere... |
| difopab 5696 | The difference of two orde... |
| inxp 5697 | The intersection of two Ca... |
| xpindi 5698 | Distributive law for Carte... |
| xpindir 5699 | Distributive law for Carte... |
| xpiindi 5700 | Distributive law for Carte... |
| xpriindi 5701 | Distributive law for Carte... |
| eliunxp 5702 | Membership in a union of C... |
| opeliunxp2 5703 | Membership in a union of C... |
| raliunxp 5704 | Write a double restricted ... |
| rexiunxp 5705 | Write a double restricted ... |
| ralxp 5706 | Universal quantification r... |
| rexxp 5707 | Existential quantification... |
| exopxfr 5708 | Transfer ordered-pair exis... |
| exopxfr2 5709 | Transfer ordered-pair exis... |
| djussxp 5710 | Disjoint union is a subset... |
| ralxpf 5711 | Version of ~ ralxp with bo... |
| rexxpf 5712 | Version of ~ rexxp with bo... |
| iunxpf 5713 | Indexed union on a Cartesi... |
| opabbi2dv 5714 | Deduce equality of a relat... |
| relop 5715 | A necessary and sufficient... |
| ideqg 5716 | For sets, the identity rel... |
| ideq 5717 | For sets, the identity rel... |
| ididg 5718 | A set is identical to itse... |
| issetid 5719 | Two ways of expressing set... |
| coss1 5720 | Subclass theorem for compo... |
| coss2 5721 | Subclass theorem for compo... |
| coeq1 5722 | Equality theorem for compo... |
| coeq2 5723 | Equality theorem for compo... |
| coeq1i 5724 | Equality inference for com... |
| coeq2i 5725 | Equality inference for com... |
| coeq1d 5726 | Equality deduction for com... |
| coeq2d 5727 | Equality deduction for com... |
| coeq12i 5728 | Equality inference for com... |
| coeq12d 5729 | Equality deduction for com... |
| nfco 5730 | Bound-variable hypothesis ... |
| brcog 5731 | Ordered pair membership in... |
| opelco2g 5732 | Ordered pair membership in... |
| brcogw 5733 | Ordered pair membership in... |
| eqbrrdva 5734 | Deduction from extensional... |
| brco 5735 | Binary relation on a compo... |
| opelco 5736 | Ordered pair membership in... |
| cnvss 5737 | Subset theorem for convers... |
| cnveq 5738 | Equality theorem for conve... |
| cnveqi 5739 | Equality inference for con... |
| cnveqd 5740 | Equality deduction for con... |
| elcnv 5741 | Membership in a converse r... |
| elcnv2 5742 | Membership in a converse r... |
| nfcnv 5743 | Bound-variable hypothesis ... |
| brcnvg 5744 | The converse of a binary r... |
| opelcnvg 5745 | Ordered-pair membership in... |
| opelcnv 5746 | Ordered-pair membership in... |
| brcnv 5747 | The converse of a binary r... |
| csbcnv 5748 | Move class substitution in... |
| csbcnvgALT 5749 | Move class substitution in... |
| cnvco 5750 | Distributive law of conver... |
| cnvuni 5751 | The converse of a class un... |
| dfdm3 5752 | Alternate definition of do... |
| dfrn2 5753 | Alternate definition of ra... |
| dfrn3 5754 | Alternate definition of ra... |
| elrn2g 5755 | Membership in a range. (C... |
| elrng 5756 | Membership in a range. (C... |
| ssrelrn 5757 | If a relation is a subset ... |
| dfdm4 5758 | Alternate definition of do... |
| dfdmf 5759 | Definition of domain, usin... |
| csbdm 5760 | Distribute proper substitu... |
| eldmg 5761 | Domain membership. Theore... |
| eldm2g 5762 | Domain membership. Theore... |
| eldm 5763 | Membership in a domain. T... |
| eldm2 5764 | Membership in a domain. T... |
| dmss 5765 | Subset theorem for domain.... |
| dmeq 5766 | Equality theorem for domai... |
| dmeqi 5767 | Equality inference for dom... |
| dmeqd 5768 | Equality deduction for dom... |
| opeldmd 5769 | Membership of first of an ... |
| opeldm 5770 | Membership of first of an ... |
| breldm 5771 | Membership of first of a b... |
| breldmg 5772 | Membership of first of a b... |
| dmun 5773 | The domain of a union is t... |
| dmin 5774 | The domain of an intersect... |
| breldmd 5775 | Membership of first of a b... |
| dmiun 5776 | The domain of an indexed u... |
| dmuni 5777 | The domain of a union. Pa... |
| dmopab 5778 | The domain of a class of o... |
| dmopabelb 5779 | A set is an element of the... |
| dmopab2rex 5780 | The domain of an ordered p... |
| dmopabss 5781 | Upper bound for the domain... |
| dmopab3 5782 | The domain of a restricted... |
| opabssxpd 5783 | An ordered-pair class abst... |
| dm0 5784 | The domain of the empty se... |
| dmi 5785 | The domain of the identity... |
| dmv 5786 | The domain of the universe... |
| dmep 5787 | The domain of the membersh... |
| domepOLD 5788 | Obsolete proof of ~ dmep a... |
| dm0rn0 5789 | An empty domain is equival... |
| rn0 5790 | The range of the empty set... |
| rnep 5791 | The range of the membershi... |
| reldm0 5792 | A relation is empty iff it... |
| dmxp 5793 | The domain of a Cartesian ... |
| dmxpid 5794 | The domain of a Cartesian ... |
| dmxpin 5795 | The domain of the intersec... |
| xpid11 5796 | The Cartesian square is a ... |
| dmcnvcnv 5797 | The domain of the double c... |
| rncnvcnv 5798 | The range of the double co... |
| elreldm 5799 | The first member of an ord... |
| rneq 5800 | Equality theorem for range... |
| rneqi 5801 | Equality inference for ran... |
| rneqd 5802 | Equality deduction for ran... |
| rnss 5803 | Subset theorem for range. ... |
| rnssi 5804 | Subclass inference for ran... |
| brelrng 5805 | The second argument of a b... |
| brelrn 5806 | The second argument of a b... |
| opelrn 5807 | Membership of second membe... |
| releldm 5808 | The first argument of a bi... |
| relelrn 5809 | The second argument of a b... |
| releldmb 5810 | Membership in a domain. (... |
| relelrnb 5811 | Membership in a range. (C... |
| releldmi 5812 | The first argument of a bi... |
| relelrni 5813 | The second argument of a b... |
| dfrnf 5814 | Definition of range, using... |
| elrn2 5815 | Membership in a range. (C... |
| elrn 5816 | Membership in a range. (C... |
| nfdm 5817 | Bound-variable hypothesis ... |
| nfrn 5818 | Bound-variable hypothesis ... |
| dmiin 5819 | Domain of an intersection.... |
| rnopab 5820 | The range of a class of or... |
| rnmpt 5821 | The range of a function in... |
| elrnmpt 5822 | The range of a function in... |
| elrnmpt1s 5823 | Elementhood in an image se... |
| elrnmpt1 5824 | Elementhood in an image se... |
| elrnmptg 5825 | Membership in the range of... |
| elrnmpti 5826 | Membership in the range of... |
| elrnmptdv 5827 | Elementhood in the range o... |
| elrnmpt2d 5828 | Elementhood in the range o... |
| dfiun3g 5829 | Alternate definition of in... |
| dfiin3g 5830 | Alternate definition of in... |
| dfiun3 5831 | Alternate definition of in... |
| dfiin3 5832 | Alternate definition of in... |
| riinint 5833 | Express a relative indexed... |
| relrn0 5834 | A relation is empty iff it... |
| dmrnssfld 5835 | The domain and range of a ... |
| dmcoss 5836 | Domain of a composition. ... |
| rncoss 5837 | Range of a composition. (... |
| dmcosseq 5838 | Domain of a composition. ... |
| dmcoeq 5839 | Domain of a composition. ... |
| rncoeq 5840 | Range of a composition. (... |
| reseq1 5841 | Equality theorem for restr... |
| reseq2 5842 | Equality theorem for restr... |
| reseq1i 5843 | Equality inference for res... |
| reseq2i 5844 | Equality inference for res... |
| reseq12i 5845 | Equality inference for res... |
| reseq1d 5846 | Equality deduction for res... |
| reseq2d 5847 | Equality deduction for res... |
| reseq12d 5848 | Equality deduction for res... |
| nfres 5849 | Bound-variable hypothesis ... |
| csbres 5850 | Distribute proper substitu... |
| res0 5851 | A restriction to the empty... |
| dfres3 5852 | Alternate definition of re... |
| opelres 5853 | Ordered pair elementhood i... |
| brres 5854 | Binary relation on a restr... |
| opelresi 5855 | Ordered pair membership in... |
| brresi 5856 | Binary relation on a restr... |
| opres 5857 | Ordered pair membership in... |
| resieq 5858 | A restricted identity rela... |
| opelidres 5859 | ` <. A , A >. ` belongs to... |
| resres 5860 | The restriction of a restr... |
| resundi 5861 | Distributive law for restr... |
| resundir 5862 | Distributive law for restr... |
| resindi 5863 | Class restriction distribu... |
| resindir 5864 | Class restriction distribu... |
| inres 5865 | Move intersection into cla... |
| resdifcom 5866 | Commutative law for restri... |
| resiun1 5867 | Distribution of restrictio... |
| resiun2 5868 | Distribution of restrictio... |
| dmres 5869 | The domain of a restrictio... |
| ssdmres 5870 | A domain restricted to a s... |
| dmresexg 5871 | The domain of a restrictio... |
| resss 5872 | A class includes its restr... |
| rescom 5873 | Commutative law for restri... |
| ssres 5874 | Subclass theorem for restr... |
| ssres2 5875 | Subclass theorem for restr... |
| relres 5876 | A restriction is a relatio... |
| resabs1 5877 | Absorption law for restric... |
| resabs1d 5878 | Absorption law for restric... |
| resabs2 5879 | Absorption law for restric... |
| residm 5880 | Idempotent law for restric... |
| resima 5881 | A restriction to an image.... |
| resima2 5882 | Image under a restricted c... |
| xpssres 5883 | Restriction of a constant ... |
| elinxp 5884 | Membership in an intersect... |
| elres 5885 | Membership in a restrictio... |
| elsnres 5886 | Membership in restriction ... |
| relssres 5887 | Simplification law for res... |
| dmressnsn 5888 | The domain of a restrictio... |
| eldmressnsn 5889 | The element of the domain ... |
| eldmeldmressn 5890 | An element of the domain (... |
| resdm 5891 | A relation restricted to i... |
| resexg 5892 | The restriction of a set i... |
| resex 5893 | The restriction of a set i... |
| resindm 5894 | When restricting a relatio... |
| resdmdfsn 5895 | Restricting a relation to ... |
| resopab 5896 | Restriction of a class abs... |
| iss 5897 | A subclass of the identity... |
| resopab2 5898 | Restriction of a class abs... |
| resmpt 5899 | Restriction of the mapping... |
| resmpt3 5900 | Unconditional restriction ... |
| resmptf 5901 | Restriction of the mapping... |
| resmptd 5902 | Restriction of the mapping... |
| dfres2 5903 | Alternate definition of th... |
| mptss 5904 | Sufficient condition for i... |
| elidinxp 5905 | Characterization of the el... |
| elidinxpid 5906 | Characterization of the el... |
| elrid 5907 | Characterization of the el... |
| idinxpres 5908 | The intersection of the id... |
| idinxpresid 5909 | The intersection of the id... |
| idssxp 5910 | A diagonal set as a subset... |
| opabresid 5911 | The restricted identity re... |
| mptresid 5912 | The restricted identity re... |
| opabresidOLD 5913 | Obsolete version of ~ opab... |
| mptresidOLD 5914 | Obsolete version of ~ mptr... |
| dmresi 5915 | The domain of a restricted... |
| restidsing 5916 | Restriction of the identit... |
| iresn0n0 5917 | The identity function rest... |
| imaeq1 5918 | Equality theorem for image... |
| imaeq2 5919 | Equality theorem for image... |
| imaeq1i 5920 | Equality theorem for image... |
| imaeq2i 5921 | Equality theorem for image... |
| imaeq1d 5922 | Equality theorem for image... |
| imaeq2d 5923 | Equality theorem for image... |
| imaeq12d 5924 | Equality theorem for image... |
| dfima2 5925 | Alternate definition of im... |
| dfima3 5926 | Alternate definition of im... |
| elimag 5927 | Membership in an image. T... |
| elima 5928 | Membership in an image. T... |
| elima2 5929 | Membership in an image. T... |
| elima3 5930 | Membership in an image. T... |
| nfima 5931 | Bound-variable hypothesis ... |
| nfimad 5932 | Deduction version of bound... |
| imadmrn 5933 | The image of the domain of... |
| imassrn 5934 | The image of a class is a ... |
| mptima 5935 | Image of a function in map... |
| imai 5936 | Image under the identity r... |
| rnresi 5937 | The range of the restricte... |
| resiima 5938 | The image of a restriction... |
| ima0 5939 | Image of the empty set. T... |
| 0ima 5940 | Image under the empty rela... |
| csbima12 5941 | Move class substitution in... |
| imadisj 5942 | A class whose image under ... |
| cnvimass 5943 | A preimage under any class... |
| cnvimarndm 5944 | The preimage of the range ... |
| imasng 5945 | The image of a singleton. ... |
| relimasn 5946 | The image of a singleton. ... |
| elrelimasn 5947 | Elementhood in the image o... |
| elimasn 5948 | Membership in an image of ... |
| elimasng 5949 | Membership in an image of ... |
| elimasni 5950 | Membership in an image of ... |
| args 5951 | Two ways to express the cl... |
| eliniseg 5952 | Membership in an initial s... |
| epini 5953 | Any set is equal to its pr... |
| iniseg 5954 | An idiom that signifies an... |
| inisegn0 5955 | Nonemptiness of an initial... |
| dffr3 5956 | Alternate definition of we... |
| dfse2 5957 | Alternate definition of se... |
| imass1 5958 | Subset theorem for image. ... |
| imass2 5959 | Subset theorem for image. ... |
| ndmima 5960 | The image of a singleton o... |
| relcnv 5961 | A converse is a relation. ... |
| relbrcnvg 5962 | When ` R ` is a relation, ... |
| eliniseg2 5963 | Eliminate the class existe... |
| relbrcnv 5964 | When ` R ` is a relation, ... |
| cotrg 5965 | Two ways of saying that th... |
| cotr 5966 | Two ways of saying a relat... |
| idrefALT 5967 | Alternate proof of ~ idref... |
| cnvsym 5968 | Two ways of saying a relat... |
| intasym 5969 | Two ways of saying a relat... |
| asymref 5970 | Two ways of saying a relat... |
| asymref2 5971 | Two ways of saying a relat... |
| intirr 5972 | Two ways of saying a relat... |
| brcodir 5973 | Two ways of saying that tw... |
| codir 5974 | Two ways of saying a relat... |
| qfto 5975 | A quantifier-free way of e... |
| xpidtr 5976 | A Cartesian square is a tr... |
| trin2 5977 | The intersection of two tr... |
| poirr2 5978 | A partial order relation i... |
| trinxp 5979 | The relation induced by a ... |
| soirri 5980 | A strict order relation is... |
| sotri 5981 | A strict order relation is... |
| son2lpi 5982 | A strict order relation ha... |
| sotri2 5983 | A transitivity relation. ... |
| sotri3 5984 | A transitivity relation. ... |
| poleloe 5985 | Express "less than or equa... |
| poltletr 5986 | Transitive law for general... |
| somin1 5987 | Property of a minimum in a... |
| somincom 5988 | Commutativity of minimum i... |
| somin2 5989 | Property of a minimum in a... |
| soltmin 5990 | Being less than a minimum,... |
| cnvopab 5991 | The converse of a class ab... |
| mptcnv 5992 | The converse of a mapping ... |
| cnv0 5993 | The converse of the empty ... |
| cnvi 5994 | The converse of the identi... |
| cnvun 5995 | The converse of a union is... |
| cnvdif 5996 | Distributive law for conve... |
| cnvin 5997 | Distributive law for conve... |
| rnun 5998 | Distributive law for range... |
| rnin 5999 | The range of an intersecti... |
| rniun 6000 | The range of an indexed un... |
| rnuni 6001 | The range of a union. Par... |
| imaundi 6002 | Distributive law for image... |
| imaundir 6003 | The image of a union. (Co... |
| dminss 6004 | An upper bound for interse... |
| imainss 6005 | An upper bound for interse... |
| inimass 6006 | The image of an intersecti... |
| inimasn 6007 | The intersection of the im... |
| cnvxp 6008 | The converse of a Cartesia... |
| xp0 6009 | The Cartesian product with... |
| xpnz 6010 | The Cartesian product of n... |
| xpeq0 6011 | At least one member of an ... |
| xpdisj1 6012 | Cartesian products with di... |
| xpdisj2 6013 | Cartesian products with di... |
| xpsndisj 6014 | Cartesian products with tw... |
| difxp 6015 | Difference of Cartesian pr... |
| difxp1 6016 | Difference law for Cartesi... |
| difxp2 6017 | Difference law for Cartesi... |
| djudisj 6018 | Disjoint unions with disjo... |
| xpdifid 6019 | The set of distinct couple... |
| resdisj 6020 | A double restriction to di... |
| rnxp 6021 | The range of a Cartesian p... |
| dmxpss 6022 | The domain of a Cartesian ... |
| rnxpss 6023 | The range of a Cartesian p... |
| rnxpid 6024 | The range of a Cartesian s... |
| ssxpb 6025 | A Cartesian product subcla... |
| xp11 6026 | The Cartesian product of n... |
| xpcan 6027 | Cancellation law for Carte... |
| xpcan2 6028 | Cancellation law for Carte... |
| ssrnres 6029 | Two ways to express surjec... |
| rninxp 6030 | Two ways to express surjec... |
| dminxp 6031 | Two ways to express totali... |
| imainrect 6032 | Image by a restricted and ... |
| xpima 6033 | Direct image by a Cartesia... |
| xpima1 6034 | Direct image by a Cartesia... |
| xpima2 6035 | Direct image by a Cartesia... |
| xpimasn 6036 | Direct image of a singleto... |
| sossfld 6037 | The base set of a strict o... |
| sofld 6038 | The base set of a nonempty... |
| cnvcnv3 6039 | The set of all ordered pai... |
| dfrel2 6040 | Alternate definition of re... |
| dfrel4v 6041 | A relation can be expresse... |
| dfrel4 6042 | A relation can be expresse... |
| cnvcnv 6043 | The double converse of a c... |
| cnvcnv2 6044 | The double converse of a c... |
| cnvcnvss 6045 | The double converse of a c... |
| cnvrescnv 6046 | Two ways to express the co... |
| cnveqb 6047 | Equality theorem for conve... |
| cnveq0 6048 | A relation empty iff its c... |
| dfrel3 6049 | Alternate definition of re... |
| elid 6050 | Characterization of the el... |
| dmresv 6051 | The domain of a universal ... |
| rnresv 6052 | The range of a universal r... |
| dfrn4 6053 | Range defined in terms of ... |
| csbrn 6054 | Distribute proper substitu... |
| rescnvcnv 6055 | The restriction of the dou... |
| cnvcnvres 6056 | The double converse of the... |
| imacnvcnv 6057 | The image of the double co... |
| dmsnn0 6058 | The domain of a singleton ... |
| rnsnn0 6059 | The range of a singleton i... |
| dmsn0 6060 | The domain of the singleto... |
| cnvsn0 6061 | The converse of the single... |
| dmsn0el 6062 | The domain of a singleton ... |
| relsn2 6063 | A singleton is a relation ... |
| dmsnopg 6064 | The domain of a singleton ... |
| dmsnopss 6065 | The domain of a singleton ... |
| dmpropg 6066 | The domain of an unordered... |
| dmsnop 6067 | The domain of a singleton ... |
| dmprop 6068 | The domain of an unordered... |
| dmtpop 6069 | The domain of an unordered... |
| cnvcnvsn 6070 | Double converse of a singl... |
| dmsnsnsn 6071 | The domain of the singleto... |
| rnsnopg 6072 | The range of a singleton o... |
| rnpropg 6073 | The range of a pair of ord... |
| cnvsng 6074 | Converse of a singleton of... |
| rnsnop 6075 | The range of a singleton o... |
| op1sta 6076 | Extract the first member o... |
| cnvsn 6077 | Converse of a singleton of... |
| op2ndb 6078 | Extract the second member ... |
| op2nda 6079 | Extract the second member ... |
| opswap 6080 | Swap the members of an ord... |
| cnvresima 6081 | An image under the convers... |
| resdm2 6082 | A class restricted to its ... |
| resdmres 6083 | Restriction to the domain ... |
| resresdm 6084 | A restriction by an arbitr... |
| imadmres 6085 | The image of the domain of... |
| mptpreima 6086 | The preimage of a function... |
| mptiniseg 6087 | Converse singleton image o... |
| dmmpt 6088 | The domain of the mapping ... |
| dmmptss 6089 | The domain of a mapping is... |
| dmmptg 6090 | The domain of the mapping ... |
| relco 6091 | A composition is a relatio... |
| dfco2 6092 | Alternate definition of a ... |
| dfco2a 6093 | Generalization of ~ dfco2 ... |
| coundi 6094 | Class composition distribu... |
| coundir 6095 | Class composition distribu... |
| cores 6096 | Restricted first member of... |
| resco 6097 | Associative law for the re... |
| imaco 6098 | Image of the composition o... |
| rnco 6099 | The range of the compositi... |
| rnco2 6100 | The range of the compositi... |
| dmco 6101 | The domain of a compositio... |
| coeq0 6102 | A composition of two relat... |
| coiun 6103 | Composition with an indexe... |
| cocnvcnv1 6104 | A composition is not affec... |
| cocnvcnv2 6105 | A composition is not affec... |
| cores2 6106 | Absorption of a reverse (p... |
| co02 6107 | Composition with the empty... |
| co01 6108 | Composition with the empty... |
| coi1 6109 | Composition with the ident... |
| coi2 6110 | Composition with the ident... |
| coires1 6111 | Composition with a restric... |
| coass 6112 | Associative law for class ... |
| relcnvtrg 6113 | General form of ~ relcnvtr... |
| relcnvtr 6114 | A relation is transitive i... |
| relssdmrn 6115 | A relation is included in ... |
| cnvssrndm 6116 | The converse is a subset o... |
| cossxp 6117 | Composition as a subset of... |
| relrelss 6118 | Two ways to describe the s... |
| unielrel 6119 | The membership relation fo... |
| relfld 6120 | The double union of a rela... |
| relresfld 6121 | Restriction of a relation ... |
| relcoi2 6122 | Composition with the ident... |
| relcoi1 6123 | Composition with the ident... |
| unidmrn 6124 | The double union of the co... |
| relcnvfld 6125 | if ` R ` is a relation, it... |
| dfdm2 6126 | Alternate definition of do... |
| unixp 6127 | The double class union of ... |
| unixp0 6128 | A Cartesian product is emp... |
| unixpid 6129 | Field of a Cartesian squar... |
| ressn 6130 | Restriction of a class to ... |
| cnviin 6131 | The converse of an interse... |
| cnvpo 6132 | The converse of a partial ... |
| cnvso 6133 | The converse of a strict o... |
| xpco 6134 | Composition of two Cartesi... |
| xpcoid 6135 | Composition of two Cartesi... |
| elsnxp 6136 | Membership in a Cartesian ... |
| reu3op 6137 | There is a unique ordered ... |
| reuop 6138 | There is a unique ordered ... |
| opreu2reurex 6139 | There is a unique ordered ... |
| opreu2reu 6140 | If there is a unique order... |
| predeq123 6143 | Equality theorem for the p... |
| predeq1 6144 | Equality theorem for the p... |
| predeq2 6145 | Equality theorem for the p... |
| predeq3 6146 | Equality theorem for the p... |
| nfpred 6147 | Bound-variable hypothesis ... |
| predpredss 6148 | If ` A ` is a subset of ` ... |
| predss 6149 | The predecessor class of `... |
| sspred 6150 | Another subset/predecessor... |
| dfpred2 6151 | An alternate definition of... |
| dfpred3 6152 | An alternate definition of... |
| dfpred3g 6153 | An alternate definition of... |
| elpredim 6154 | Membership in a predecesso... |
| elpred 6155 | Membership in a predecesso... |
| elpredg 6156 | Membership in a predecesso... |
| predasetex 6157 | The predecessor class exis... |
| dffr4 6158 | Alternate definition of we... |
| predel 6159 | Membership in the predeces... |
| predpo 6160 | Property of the precessor ... |
| predso 6161 | Property of the predecesso... |
| predbrg 6162 | Closed form of ~ elpredim ... |
| setlikespec 6163 | If ` R ` is set-like in ` ... |
| predidm 6164 | Idempotent law for the pre... |
| predin 6165 | Intersection law for prede... |
| predun 6166 | Union law for predecessor ... |
| preddif 6167 | Difference law for predece... |
| predep 6168 | The predecessor under the ... |
| preddowncl 6169 | A property of classes that... |
| predpoirr 6170 | Given a partial ordering, ... |
| predfrirr 6171 | Given a well-founded relat... |
| pred0 6172 | The predecessor class over... |
| tz6.26 6173 | All nonempty subclasses of... |
| tz6.26i 6174 | All nonempty subclasses of... |
| wfi 6175 | The Principle of Well-Foun... |
| wfii 6176 | The Principle of Well-Foun... |
| wfisg 6177 | Well-Founded Induction Sch... |
| wfis 6178 | Well-Founded Induction Sch... |
| wfis2fg 6179 | Well-Founded Induction Sch... |
| wfis2f 6180 | Well Founded Induction sch... |
| wfis2g 6181 | Well-Founded Induction Sch... |
| wfis2 6182 | Well Founded Induction sch... |
| wfis3 6183 | Well Founded Induction sch... |
| ordeq 6192 | Equality theorem for the o... |
| elong 6193 | An ordinal number is an or... |
| elon 6194 | An ordinal number is an or... |
| eloni 6195 | An ordinal number has the ... |
| elon2 6196 | An ordinal number is an or... |
| limeq 6197 | Equality theorem for the l... |
| ordwe 6198 | Membership well-orders eve... |
| ordtr 6199 | An ordinal class is transi... |
| ordfr 6200 | Membership is well-founded... |
| ordelss 6201 | An element of an ordinal c... |
| trssord 6202 | A transitive subclass of a... |
| ordirr 6203 | No ordinal class is a memb... |
| nordeq 6204 | A member of an ordinal cla... |
| ordn2lp 6205 | An ordinal class cannot be... |
| tz7.5 6206 | A nonempty subclass of an ... |
| ordelord 6207 | An element of an ordinal c... |
| tron 6208 | The class of all ordinal n... |
| ordelon 6209 | An element of an ordinal c... |
| onelon 6210 | An element of an ordinal n... |
| tz7.7 6211 | A transitive class belongs... |
| ordelssne 6212 | For ordinal classes, membe... |
| ordelpss 6213 | For ordinal classes, membe... |
| ordsseleq 6214 | For ordinal classes, inclu... |
| ordin 6215 | The intersection of two or... |
| onin 6216 | The intersection of two or... |
| ordtri3or 6217 | A trichotomy law for ordin... |
| ordtri1 6218 | A trichotomy law for ordin... |
| ontri1 6219 | A trichotomy law for ordin... |
| ordtri2 6220 | A trichotomy law for ordin... |
| ordtri3 6221 | A trichotomy law for ordin... |
| ordtri4 6222 | A trichotomy law for ordin... |
| orddisj 6223 | An ordinal class and its s... |
| onfr 6224 | The ordinal class is well-... |
| onelpss 6225 | Relationship between membe... |
| onsseleq 6226 | Relationship between subse... |
| onelss 6227 | An element of an ordinal n... |
| ordtr1 6228 | Transitive law for ordinal... |
| ordtr2 6229 | Transitive law for ordinal... |
| ordtr3 6230 | Transitive law for ordinal... |
| ontr1 6231 | Transitive law for ordinal... |
| ontr2 6232 | Transitive law for ordinal... |
| ordunidif 6233 | The union of an ordinal st... |
| ordintdif 6234 | If ` B ` is smaller than `... |
| onintss 6235 | If a property is true for ... |
| oneqmini 6236 | A way to show that an ordi... |
| ord0 6237 | The empty set is an ordina... |
| 0elon 6238 | The empty set is an ordina... |
| ord0eln0 6239 | A nonempty ordinal contain... |
| on0eln0 6240 | An ordinal number contains... |
| dflim2 6241 | An alternate definition of... |
| inton 6242 | The intersection of the cl... |
| nlim0 6243 | The empty set is not a lim... |
| limord 6244 | A limit ordinal is ordinal... |
| limuni 6245 | A limit ordinal is its own... |
| limuni2 6246 | The union of a limit ordin... |
| 0ellim 6247 | A limit ordinal contains t... |
| limelon 6248 | A limit ordinal class that... |
| onn0 6249 | The class of all ordinal n... |
| suceq 6250 | Equality of successors. (... |
| elsuci 6251 | Membership in a successor.... |
| elsucg 6252 | Membership in a successor.... |
| elsuc2g 6253 | Variant of membership in a... |
| elsuc 6254 | Membership in a successor.... |
| elsuc2 6255 | Membership in a successor.... |
| nfsuc 6256 | Bound-variable hypothesis ... |
| elelsuc 6257 | Membership in a successor.... |
| sucel 6258 | Membership of a successor ... |
| suc0 6259 | The successor of the empty... |
| sucprc 6260 | A proper class is its own ... |
| unisuc 6261 | A transitive class is equa... |
| sssucid 6262 | A class is included in its... |
| sucidg 6263 | Part of Proposition 7.23 o... |
| sucid 6264 | A set belongs to its succe... |
| nsuceq0 6265 | No successor is empty. (C... |
| eqelsuc 6266 | A set belongs to the succe... |
| iunsuc 6267 | Inductive definition for t... |
| suctr 6268 | The successor of a transit... |
| trsuc 6269 | A set whose successor belo... |
| trsucss 6270 | A member of the successor ... |
| ordsssuc 6271 | An ordinal is a subset of ... |
| onsssuc 6272 | A subset of an ordinal num... |
| ordsssuc2 6273 | An ordinal subset of an or... |
| onmindif 6274 | When its successor is subt... |
| ordnbtwn 6275 | There is no set between an... |
| onnbtwn 6276 | There is no set between an... |
| sucssel 6277 | A set whose successor is a... |
| orddif 6278 | Ordinal derived from its s... |
| orduniss 6279 | An ordinal class includes ... |
| ordtri2or 6280 | A trichotomy law for ordin... |
| ordtri2or2 6281 | A trichotomy law for ordin... |
| ordtri2or3 6282 | A consequence of total ord... |
| ordelinel 6283 | The intersection of two or... |
| ordssun 6284 | Property of a subclass of ... |
| ordequn 6285 | The maximum (i.e. union) o... |
| ordun 6286 | The maximum (i.e. union) o... |
| ordunisssuc 6287 | A subclass relationship fo... |
| suc11 6288 | The successor operation be... |
| onordi 6289 | An ordinal number is an or... |
| ontrci 6290 | An ordinal number is a tra... |
| onirri 6291 | An ordinal number is not a... |
| oneli 6292 | A member of an ordinal num... |
| onelssi 6293 | A member of an ordinal num... |
| onssneli 6294 | An ordering law for ordina... |
| onssnel2i 6295 | An ordering law for ordina... |
| onelini 6296 | An element of an ordinal n... |
| oneluni 6297 | An ordinal number equals i... |
| onunisuci 6298 | An ordinal number is equal... |
| onsseli 6299 | Subset is equivalent to me... |
| onun2i 6300 | The union of two ordinal n... |
| unizlim 6301 | An ordinal equal to its ow... |
| on0eqel 6302 | An ordinal number either e... |
| snsn0non 6303 | The singleton of the singl... |
| onxpdisj 6304 | Ordinal numbers and ordere... |
| onnev 6305 | The class of ordinal numbe... |
| iotajust 6307 | Soundness justification th... |
| dfiota2 6309 | Alternate definition for d... |
| nfiota1 6310 | Bound-variable hypothesis ... |
| nfiotadw 6311 | Deduction version of ~ nfi... |
| nfiotaw 6312 | Bound-variable hypothesis ... |
| nfiotad 6313 | Deduction version of ~ nfi... |
| nfiota 6314 | Bound-variable hypothesis ... |
| cbviotaw 6315 | Change bound variables in ... |
| cbviotavw 6316 | Change bound variables in ... |
| cbviota 6317 | Change bound variables in ... |
| cbviotav 6318 | Change bound variables in ... |
| sb8iota 6319 | Variable substitution in d... |
| iotaeq 6320 | Equality theorem for descr... |
| iotabi 6321 | Equivalence theorem for de... |
| uniabio 6322 | Part of Theorem 8.17 in [Q... |
| iotaval 6323 | Theorem 8.19 in [Quine] p.... |
| iotauni 6324 | Equivalence between two di... |
| iotaint 6325 | Equivalence between two di... |
| iota1 6326 | Property of iota. (Contri... |
| iotanul 6327 | Theorem 8.22 in [Quine] p.... |
| iotassuni 6328 | The ` iota ` class is a su... |
| iotaex 6329 | Theorem 8.23 in [Quine] p.... |
| iota4 6330 | Theorem *14.22 in [Whitehe... |
| iota4an 6331 | Theorem *14.23 in [Whitehe... |
| iota5 6332 | A method for computing iot... |
| iotabidv 6333 | Formula-building deduction... |
| iotabii 6334 | Formula-building deduction... |
| iotacl 6335 | Membership law for descrip... |
| iota2df 6336 | A condition that allows us... |
| iota2d 6337 | A condition that allows us... |
| iota2 6338 | The unique element such th... |
| iotan0 6339 | Representation of "the uni... |
| sniota 6340 | A class abstraction with a... |
| dfiota4 6341 | The ` iota ` operation usi... |
| csbiota 6342 | Class substitution within ... |
| dffun2 6359 | Alternate definition of a ... |
| dffun3 6360 | Alternate definition of fu... |
| dffun4 6361 | Alternate definition of a ... |
| dffun5 6362 | Alternate definition of fu... |
| dffun6f 6363 | Definition of function, us... |
| dffun6 6364 | Alternate definition of a ... |
| funmo 6365 | A function has at most one... |
| funrel 6366 | A function is a relation. ... |
| 0nelfun 6367 | A function does not contai... |
| funss 6368 | Subclass theorem for funct... |
| funeq 6369 | Equality theorem for funct... |
| funeqi 6370 | Equality inference for the... |
| funeqd 6371 | Equality deduction for the... |
| nffun 6372 | Bound-variable hypothesis ... |
| sbcfung 6373 | Distribute proper substitu... |
| funeu 6374 | There is exactly one value... |
| funeu2 6375 | There is exactly one value... |
| dffun7 6376 | Alternate definition of a ... |
| dffun8 6377 | Alternate definition of a ... |
| dffun9 6378 | Alternate definition of a ... |
| funfn 6379 | A class is a function if a... |
| funfnd 6380 | A function is a function o... |
| funi 6381 | The identity relation is a... |
| nfunv 6382 | The universal class is not... |
| funopg 6383 | A Kuratowski ordered pair ... |
| funopab 6384 | A class of ordered pairs i... |
| funopabeq 6385 | A class of ordered pairs o... |
| funopab4 6386 | A class of ordered pairs o... |
| funmpt 6387 | A function in maps-to nota... |
| funmpt2 6388 | Functionality of a class g... |
| funco 6389 | The composition of two fun... |
| funresfunco 6390 | Composition of two functio... |
| funres 6391 | A restriction of a functio... |
| funssres 6392 | The restriction of a funct... |
| fun2ssres 6393 | Equality of restrictions o... |
| funun 6394 | The union of functions wit... |
| fununmo 6395 | If the union of classes is... |
| fununfun 6396 | If the union of classes is... |
| fundif 6397 | A function with removed el... |
| funcnvsn 6398 | The converse singleton of ... |
| funsng 6399 | A singleton of an ordered ... |
| fnsng 6400 | Functionality and domain o... |
| funsn 6401 | A singleton of an ordered ... |
| funprg 6402 | A set of two pairs is a fu... |
| funtpg 6403 | A set of three pairs is a ... |
| funpr 6404 | A function with a domain o... |
| funtp 6405 | A function with a domain o... |
| fnsn 6406 | Functionality and domain o... |
| fnprg 6407 | Function with a domain of ... |
| fntpg 6408 | Function with a domain of ... |
| fntp 6409 | A function with a domain o... |
| funcnvpr 6410 | The converse pair of order... |
| funcnvtp 6411 | The converse triple of ord... |
| funcnvqp 6412 | The converse quadruple of ... |
| fun0 6413 | The empty set is a functio... |
| funcnv0 6414 | The converse of the empty ... |
| funcnvcnv 6415 | The double converse of a f... |
| funcnv2 6416 | A simpler equivalence for ... |
| funcnv 6417 | The converse of a class is... |
| funcnv3 6418 | A condition showing a clas... |
| fun2cnv 6419 | The double converse of a c... |
| svrelfun 6420 | A single-valued relation i... |
| fncnv 6421 | Single-rootedness (see ~ f... |
| fun11 6422 | Two ways of stating that `... |
| fununi 6423 | The union of a chain (with... |
| funin 6424 | The intersection with a fu... |
| funres11 6425 | The restriction of a one-t... |
| funcnvres 6426 | The converse of a restrict... |
| cnvresid 6427 | Converse of a restricted i... |
| funcnvres2 6428 | The converse of a restrict... |
| funimacnv 6429 | The image of the preimage ... |
| funimass1 6430 | A kind of contraposition l... |
| funimass2 6431 | A kind of contraposition l... |
| imadif 6432 | The image of a difference ... |
| imain 6433 | The image of an intersecti... |
| funimaexg 6434 | Axiom of Replacement using... |
| funimaex 6435 | The image of a set under a... |
| isarep1 6436 | Part of a study of the Axi... |
| isarep2 6437 | Part of a study of the Axi... |
| fneq1 6438 | Equality theorem for funct... |
| fneq2 6439 | Equality theorem for funct... |
| fneq1d 6440 | Equality deduction for fun... |
| fneq2d 6441 | Equality deduction for fun... |
| fneq12d 6442 | Equality deduction for fun... |
| fneq12 6443 | Equality theorem for funct... |
| fneq1i 6444 | Equality inference for fun... |
| fneq2i 6445 | Equality inference for fun... |
| nffn 6446 | Bound-variable hypothesis ... |
| fnfun 6447 | A function with domain is ... |
| fnrel 6448 | A function with domain is ... |
| fndm 6449 | The domain of a function. ... |
| fndmd 6450 | The domain of a function. ... |
| funfni 6451 | Inference to convert a fun... |
| fndmu 6452 | A function has a unique do... |
| fnbr 6453 | The first argument of bina... |
| fnop 6454 | The first argument of an o... |
| fneu 6455 | There is exactly one value... |
| fneu2 6456 | There is exactly one value... |
| fnun 6457 | The union of two functions... |
| fnunsn 6458 | Extension of a function wi... |
| fnco 6459 | Composition of two functio... |
| fnresdm 6460 | A function does not change... |
| fnresdisj 6461 | A function restricted to a... |
| 2elresin 6462 | Membership in two function... |
| fnssresb 6463 | Restriction of a function ... |
| fnssres 6464 | Restriction of a function ... |
| fnssresd 6465 | Restriction of a function ... |
| fnresin1 6466 | Restriction of a function'... |
| fnresin2 6467 | Restriction of a function'... |
| fnres 6468 | An equivalence for functio... |
| idfn 6469 | The identity relation is a... |
| fnresi 6470 | The restricted identity re... |
| fnresiOLD 6471 | Obsolete proof of ~ fnresi... |
| fnima 6472 | The image of a function's ... |
| fn0 6473 | A function with empty doma... |
| fnimadisj 6474 | A class that is disjoint w... |
| fnimaeq0 6475 | Images under a function ne... |
| dfmpt3 6476 | Alternate definition for t... |
| mptfnf 6477 | The maps-to notation defin... |
| fnmptf 6478 | The maps-to notation defin... |
| fnopabg 6479 | Functionality and domain o... |
| fnopab 6480 | Functionality and domain o... |
| mptfng 6481 | The maps-to notation defin... |
| fnmpt 6482 | The maps-to notation defin... |
| fnmptd 6483 | The maps-to notation defin... |
| mpt0 6484 | A mapping operation with e... |
| fnmpti 6485 | Functionality and domain o... |
| dmmpti 6486 | Domain of the mapping oper... |
| dmmptd 6487 | The domain of the mapping ... |
| mptun 6488 | Union of mappings which ar... |
| feq1 6489 | Equality theorem for funct... |
| feq2 6490 | Equality theorem for funct... |
| feq3 6491 | Equality theorem for funct... |
| feq23 6492 | Equality theorem for funct... |
| feq1d 6493 | Equality deduction for fun... |
| feq2d 6494 | Equality deduction for fun... |
| feq3d 6495 | Equality deduction for fun... |
| feq12d 6496 | Equality deduction for fun... |
| feq123d 6497 | Equality deduction for fun... |
| feq123 6498 | Equality theorem for funct... |
| feq1i 6499 | Equality inference for fun... |
| feq2i 6500 | Equality inference for fun... |
| feq12i 6501 | Equality inference for fun... |
| feq23i 6502 | Equality inference for fun... |
| feq23d 6503 | Equality deduction for fun... |
| nff 6504 | Bound-variable hypothesis ... |
| sbcfng 6505 | Distribute proper substitu... |
| sbcfg 6506 | Distribute proper substitu... |
| elimf 6507 | Eliminate a mapping hypoth... |
| ffn 6508 | A mapping is a function wi... |
| ffnd 6509 | A mapping is a function wi... |
| dffn2 6510 | Any function is a mapping ... |
| ffun 6511 | A mapping is a function. ... |
| ffund 6512 | A mapping is a function, d... |
| frel 6513 | A mapping is a relation. ... |
| frn 6514 | The range of a mapping. (... |
| frnd 6515 | Deduction form of ~ frn . ... |
| fdm 6516 | The domain of a mapping. ... |
| fdmd 6517 | Deduction form of ~ fdm . ... |
| fdmi 6518 | Inference associated with ... |
| dffn3 6519 | A function maps to its ran... |
| ffrn 6520 | A function maps to its ran... |
| fss 6521 | Expanding the codomain of ... |
| fssd 6522 | Expanding the codomain of ... |
| fssdmd 6523 | Expressing that a class is... |
| fssdm 6524 | Expressing that a class is... |
| fco 6525 | Composition of two mapping... |
| fcod 6526 | Composition of two mapping... |
| fco2 6527 | Functionality of a composi... |
| fssxp 6528 | A mapping is a class of or... |
| funssxp 6529 | Two ways of specifying a p... |
| ffdm 6530 | A mapping is a partial fun... |
| ffdmd 6531 | The domain of a function. ... |
| fdmrn 6532 | A different way to write `... |
| opelf 6533 | The members of an ordered ... |
| fun 6534 | The union of two functions... |
| fun2 6535 | The union of two functions... |
| fun2d 6536 | The union of functions wit... |
| fnfco 6537 | Composition of two functio... |
| fssres 6538 | Restriction of a function ... |
| fssresd 6539 | Restriction of a function ... |
| fssres2 6540 | Restriction of a restricte... |
| fresin 6541 | An identity for the mappin... |
| resasplit 6542 | If two functions agree on ... |
| fresaun 6543 | The union of two functions... |
| fresaunres2 6544 | From the union of two func... |
| fresaunres1 6545 | From the union of two func... |
| fcoi1 6546 | Composition of a mapping a... |
| fcoi2 6547 | Composition of restricted ... |
| feu 6548 | There is exactly one value... |
| fimass 6549 | The image of a class is a ... |
| fcnvres 6550 | The converse of a restrict... |
| fimacnvdisj 6551 | The preimage of a class di... |
| fint 6552 | Function into an intersect... |
| fin 6553 | Mapping into an intersecti... |
| f0 6554 | The empty function. (Cont... |
| f00 6555 | A class is a function with... |
| f0bi 6556 | A function with empty doma... |
| f0dom0 6557 | A function is empty iff it... |
| f0rn0 6558 | If there is no element in ... |
| fconst 6559 | A Cartesian product with a... |
| fconstg 6560 | A Cartesian product with a... |
| fnconstg 6561 | A Cartesian product with a... |
| fconst6g 6562 | Constant function with loo... |
| fconst6 6563 | A constant function as a m... |
| f1eq1 6564 | Equality theorem for one-t... |
| f1eq2 6565 | Equality theorem for one-t... |
| f1eq3 6566 | Equality theorem for one-t... |
| nff1 6567 | Bound-variable hypothesis ... |
| dff12 6568 | Alternate definition of a ... |
| f1f 6569 | A one-to-one mapping is a ... |
| f1fn 6570 | A one-to-one mapping is a ... |
| f1fun 6571 | A one-to-one mapping is a ... |
| f1rel 6572 | A one-to-one onto mapping ... |
| f1dm 6573 | The domain of a one-to-one... |
| f1ss 6574 | A function that is one-to-... |
| f1ssr 6575 | A function that is one-to-... |
| f1ssres 6576 | A function that is one-to-... |
| f1resf1 6577 | The restriction of an inje... |
| f1cnvcnv 6578 | Two ways to express that a... |
| f1co 6579 | Composition of one-to-one ... |
| foeq1 6580 | Equality theorem for onto ... |
| foeq2 6581 | Equality theorem for onto ... |
| foeq3 6582 | Equality theorem for onto ... |
| nffo 6583 | Bound-variable hypothesis ... |
| fof 6584 | An onto mapping is a mappi... |
| fofun 6585 | An onto mapping is a funct... |
| fofn 6586 | An onto mapping is a funct... |
| forn 6587 | The codomain of an onto fu... |
| dffo2 6588 | Alternate definition of an... |
| foima 6589 | The image of the domain of... |
| dffn4 6590 | A function maps onto its r... |
| funforn 6591 | A function maps its domain... |
| fodmrnu 6592 | An onto function has uniqu... |
| fimadmfo 6593 | A function is a function o... |
| fores 6594 | Restriction of an onto fun... |
| fimadmfoALT 6595 | Alternate proof of ~ fimad... |
| foco 6596 | Composition of onto functi... |
| foconst 6597 | A nonzero constant functio... |
| f1oeq1 6598 | Equality theorem for one-t... |
| f1oeq2 6599 | Equality theorem for one-t... |
| f1oeq3 6600 | Equality theorem for one-t... |
| f1oeq23 6601 | Equality theorem for one-t... |
| f1eq123d 6602 | Equality deduction for one... |
| foeq123d 6603 | Equality deduction for ont... |
| f1oeq123d 6604 | Equality deduction for one... |
| f1oeq2d 6605 | Equality deduction for one... |
| f1oeq3d 6606 | Equality deduction for one... |
| nff1o 6607 | Bound-variable hypothesis ... |
| f1of1 6608 | A one-to-one onto mapping ... |
| f1of 6609 | A one-to-one onto mapping ... |
| f1ofn 6610 | A one-to-one onto mapping ... |
| f1ofun 6611 | A one-to-one onto mapping ... |
| f1orel 6612 | A one-to-one onto mapping ... |
| f1odm 6613 | The domain of a one-to-one... |
| dff1o2 6614 | Alternate definition of on... |
| dff1o3 6615 | Alternate definition of on... |
| f1ofo 6616 | A one-to-one onto function... |
| dff1o4 6617 | Alternate definition of on... |
| dff1o5 6618 | Alternate definition of on... |
| f1orn 6619 | A one-to-one function maps... |
| f1f1orn 6620 | A one-to-one function maps... |
| f1ocnv 6621 | The converse of a one-to-o... |
| f1ocnvb 6622 | A relation is a one-to-one... |
| f1ores 6623 | The restriction of a one-t... |
| f1orescnv 6624 | The converse of a one-to-o... |
| f1imacnv 6625 | Preimage of an image. (Co... |
| foimacnv 6626 | A reverse version of ~ f1i... |
| foun 6627 | The union of two onto func... |
| f1oun 6628 | The union of two one-to-on... |
| resdif 6629 | The restriction of a one-t... |
| resin 6630 | The restriction of a one-t... |
| f1oco 6631 | Composition of one-to-one ... |
| f1cnv 6632 | The converse of an injecti... |
| funcocnv2 6633 | Composition with the conve... |
| fococnv2 6634 | The composition of an onto... |
| f1ococnv2 6635 | The composition of a one-t... |
| f1cocnv2 6636 | Composition of an injectiv... |
| f1ococnv1 6637 | The composition of a one-t... |
| f1cocnv1 6638 | Composition of an injectiv... |
| funcoeqres 6639 | Express a constraint on a ... |
| f1ssf1 6640 | A subset of an injective f... |
| f10 6641 | The empty set maps one-to-... |
| f10d 6642 | The empty set maps one-to-... |
| f1o00 6643 | One-to-one onto mapping of... |
| fo00 6644 | Onto mapping of the empty ... |
| f1o0 6645 | One-to-one onto mapping of... |
| f1oi 6646 | A restriction of the ident... |
| f1ovi 6647 | The identity relation is a... |
| f1osn 6648 | A singleton of an ordered ... |
| f1osng 6649 | A singleton of an ordered ... |
| f1sng 6650 | A singleton of an ordered ... |
| fsnd 6651 | A singleton of an ordered ... |
| f1oprswap 6652 | A two-element swap is a bi... |
| f1oprg 6653 | An unordered pair of order... |
| tz6.12-2 6654 | Function value when ` F ` ... |
| fveu 6655 | The value of a function at... |
| brprcneu 6656 | If ` A ` is a proper class... |
| fvprc 6657 | A function's value at a pr... |
| rnfvprc 6658 | The range of a function va... |
| fv2 6659 | Alternate definition of fu... |
| dffv3 6660 | A definition of function v... |
| dffv4 6661 | The previous definition of... |
| elfv 6662 | Membership in a function v... |
| fveq1 6663 | Equality theorem for funct... |
| fveq2 6664 | Equality theorem for funct... |
| fveq1i 6665 | Equality inference for fun... |
| fveq1d 6666 | Equality deduction for fun... |
| fveq2i 6667 | Equality inference for fun... |
| fveq2d 6668 | Equality deduction for fun... |
| 2fveq3 6669 | Equality theorem for neste... |
| fveq12i 6670 | Equality deduction for fun... |
| fveq12d 6671 | Equality deduction for fun... |
| fveqeq2d 6672 | Equality deduction for fun... |
| fveqeq2 6673 | Equality deduction for fun... |
| nffv 6674 | Bound-variable hypothesis ... |
| nffvmpt1 6675 | Bound-variable hypothesis ... |
| nffvd 6676 | Deduction version of bound... |
| fvex 6677 | The value of a class exist... |
| fvexi 6678 | The value of a class exist... |
| fvexd 6679 | The value of a class exist... |
| fvif 6680 | Move a conditional outside... |
| iffv 6681 | Move a conditional outside... |
| fv3 6682 | Alternate definition of th... |
| fvres 6683 | The value of a restricted ... |
| fvresd 6684 | The value of a restricted ... |
| funssfv 6685 | The value of a member of t... |
| tz6.12-1 6686 | Function value. Theorem 6... |
| tz6.12 6687 | Function value. Theorem 6... |
| tz6.12f 6688 | Function value, using boun... |
| tz6.12c 6689 | Corollary of Theorem 6.12(... |
| tz6.12i 6690 | Corollary of Theorem 6.12(... |
| fvbr0 6691 | Two possibilities for the ... |
| fvrn0 6692 | A function value is a memb... |
| fvssunirn 6693 | The result of a function v... |
| ndmfv 6694 | The value of a class outsi... |
| ndmfvrcl 6695 | Reverse closure law for fu... |
| elfvdm 6696 | If a function value has a ... |
| elfvex 6697 | If a function value has a ... |
| elfvexd 6698 | If a function value has a ... |
| eliman0 6699 | A nonempty function value ... |
| nfvres 6700 | The value of a non-member ... |
| nfunsn 6701 | If the restriction of a cl... |
| fvfundmfvn0 6702 | If the "value of a class" ... |
| 0fv 6703 | Function value of the empt... |
| fv2prc 6704 | A function value of a func... |
| elfv2ex 6705 | If a function value of a f... |
| fveqres 6706 | Equal values imply equal v... |
| csbfv12 6707 | Move class substitution in... |
| csbfv2g 6708 | Move class substitution in... |
| csbfv 6709 | Substitution for a functio... |
| funbrfv 6710 | The second argument of a b... |
| funopfv 6711 | The second element in an o... |
| fnbrfvb 6712 | Equivalence of function va... |
| fnopfvb 6713 | Equivalence of function va... |
| funbrfvb 6714 | Equivalence of function va... |
| funopfvb 6715 | Equivalence of function va... |
| fnbrfvb2 6716 | Version of ~ fnbrfvb for f... |
| funbrfv2b 6717 | Function value in terms of... |
| dffn5 6718 | Representation of a functi... |
| fnrnfv 6719 | The range of a function ex... |
| fvelrnb 6720 | A member of a function's r... |
| foelrni 6721 | A member of a surjective f... |
| dfimafn 6722 | Alternate definition of th... |
| dfimafn2 6723 | Alternate definition of th... |
| funimass4 6724 | Membership relation for th... |
| fvelima 6725 | Function value in an image... |
| fvelimad 6726 | Function value in an image... |
| feqmptd 6727 | Deduction form of ~ dffn5 ... |
| feqresmpt 6728 | Express a restricted funct... |
| feqmptdf 6729 | Deduction form of ~ dffn5f... |
| dffn5f 6730 | Representation of a functi... |
| fvelimab 6731 | Function value in an image... |
| fvelimabd 6732 | Deduction form of ~ fvelim... |
| unima 6733 | Image of a union. (Contri... |
| fvi 6734 | The value of the identity ... |
| fviss 6735 | The value of the identity ... |
| fniinfv 6736 | The indexed intersection o... |
| fnsnfv 6737 | Singleton of function valu... |
| opabiotafun 6738 | Define a function whose va... |
| opabiotadm 6739 | Define a function whose va... |
| opabiota 6740 | Define a function whose va... |
| fnimapr 6741 | The image of a pair under ... |
| ssimaex 6742 | The existence of a subimag... |
| ssimaexg 6743 | The existence of a subimag... |
| funfv 6744 | A simplified expression fo... |
| funfv2 6745 | The value of a function. ... |
| funfv2f 6746 | The value of a function. ... |
| fvun 6747 | Value of the union of two ... |
| fvun1 6748 | The value of a union when ... |
| fvun2 6749 | The value of a union when ... |
| dffv2 6750 | Alternate definition of fu... |
| dmfco 6751 | Domains of a function comp... |
| fvco2 6752 | Value of a function compos... |
| fvco 6753 | Value of a function compos... |
| fvco3 6754 | Value of a function compos... |
| fvco3d 6755 | Value of a function compos... |
| fvco4i 6756 | Conditions for a compositi... |
| fvopab3g 6757 | Value of a function given ... |
| fvopab3ig 6758 | Value of a function given ... |
| brfvopabrbr 6759 | The binary relation of a f... |
| fvmptg 6760 | Value of a function given ... |
| fvmpti 6761 | Value of a function given ... |
| fvmpt 6762 | Value of a function given ... |
| fvmpt2f 6763 | Value of a function given ... |
| fvtresfn 6764 | Functionality of a tuple-r... |
| fvmpts 6765 | Value of a function given ... |
| fvmpt3 6766 | Value of a function given ... |
| fvmpt3i 6767 | Value of a function given ... |
| fvmptdf 6768 | Deduction version of ~ fvm... |
| fvmptd 6769 | Deduction version of ~ fvm... |
| fvmptd2 6770 | Deduction version of ~ fvm... |
| mptrcl 6771 | Reverse closure for a mapp... |
| fvmpt2i 6772 | Value of a function given ... |
| fvmpt2 6773 | Value of a function given ... |
| fvmptss 6774 | If all the values of the m... |
| fvmpt2d 6775 | Deduction version of ~ fvm... |
| fvmptex 6776 | Express a function ` F ` w... |
| fvmptd3f 6777 | Alternate deduction versio... |
| fvmptd2f 6778 | Alternate deduction versio... |
| fvmptdv 6779 | Alternate deduction versio... |
| fvmptdv2 6780 | Alternate deduction versio... |
| mpteqb 6781 | Bidirectional equality the... |
| fvmptt 6782 | Closed theorem form of ~ f... |
| fvmptf 6783 | Value of a function given ... |
| fvmptnf 6784 | The value of a function gi... |
| fvmptd3 6785 | Deduction version of ~ fvm... |
| fvmptn 6786 | This somewhat non-intuitiv... |
| fvmptss2 6787 | A mapping always evaluates... |
| elfvmptrab1w 6788 | Implications for the value... |
| elfvmptrab1 6789 | Implications for the value... |
| elfvmptrab 6790 | Implications for the value... |
| fvopab4ndm 6791 | Value of a function given ... |
| fvmptndm 6792 | Value of a function given ... |
| fvmptrabfv 6793 | Value of a function mappin... |
| fvopab5 6794 | The value of a function th... |
| fvopab6 6795 | Value of a function given ... |
| eqfnfv 6796 | Equality of functions is d... |
| eqfnfv2 6797 | Equality of functions is d... |
| eqfnfv3 6798 | Derive equality of functio... |
| eqfnfvd 6799 | Deduction for equality of ... |
| eqfnfv2f 6800 | Equality of functions is d... |
| eqfunfv 6801 | Equality of functions is d... |
| fvreseq0 6802 | Equality of restricted fun... |
| fvreseq1 6803 | Equality of a function res... |
| fvreseq 6804 | Equality of restricted fun... |
| fnmptfvd 6805 | A function with a given do... |
| fndmdif 6806 | Two ways to express the lo... |
| fndmdifcom 6807 | The difference set between... |
| fndmdifeq0 6808 | The difference set of two ... |
| fndmin 6809 | Two ways to express the lo... |
| fneqeql 6810 | Two functions are equal if... |
| fneqeql2 6811 | Two functions are equal if... |
| fnreseql 6812 | Two functions are equal on... |
| chfnrn 6813 | The range of a choice func... |
| funfvop 6814 | Ordered pair with function... |
| funfvbrb 6815 | Two ways to say that ` A `... |
| fvimacnvi 6816 | A member of a preimage is ... |
| fvimacnv 6817 | The argument of a function... |
| funimass3 6818 | A kind of contraposition l... |
| funimass5 6819 | A subclass of a preimage i... |
| funconstss 6820 | Two ways of specifying tha... |
| fvimacnvALT 6821 | Alternate proof of ~ fvima... |
| elpreima 6822 | Membership in the preimage... |
| elpreimad 6823 | Membership in the preimage... |
| fniniseg 6824 | Membership in the preimage... |
| fncnvima2 6825 | Inverse images under funct... |
| fniniseg2 6826 | Inverse point images under... |
| unpreima 6827 | Preimage of a union. (Con... |
| inpreima 6828 | Preimage of an intersectio... |
| difpreima 6829 | Preimage of a difference. ... |
| respreima 6830 | The preimage of a restrict... |
| iinpreima 6831 | Preimage of an intersectio... |
| intpreima 6832 | Preimage of an intersectio... |
| fimacnv 6833 | The preimage of the codoma... |
| fimacnvinrn 6834 | Taking the converse image ... |
| fimacnvinrn2 6835 | Taking the converse image ... |
| fvn0ssdmfun 6836 | If a class' function value... |
| fnopfv 6837 | Ordered pair with function... |
| fvelrn 6838 | A function's value belongs... |
| nelrnfvne 6839 | A function value cannot be... |
| fveqdmss 6840 | If the empty set is not co... |
| fveqressseq 6841 | If the empty set is not co... |
| fnfvelrn 6842 | A function's value belongs... |
| ffvelrn 6843 | A function's value belongs... |
| ffvelrni 6844 | A function's value belongs... |
| ffvelrnda 6845 | A function's value belongs... |
| ffvelrnd 6846 | A function's value belongs... |
| rexrn 6847 | Restricted existential qua... |
| ralrn 6848 | Restricted universal quant... |
| elrnrexdm 6849 | For any element in the ran... |
| elrnrexdmb 6850 | For any element in the ran... |
| eldmrexrn 6851 | For any element in the dom... |
| eldmrexrnb 6852 | For any element in the dom... |
| fvcofneq 6853 | The values of two function... |
| ralrnmptw 6854 | A restricted quantifier ov... |
| rexrnmptw 6855 | A restricted quantifier ov... |
| ralrnmpt 6856 | A restricted quantifier ov... |
| rexrnmpt 6857 | A restricted quantifier ov... |
| f0cli 6858 | Unconditional closure of a... |
| dff2 6859 | Alternate definition of a ... |
| dff3 6860 | Alternate definition of a ... |
| dff4 6861 | Alternate definition of a ... |
| dffo3 6862 | An onto mapping expressed ... |
| dffo4 6863 | Alternate definition of an... |
| dffo5 6864 | Alternate definition of an... |
| exfo 6865 | A relation equivalent to t... |
| foelrn 6866 | Property of a surjective f... |
| foco2 6867 | If a composition of two fu... |
| fmpt 6868 | Functionality of the mappi... |
| f1ompt 6869 | Express bijection for a ma... |
| fmpti 6870 | Functionality of the mappi... |
| fvmptelrn 6871 | The value of a function at... |
| fmptd 6872 | Domain and codomain of the... |
| fmpttd 6873 | Version of ~ fmptd with in... |
| fmpt3d 6874 | Domain and codomain of the... |
| fmptdf 6875 | A version of ~ fmptd using... |
| ffnfv 6876 | A function maps to a class... |
| ffnfvf 6877 | A function maps to a class... |
| fnfvrnss 6878 | An upper bound for range d... |
| frnssb 6879 | A function is a function i... |
| rnmptss 6880 | The range of an operation ... |
| fmpt2d 6881 | Domain and codomain of the... |
| ffvresb 6882 | A necessary and sufficient... |
| f1oresrab 6883 | Build a bijection between ... |
| f1ossf1o 6884 | Restricting a bijection, w... |
| fmptco 6885 | Composition of two functio... |
| fmptcof 6886 | Version of ~ fmptco where ... |
| fmptcos 6887 | Composition of two functio... |
| cofmpt 6888 | Express composition of a m... |
| fcompt 6889 | Express composition of two... |
| fcoconst 6890 | Composition with a constan... |
| fsn 6891 | A function maps a singleto... |
| fsn2 6892 | A function that maps a sin... |
| fsng 6893 | A function maps a singleto... |
| fsn2g 6894 | A function that maps a sin... |
| xpsng 6895 | The Cartesian product of t... |
| xpprsng 6896 | The Cartesian product of a... |
| xpsn 6897 | The Cartesian product of t... |
| f1o2sn 6898 | A singleton consisting in ... |
| residpr 6899 | Restriction of the identit... |
| dfmpt 6900 | Alternate definition for t... |
| fnasrn 6901 | A function expressed as th... |
| idref 6902 | Two ways to state that a r... |
| funiun 6903 | A function is a union of s... |
| funopsn 6904 | If a function is an ordere... |
| funop 6905 | An ordered pair is a funct... |
| funopdmsn 6906 | The domain of a function w... |
| funsndifnop 6907 | A singleton of an ordered ... |
| funsneqopb 6908 | A singleton of an ordered ... |
| ressnop0 6909 | If ` A ` is not in ` C ` ,... |
| fpr 6910 | A function with a domain o... |
| fprg 6911 | A function with a domain o... |
| ftpg 6912 | A function with a domain o... |
| ftp 6913 | A function with a domain o... |
| fnressn 6914 | A function restricted to a... |
| funressn 6915 | A function restricted to a... |
| fressnfv 6916 | The value of a function re... |
| fvrnressn 6917 | If the value of a function... |
| fvressn 6918 | The value of a function re... |
| fvn0fvelrn 6919 | If the value of a function... |
| fvconst 6920 | The value of a constant fu... |
| fnsnr 6921 | If a class belongs to a fu... |
| fnsnb 6922 | A function whose domain is... |
| fmptsn 6923 | Express a singleton functi... |
| fmptsng 6924 | Express a singleton functi... |
| fmptsnd 6925 | Express a singleton functi... |
| fmptap 6926 | Append an additional value... |
| fmptapd 6927 | Append an additional value... |
| fmptpr 6928 | Express a pair function in... |
| fvresi 6929 | The value of a restricted ... |
| fninfp 6930 | Express the class of fixed... |
| fnelfp 6931 | Property of a fixed point ... |
| fndifnfp 6932 | Express the class of non-f... |
| fnelnfp 6933 | Property of a non-fixed po... |
| fnnfpeq0 6934 | A function is the identity... |
| fvunsn 6935 | Remove an ordered pair not... |
| fvsng 6936 | The value of a singleton o... |
| fvsn 6937 | The value of a singleton o... |
| fvsnun1 6938 | The value of a function wi... |
| fvsnun2 6939 | The value of a function wi... |
| fnsnsplit 6940 | Split a function into a si... |
| fsnunf 6941 | Adjoining a point to a fun... |
| fsnunf2 6942 | Adjoining a point to a pun... |
| fsnunfv 6943 | Recover the added point fr... |
| fsnunres 6944 | Recover the original funct... |
| funresdfunsn 6945 | Restricting a function to ... |
| fvpr1 6946 | The value of a function wi... |
| fvpr2 6947 | The value of a function wi... |
| fvpr1g 6948 | The value of a function wi... |
| fvpr2g 6949 | The value of a function wi... |
| fprb 6950 | A condition for functionho... |
| fvtp1 6951 | The first value of a funct... |
| fvtp2 6952 | The second value of a func... |
| fvtp3 6953 | The third value of a funct... |
| fvtp1g 6954 | The value of a function wi... |
| fvtp2g 6955 | The value of a function wi... |
| fvtp3g 6956 | The value of a function wi... |
| tpres 6957 | An unordered triple of ord... |
| fvconst2g 6958 | The value of a constant fu... |
| fconst2g 6959 | A constant function expres... |
| fvconst2 6960 | The value of a constant fu... |
| fconst2 6961 | A constant function expres... |
| fconst5 6962 | Two ways to express that a... |
| rnmptc 6963 | Range of a constant functi... |
| rnmptcOLD 6964 | Range of a constant functi... |
| fnprb 6965 | A function whose domain ha... |
| fntpb 6966 | A function whose domain ha... |
| fnpr2g 6967 | A function whose domain ha... |
| fpr2g 6968 | A function that maps a pai... |
| fconstfv 6969 | A constant function expres... |
| fconst3 6970 | Two ways to express a cons... |
| fconst4 6971 | Two ways to express a cons... |
| resfunexg 6972 | The restriction of a funct... |
| resiexd 6973 | The restriction of the ide... |
| fnex 6974 | If the domain of a functio... |
| fnexd 6975 | If the domain of a functio... |
| funex 6976 | If the domain of a functio... |
| opabex 6977 | Existence of a function ex... |
| mptexg 6978 | If the domain of a functio... |
| mptexgf 6979 | If the domain of a functio... |
| mptex 6980 | If the domain of a functio... |
| mptexd 6981 | If the domain of a functio... |
| mptrabex 6982 | If the domain of a functio... |
| fex 6983 | If the domain of a mapping... |
| mptfvmpt 6984 | A function in maps-to nota... |
| eufnfv 6985 | A function is uniquely det... |
| funfvima 6986 | A function's value in a pr... |
| funfvima2 6987 | A function's value in an i... |
| funfvima2d 6988 | A function's value in a pr... |
| fnfvima 6989 | The function value of an o... |
| fnfvimad 6990 | A function's value belongs... |
| resfvresima 6991 | The value of the function ... |
| funfvima3 6992 | A class including a functi... |
| rexima 6993 | Existential quantification... |
| ralima 6994 | Universal quantification u... |
| fvclss 6995 | Upper bound for the class ... |
| elabrex 6996 | Elementhood in an image se... |
| abrexco 6997 | Composition of two image m... |
| imaiun 6998 | The image of an indexed un... |
| imauni 6999 | The image of a union is th... |
| fniunfv 7000 | The indexed union of a fun... |
| funiunfv 7001 | The indexed union of a fun... |
| funiunfvf 7002 | The indexed union of a fun... |
| eluniima 7003 | Membership in the union of... |
| elunirn 7004 | Membership in the union of... |
| elunirnALT 7005 | Alternate proof of ~ eluni... |
| fnunirn 7006 | Membership in a union of s... |
| dff13 7007 | A one-to-one function in t... |
| dff13f 7008 | A one-to-one function in t... |
| f1veqaeq 7009 | If the values of a one-to-... |
| f1cofveqaeq 7010 | If the values of a composi... |
| f1cofveqaeqALT 7011 | Alternate proof of ~ f1cof... |
| 2f1fvneq 7012 | If two one-to-one function... |
| f1mpt 7013 | Express injection for a ma... |
| f1fveq 7014 | Equality of function value... |
| f1elima 7015 | Membership in the image of... |
| f1imass 7016 | Taking images under a one-... |
| f1imaeq 7017 | Taking images under a one-... |
| f1imapss 7018 | Taking images under a one-... |
| fpropnf1 7019 | A function, given by an un... |
| f1dom3fv3dif 7020 | The function values for a ... |
| f1dom3el3dif 7021 | The range of a 1-1 functio... |
| dff14a 7022 | A one-to-one function in t... |
| dff14b 7023 | A one-to-one function in t... |
| f12dfv 7024 | A one-to-one function with... |
| f13dfv 7025 | A one-to-one function with... |
| dff1o6 7026 | A one-to-one onto function... |
| f1ocnvfv1 7027 | The converse value of the ... |
| f1ocnvfv2 7028 | The value of the converse ... |
| f1ocnvfv 7029 | Relationship between the v... |
| f1ocnvfvb 7030 | Relationship between the v... |
| nvof1o 7031 | An involution is a bijecti... |
| nvocnv 7032 | The converse of an involut... |
| fsnex 7033 | Relate a function with a s... |
| f1prex 7034 | Relate a one-to-one functi... |
| f1ocnvdm 7035 | The value of the converse ... |
| f1ocnvfvrneq 7036 | If the values of a one-to-... |
| fcof1 7037 | An application is injectiv... |
| fcofo 7038 | An application is surjecti... |
| cbvfo 7039 | Change bound variable betw... |
| cbvexfo 7040 | Change bound variable betw... |
| cocan1 7041 | An injection is left-cance... |
| cocan2 7042 | A surjection is right-canc... |
| fcof1oinvd 7043 | Show that a function is th... |
| fcof1od 7044 | A function is bijective if... |
| 2fcoidinvd 7045 | Show that a function is th... |
| fcof1o 7046 | Show that two functions ar... |
| 2fvcoidd 7047 | Show that the composition ... |
| 2fvidf1od 7048 | A function is bijective if... |
| 2fvidinvd 7049 | Show that two functions ar... |
| foeqcnvco 7050 | Condition for function equ... |
| f1eqcocnv 7051 | Condition for function equ... |
| fveqf1o 7052 | Given a bijection ` F ` , ... |
| nf1const 7053 | A constant function from a... |
| nf1oconst 7054 | A constant function from a... |
| fliftrel 7055 | ` F ` , a function lift, i... |
| fliftel 7056 | Elementhood in the relatio... |
| fliftel1 7057 | Elementhood in the relatio... |
| fliftcnv 7058 | Converse of the relation `... |
| fliftfun 7059 | The function ` F ` is the ... |
| fliftfund 7060 | The function ` F ` is the ... |
| fliftfuns 7061 | The function ` F ` is the ... |
| fliftf 7062 | The domain and range of th... |
| fliftval 7063 | The value of the function ... |
| isoeq1 7064 | Equality theorem for isomo... |
| isoeq2 7065 | Equality theorem for isomo... |
| isoeq3 7066 | Equality theorem for isomo... |
| isoeq4 7067 | Equality theorem for isomo... |
| isoeq5 7068 | Equality theorem for isomo... |
| nfiso 7069 | Bound-variable hypothesis ... |
| isof1o 7070 | An isomorphism is a one-to... |
| isof1oidb 7071 | A function is a bijection ... |
| isof1oopb 7072 | A function is a bijection ... |
| isorel 7073 | An isomorphism connects bi... |
| soisores 7074 | Express the condition of i... |
| soisoi 7075 | Infer isomorphism from one... |
| isoid 7076 | Identity law for isomorphi... |
| isocnv 7077 | Converse law for isomorphi... |
| isocnv2 7078 | Converse law for isomorphi... |
| isocnv3 7079 | Complementation law for is... |
| isores2 7080 | An isomorphism from one we... |
| isores1 7081 | An isomorphism from one we... |
| isores3 7082 | Induced isomorphism on a s... |
| isotr 7083 | Composition (transitive) l... |
| isomin 7084 | Isomorphisms preserve mini... |
| isoini 7085 | Isomorphisms preserve init... |
| isoini2 7086 | Isomorphisms are isomorphi... |
| isofrlem 7087 | Lemma for ~ isofr . (Cont... |
| isoselem 7088 | Lemma for ~ isose . (Cont... |
| isofr 7089 | An isomorphism preserves w... |
| isose 7090 | An isomorphism preserves s... |
| isofr2 7091 | A weak form of ~ isofr tha... |
| isopolem 7092 | Lemma for ~ isopo . (Cont... |
| isopo 7093 | An isomorphism preserves p... |
| isosolem 7094 | Lemma for ~ isoso . (Cont... |
| isoso 7095 | An isomorphism preserves s... |
| isowe 7096 | An isomorphism preserves w... |
| isowe2 7097 | A weak form of ~ isowe tha... |
| f1oiso 7098 | Any one-to-one onto functi... |
| f1oiso2 7099 | Any one-to-one onto functi... |
| f1owe 7100 | Well-ordering of isomorphi... |
| weniso 7101 | A set-like well-ordering h... |
| weisoeq 7102 | Thus, there is at most one... |
| weisoeq2 7103 | Thus, there is at most one... |
| knatar 7104 | The Knaster-Tarski theorem... |
| canth 7105 | No set ` A ` is equinumero... |
| ncanth 7106 | Cantor's theorem fails for... |
| riotaeqdv 7109 | Formula-building deduction... |
| riotabidv 7110 | Formula-building deduction... |
| riotaeqbidv 7111 | Equality deduction for res... |
| riotaex 7112 | Restricted iota is a set. ... |
| riotav 7113 | An iota restricted to the ... |
| riotauni 7114 | Restricted iota in terms o... |
| nfriota1 7115 | The abstraction variable i... |
| nfriotadw 7116 | Deduction version of ~ nfr... |
| cbvriotaw 7117 | Change bound variable in a... |
| cbvriotavw 7118 | Change bound variable in a... |
| nfriotad 7119 | Deduction version of ~ nfr... |
| nfriota 7120 | A variable not free in a w... |
| cbvriota 7121 | Change bound variable in a... |
| cbvriotav 7122 | Change bound variable in a... |
| csbriota 7123 | Interchange class substitu... |
| riotacl2 7124 | Membership law for "the un... |
| riotacl 7125 | Closure of restricted iota... |
| riotasbc 7126 | Substitution law for descr... |
| riotabidva 7127 | Equivalent wff's yield equ... |
| riotabiia 7128 | Equivalent wff's yield equ... |
| riota1 7129 | Property of restricted iot... |
| riota1a 7130 | Property of iota. (Contri... |
| riota2df 7131 | A deduction version of ~ r... |
| riota2f 7132 | This theorem shows a condi... |
| riota2 7133 | This theorem shows a condi... |
| riotaeqimp 7134 | If two restricted iota des... |
| riotaprop 7135 | Properties of a restricted... |
| riota5f 7136 | A method for computing res... |
| riota5 7137 | A method for computing res... |
| riotass2 7138 | Restriction of a unique el... |
| riotass 7139 | Restriction of a unique el... |
| moriotass 7140 | Restriction of a unique el... |
| snriota 7141 | A restricted class abstrac... |
| riotaxfrd 7142 | Change the variable ` x ` ... |
| eusvobj2 7143 | Specify the same property ... |
| eusvobj1 7144 | Specify the same object in... |
| f1ofveu 7145 | There is one domain elemen... |
| f1ocnvfv3 7146 | Value of the converse of a... |
| riotaund 7147 | Restricted iota equals the... |
| riotassuni 7148 | The restricted iota class ... |
| riotaclb 7149 | Bidirectional closure of r... |
| oveq 7156 | Equality theorem for opera... |
| oveq1 7157 | Equality theorem for opera... |
| oveq2 7158 | Equality theorem for opera... |
| oveq12 7159 | Equality theorem for opera... |
| oveq1i 7160 | Equality inference for ope... |
| oveq2i 7161 | Equality inference for ope... |
| oveq12i 7162 | Equality inference for ope... |
| oveqi 7163 | Equality inference for ope... |
| oveq123i 7164 | Equality inference for ope... |
| oveq1d 7165 | Equality deduction for ope... |
| oveq2d 7166 | Equality deduction for ope... |
| oveqd 7167 | Equality deduction for ope... |
| oveq12d 7168 | Equality deduction for ope... |
| oveqan12d 7169 | Equality deduction for ope... |
| oveqan12rd 7170 | Equality deduction for ope... |
| oveq123d 7171 | Equality deduction for ope... |
| fvoveq1d 7172 | Equality deduction for nes... |
| fvoveq1 7173 | Equality theorem for neste... |
| ovanraleqv 7174 | Equality theorem for a con... |
| imbrov2fvoveq 7175 | Equality theorem for neste... |
| ovrspc2v 7176 | If an operation value is e... |
| oveqrspc2v 7177 | Restricted specialization ... |
| oveqdr 7178 | Equality of two operations... |
| nfovd 7179 | Deduction version of bound... |
| nfov 7180 | Bound-variable hypothesis ... |
| oprabidw 7181 | The law of concretion. Sp... |
| oprabid 7182 | The law of concretion. Sp... |
| ovex 7183 | The result of an operation... |
| ovexi 7184 | The result of an operation... |
| ovexd 7185 | The result of an operation... |
| ovssunirn 7186 | The result of an operation... |
| 0ov 7187 | Operation value of the emp... |
| ovprc 7188 | The value of an operation ... |
| ovprc1 7189 | The value of an operation ... |
| ovprc2 7190 | The value of an operation ... |
| ovrcl 7191 | Reverse closure for an ope... |
| csbov123 7192 | Move class substitution in... |
| csbov 7193 | Move class substitution in... |
| csbov12g 7194 | Move class substitution in... |
| csbov1g 7195 | Move class substitution in... |
| csbov2g 7196 | Move class substitution in... |
| rspceov 7197 | A frequently used special ... |
| elovimad 7198 | Elementhood of the image s... |
| fnbrovb 7199 | Value of a binary operatio... |
| fnotovb 7200 | Equivalence of operation v... |
| opabbrex 7201 | A collection of ordered pa... |
| opabresex2d 7202 | Restrictions of a collecti... |
| fvmptopab 7203 | The function value of a ma... |
| f1opr 7204 | Condition for an operation... |
| brfvopab 7205 | The classes involved in a ... |
| dfoprab2 7206 | Class abstraction for oper... |
| reloprab 7207 | An operation class abstrac... |
| oprabv 7208 | If a pair and a class are ... |
| nfoprab1 7209 | The abstraction variables ... |
| nfoprab2 7210 | The abstraction variables ... |
| nfoprab3 7211 | The abstraction variables ... |
| nfoprab 7212 | Bound-variable hypothesis ... |
| oprabbid 7213 | Equivalent wff's yield equ... |
| oprabbidv 7214 | Equivalent wff's yield equ... |
| oprabbii 7215 | Equivalent wff's yield equ... |
| ssoprab2 7216 | Equivalence of ordered pai... |
| ssoprab2b 7217 | Equivalence of ordered pai... |
| eqoprab2bw 7218 | Equivalence of ordered pai... |
| eqoprab2b 7219 | Equivalence of ordered pai... |
| mpoeq123 7220 | An equality theorem for th... |
| mpoeq12 7221 | An equality theorem for th... |
| mpoeq123dva 7222 | An equality deduction for ... |
| mpoeq123dv 7223 | An equality deduction for ... |
| mpoeq123i 7224 | An equality inference for ... |
| mpoeq3dva 7225 | Slightly more general equa... |
| mpoeq3ia 7226 | An equality inference for ... |
| mpoeq3dv 7227 | An equality deduction for ... |
| nfmpo1 7228 | Bound-variable hypothesis ... |
| nfmpo2 7229 | Bound-variable hypothesis ... |
| nfmpo 7230 | Bound-variable hypothesis ... |
| 0mpo0 7231 | A mapping operation with e... |
| mpo0v 7232 | A mapping operation with e... |
| mpo0 7233 | A mapping operation with e... |
| oprab4 7234 | Two ways to state the doma... |
| cbvoprab1 7235 | Rule used to change first ... |
| cbvoprab2 7236 | Change the second bound va... |
| cbvoprab12 7237 | Rule used to change first ... |
| cbvoprab12v 7238 | Rule used to change first ... |
| cbvoprab3 7239 | Rule used to change the th... |
| cbvoprab3v 7240 | Rule used to change the th... |
| cbvmpox 7241 | Rule to change the bound v... |
| cbvmpo 7242 | Rule to change the bound v... |
| cbvmpov 7243 | Rule to change the bound v... |
| elimdelov 7244 | Eliminate a hypothesis whi... |
| ovif 7245 | Move a conditional outside... |
| ovif2 7246 | Move a conditional outside... |
| ovif12 7247 | Move a conditional outside... |
| ifov 7248 | Move a conditional outside... |
| dmoprab 7249 | The domain of an operation... |
| dmoprabss 7250 | The domain of an operation... |
| rnoprab 7251 | The range of an operation ... |
| rnoprab2 7252 | The range of a restricted ... |
| reldmoprab 7253 | The domain of an operation... |
| oprabss 7254 | Structure of an operation ... |
| eloprabga 7255 | The law of concretion for ... |
| eloprabg 7256 | The law of concretion for ... |
| ssoprab2i 7257 | Inference of operation cla... |
| mpov 7258 | Operation with universal d... |
| mpomptx 7259 | Express a two-argument fun... |
| mpompt 7260 | Express a two-argument fun... |
| mpodifsnif 7261 | A mapping with two argumen... |
| mposnif 7262 | A mapping with two argumen... |
| fconstmpo 7263 | Representation of a consta... |
| resoprab 7264 | Restriction of an operatio... |
| resoprab2 7265 | Restriction of an operator... |
| resmpo 7266 | Restriction of the mapping... |
| funoprabg 7267 | "At most one" is a suffici... |
| funoprab 7268 | "At most one" is a suffici... |
| fnoprabg 7269 | Functionality and domain o... |
| mpofun 7270 | The maps-to notation for a... |
| fnoprab 7271 | Functionality and domain o... |
| ffnov 7272 | An operation maps to a cla... |
| fovcl 7273 | Closure law for an operati... |
| eqfnov 7274 | Equality of two operations... |
| eqfnov2 7275 | Two operators with the sam... |
| fnov 7276 | Representation of a functi... |
| mpo2eqb 7277 | Bidirectional equality the... |
| rnmpo 7278 | The range of an operation ... |
| reldmmpo 7279 | The domain of an operation... |
| elrnmpog 7280 | Membership in the range of... |
| elrnmpo 7281 | Membership in the range of... |
| elrnmpores 7282 | Membership in the range of... |
| ralrnmpo 7283 | A restricted quantifier ov... |
| rexrnmpo 7284 | A restricted quantifier ov... |
| ovid 7285 | The value of an operation ... |
| ovidig 7286 | The value of an operation ... |
| ovidi 7287 | The value of an operation ... |
| ov 7288 | The value of an operation ... |
| ovigg 7289 | The value of an operation ... |
| ovig 7290 | The value of an operation ... |
| ovmpt4g 7291 | Value of a function given ... |
| ovmpos 7292 | Value of a function given ... |
| ov2gf 7293 | The value of an operation ... |
| ovmpodxf 7294 | Value of an operation give... |
| ovmpodx 7295 | Value of an operation give... |
| ovmpod 7296 | Value of an operation give... |
| ovmpox 7297 | The value of an operation ... |
| ovmpoga 7298 | Value of an operation give... |
| ovmpoa 7299 | Value of an operation give... |
| ovmpodf 7300 | Alternate deduction versio... |
| ovmpodv 7301 | Alternate deduction versio... |
| ovmpodv2 7302 | Alternate deduction versio... |
| ovmpog 7303 | Value of an operation give... |
| ovmpo 7304 | Value of an operation give... |
| ov3 7305 | The value of an operation ... |
| ov6g 7306 | The value of an operation ... |
| ovg 7307 | The value of an operation ... |
| ovres 7308 | The value of a restricted ... |
| ovresd 7309 | Lemma for converting metri... |
| oprres 7310 | The restriction of an oper... |
| oprssov 7311 | The value of a member of t... |
| fovrn 7312 | An operation's value belon... |
| fovrnda 7313 | An operation's value belon... |
| fovrnd 7314 | An operation's value belon... |
| fnrnov 7315 | The range of an operation ... |
| foov 7316 | An onto mapping of an oper... |
| fnovrn 7317 | An operation's value belon... |
| ovelrn 7318 | A member of an operation's... |
| funimassov 7319 | Membership relation for th... |
| ovelimab 7320 | Operation value in an imag... |
| ovima0 7321 | An operation value is a me... |
| ovconst2 7322 | The value of a constant op... |
| oprssdm 7323 | Domain of closure of an op... |
| nssdmovg 7324 | The value of an operation ... |
| ndmovg 7325 | The value of an operation ... |
| ndmov 7326 | The value of an operation ... |
| ndmovcl 7327 | The closure of an operatio... |
| ndmovrcl 7328 | Reverse closure law, when ... |
| ndmovcom 7329 | Any operation is commutati... |
| ndmovass 7330 | Any operation is associati... |
| ndmovdistr 7331 | Any operation is distribut... |
| ndmovord 7332 | Elimination of redundant a... |
| ndmovordi 7333 | Elimination of redundant a... |
| caovclg 7334 | Convert an operation closu... |
| caovcld 7335 | Convert an operation closu... |
| caovcl 7336 | Convert an operation closu... |
| caovcomg 7337 | Convert an operation commu... |
| caovcomd 7338 | Convert an operation commu... |
| caovcom 7339 | Convert an operation commu... |
| caovassg 7340 | Convert an operation assoc... |
| caovassd 7341 | Convert an operation assoc... |
| caovass 7342 | Convert an operation assoc... |
| caovcang 7343 | Convert an operation cance... |
| caovcand 7344 | Convert an operation cance... |
| caovcanrd 7345 | Commute the arguments of a... |
| caovcan 7346 | Convert an operation cance... |
| caovordig 7347 | Convert an operation order... |
| caovordid 7348 | Convert an operation order... |
| caovordg 7349 | Convert an operation order... |
| caovordd 7350 | Convert an operation order... |
| caovord2d 7351 | Operation ordering law wit... |
| caovord3d 7352 | Ordering law. (Contribute... |
| caovord 7353 | Convert an operation order... |
| caovord2 7354 | Operation ordering law wit... |
| caovord3 7355 | Ordering law. (Contribute... |
| caovdig 7356 | Convert an operation distr... |
| caovdid 7357 | Convert an operation distr... |
| caovdir2d 7358 | Convert an operation distr... |
| caovdirg 7359 | Convert an operation rever... |
| caovdird 7360 | Convert an operation distr... |
| caovdi 7361 | Convert an operation distr... |
| caov32d 7362 | Rearrange arguments in a c... |
| caov12d 7363 | Rearrange arguments in a c... |
| caov31d 7364 | Rearrange arguments in a c... |
| caov13d 7365 | Rearrange arguments in a c... |
| caov4d 7366 | Rearrange arguments in a c... |
| caov411d 7367 | Rearrange arguments in a c... |
| caov42d 7368 | Rearrange arguments in a c... |
| caov32 7369 | Rearrange arguments in a c... |
| caov12 7370 | Rearrange arguments in a c... |
| caov31 7371 | Rearrange arguments in a c... |
| caov13 7372 | Rearrange arguments in a c... |
| caov4 7373 | Rearrange arguments in a c... |
| caov411 7374 | Rearrange arguments in a c... |
| caov42 7375 | Rearrange arguments in a c... |
| caovdir 7376 | Reverse distributive law. ... |
| caovdilem 7377 | Lemma used by real number ... |
| caovlem2 7378 | Lemma used in real number ... |
| caovmo 7379 | Uniqueness of inverse elem... |
| mpondm0 7380 | The value of an operation ... |
| elmpocl 7381 | If a two-parameter class i... |
| elmpocl1 7382 | If a two-parameter class i... |
| elmpocl2 7383 | If a two-parameter class i... |
| elovmpo 7384 | Utility lemma for two-para... |
| elovmporab 7385 | Implications for the value... |
| elovmporab1w 7386 | Implications for the value... |
| elovmporab1 7387 | Implications for the value... |
| 2mpo0 7388 | If the operation value of ... |
| relmptopab 7389 | Any function to sets of or... |
| f1ocnvd 7390 | Describe an implicit one-t... |
| f1od 7391 | Describe an implicit one-t... |
| f1ocnv2d 7392 | Describe an implicit one-t... |
| f1o2d 7393 | Describe an implicit one-t... |
| f1opw2 7394 | A one-to-one mapping induc... |
| f1opw 7395 | A one-to-one mapping induc... |
| elovmpt3imp 7396 | If the value of a function... |
| ovmpt3rab1 7397 | The value of an operation ... |
| ovmpt3rabdm 7398 | If the value of a function... |
| elovmpt3rab1 7399 | Implications for the value... |
| elovmpt3rab 7400 | Implications for the value... |
| ofeq 7405 | Equality theorem for funct... |
| ofreq 7406 | Equality theorem for funct... |
| ofexg 7407 | A function operation restr... |
| nfof 7408 | Hypothesis builder for fun... |
| nfofr 7409 | Hypothesis builder for fun... |
| offval 7410 | Value of an operation appl... |
| ofrfval 7411 | Value of a relation applie... |
| ofval 7412 | Evaluate a function operat... |
| ofrval 7413 | Exhibit a function relatio... |
| offn 7414 | The function operation pro... |
| offval2f 7415 | The function operation exp... |
| ofmresval 7416 | Value of a restriction of ... |
| fnfvof 7417 | Function value of a pointw... |
| off 7418 | The function operation pro... |
| ofres 7419 | Restrict the operands of a... |
| offval2 7420 | The function operation exp... |
| ofrfval2 7421 | The function relation acti... |
| ofmpteq 7422 | Value of a pointwise opera... |
| ofco 7423 | The composition of a funct... |
| offveq 7424 | Convert an identity of the... |
| offveqb 7425 | Equivalent expressions for... |
| ofc1 7426 | Left operation by a consta... |
| ofc2 7427 | Right operation by a const... |
| ofc12 7428 | Function operation on two ... |
| caofref 7429 | Transfer a reflexive law t... |
| caofinvl 7430 | Transfer a left inverse la... |
| caofid0l 7431 | Transfer a left identity l... |
| caofid0r 7432 | Transfer a right identity ... |
| caofid1 7433 | Transfer a right absorptio... |
| caofid2 7434 | Transfer a right absorptio... |
| caofcom 7435 | Transfer a commutative law... |
| caofrss 7436 | Transfer a relation subset... |
| caofass 7437 | Transfer an associative la... |
| caoftrn 7438 | Transfer a transitivity la... |
| caofdi 7439 | Transfer a distributive la... |
| caofdir 7440 | Transfer a reverse distrib... |
| caonncan 7441 | Transfer ~ nncan -shaped l... |
| relrpss 7444 | The proper subset relation... |
| brrpssg 7445 | The proper subset relation... |
| brrpss 7446 | The proper subset relation... |
| porpss 7447 | Every class is partially o... |
| sorpss 7448 | Express strict ordering un... |
| sorpssi 7449 | Property of a chain of set... |
| sorpssun 7450 | A chain of sets is closed ... |
| sorpssin 7451 | A chain of sets is closed ... |
| sorpssuni 7452 | In a chain of sets, a maxi... |
| sorpssint 7453 | In a chain of sets, a mini... |
| sorpsscmpl 7454 | The componentwise compleme... |
| zfun 7456 | Axiom of Union expressed w... |
| axun2 7457 | A variant of the Axiom of ... |
| uniex2 7458 | The Axiom of Union using t... |
| vuniex 7459 | The union of a setvar is a... |
| uniexg 7460 | The ZF Axiom of Union in c... |
| uniex 7461 | The Axiom of Union in clas... |
| uniexd 7462 | Deduction version of the Z... |
| unex 7463 | The union of two sets is a... |
| tpex 7464 | An unordered triple of cla... |
| unexb 7465 | Existence of union is equi... |
| unexg 7466 | A union of two sets is a s... |
| xpexg 7467 | The Cartesian product of t... |
| xpexd 7468 | The Cartesian product of t... |
| 3xpexg 7469 | The Cartesian product of t... |
| xpex 7470 | The Cartesian product of t... |
| sqxpexg 7471 | The Cartesian square of a ... |
| abnexg 7472 | Sufficient condition for a... |
| abnex 7473 | Sufficient condition for a... |
| snnex 7474 | The class of all singleton... |
| pwnex 7475 | The class of all power set... |
| difex2 7476 | If the subtrahend of a cla... |
| difsnexi 7477 | If the difference of a cla... |
| uniuni 7478 | Expression for double unio... |
| uniexr 7479 | Converse of the Axiom of U... |
| uniexb 7480 | The Axiom of Union and its... |
| pwexr 7481 | Converse of the Axiom of P... |
| pwexb 7482 | The Axiom of Power Sets an... |
| elpwpwel 7483 | A class belongs to a doubl... |
| eldifpw 7484 | Membership in a power clas... |
| elpwun 7485 | Membership in the power cl... |
| pwuncl 7486 | Power classes are closed u... |
| iunpw 7487 | An indexed union of a powe... |
| fr3nr 7488 | A well-founded relation ha... |
| epne3 7489 | A well-founded class conta... |
| dfwe2 7490 | Alternate definition of we... |
| epweon 7491 | The membership relation we... |
| ordon 7492 | The class of all ordinal n... |
| onprc 7493 | No set contains all ordina... |
| ssorduni 7494 | The union of a class of or... |
| ssonuni 7495 | The union of a set of ordi... |
| ssonunii 7496 | The union of a set of ordi... |
| ordeleqon 7497 | A way to express the ordin... |
| ordsson 7498 | Any ordinal class is a sub... |
| onss 7499 | An ordinal number is a sub... |
| predon 7500 | The predecessor of an ordi... |
| ssonprc 7501 | Two ways of saying a class... |
| onuni 7502 | The union of an ordinal nu... |
| orduni 7503 | The union of an ordinal cl... |
| onint 7504 | The intersection (infimum)... |
| onint0 7505 | The intersection of a clas... |
| onssmin 7506 | A nonempty class of ordina... |
| onminesb 7507 | If a property is true for ... |
| onminsb 7508 | If a property is true for ... |
| oninton 7509 | The intersection of a none... |
| onintrab 7510 | The intersection of a clas... |
| onintrab2 7511 | An existence condition equ... |
| onnmin 7512 | No member of a set of ordi... |
| onnminsb 7513 | An ordinal number smaller ... |
| oneqmin 7514 | A way to show that an ordi... |
| uniordint 7515 | The union of a set of ordi... |
| onminex 7516 | If a wff is true for an or... |
| sucon 7517 | The class of all ordinal n... |
| sucexb 7518 | A successor exists iff its... |
| sucexg 7519 | The successor of a set is ... |
| sucex 7520 | The successor of a set is ... |
| onmindif2 7521 | The minimum of a class of ... |
| suceloni 7522 | The successor of an ordina... |
| ordsuc 7523 | The successor of an ordina... |
| ordpwsuc 7524 | The collection of ordinals... |
| onpwsuc 7525 | The collection of ordinal ... |
| sucelon 7526 | The successor of an ordina... |
| ordsucss 7527 | The successor of an elemen... |
| onpsssuc 7528 | An ordinal number is a pro... |
| ordelsuc 7529 | A set belongs to an ordina... |
| onsucmin 7530 | The successor of an ordina... |
| ordsucelsuc 7531 | Membership is inherited by... |
| ordsucsssuc 7532 | The subclass relationship ... |
| ordsucuniel 7533 | Given an element ` A ` of ... |
| ordsucun 7534 | The successor of the maxim... |
| ordunpr 7535 | The maximum of two ordinal... |
| ordunel 7536 | The maximum of two ordinal... |
| onsucuni 7537 | A class of ordinal numbers... |
| ordsucuni 7538 | An ordinal class is a subc... |
| orduniorsuc 7539 | An ordinal class is either... |
| unon 7540 | The class of all ordinal n... |
| ordunisuc 7541 | An ordinal class is equal ... |
| orduniss2 7542 | The union of the ordinal s... |
| onsucuni2 7543 | A successor ordinal is the... |
| 0elsuc 7544 | The successor of an ordina... |
| limon 7545 | The class of ordinal numbe... |
| onssi 7546 | An ordinal number is a sub... |
| onsuci 7547 | The successor of an ordina... |
| onuniorsuci 7548 | An ordinal number is eithe... |
| onuninsuci 7549 | A limit ordinal is not a s... |
| onsucssi 7550 | A set belongs to an ordina... |
| nlimsucg 7551 | A successor is not a limit... |
| orduninsuc 7552 | An ordinal equal to its un... |
| ordunisuc2 7553 | An ordinal equal to its un... |
| ordzsl 7554 | An ordinal is zero, a succ... |
| onzsl 7555 | An ordinal number is zero,... |
| dflim3 7556 | An alternate definition of... |
| dflim4 7557 | An alternate definition of... |
| limsuc 7558 | The successor of a member ... |
| limsssuc 7559 | A class includes a limit o... |
| nlimon 7560 | Two ways to express the cl... |
| limuni3 7561 | The union of a nonempty cl... |
| tfi 7562 | The Principle of Transfini... |
| tfis 7563 | Transfinite Induction Sche... |
| tfis2f 7564 | Transfinite Induction Sche... |
| tfis2 7565 | Transfinite Induction Sche... |
| tfis3 7566 | Transfinite Induction Sche... |
| tfisi 7567 | A transfinite induction sc... |
| tfinds 7568 | Principle of Transfinite I... |
| tfindsg 7569 | Transfinite Induction (inf... |
| tfindsg2 7570 | Transfinite Induction (inf... |
| tfindes 7571 | Transfinite Induction with... |
| tfinds2 7572 | Transfinite Induction (inf... |
| tfinds3 7573 | Principle of Transfinite I... |
| dfom2 7576 | An alternate definition of... |
| elom 7577 | Membership in omega. The ... |
| omsson 7578 | Omega is a subset of ` On ... |
| limomss 7579 | The class of natural numbe... |
| nnon 7580 | A natural number is an ord... |
| nnoni 7581 | A natural number is an ord... |
| nnord 7582 | A natural number is ordina... |
| ordom 7583 | Omega is ordinal. Theorem... |
| elnn 7584 | A member of a natural numb... |
| omon 7585 | The class of natural numbe... |
| omelon2 7586 | Omega is an ordinal number... |
| nnlim 7587 | A natural number is not a ... |
| omssnlim 7588 | The class of natural numbe... |
| limom 7589 | Omega is a limit ordinal. ... |
| peano2b 7590 | A class belongs to omega i... |
| nnsuc 7591 | A nonzero natural number i... |
| omsucne 7592 | A natural number is not th... |
| ssnlim 7593 | An ordinal subclass of non... |
| omsinds 7594 | Strong (or "total") induct... |
| peano1 7595 | Zero is a natural number. ... |
| peano2 7596 | The successor of any natur... |
| peano3 7597 | The successor of any natur... |
| peano4 7598 | Two natural numbers are eq... |
| peano5 7599 | The induction postulate: a... |
| nn0suc 7600 | A natural number is either... |
| find 7601 | The Principle of Finite In... |
| finds 7602 | Principle of Finite Induct... |
| findsg 7603 | Principle of Finite Induct... |
| finds2 7604 | Principle of Finite Induct... |
| finds1 7605 | Principle of Finite Induct... |
| findes 7606 | Finite induction with expl... |
| dmexg 7607 | The domain of a set is a s... |
| rnexg 7608 | The range of a set is a se... |
| dmexd 7609 | The domain of a set is a s... |
| dmex 7610 | The domain of a set is a s... |
| rnex 7611 | The range of a set is a se... |
| iprc 7612 | The identity function is a... |
| resiexg 7613 | The existence of a restric... |
| imaexg 7614 | The image of a set is a se... |
| imaex 7615 | The image of a set is a se... |
| exse2 7616 | Any set relation is set-li... |
| xpexr 7617 | If a Cartesian product is ... |
| xpexr2 7618 | If a nonempty Cartesian pr... |
| xpexcnv 7619 | A condition where the conv... |
| soex 7620 | If the relation in a stric... |
| elxp4 7621 | Membership in a Cartesian ... |
| elxp5 7622 | Membership in a Cartesian ... |
| cnvexg 7623 | The converse of a set is a... |
| cnvex 7624 | The converse of a set is a... |
| relcnvexb 7625 | A relation is a set iff it... |
| f1oexrnex 7626 | If the range of a 1-1 onto... |
| f1oexbi 7627 | There is a one-to-one onto... |
| coexg 7628 | The composition of two set... |
| coex 7629 | The composition of two set... |
| funcnvuni 7630 | The union of a chain (with... |
| fun11uni 7631 | The union of a chain (with... |
| fex2 7632 | A function with bounded do... |
| fabexg 7633 | Existence of a set of func... |
| fabex 7634 | Existence of a set of func... |
| dmfex 7635 | If a mapping is a set, its... |
| f1oabexg 7636 | The class of all 1-1-onto ... |
| fiunlem 7637 | Lemma for ~ fiun and ~ f1i... |
| fiun 7638 | The union of a chain (with... |
| f1iun 7639 | The union of a chain (with... |
| fviunfun 7640 | The function value of an i... |
| ffoss 7641 | Relationship between a map... |
| f11o 7642 | Relationship between one-t... |
| resfunexgALT 7643 | Alternate proof of ~ resfu... |
| cofunexg 7644 | Existence of a composition... |
| cofunex2g 7645 | Existence of a composition... |
| fnexALT 7646 | Alternate proof of ~ fnex ... |
| funexw 7647 | Weak version of ~ funex th... |
| mptexw 7648 | Weak version of ~ mptex th... |
| funrnex 7649 | If the domain of a functio... |
| zfrep6 7650 | A version of the Axiom of ... |
| fornex 7651 | If the domain of an onto f... |
| f1dmex 7652 | If the codomain of a one-t... |
| f1ovv 7653 | The range of a 1-1 onto fu... |
| fvclex 7654 | Existence of the class of ... |
| fvresex 7655 | Existence of the class of ... |
| abrexexg 7656 | Existence of a class abstr... |
| abrexex 7657 | Existence of a class abstr... |
| iunexg 7658 | The existence of an indexe... |
| abrexex2g 7659 | Existence of an existentia... |
| opabex3d 7660 | Existence of an ordered pa... |
| opabex3rd 7661 | Existence of an ordered pa... |
| opabex3 7662 | Existence of an ordered pa... |
| iunex 7663 | The existence of an indexe... |
| abrexex2 7664 | Existence of an existentia... |
| abexssex 7665 | Existence of a class abstr... |
| abexex 7666 | A condition where a class ... |
| f1oweALT 7667 | Alternate proof of ~ f1owe... |
| wemoiso 7668 | Thus, there is at most one... |
| wemoiso2 7669 | Thus, there is at most one... |
| oprabexd 7670 | Existence of an operator a... |
| oprabex 7671 | Existence of an operation ... |
| oprabex3 7672 | Existence of an operation ... |
| oprabrexex2 7673 | Existence of an existentia... |
| ab2rexex 7674 | Existence of a class abstr... |
| ab2rexex2 7675 | Existence of an existentia... |
| xpexgALT 7676 | Alternate proof of ~ xpexg... |
| offval3 7677 | General value of ` ( F oF ... |
| offres 7678 | Pointwise combination comm... |
| ofmres 7679 | Equivalent expressions for... |
| ofmresex 7680 | Existence of a restriction... |
| 1stval 7685 | The value of the function ... |
| 2ndval 7686 | The value of the function ... |
| 1stnpr 7687 | Value of the first-member ... |
| 2ndnpr 7688 | Value of the second-member... |
| 1st0 7689 | The value of the first-mem... |
| 2nd0 7690 | The value of the second-me... |
| op1st 7691 | Extract the first member o... |
| op2nd 7692 | Extract the second member ... |
| op1std 7693 | Extract the first member o... |
| op2ndd 7694 | Extract the second member ... |
| op1stg 7695 | Extract the first member o... |
| op2ndg 7696 | Extract the second member ... |
| ot1stg 7697 | Extract the first member o... |
| ot2ndg 7698 | Extract the second member ... |
| ot3rdg 7699 | Extract the third member o... |
| 1stval2 7700 | Alternate value of the fun... |
| 2ndval2 7701 | Alternate value of the fun... |
| oteqimp 7702 | The components of an order... |
| fo1st 7703 | The ` 1st ` function maps ... |
| fo2nd 7704 | The ` 2nd ` function maps ... |
| br1steqg 7705 | Uniqueness condition for t... |
| br2ndeqg 7706 | Uniqueness condition for t... |
| f1stres 7707 | Mapping of a restriction o... |
| f2ndres 7708 | Mapping of a restriction o... |
| fo1stres 7709 | Onto mapping of a restrict... |
| fo2ndres 7710 | Onto mapping of a restrict... |
| 1st2val 7711 | Value of an alternate defi... |
| 2nd2val 7712 | Value of an alternate defi... |
| 1stcof 7713 | Composition of the first m... |
| 2ndcof 7714 | Composition of the second ... |
| xp1st 7715 | Location of the first elem... |
| xp2nd 7716 | Location of the second ele... |
| elxp6 7717 | Membership in a Cartesian ... |
| elxp7 7718 | Membership in a Cartesian ... |
| eqopi 7719 | Equality with an ordered p... |
| xp2 7720 | Representation of Cartesia... |
| unielxp 7721 | The membership relation fo... |
| 1st2nd2 7722 | Reconstruction of a member... |
| 1st2ndb 7723 | Reconstruction of an order... |
| xpopth 7724 | An ordered pair theorem fo... |
| eqop 7725 | Two ways to express equali... |
| eqop2 7726 | Two ways to express equali... |
| op1steq 7727 | Two ways of expressing tha... |
| opreuopreu 7728 | There is a unique ordered ... |
| el2xptp 7729 | A member of a nested Carte... |
| el2xptp0 7730 | A member of a nested Carte... |
| 2nd1st 7731 | Swap the members of an ord... |
| 1st2nd 7732 | Reconstruction of a member... |
| 1stdm 7733 | The first ordered pair com... |
| 2ndrn 7734 | The second ordered pair co... |
| 1st2ndbr 7735 | Express an element of a re... |
| releldm2 7736 | Two ways of expressing mem... |
| reldm 7737 | An expression for the doma... |
| releldmdifi 7738 | One way of expressing memb... |
| funfv1st2nd 7739 | The function value for the... |
| funelss 7740 | If the first component of ... |
| funeldmdif 7741 | Two ways of expressing mem... |
| sbcopeq1a 7742 | Equality theorem for subst... |
| csbopeq1a 7743 | Equality theorem for subst... |
| dfopab2 7744 | A way to define an ordered... |
| dfoprab3s 7745 | A way to define an operati... |
| dfoprab3 7746 | Operation class abstractio... |
| dfoprab4 7747 | Operation class abstractio... |
| dfoprab4f 7748 | Operation class abstractio... |
| opabex2 7749 | Condition for an operation... |
| opabn1stprc 7750 | An ordered-pair class abst... |
| opiota 7751 | The property of a uniquely... |
| cnvoprab 7752 | The converse of a class ab... |
| dfxp3 7753 | Define the Cartesian produ... |
| elopabi 7754 | A consequence of membershi... |
| eloprabi 7755 | A consequence of membershi... |
| mpomptsx 7756 | Express a two-argument fun... |
| mpompts 7757 | Express a two-argument fun... |
| dmmpossx 7758 | The domain of a mapping is... |
| fmpox 7759 | Functionality, domain and ... |
| fmpo 7760 | Functionality, domain and ... |
| fnmpo 7761 | Functionality and domain o... |
| fnmpoi 7762 | Functionality and domain o... |
| dmmpo 7763 | Domain of a class given by... |
| ovmpoelrn 7764 | An operation's value belon... |
| dmmpoga 7765 | Domain of an operation giv... |
| dmmpog 7766 | Domain of an operation giv... |
| mpoexxg 7767 | Existence of an operation ... |
| mpoexg 7768 | Existence of an operation ... |
| mpoexga 7769 | If the domain of an operat... |
| mpoexw 7770 | Weak version of ~ mpoex th... |
| mpoex 7771 | If the domain of an operat... |
| mptmpoopabbrd 7772 | The operation value of a f... |
| mptmpoopabovd 7773 | The operation value of a f... |
| el2mpocsbcl 7774 | If the operation value of ... |
| el2mpocl 7775 | If the operation value of ... |
| fnmpoovd 7776 | A function with a Cartesia... |
| offval22 7777 | The function operation exp... |
| brovpreldm 7778 | If a binary relation holds... |
| bropopvvv 7779 | If a binary relation holds... |
| bropfvvvvlem 7780 | Lemma for ~ bropfvvvv . (... |
| bropfvvvv 7781 | If a binary relation holds... |
| ovmptss 7782 | If all the values of the m... |
| relmpoopab 7783 | Any function to sets of or... |
| fmpoco 7784 | Composition of two functio... |
| oprabco 7785 | Composition of a function ... |
| oprab2co 7786 | Composition of operator ab... |
| df1st2 7787 | An alternate possible defi... |
| df2nd2 7788 | An alternate possible defi... |
| 1stconst 7789 | The mapping of a restricti... |
| 2ndconst 7790 | The mapping of a restricti... |
| dfmpo 7791 | Alternate definition for t... |
| mposn 7792 | An operation (in maps-to n... |
| curry1 7793 | Composition with ` ``' ( 2... |
| curry1val 7794 | The value of a curried fun... |
| curry1f 7795 | Functionality of a curried... |
| curry2 7796 | Composition with ` ``' ( 1... |
| curry2f 7797 | Functionality of a curried... |
| curry2val 7798 | The value of a curried fun... |
| cnvf1olem 7799 | Lemma for ~ cnvf1o . (Con... |
| cnvf1o 7800 | Describe a function that m... |
| fparlem1 7801 | Lemma for ~ fpar . (Contr... |
| fparlem2 7802 | Lemma for ~ fpar . (Contr... |
| fparlem3 7803 | Lemma for ~ fpar . (Contr... |
| fparlem4 7804 | Lemma for ~ fpar . (Contr... |
| fpar 7805 | Merge two functions in par... |
| fsplit 7806 | A function that can be use... |
| fsplitOLD 7807 | Obsolete proof of ~ fsplit... |
| fsplitfpar 7808 | Merge two functions with a... |
| offsplitfpar 7809 | Express the function opera... |
| f2ndf 7810 | The ` 2nd ` (second compon... |
| fo2ndf 7811 | The ` 2nd ` (second compon... |
| f1o2ndf1 7812 | The ` 2nd ` (second compon... |
| algrflem 7813 | Lemma for ~ algrf and rela... |
| frxp 7814 | A lexicographical ordering... |
| xporderlem 7815 | Lemma for lexicographical ... |
| poxp 7816 | A lexicographical ordering... |
| soxp 7817 | A lexicographical ordering... |
| wexp 7818 | A lexicographical ordering... |
| fnwelem 7819 | Lemma for ~ fnwe . (Contr... |
| fnwe 7820 | A variant on lexicographic... |
| fnse 7821 | Condition for the well-ord... |
| fvproj 7822 | Value of a function on ord... |
| fimaproj 7823 | Image of a cartesian produ... |
| suppval 7826 | The value of the operation... |
| supp0prc 7827 | The support of a class is ... |
| suppvalbr 7828 | The value of the operation... |
| supp0 7829 | The support of the empty s... |
| suppval1 7830 | The value of the operation... |
| suppvalfn 7831 | The value of the operation... |
| elsuppfn 7832 | An element of the support ... |
| cnvimadfsn 7833 | The support of functions "... |
| suppimacnvss 7834 | The support of functions "... |
| suppimacnv 7835 | Support sets of functions ... |
| frnsuppeq 7836 | Two ways of writing the su... |
| suppssdm 7837 | The support of a function ... |
| suppsnop 7838 | The support of a singleton... |
| snopsuppss 7839 | The support of a singleton... |
| fvn0elsupp 7840 | If the function value for ... |
| fvn0elsuppb 7841 | The function value for a g... |
| rexsupp 7842 | Existential quantification... |
| ressuppss 7843 | The support of the restric... |
| suppun 7844 | The support of a class/fun... |
| ressuppssdif 7845 | The support of the restric... |
| mptsuppdifd 7846 | The support of a function ... |
| mptsuppd 7847 | The support of a function ... |
| extmptsuppeq 7848 | The support of an extended... |
| suppfnss 7849 | The support of a function ... |
| funsssuppss 7850 | The support of a function ... |
| fnsuppres 7851 | Two ways to express restri... |
| fnsuppeq0 7852 | The support of a function ... |
| fczsupp0 7853 | The support of a constant ... |
| suppss 7854 | Show that the support of a... |
| suppssr 7855 | A function is zero outside... |
| suppssov1 7856 | Formula building theorem f... |
| suppssof1 7857 | Formula building theorem f... |
| suppss2 7858 | Show that the support of a... |
| suppsssn 7859 | Show that the support of a... |
| suppssfv 7860 | Formula building theorem f... |
| suppofssd 7861 | Condition for the support ... |
| suppofss1d 7862 | Condition for the support ... |
| suppofss2d 7863 | Condition for the support ... |
| suppco 7864 | The support of the composi... |
| suppcofnd 7865 | The support of the composi... |
| supp0cosupp0 7866 | The support of the composi... |
| supp0cosupp0OLD 7867 | Obsolete version of ~ supp... |
| imacosupp 7868 | The image of the support o... |
| imacosuppOLD 7869 | Obsolete version of ~ imac... |
| opeliunxp2f 7870 | Membership in a union of C... |
| mpoxeldm 7871 | If there is an element of ... |
| mpoxneldm 7872 | If the first argument of a... |
| mpoxopn0yelv 7873 | If there is an element of ... |
| mpoxopynvov0g 7874 | If the second argument of ... |
| mpoxopxnop0 7875 | If the first argument of a... |
| mpoxopx0ov0 7876 | If the first argument of a... |
| mpoxopxprcov0 7877 | If the components of the f... |
| mpoxopynvov0 7878 | If the second argument of ... |
| mpoxopoveq 7879 | Value of an operation give... |
| mpoxopovel 7880 | Element of the value of an... |
| mpoxopoveqd 7881 | Value of an operation give... |
| brovex 7882 | A binary relation of the v... |
| brovmpoex 7883 | A binary relation of the v... |
| sprmpod 7884 | The extension of a binary ... |
| tposss 7887 | Subset theorem for transpo... |
| tposeq 7888 | Equality theorem for trans... |
| tposeqd 7889 | Equality theorem for trans... |
| tposssxp 7890 | The transposition is a sub... |
| reltpos 7891 | The transposition is a rel... |
| brtpos2 7892 | Value of the transposition... |
| brtpos0 7893 | The behavior of ` tpos ` w... |
| reldmtpos 7894 | Necessary and sufficient c... |
| brtpos 7895 | The transposition swaps ar... |
| ottpos 7896 | The transposition swaps th... |
| relbrtpos 7897 | The transposition swaps ar... |
| dmtpos 7898 | The domain of ` tpos F ` w... |
| rntpos 7899 | The range of ` tpos F ` wh... |
| tposexg 7900 | The transposition of a set... |
| ovtpos 7901 | The transposition swaps th... |
| tposfun 7902 | The transposition of a fun... |
| dftpos2 7903 | Alternate definition of ` ... |
| dftpos3 7904 | Alternate definition of ` ... |
| dftpos4 7905 | Alternate definition of ` ... |
| tpostpos 7906 | Value of the double transp... |
| tpostpos2 7907 | Value of the double transp... |
| tposfn2 7908 | The domain of a transposit... |
| tposfo2 7909 | Condition for a surjective... |
| tposf2 7910 | The domain and range of a ... |
| tposf12 7911 | Condition for an injective... |
| tposf1o2 7912 | Condition of a bijective t... |
| tposfo 7913 | The domain and range of a ... |
| tposf 7914 | The domain and range of a ... |
| tposfn 7915 | Functionality of a transpo... |
| tpos0 7916 | Transposition of the empty... |
| tposco 7917 | Transposition of a composi... |
| tpossym 7918 | Two ways to say a function... |
| tposeqi 7919 | Equality theorem for trans... |
| tposex 7920 | A transposition is a set. ... |
| nftpos 7921 | Hypothesis builder for tra... |
| tposoprab 7922 | Transposition of a class o... |
| tposmpo 7923 | Transposition of a two-arg... |
| tposconst 7924 | The transposition of a con... |
| mpocurryd 7929 | The currying of an operati... |
| mpocurryvald 7930 | The value of a curried ope... |
| fvmpocurryd 7931 | The value of the value of ... |
| pwuninel2 7934 | Direct proof of ~ pwuninel... |
| pwuninel 7935 | The power set of the union... |
| undefval 7936 | Value of the undefined val... |
| undefnel2 7937 | The undefined value genera... |
| undefnel 7938 | The undefined value genera... |
| undefne0 7939 | The undefined value genera... |
| wrecseq123 7942 | General equality theorem f... |
| nfwrecs 7943 | Bound-variable hypothesis ... |
| wrecseq1 7944 | Equality theorem for the w... |
| wrecseq2 7945 | Equality theorem for the w... |
| wrecseq3 7946 | Equality theorem for the w... |
| wfr3g 7947 | Functions defined by well-... |
| wfrlem1 7948 | Lemma for well-founded rec... |
| wfrlem2 7949 | Lemma for well-founded rec... |
| wfrlem3 7950 | Lemma for well-founded rec... |
| wfrlem3a 7951 | Lemma for well-founded rec... |
| wfrlem4 7952 | Lemma for well-founded rec... |
| wfrlem5 7953 | Lemma for well-founded rec... |
| wfrrel 7954 | The well-founded recursion... |
| wfrdmss 7955 | The domain of the well-fou... |
| wfrlem8 7956 | Lemma for well-founded rec... |
| wfrdmcl 7957 | Given ` F = wrecs ( R , A ... |
| wfrlem10 7958 | Lemma for well-founded rec... |
| wfrfun 7959 | The well-founded function ... |
| wfrlem12 7960 | Lemma for well-founded rec... |
| wfrlem13 7961 | Lemma for well-founded rec... |
| wfrlem14 7962 | Lemma for well-founded rec... |
| wfrlem15 7963 | Lemma for well-founded rec... |
| wfrlem16 7964 | Lemma for well-founded rec... |
| wfrlem17 7965 | Without using ~ ax-rep , s... |
| wfr2a 7966 | A weak version of ~ wfr2 w... |
| wfr1 7967 | The Principle of Well-Foun... |
| wfr2 7968 | The Principle of Well-Foun... |
| wfr3 7969 | The principle of Well-Foun... |
| iunon 7970 | The indexed union of a set... |
| iinon 7971 | The nonempty indexed inter... |
| onfununi 7972 | A property of functions on... |
| onovuni 7973 | A variant of ~ onfununi fo... |
| onoviun 7974 | A variant of ~ onovuni wit... |
| onnseq 7975 | There are no length ` _om ... |
| dfsmo2 7978 | Alternate definition of a ... |
| issmo 7979 | Conditions for which ` A `... |
| issmo2 7980 | Alternate definition of a ... |
| smoeq 7981 | Equality theorem for stric... |
| smodm 7982 | The domain of a strictly m... |
| smores 7983 | A strictly monotone functi... |
| smores3 7984 | A strictly monotone functi... |
| smores2 7985 | A strictly monotone ordina... |
| smodm2 7986 | The domain of a strictly m... |
| smofvon2 7987 | The function values of a s... |
| iordsmo 7988 | The identity relation rest... |
| smo0 7989 | The null set is a strictly... |
| smofvon 7990 | If ` B ` is a strictly mon... |
| smoel 7991 | If ` x ` is less than ` y ... |
| smoiun 7992 | The value of a strictly mo... |
| smoiso 7993 | If ` F ` is an isomorphism... |
| smoel2 7994 | A strictly monotone ordina... |
| smo11 7995 | A strictly monotone ordina... |
| smoord 7996 | A strictly monotone ordina... |
| smoword 7997 | A strictly monotone ordina... |
| smogt 7998 | A strictly monotone ordina... |
| smorndom 7999 | The range of a strictly mo... |
| smoiso2 8000 | The strictly monotone ordi... |
| dfrecs3 8003 | The old definition of tran... |
| recseq 8004 | Equality theorem for ` rec... |
| nfrecs 8005 | Bound-variable hypothesis ... |
| tfrlem1 8006 | A technical lemma for tran... |
| tfrlem3a 8007 | Lemma for transfinite recu... |
| tfrlem3 8008 | Lemma for transfinite recu... |
| tfrlem4 8009 | Lemma for transfinite recu... |
| tfrlem5 8010 | Lemma for transfinite recu... |
| recsfval 8011 | Lemma for transfinite recu... |
| tfrlem6 8012 | Lemma for transfinite recu... |
| tfrlem7 8013 | Lemma for transfinite recu... |
| tfrlem8 8014 | Lemma for transfinite recu... |
| tfrlem9 8015 | Lemma for transfinite recu... |
| tfrlem9a 8016 | Lemma for transfinite recu... |
| tfrlem10 8017 | Lemma for transfinite recu... |
| tfrlem11 8018 | Lemma for transfinite recu... |
| tfrlem12 8019 | Lemma for transfinite recu... |
| tfrlem13 8020 | Lemma for transfinite recu... |
| tfrlem14 8021 | Lemma for transfinite recu... |
| tfrlem15 8022 | Lemma for transfinite recu... |
| tfrlem16 8023 | Lemma for finite recursion... |
| tfr1a 8024 | A weak version of ~ tfr1 w... |
| tfr2a 8025 | A weak version of ~ tfr2 w... |
| tfr2b 8026 | Without assuming ~ ax-rep ... |
| tfr1 8027 | Principle of Transfinite R... |
| tfr2 8028 | Principle of Transfinite R... |
| tfr3 8029 | Principle of Transfinite R... |
| tfr1ALT 8030 | Alternate proof of ~ tfr1 ... |
| tfr2ALT 8031 | Alternate proof of ~ tfr2 ... |
| tfr3ALT 8032 | Alternate proof of ~ tfr3 ... |
| recsfnon 8033 | Strong transfinite recursi... |
| recsval 8034 | Strong transfinite recursi... |
| tz7.44lem1 8035 | ` G ` is a function. Lemm... |
| tz7.44-1 8036 | The value of ` F ` at ` (/... |
| tz7.44-2 8037 | The value of ` F ` at a su... |
| tz7.44-3 8038 | The value of ` F ` at a li... |
| rdgeq1 8041 | Equality theorem for the r... |
| rdgeq2 8042 | Equality theorem for the r... |
| rdgeq12 8043 | Equality theorem for the r... |
| nfrdg 8044 | Bound-variable hypothesis ... |
| rdglem1 8045 | Lemma used with the recurs... |
| rdgfun 8046 | The recursive definition g... |
| rdgdmlim 8047 | The domain of the recursiv... |
| rdgfnon 8048 | The recursive definition g... |
| rdgvalg 8049 | Value of the recursive def... |
| rdgval 8050 | Value of the recursive def... |
| rdg0 8051 | The initial value of the r... |
| rdgseg 8052 | The initial segments of th... |
| rdgsucg 8053 | The value of the recursive... |
| rdgsuc 8054 | The value of the recursive... |
| rdglimg 8055 | The value of the recursive... |
| rdglim 8056 | The value of the recursive... |
| rdg0g 8057 | The initial value of the r... |
| rdgsucmptf 8058 | The value of the recursive... |
| rdgsucmptnf 8059 | The value of the recursive... |
| rdgsucmpt2 8060 | This version of ~ rdgsucmp... |
| rdgsucmpt 8061 | The value of the recursive... |
| rdglim2 8062 | The value of the recursive... |
| rdglim2a 8063 | The value of the recursive... |
| frfnom 8064 | The function generated by ... |
| fr0g 8065 | The initial value resultin... |
| frsuc 8066 | The successor value result... |
| frsucmpt 8067 | The successor value result... |
| frsucmptn 8068 | The value of the finite re... |
| frsucmpt2w 8069 | Version of ~ frsucmpt2 wit... |
| frsucmpt2 8070 | The successor value result... |
| tz7.48lem 8071 | A way of showing an ordina... |
| tz7.48-2 8072 | Proposition 7.48(2) of [Ta... |
| tz7.48-1 8073 | Proposition 7.48(1) of [Ta... |
| tz7.48-3 8074 | Proposition 7.48(3) of [Ta... |
| tz7.49 8075 | Proposition 7.49 of [Takeu... |
| tz7.49c 8076 | Corollary of Proposition 7... |
| seqomlem0 8079 | Lemma for ` seqom ` . Cha... |
| seqomlem1 8080 | Lemma for ` seqom ` . The... |
| seqomlem2 8081 | Lemma for ` seqom ` . (Co... |
| seqomlem3 8082 | Lemma for ` seqom ` . (Co... |
| seqomlem4 8083 | Lemma for ` seqom ` . (Co... |
| seqomeq12 8084 | Equality theorem for ` seq... |
| fnseqom 8085 | An index-aware recursive d... |
| seqom0g 8086 | Value of an index-aware re... |
| seqomsuc 8087 | Value of an index-aware re... |
| omsucelsucb 8088 | Membership is inherited by... |
| 1on 8103 | Ordinal 1 is an ordinal nu... |
| 1oex 8104 | Ordinal 1 is a set. (Cont... |
| 2on 8105 | Ordinal 2 is an ordinal nu... |
| 2oex 8106 | ` 2o ` is a set. (Contrib... |
| 2on0 8107 | Ordinal two is not zero. ... |
| 3on 8108 | Ordinal 3 is an ordinal nu... |
| 4on 8109 | Ordinal 3 is an ordinal nu... |
| df1o2 8110 | Expanded value of the ordi... |
| df2o3 8111 | Expanded value of the ordi... |
| df2o2 8112 | Expanded value of the ordi... |
| 1n0 8113 | Ordinal one is not equal t... |
| xp01disj 8114 | Cartesian products with th... |
| xp01disjl 8115 | Cartesian products with th... |
| ordgt0ge1 8116 | Two ways to express that a... |
| ordge1n0 8117 | An ordinal greater than or... |
| el1o 8118 | Membership in ordinal one.... |
| dif1o 8119 | Two ways to say that ` A `... |
| ondif1 8120 | Two ways to say that ` A `... |
| ondif2 8121 | Two ways to say that ` A `... |
| 2oconcl 8122 | Closure of the pair swappi... |
| 0lt1o 8123 | Ordinal zero is less than ... |
| dif20el 8124 | An ordinal greater than on... |
| 0we1 8125 | The empty set is a well-or... |
| brwitnlem 8126 | Lemma for relations which ... |
| fnoa 8127 | Functionality and domain o... |
| fnom 8128 | Functionality and domain o... |
| fnoe 8129 | Functionality and domain o... |
| oav 8130 | Value of ordinal addition.... |
| omv 8131 | Value of ordinal multiplic... |
| oe0lem 8132 | A helper lemma for ~ oe0 a... |
| oev 8133 | Value of ordinal exponenti... |
| oevn0 8134 | Value of ordinal exponenti... |
| oa0 8135 | Addition with zero. Propo... |
| om0 8136 | Ordinal multiplication wit... |
| oe0m 8137 | Ordinal exponentiation wit... |
| om0x 8138 | Ordinal multiplication wit... |
| oe0m0 8139 | Ordinal exponentiation wit... |
| oe0m1 8140 | Ordinal exponentiation wit... |
| oe0 8141 | Ordinal exponentiation wit... |
| oev2 8142 | Alternate value of ordinal... |
| oasuc 8143 | Addition with successor. ... |
| oesuclem 8144 | Lemma for ~ oesuc . (Cont... |
| omsuc 8145 | Multiplication with succes... |
| oesuc 8146 | Ordinal exponentiation wit... |
| onasuc 8147 | Addition with successor. ... |
| onmsuc 8148 | Multiplication with succes... |
| onesuc 8149 | Exponentiation with a succ... |
| oa1suc 8150 | Addition with 1 is same as... |
| oalim 8151 | Ordinal addition with a li... |
| omlim 8152 | Ordinal multiplication wit... |
| oelim 8153 | Ordinal exponentiation wit... |
| oacl 8154 | Closure law for ordinal ad... |
| omcl 8155 | Closure law for ordinal mu... |
| oecl 8156 | Closure law for ordinal ex... |
| oa0r 8157 | Ordinal addition with zero... |
| om0r 8158 | Ordinal multiplication wit... |
| o1p1e2 8159 | 1 + 1 = 2 for ordinal numb... |
| o2p2e4 8160 | 2 + 2 = 4 for ordinal numb... |
| o2p2e4OLD 8161 | 2 + 2 = 4 for ordinal numb... |
| om1 8162 | Ordinal multiplication wit... |
| om1r 8163 | Ordinal multiplication wit... |
| oe1 8164 | Ordinal exponentiation wit... |
| oe1m 8165 | Ordinal exponentiation wit... |
| oaordi 8166 | Ordering property of ordin... |
| oaord 8167 | Ordering property of ordin... |
| oacan 8168 | Left cancellation law for ... |
| oaword 8169 | Weak ordering property of ... |
| oawordri 8170 | Weak ordering property of ... |
| oaord1 8171 | An ordinal is less than it... |
| oaword1 8172 | An ordinal is less than or... |
| oaword2 8173 | An ordinal is less than or... |
| oawordeulem 8174 | Lemma for ~ oawordex . (C... |
| oawordeu 8175 | Existence theorem for weak... |
| oawordexr 8176 | Existence theorem for weak... |
| oawordex 8177 | Existence theorem for weak... |
| oaordex 8178 | Existence theorem for orde... |
| oa00 8179 | An ordinal sum is zero iff... |
| oalimcl 8180 | The ordinal sum with a lim... |
| oaass 8181 | Ordinal addition is associ... |
| oarec 8182 | Recursive definition of or... |
| oaf1o 8183 | Left addition by a constan... |
| oacomf1olem 8184 | Lemma for ~ oacomf1o . (C... |
| oacomf1o 8185 | Define a bijection from ` ... |
| omordi 8186 | Ordering property of ordin... |
| omord2 8187 | Ordering property of ordin... |
| omord 8188 | Ordering property of ordin... |
| omcan 8189 | Left cancellation law for ... |
| omword 8190 | Weak ordering property of ... |
| omwordi 8191 | Weak ordering property of ... |
| omwordri 8192 | Weak ordering property of ... |
| omword1 8193 | An ordinal is less than or... |
| omword2 8194 | An ordinal is less than or... |
| om00 8195 | The product of two ordinal... |
| om00el 8196 | The product of two nonzero... |
| omordlim 8197 | Ordering involving the pro... |
| omlimcl 8198 | The product of any nonzero... |
| odi 8199 | Distributive law for ordin... |
| omass 8200 | Multiplication of ordinal ... |
| oneo 8201 | If an ordinal number is ev... |
| omeulem1 8202 | Lemma for ~ omeu : existen... |
| omeulem2 8203 | Lemma for ~ omeu : uniquen... |
| omopth2 8204 | An ordered pair-like theor... |
| omeu 8205 | The division algorithm for... |
| oen0 8206 | Ordinal exponentiation wit... |
| oeordi 8207 | Ordering law for ordinal e... |
| oeord 8208 | Ordering property of ordin... |
| oecan 8209 | Left cancellation law for ... |
| oeword 8210 | Weak ordering property of ... |
| oewordi 8211 | Weak ordering property of ... |
| oewordri 8212 | Weak ordering property of ... |
| oeworde 8213 | Ordinal exponentiation com... |
| oeordsuc 8214 | Ordering property of ordin... |
| oelim2 8215 | Ordinal exponentiation wit... |
| oeoalem 8216 | Lemma for ~ oeoa . (Contr... |
| oeoa 8217 | Sum of exponents law for o... |
| oeoelem 8218 | Lemma for ~ oeoe . (Contr... |
| oeoe 8219 | Product of exponents law f... |
| oelimcl 8220 | The ordinal exponential wi... |
| oeeulem 8221 | Lemma for ~ oeeu . (Contr... |
| oeeui 8222 | The division algorithm for... |
| oeeu 8223 | The division algorithm for... |
| nna0 8224 | Addition with zero. Theor... |
| nnm0 8225 | Multiplication with zero. ... |
| nnasuc 8226 | Addition with successor. ... |
| nnmsuc 8227 | Multiplication with succes... |
| nnesuc 8228 | Exponentiation with a succ... |
| nna0r 8229 | Addition to zero. Remark ... |
| nnm0r 8230 | Multiplication with zero. ... |
| nnacl 8231 | Closure of addition of nat... |
| nnmcl 8232 | Closure of multiplication ... |
| nnecl 8233 | Closure of exponentiation ... |
| nnacli 8234 | ` _om ` is closed under ad... |
| nnmcli 8235 | ` _om ` is closed under mu... |
| nnarcl 8236 | Reverse closure law for ad... |
| nnacom 8237 | Addition of natural number... |
| nnaordi 8238 | Ordering property of addit... |
| nnaord 8239 | Ordering property of addit... |
| nnaordr 8240 | Ordering property of addit... |
| nnawordi 8241 | Adding to both sides of an... |
| nnaass 8242 | Addition of natural number... |
| nndi 8243 | Distributive law for natur... |
| nnmass 8244 | Multiplication of natural ... |
| nnmsucr 8245 | Multiplication with succes... |
| nnmcom 8246 | Multiplication of natural ... |
| nnaword 8247 | Weak ordering property of ... |
| nnacan 8248 | Cancellation law for addit... |
| nnaword1 8249 | Weak ordering property of ... |
| nnaword2 8250 | Weak ordering property of ... |
| nnmordi 8251 | Ordering property of multi... |
| nnmord 8252 | Ordering property of multi... |
| nnmword 8253 | Weak ordering property of ... |
| nnmcan 8254 | Cancellation law for multi... |
| nnmwordi 8255 | Weak ordering property of ... |
| nnmwordri 8256 | Weak ordering property of ... |
| nnawordex 8257 | Equivalence for weak order... |
| nnaordex 8258 | Equivalence for ordering. ... |
| 1onn 8259 | One is a natural number. ... |
| 2onn 8260 | The ordinal 2 is a natural... |
| 3onn 8261 | The ordinal 3 is a natural... |
| 4onn 8262 | The ordinal 4 is a natural... |
| 1one2o 8263 | Ordinal one is not ordinal... |
| oaabslem 8264 | Lemma for ~ oaabs . (Cont... |
| oaabs 8265 | Ordinal addition absorbs a... |
| oaabs2 8266 | The absorption law ~ oaabs... |
| omabslem 8267 | Lemma for ~ omabs . (Cont... |
| omabs 8268 | Ordinal multiplication is ... |
| nnm1 8269 | Multiply an element of ` _... |
| nnm2 8270 | Multiply an element of ` _... |
| nn2m 8271 | Multiply an element of ` _... |
| nnneo 8272 | If a natural number is eve... |
| nneob 8273 | A natural number is even i... |
| omsmolem 8274 | Lemma for ~ omsmo . (Cont... |
| omsmo 8275 | A strictly monotonic ordin... |
| omopthlem1 8276 | Lemma for ~ omopthi . (Co... |
| omopthlem2 8277 | Lemma for ~ omopthi . (Co... |
| omopthi 8278 | An ordered pair theorem fo... |
| omopth 8279 | An ordered pair theorem fo... |
| dfer2 8284 | Alternate definition of eq... |
| dfec2 8286 | Alternate definition of ` ... |
| ecexg 8287 | An equivalence class modul... |
| ecexr 8288 | A nonempty equivalence cla... |
| ereq1 8290 | Equality theorem for equiv... |
| ereq2 8291 | Equality theorem for equiv... |
| errel 8292 | An equivalence relation is... |
| erdm 8293 | The domain of an equivalen... |
| ercl 8294 | Elementhood in the field o... |
| ersym 8295 | An equivalence relation is... |
| ercl2 8296 | Elementhood in the field o... |
| ersymb 8297 | An equivalence relation is... |
| ertr 8298 | An equivalence relation is... |
| ertrd 8299 | A transitivity relation fo... |
| ertr2d 8300 | A transitivity relation fo... |
| ertr3d 8301 | A transitivity relation fo... |
| ertr4d 8302 | A transitivity relation fo... |
| erref 8303 | An equivalence relation is... |
| ercnv 8304 | The converse of an equival... |
| errn 8305 | The range and domain of an... |
| erssxp 8306 | An equivalence relation is... |
| erex 8307 | An equivalence relation is... |
| erexb 8308 | An equivalence relation is... |
| iserd 8309 | A reflexive, symmetric, tr... |
| iseri 8310 | A reflexive, symmetric, tr... |
| iseriALT 8311 | Alternate proof of ~ iseri... |
| brdifun 8312 | Evaluate the incomparabili... |
| swoer 8313 | Incomparability under a st... |
| swoord1 8314 | The incomparability equiva... |
| swoord2 8315 | The incomparability equiva... |
| swoso 8316 | If the incomparability rel... |
| eqerlem 8317 | Lemma for ~ eqer . (Contr... |
| eqer 8318 | Equivalence relation invol... |
| ider 8319 | The identity relation is a... |
| 0er 8320 | The empty set is an equiva... |
| eceq1 8321 | Equality theorem for equiv... |
| eceq1d 8322 | Equality theorem for equiv... |
| eceq2 8323 | Equality theorem for equiv... |
| eceq2i 8324 | Equality theorem for the `... |
| eceq2d 8325 | Equality theorem for the `... |
| elecg 8326 | Membership in an equivalen... |
| elec 8327 | Membership in an equivalen... |
| relelec 8328 | Membership in an equivalen... |
| ecss 8329 | An equivalence class is a ... |
| ecdmn0 8330 | A representative of a none... |
| ereldm 8331 | Equality of equivalence cl... |
| erth 8332 | Basic property of equivale... |
| erth2 8333 | Basic property of equivale... |
| erthi 8334 | Basic property of equivale... |
| erdisj 8335 | Equivalence classes do not... |
| ecidsn 8336 | An equivalence class modul... |
| qseq1 8337 | Equality theorem for quoti... |
| qseq2 8338 | Equality theorem for quoti... |
| qseq2i 8339 | Equality theorem for quoti... |
| qseq2d 8340 | Equality theorem for quoti... |
| qseq12 8341 | Equality theorem for quoti... |
| elqsg 8342 | Closed form of ~ elqs . (... |
| elqs 8343 | Membership in a quotient s... |
| elqsi 8344 | Membership in a quotient s... |
| elqsecl 8345 | Membership in a quotient s... |
| ecelqsg 8346 | Membership of an equivalen... |
| ecelqsi 8347 | Membership of an equivalen... |
| ecopqsi 8348 | "Closure" law for equivale... |
| qsexg 8349 | A quotient set exists. (C... |
| qsex 8350 | A quotient set exists. (C... |
| uniqs 8351 | The union of a quotient se... |
| qsss 8352 | A quotient set is a set of... |
| uniqs2 8353 | The union of a quotient se... |
| snec 8354 | The singleton of an equiva... |
| ecqs 8355 | Equivalence class in terms... |
| ecid 8356 | A set is equal to its cose... |
| qsid 8357 | A set is equal to its quot... |
| ectocld 8358 | Implicit substitution of c... |
| ectocl 8359 | Implicit substitution of c... |
| elqsn0 8360 | A quotient set does not co... |
| ecelqsdm 8361 | Membership of an equivalen... |
| xpider 8362 | A Cartesian square is an e... |
| iiner 8363 | The intersection of a none... |
| riiner 8364 | The relative intersection ... |
| erinxp 8365 | A restricted equivalence r... |
| ecinxp 8366 | Restrict the relation in a... |
| qsinxp 8367 | Restrict the equivalence r... |
| qsdisj 8368 | Members of a quotient set ... |
| qsdisj2 8369 | A quotient set is a disjoi... |
| qsel 8370 | If an element of a quotien... |
| uniinqs 8371 | Class union distributes ov... |
| qliftlem 8372 | ` F ` , a function lift, i... |
| qliftrel 8373 | ` F ` , a function lift, i... |
| qliftel 8374 | Elementhood in the relatio... |
| qliftel1 8375 | Elementhood in the relatio... |
| qliftfun 8376 | The function ` F ` is the ... |
| qliftfund 8377 | The function ` F ` is the ... |
| qliftfuns 8378 | The function ` F ` is the ... |
| qliftf 8379 | The domain and range of th... |
| qliftval 8380 | The value of the function ... |
| ecoptocl 8381 | Implicit substitution of c... |
| 2ecoptocl 8382 | Implicit substitution of c... |
| 3ecoptocl 8383 | Implicit substitution of c... |
| brecop 8384 | Binary relation on a quoti... |
| brecop2 8385 | Binary relation on a quoti... |
| eroveu 8386 | Lemma for ~ erov and ~ ero... |
| erovlem 8387 | Lemma for ~ erov and ~ ero... |
| erov 8388 | The value of an operation ... |
| eroprf 8389 | Functionality of an operat... |
| erov2 8390 | The value of an operation ... |
| eroprf2 8391 | Functionality of an operat... |
| ecopoveq 8392 | This is the first of sever... |
| ecopovsym 8393 | Assuming the operation ` F... |
| ecopovtrn 8394 | Assuming that operation ` ... |
| ecopover 8395 | Assuming that operation ` ... |
| eceqoveq 8396 | Equality of equivalence re... |
| ecovcom 8397 | Lemma used to transfer a c... |
| ecovass 8398 | Lemma used to transfer an ... |
| ecovdi 8399 | Lemma used to transfer a d... |
| mapprc 8404 | When ` A ` is a proper cla... |
| pmex 8405 | The class of all partial f... |
| mapex 8406 | The class of all functions... |
| fnmap 8407 | Set exponentiation has a u... |
| fnpm 8408 | Partial function exponenti... |
| reldmmap 8409 | Set exponentiation is a we... |
| mapvalg 8410 | The value of set exponenti... |
| pmvalg 8411 | The value of the partial m... |
| mapval 8412 | The value of set exponenti... |
| elmapg 8413 | Membership relation for se... |
| elmapd 8414 | Deduction form of ~ elmapg... |
| mapdm0 8415 | The empty set is the only ... |
| elpmg 8416 | The predicate "is a partia... |
| elpm2g 8417 | The predicate "is a partia... |
| elpm2r 8418 | Sufficient condition for b... |
| elpmi 8419 | A partial function is a fu... |
| pmfun 8420 | A partial function is a fu... |
| elmapex 8421 | Eliminate antecedent for m... |
| elmapi 8422 | A mapping is a function, f... |
| elmapfn 8423 | A mapping is a function wi... |
| elmapfun 8424 | A mapping is always a func... |
| elmapssres 8425 | A restricted mapping is a ... |
| fpmg 8426 | A total function is a part... |
| pmss12g 8427 | Subset relation for the se... |
| pmresg 8428 | Elementhood of a restricte... |
| elmap 8429 | Membership relation for se... |
| mapval2 8430 | Alternate expression for t... |
| elpm 8431 | The predicate "is a partia... |
| elpm2 8432 | The predicate "is a partia... |
| fpm 8433 | A total function is a part... |
| mapsspm 8434 | Set exponentiation is a su... |
| pmsspw 8435 | Partial maps are a subset ... |
| mapsspw 8436 | Set exponentiation is a su... |
| mapfvd 8437 | The value of a function th... |
| elmapresaun 8438 | ~ fresaun transposed to ma... |
| fvmptmap 8439 | Special case of ~ fvmpt fo... |
| map0e 8440 | Set exponentiation with an... |
| map0b 8441 | Set exponentiation with an... |
| map0g 8442 | Set exponentiation is empt... |
| 0map0sn0 8443 | The set of mappings of the... |
| mapsnd 8444 | The value of set exponenti... |
| map0 8445 | Set exponentiation is empt... |
| mapsn 8446 | The value of set exponenti... |
| mapss 8447 | Subset inheritance for set... |
| fdiagfn 8448 | Functionality of the diago... |
| fvdiagfn 8449 | Functionality of the diago... |
| mapsnconst 8450 | Every singleton map is a c... |
| mapsncnv 8451 | Expression for the inverse... |
| mapsnf1o2 8452 | Explicit bijection between... |
| mapsnf1o3 8453 | Explicit bijection in the ... |
| ralxpmap 8454 | Quantification over functi... |
| dfixp 8457 | Eliminate the expression `... |
| ixpsnval 8458 | The value of an infinite C... |
| elixp2 8459 | Membership in an infinite ... |
| fvixp 8460 | Projection of a factor of ... |
| ixpfn 8461 | A nuple is a function. (C... |
| elixp 8462 | Membership in an infinite ... |
| elixpconst 8463 | Membership in an infinite ... |
| ixpconstg 8464 | Infinite Cartesian product... |
| ixpconst 8465 | Infinite Cartesian product... |
| ixpeq1 8466 | Equality theorem for infin... |
| ixpeq1d 8467 | Equality theorem for infin... |
| ss2ixp 8468 | Subclass theorem for infin... |
| ixpeq2 8469 | Equality theorem for infin... |
| ixpeq2dva 8470 | Equality theorem for infin... |
| ixpeq2dv 8471 | Equality theorem for infin... |
| cbvixp 8472 | Change bound variable in a... |
| cbvixpv 8473 | Change bound variable in a... |
| nfixpw 8474 | Bound-variable hypothesis ... |
| nfixp 8475 | Bound-variable hypothesis ... |
| nfixp1 8476 | The index variable in an i... |
| ixpprc 8477 | A cartesian product of pro... |
| ixpf 8478 | A member of an infinite Ca... |
| uniixp 8479 | The union of an infinite C... |
| ixpexg 8480 | The existence of an infini... |
| ixpin 8481 | The intersection of two in... |
| ixpiin 8482 | The indexed intersection o... |
| ixpint 8483 | The intersection of a coll... |
| ixp0x 8484 | An infinite Cartesian prod... |
| ixpssmap2g 8485 | An infinite Cartesian prod... |
| ixpssmapg 8486 | An infinite Cartesian prod... |
| 0elixp 8487 | Membership of the empty se... |
| ixpn0 8488 | The infinite Cartesian pro... |
| ixp0 8489 | The infinite Cartesian pro... |
| ixpssmap 8490 | An infinite Cartesian prod... |
| resixp 8491 | Restriction of an element ... |
| undifixp 8492 | Union of two projections o... |
| mptelixpg 8493 | Condition for an explicit ... |
| resixpfo 8494 | Restriction of elements of... |
| elixpsn 8495 | Membership in a class of s... |
| ixpsnf1o 8496 | A bijection between a clas... |
| mapsnf1o 8497 | A bijection between a set ... |
| boxriin 8498 | A rectangular subset of a ... |
| boxcutc 8499 | The relative complement of... |
| relen 8508 | Equinumerosity is a relati... |
| reldom 8509 | Dominance is a relation. ... |
| relsdom 8510 | Strict dominance is a rela... |
| encv 8511 | If two classes are equinum... |
| bren 8512 | Equinumerosity relation. ... |
| brdomg 8513 | Dominance relation. (Cont... |
| brdomi 8514 | Dominance relation. (Cont... |
| brdom 8515 | Dominance relation. (Cont... |
| domen 8516 | Dominance in terms of equi... |
| domeng 8517 | Dominance in terms of equi... |
| ctex 8518 | A countable set is a set. ... |
| f1oen3g 8519 | The domain and range of a ... |
| f1oen2g 8520 | The domain and range of a ... |
| f1dom2g 8521 | The domain of a one-to-one... |
| f1oeng 8522 | The domain and range of a ... |
| f1domg 8523 | The domain of a one-to-one... |
| f1oen 8524 | The domain and range of a ... |
| f1dom 8525 | The domain of a one-to-one... |
| brsdom 8526 | Strict dominance relation,... |
| isfi 8527 | Express " ` A ` is finite.... |
| enssdom 8528 | Equinumerosity implies dom... |
| dfdom2 8529 | Alternate definition of do... |
| endom 8530 | Equinumerosity implies dom... |
| sdomdom 8531 | Strict dominance implies d... |
| sdomnen 8532 | Strict dominance implies n... |
| brdom2 8533 | Dominance in terms of stri... |
| bren2 8534 | Equinumerosity expressed i... |
| enrefg 8535 | Equinumerosity is reflexiv... |
| enref 8536 | Equinumerosity is reflexiv... |
| eqeng 8537 | Equality implies equinumer... |
| domrefg 8538 | Dominance is reflexive. (... |
| en2d 8539 | Equinumerosity inference f... |
| en3d 8540 | Equinumerosity inference f... |
| en2i 8541 | Equinumerosity inference f... |
| en3i 8542 | Equinumerosity inference f... |
| dom2lem 8543 | A mapping (first hypothesi... |
| dom2d 8544 | A mapping (first hypothesi... |
| dom3d 8545 | A mapping (first hypothesi... |
| dom2 8546 | A mapping (first hypothesi... |
| dom3 8547 | A mapping (first hypothesi... |
| idssen 8548 | Equality implies equinumer... |
| ssdomg 8549 | A set dominates its subset... |
| ener 8550 | Equinumerosity is an equiv... |
| ensymb 8551 | Symmetry of equinumerosity... |
| ensym 8552 | Symmetry of equinumerosity... |
| ensymi 8553 | Symmetry of equinumerosity... |
| ensymd 8554 | Symmetry of equinumerosity... |
| entr 8555 | Transitivity of equinumero... |
| domtr 8556 | Transitivity of dominance ... |
| entri 8557 | A chained equinumerosity i... |
| entr2i 8558 | A chained equinumerosity i... |
| entr3i 8559 | A chained equinumerosity i... |
| entr4i 8560 | A chained equinumerosity i... |
| endomtr 8561 | Transitivity of equinumero... |
| domentr 8562 | Transitivity of dominance ... |
| f1imaeng 8563 | A one-to-one function's im... |
| f1imaen2g 8564 | A one-to-one function's im... |
| f1imaen 8565 | A one-to-one function's im... |
| en0 8566 | The empty set is equinumer... |
| ensn1 8567 | A singleton is equinumerou... |
| ensn1g 8568 | A singleton is equinumerou... |
| enpr1g 8569 | ` { A , A } ` has only one... |
| en1 8570 | A set is equinumerous to o... |
| en1b 8571 | A set is equinumerous to o... |
| reuen1 8572 | Two ways to express "exact... |
| euen1 8573 | Two ways to express "exact... |
| euen1b 8574 | Two ways to express " ` A ... |
| en1uniel 8575 | A singleton contains its s... |
| 2dom 8576 | A set that dominates ordin... |
| fundmen 8577 | A function is equinumerous... |
| fundmeng 8578 | A function is equinumerous... |
| cnven 8579 | A relational set is equinu... |
| cnvct 8580 | If a set is countable, so ... |
| fndmeng 8581 | A function is equinumerate... |
| mapsnend 8582 | Set exponentiation to a si... |
| mapsnen 8583 | Set exponentiation to a si... |
| snmapen 8584 | Set exponentiation: a sing... |
| snmapen1 8585 | Set exponentiation: a sing... |
| map1 8586 | Set exponentiation: ordina... |
| en2sn 8587 | Two singletons are equinum... |
| snfi 8588 | A singleton is finite. (C... |
| fiprc 8589 | The class of finite sets i... |
| unen 8590 | Equinumerosity of union of... |
| enpr2d 8591 | A pair with distinct eleme... |
| ssct 8592 | Any subset of a countable ... |
| difsnen 8593 | All decrements of a set ar... |
| domdifsn 8594 | Dominance over a set with ... |
| xpsnen 8595 | A set is equinumerous to i... |
| xpsneng 8596 | A set is equinumerous to i... |
| xp1en 8597 | One times a cardinal numbe... |
| endisj 8598 | Any two sets are equinumer... |
| undom 8599 | Dominance law for union. ... |
| xpcomf1o 8600 | The canonical bijection fr... |
| xpcomco 8601 | Composition with the bijec... |
| xpcomen 8602 | Commutative law for equinu... |
| xpcomeng 8603 | Commutative law for equinu... |
| xpsnen2g 8604 | A set is equinumerous to i... |
| xpassen 8605 | Associative law for equinu... |
| xpdom2 8606 | Dominance law for Cartesia... |
| xpdom2g 8607 | Dominance law for Cartesia... |
| xpdom1g 8608 | Dominance law for Cartesia... |
| xpdom3 8609 | A set is dominated by its ... |
| xpdom1 8610 | Dominance law for Cartesia... |
| domunsncan 8611 | A singleton cancellation l... |
| omxpenlem 8612 | Lemma for ~ omxpen . (Con... |
| omxpen 8613 | The cardinal and ordinal p... |
| omf1o 8614 | Construct an explicit bije... |
| pw2f1olem 8615 | Lemma for ~ pw2f1o . (Con... |
| pw2f1o 8616 | The power set of a set is ... |
| pw2eng 8617 | The power set of a set is ... |
| pw2en 8618 | The power set of a set is ... |
| fopwdom 8619 | Covering implies injection... |
| enfixsn 8620 | Given two equipollent sets... |
| sbthlem1 8621 | Lemma for ~ sbth . (Contr... |
| sbthlem2 8622 | Lemma for ~ sbth . (Contr... |
| sbthlem3 8623 | Lemma for ~ sbth . (Contr... |
| sbthlem4 8624 | Lemma for ~ sbth . (Contr... |
| sbthlem5 8625 | Lemma for ~ sbth . (Contr... |
| sbthlem6 8626 | Lemma for ~ sbth . (Contr... |
| sbthlem7 8627 | Lemma for ~ sbth . (Contr... |
| sbthlem8 8628 | Lemma for ~ sbth . (Contr... |
| sbthlem9 8629 | Lemma for ~ sbth . (Contr... |
| sbthlem10 8630 | Lemma for ~ sbth . (Contr... |
| sbth 8631 | Schroeder-Bernstein Theore... |
| sbthb 8632 | Schroeder-Bernstein Theore... |
| sbthcl 8633 | Schroeder-Bernstein Theore... |
| dfsdom2 8634 | Alternate definition of st... |
| brsdom2 8635 | Alternate definition of st... |
| sdomnsym 8636 | Strict dominance is asymme... |
| domnsym 8637 | Theorem 22(i) of [Suppes] ... |
| 0domg 8638 | Any set dominates the empt... |
| dom0 8639 | A set dominated by the emp... |
| 0sdomg 8640 | A set strictly dominates t... |
| 0dom 8641 | Any set dominates the empt... |
| 0sdom 8642 | A set strictly dominates t... |
| sdom0 8643 | The empty set does not str... |
| sdomdomtr 8644 | Transitivity of strict dom... |
| sdomentr 8645 | Transitivity of strict dom... |
| domsdomtr 8646 | Transitivity of dominance ... |
| ensdomtr 8647 | Transitivity of equinumero... |
| sdomirr 8648 | Strict dominance is irrefl... |
| sdomtr 8649 | Strict dominance is transi... |
| sdomn2lp 8650 | Strict dominance has no 2-... |
| enen1 8651 | Equality-like theorem for ... |
| enen2 8652 | Equality-like theorem for ... |
| domen1 8653 | Equality-like theorem for ... |
| domen2 8654 | Equality-like theorem for ... |
| sdomen1 8655 | Equality-like theorem for ... |
| sdomen2 8656 | Equality-like theorem for ... |
| domtriord 8657 | Dominance is trichotomous ... |
| sdomel 8658 | Strict dominance implies o... |
| sdomdif 8659 | The difference of a set fr... |
| onsdominel 8660 | An ordinal with more eleme... |
| domunsn 8661 | Dominance over a set with ... |
| fodomr 8662 | There exists a mapping fro... |
| pwdom 8663 | Injection of sets implies ... |
| canth2 8664 | Cantor's Theorem. No set ... |
| canth2g 8665 | Cantor's theorem with the ... |
| 2pwuninel 8666 | The power set of the power... |
| 2pwne 8667 | No set equals the power se... |
| disjen 8668 | A stronger form of ~ pwuni... |
| disjenex 8669 | Existence version of ~ dis... |
| domss2 8670 | A corollary of ~ disjenex ... |
| domssex2 8671 | A corollary of ~ disjenex ... |
| domssex 8672 | Weakening of ~ domssex to ... |
| xpf1o 8673 | Construct a bijection on a... |
| xpen 8674 | Equinumerosity law for Car... |
| mapen 8675 | Two set exponentiations ar... |
| mapdom1 8676 | Order-preserving property ... |
| mapxpen 8677 | Equinumerosity law for dou... |
| xpmapenlem 8678 | Lemma for ~ xpmapen . (Co... |
| xpmapen 8679 | Equinumerosity law for set... |
| mapunen 8680 | Equinumerosity law for set... |
| map2xp 8681 | A cardinal power with expo... |
| mapdom2 8682 | Order-preserving property ... |
| mapdom3 8683 | Set exponentiation dominat... |
| pwen 8684 | If two sets are equinumero... |
| ssenen 8685 | Equinumerosity of equinume... |
| limenpsi 8686 | A limit ordinal is equinum... |
| limensuci 8687 | A limit ordinal is equinum... |
| limensuc 8688 | A limit ordinal is equinum... |
| infensuc 8689 | Any infinite ordinal is eq... |
| phplem1 8690 | Lemma for Pigeonhole Princ... |
| phplem2 8691 | Lemma for Pigeonhole Princ... |
| phplem3 8692 | Lemma for Pigeonhole Princ... |
| phplem4 8693 | Lemma for Pigeonhole Princ... |
| nneneq 8694 | Two equinumerous natural n... |
| php 8695 | Pigeonhole Principle. A n... |
| php2 8696 | Corollary of Pigeonhole Pr... |
| php3 8697 | Corollary of Pigeonhole Pr... |
| php4 8698 | Corollary of the Pigeonhol... |
| php5 8699 | Corollary of the Pigeonhol... |
| phpeqd 8700 | Corollary of the Pigeonhol... |
| snnen2o 8701 | A singleton ` { A } ` is n... |
| onomeneq 8702 | An ordinal number equinume... |
| onfin 8703 | An ordinal number is finit... |
| onfin2 8704 | A set is a natural number ... |
| nnfi 8705 | Natural numbers are finite... |
| nndomo 8706 | Cardinal ordering agrees w... |
| nnsdomo 8707 | Cardinal ordering agrees w... |
| sucdom2 8708 | Strict dominance of a set ... |
| sucdom 8709 | Strict dominance of a set ... |
| 0sdom1dom 8710 | Strict dominance over zero... |
| 1sdom2 8711 | Ordinal 1 is strictly domi... |
| sdom1 8712 | A set has less than one me... |
| modom 8713 | Two ways to express "at mo... |
| modom2 8714 | Two ways to express "at mo... |
| 1sdom 8715 | A set that strictly domina... |
| unxpdomlem1 8716 | Lemma for ~ unxpdom . (Tr... |
| unxpdomlem2 8717 | Lemma for ~ unxpdom . (Co... |
| unxpdomlem3 8718 | Lemma for ~ unxpdom . (Co... |
| unxpdom 8719 | Cartesian product dominate... |
| unxpdom2 8720 | Corollary of ~ unxpdom . ... |
| sucxpdom 8721 | Cartesian product dominate... |
| pssinf 8722 | A set equinumerous to a pr... |
| fisseneq 8723 | A finite set is equal to i... |
| ominf 8724 | The set of natural numbers... |
| isinf 8725 | Any set that is not finite... |
| fineqvlem 8726 | Lemma for ~ fineqv . (Con... |
| fineqv 8727 | If the Axiom of Infinity i... |
| enfi 8728 | Equinumerous sets have the... |
| enfii 8729 | A set equinumerous to a fi... |
| pssnn 8730 | A proper subset of a natur... |
| ssnnfi 8731 | A subset of a natural numb... |
| ssfi 8732 | A subset of a finite set i... |
| domfi 8733 | A set dominated by a finit... |
| xpfir 8734 | The components of a nonemp... |
| ssfid 8735 | A subset of a finite set i... |
| infi 8736 | The intersection of two se... |
| rabfi 8737 | A restricted class built f... |
| finresfin 8738 | The restriction of a finit... |
| f1finf1o 8739 | Any injection from one fin... |
| 0fin 8740 | The empty set is finite. ... |
| nfielex 8741 | If a class is not finite, ... |
| en1eqsn 8742 | A set with one element is ... |
| en1eqsnbi 8743 | A set containing an elemen... |
| diffi 8744 | If ` A ` is finite, ` ( A ... |
| dif1en 8745 | If a set ` A ` is equinume... |
| enp1ilem 8746 | Lemma for uses of ~ enp1i ... |
| enp1i 8747 | Proof induction for ~ en2i... |
| en2 8748 | A set equinumerous to ordi... |
| en3 8749 | A set equinumerous to ordi... |
| en4 8750 | A set equinumerous to ordi... |
| findcard 8751 | Schema for induction on th... |
| findcard2 8752 | Schema for induction on th... |
| findcard2s 8753 | Variation of ~ findcard2 r... |
| findcard2d 8754 | Deduction version of ~ fin... |
| findcard3 8755 | Schema for strong inductio... |
| ac6sfi 8756 | A version of ~ ac6s for fi... |
| frfi 8757 | A partial order is well-fo... |
| fimax2g 8758 | A finite set has a maximum... |
| fimaxg 8759 | A finite set has a maximum... |
| fisupg 8760 | Lemma showing existence an... |
| wofi 8761 | A total order on a finite ... |
| ordunifi 8762 | The maximum of a finite co... |
| nnunifi 8763 | The union (supremum) of a ... |
| unblem1 8764 | Lemma for ~ unbnn . After... |
| unblem2 8765 | Lemma for ~ unbnn . The v... |
| unblem3 8766 | Lemma for ~ unbnn . The v... |
| unblem4 8767 | Lemma for ~ unbnn . The f... |
| unbnn 8768 | Any unbounded subset of na... |
| unbnn2 8769 | Version of ~ unbnn that do... |
| isfinite2 8770 | Any set strictly dominated... |
| nnsdomg 8771 | Omega strictly dominates a... |
| isfiniteg 8772 | A set is finite iff it is ... |
| infsdomnn 8773 | An infinite set strictly d... |
| infn0 8774 | An infinite set is not emp... |
| fin2inf 8775 | This (useless) theorem, wh... |
| unfilem1 8776 | Lemma for proving that the... |
| unfilem2 8777 | Lemma for proving that the... |
| unfilem3 8778 | Lemma for proving that the... |
| unfi 8779 | The union of two finite se... |
| unfir 8780 | If a union is finite, the ... |
| unfi2 8781 | The union of two finite se... |
| difinf 8782 | An infinite set ` A ` minu... |
| xpfi 8783 | The Cartesian product of t... |
| 3xpfi 8784 | The Cartesian product of t... |
| domunfican 8785 | A finite set union cancell... |
| infcntss 8786 | Every infinite set has a d... |
| prfi 8787 | An unordered pair is finit... |
| tpfi 8788 | An unordered triple is fin... |
| fiint 8789 | Equivalent ways of stating... |
| fnfi 8790 | A version of ~ fnex for fi... |
| fodomfi 8791 | An onto function implies d... |
| fodomfib 8792 | Equivalence of an onto map... |
| fofinf1o 8793 | Any surjection from one fi... |
| rneqdmfinf1o 8794 | Any function from a finite... |
| fidomdm 8795 | Any finite set dominates i... |
| dmfi 8796 | The domain of a finite set... |
| fundmfibi 8797 | A function is finite if an... |
| resfnfinfin 8798 | The restriction of a funct... |
| residfi 8799 | A restricted identity func... |
| cnvfi 8800 | If a set is finite, its co... |
| rnfi 8801 | The range of a finite set ... |
| f1dmvrnfibi 8802 | A one-to-one function whos... |
| f1vrnfibi 8803 | A one-to-one function whic... |
| fofi 8804 | If a function has a finite... |
| f1fi 8805 | If a 1-to-1 function has a... |
| iunfi 8806 | The finite union of finite... |
| unifi 8807 | The finite union of finite... |
| unifi2 8808 | The finite union of finite... |
| infssuni 8809 | If an infinite set ` A ` i... |
| unirnffid 8810 | The union of the range of ... |
| imafi 8811 | Images of finite sets are ... |
| pwfilem 8812 | Lemma for ~ pwfi . (Contr... |
| pwfi 8813 | The power set of a finite ... |
| mapfi 8814 | Set exponentiation of fini... |
| ixpfi 8815 | A Cartesian product of fin... |
| ixpfi2 8816 | A Cartesian product of fin... |
| mptfi 8817 | A finite mapping set is fi... |
| abrexfi 8818 | An image set from a finite... |
| cnvimamptfin 8819 | A preimage of a mapping wi... |
| elfpw 8820 | Membership in a class of f... |
| unifpw 8821 | A set is the union of its ... |
| f1opwfi 8822 | A one-to-one mapping induc... |
| fissuni 8823 | A finite subset of a union... |
| fipreima 8824 | Given a finite subset ` A ... |
| finsschain 8825 | A finite subset of the uni... |
| indexfi 8826 | If for every element of a ... |
| relfsupp 8829 | The property of a function... |
| relprcnfsupp 8830 | A proper class is never fi... |
| isfsupp 8831 | The property of a class to... |
| funisfsupp 8832 | The property of a function... |
| fsuppimp 8833 | Implications of a class be... |
| fsuppimpd 8834 | A finitely supported funct... |
| fisuppfi 8835 | A function on a finite set... |
| fdmfisuppfi 8836 | The support of a function ... |
| fdmfifsupp 8837 | A function with a finite d... |
| fsuppmptdm 8838 | A mapping with a finite do... |
| fndmfisuppfi 8839 | The support of a function ... |
| fndmfifsupp 8840 | A function with a finite d... |
| suppeqfsuppbi 8841 | If two functions have the ... |
| suppssfifsupp 8842 | If the support of a functi... |
| fsuppsssupp 8843 | If the support of a functi... |
| fsuppxpfi 8844 | The cartesian product of t... |
| fczfsuppd 8845 | A constant function with v... |
| fsuppun 8846 | The union of two finitely ... |
| fsuppunfi 8847 | The union of the support o... |
| fsuppunbi 8848 | If the union of two classe... |
| 0fsupp 8849 | The empty set is a finitel... |
| snopfsupp 8850 | A singleton containing an ... |
| funsnfsupp 8851 | Finite support for a funct... |
| fsuppres 8852 | The restriction of a finit... |
| ressuppfi 8853 | If the support of the rest... |
| resfsupp 8854 | If the restriction of a fu... |
| resfifsupp 8855 | The restriction of a funct... |
| frnfsuppbi 8856 | Two ways of saying that a ... |
| fsuppmptif 8857 | A function mapping an argu... |
| fsuppcolem 8858 | Lemma for ~ fsuppco . For... |
| fsuppco 8859 | The composition of a 1-1 f... |
| fsuppco2 8860 | The composition of a funct... |
| fsuppcor 8861 | The composition of a funct... |
| mapfienlem1 8862 | Lemma 1 for ~ mapfien . (... |
| mapfienlem2 8863 | Lemma 2 for ~ mapfien . (... |
| mapfienlem3 8864 | Lemma 3 for ~ mapfien . (... |
| mapfien 8865 | A bijection of the base se... |
| mapfien2 8866 | Equinumerousity relation f... |
| sniffsupp 8867 | A function mapping all but... |
| fival 8870 | The set of all the finite ... |
| elfi 8871 | Specific properties of an ... |
| elfi2 8872 | The empty intersection nee... |
| elfir 8873 | Sufficient condition for a... |
| intrnfi 8874 | Sufficient condition for t... |
| iinfi 8875 | An indexed intersection of... |
| inelfi 8876 | The intersection of two se... |
| ssfii 8877 | Any element of a set ` A `... |
| fi0 8878 | The set of finite intersec... |
| fieq0 8879 | A set is empty iff the cla... |
| fiin 8880 | The elements of ` ( fi `` ... |
| dffi2 8881 | The set of finite intersec... |
| fiss 8882 | Subset relationship for fu... |
| inficl 8883 | A set which is closed unde... |
| fipwuni 8884 | The set of finite intersec... |
| fisn 8885 | A singleton is closed unde... |
| fiuni 8886 | The union of the finite in... |
| fipwss 8887 | If a set is a family of su... |
| elfiun 8888 | A finite intersection of e... |
| dffi3 8889 | The set of finite intersec... |
| fifo 8890 | Describe a surjection from... |
| marypha1lem 8891 | Core induction for Philip ... |
| marypha1 8892 | (Philip) Hall's marriage t... |
| marypha2lem1 8893 | Lemma for ~ marypha2 . Pr... |
| marypha2lem2 8894 | Lemma for ~ marypha2 . Pr... |
| marypha2lem3 8895 | Lemma for ~ marypha2 . Pr... |
| marypha2lem4 8896 | Lemma for ~ marypha2 . Pr... |
| marypha2 8897 | Version of ~ marypha1 usin... |
| dfsup2 8902 | Quantifier free definition... |
| supeq1 8903 | Equality theorem for supre... |
| supeq1d 8904 | Equality deduction for sup... |
| supeq1i 8905 | Equality inference for sup... |
| supeq2 8906 | Equality theorem for supre... |
| supeq3 8907 | Equality theorem for supre... |
| supeq123d 8908 | Equality deduction for sup... |
| nfsup 8909 | Hypothesis builder for sup... |
| supmo 8910 | Any class ` B ` has at mos... |
| supexd 8911 | A supremum is a set. (Con... |
| supeu 8912 | A supremum is unique. Sim... |
| supval2 8913 | Alternate expression for t... |
| eqsup 8914 | Sufficient condition for a... |
| eqsupd 8915 | Sufficient condition for a... |
| supcl 8916 | A supremum belongs to its ... |
| supub 8917 | A supremum is an upper bou... |
| suplub 8918 | A supremum is the least up... |
| suplub2 8919 | Bidirectional form of ~ su... |
| supnub 8920 | An upper bound is not less... |
| supex 8921 | A supremum is a set. (Con... |
| sup00 8922 | The supremum under an empt... |
| sup0riota 8923 | The supremum of an empty s... |
| sup0 8924 | The supremum of an empty s... |
| supmax 8925 | The greatest element of a ... |
| fisup2g 8926 | A finite set satisfies the... |
| fisupcl 8927 | A nonempty finite set cont... |
| supgtoreq 8928 | The supremum of a finite s... |
| suppr 8929 | The supremum of a pair. (... |
| supsn 8930 | The supremum of a singleto... |
| supisolem 8931 | Lemma for ~ supiso . (Con... |
| supisoex 8932 | Lemma for ~ supiso . (Con... |
| supiso 8933 | Image of a supremum under ... |
| infeq1 8934 | Equality theorem for infim... |
| infeq1d 8935 | Equality deduction for inf... |
| infeq1i 8936 | Equality inference for inf... |
| infeq2 8937 | Equality theorem for infim... |
| infeq3 8938 | Equality theorem for infim... |
| infeq123d 8939 | Equality deduction for inf... |
| nfinf 8940 | Hypothesis builder for inf... |
| infexd 8941 | An infimum is a set. (Con... |
| eqinf 8942 | Sufficient condition for a... |
| eqinfd 8943 | Sufficient condition for a... |
| infval 8944 | Alternate expression for t... |
| infcllem 8945 | Lemma for ~ infcl , ~ infl... |
| infcl 8946 | An infimum belongs to its ... |
| inflb 8947 | An infimum is a lower boun... |
| infglb 8948 | An infimum is the greatest... |
| infglbb 8949 | Bidirectional form of ~ in... |
| infnlb 8950 | A lower bound is not great... |
| infex 8951 | An infimum is a set. (Con... |
| infmin 8952 | The smallest element of a ... |
| infmo 8953 | Any class ` B ` has at mos... |
| infeu 8954 | An infimum is unique. (Co... |
| fimin2g 8955 | A finite set has a minimum... |
| fiming 8956 | A finite set has a minimum... |
| fiinfg 8957 | Lemma showing existence an... |
| fiinf2g 8958 | A finite set satisfies the... |
| fiinfcl 8959 | A nonempty finite set cont... |
| infltoreq 8960 | The infimum of a finite se... |
| infpr 8961 | The infimum of a pair. (C... |
| infsupprpr 8962 | The infimum of a proper pa... |
| infsn 8963 | The infimum of a singleton... |
| inf00 8964 | The infimum regarding an e... |
| infempty 8965 | The infimum of an empty se... |
| infiso 8966 | Image of an infimum under ... |
| dfoi 8969 | Rewrite ~ df-oi with abbre... |
| oieq1 8970 | Equality theorem for ordin... |
| oieq2 8971 | Equality theorem for ordin... |
| nfoi 8972 | Hypothesis builder for ord... |
| ordiso2 8973 | Generalize ~ ordiso to pro... |
| ordiso 8974 | Order-isomorphic ordinal n... |
| ordtypecbv 8975 | Lemma for ~ ordtype . (Co... |
| ordtypelem1 8976 | Lemma for ~ ordtype . (Co... |
| ordtypelem2 8977 | Lemma for ~ ordtype . (Co... |
| ordtypelem3 8978 | Lemma for ~ ordtype . (Co... |
| ordtypelem4 8979 | Lemma for ~ ordtype . (Co... |
| ordtypelem5 8980 | Lemma for ~ ordtype . (Co... |
| ordtypelem6 8981 | Lemma for ~ ordtype . (Co... |
| ordtypelem7 8982 | Lemma for ~ ordtype . ` ra... |
| ordtypelem8 8983 | Lemma for ~ ordtype . (Co... |
| ordtypelem9 8984 | Lemma for ~ ordtype . Eit... |
| ordtypelem10 8985 | Lemma for ~ ordtype . Usi... |
| oi0 8986 | Definition of the ordinal ... |
| oicl 8987 | The order type of the well... |
| oif 8988 | The order isomorphism of t... |
| oiiso2 8989 | The order isomorphism of t... |
| ordtype 8990 | For any set-like well-orde... |
| oiiniseg 8991 | ` ran F ` is an initial se... |
| ordtype2 8992 | For any set-like well-orde... |
| oiexg 8993 | The order isomorphism on a... |
| oion 8994 | The order type of the well... |
| oiiso 8995 | The order isomorphism of t... |
| oien 8996 | The order type of a well-o... |
| oieu 8997 | Uniqueness of the unique o... |
| oismo 8998 | When ` A ` is a subclass o... |
| oiid 8999 | The order type of an ordin... |
| hartogslem1 9000 | Lemma for ~ hartogs . (Co... |
| hartogslem2 9001 | Lemma for ~ hartogs . (Co... |
| hartogs 9002 | Given any set, the Hartogs... |
| wofib 9003 | The only sets which are we... |
| wemaplem1 9004 | Value of the lexicographic... |
| wemaplem2 9005 | Lemma for ~ wemapso . Tra... |
| wemaplem3 9006 | Lemma for ~ wemapso . Tra... |
| wemappo 9007 | Construct lexicographic or... |
| wemapsolem 9008 | Lemma for ~ wemapso . (Co... |
| wemapso 9009 | Construct lexicographic or... |
| wemapso2lem 9010 | Lemma for ~ wemapso2 . (C... |
| wemapso2 9011 | An alternative to having a... |
| card2on 9012 | The alternate definition o... |
| card2inf 9013 | The alternate definition o... |
| harf 9018 | Functionality of the Harto... |
| harcl 9019 | Closure of the Hartogs fun... |
| harval 9020 | Function value of the Hart... |
| elharval 9021 | The Hartogs number of a se... |
| harndom 9022 | The Hartogs number of a se... |
| harword 9023 | Weak ordering property of ... |
| relwdom 9024 | Weak dominance is a relati... |
| brwdom 9025 | Property of weak dominance... |
| brwdomi 9026 | Property of weak dominance... |
| brwdomn0 9027 | Weak dominance over nonemp... |
| 0wdom 9028 | Any set weakly dominates t... |
| fowdom 9029 | An onto function implies w... |
| wdomref 9030 | Reflexivity of weak domina... |
| brwdom2 9031 | Alternate characterization... |
| domwdom 9032 | Weak dominance is implied ... |
| wdomtr 9033 | Transitivity of weak domin... |
| wdomen1 9034 | Equality-like theorem for ... |
| wdomen2 9035 | Equality-like theorem for ... |
| wdompwdom 9036 | Weak dominance strengthens... |
| canthwdom 9037 | Cantor's Theorem, stated u... |
| wdom2d 9038 | Deduce weak dominance from... |
| wdomd 9039 | Deduce weak dominance from... |
| brwdom3 9040 | Condition for weak dominan... |
| brwdom3i 9041 | Weak dominance implies exi... |
| unwdomg 9042 | Weak dominance of a (disjo... |
| xpwdomg 9043 | Weak dominance of a Cartes... |
| wdomima2g 9044 | A set is weakly dominant o... |
| wdomimag 9045 | A set is weakly dominant o... |
| unxpwdom2 9046 | Lemma for ~ unxpwdom . (C... |
| unxpwdom 9047 | If a Cartesian product is ... |
| harwdom 9048 | The Hartogs function is we... |
| ixpiunwdom 9049 | Describe an onto function ... |
| axreg2 9051 | Axiom of Regularity expres... |
| zfregcl 9052 | The Axiom of Regularity wi... |
| zfreg 9053 | The Axiom of Regularity us... |
| elirrv 9054 | The membership relation is... |
| elirr 9055 | No class is a member of it... |
| elneq 9056 | A class is not equal to an... |
| nelaneq 9057 | A class is not an element ... |
| epinid0 9058 | The membership (epsilon) r... |
| sucprcreg 9059 | A class is equal to its su... |
| ruv 9060 | The Russell class is equal... |
| ruALT 9061 | Alternate proof of ~ ru , ... |
| zfregfr 9062 | The membership relation is... |
| en2lp 9063 | No class has 2-cycle membe... |
| elnanel 9064 | Two classes are not elemen... |
| cnvepnep 9065 | The membership (epsilon) r... |
| epnsym 9066 | The membership (epsilon) r... |
| elnotel 9067 | A class cannot be an eleme... |
| elnel 9068 | A class cannot be an eleme... |
| en3lplem1 9069 | Lemma for ~ en3lp . (Cont... |
| en3lplem2 9070 | Lemma for ~ en3lp . (Cont... |
| en3lp 9071 | No class has 3-cycle membe... |
| preleqg 9072 | Equality of two unordered ... |
| preleq 9073 | Equality of two unordered ... |
| preleqALT 9074 | Alternate proof of ~ prele... |
| opthreg 9075 | Theorem for alternate repr... |
| suc11reg 9076 | The successor operation be... |
| dford2 9077 | Assuming ~ ax-reg , an ord... |
| inf0 9078 | Our Axiom of Infinity deri... |
| inf1 9079 | Variation of Axiom of Infi... |
| inf2 9080 | Variation of Axiom of Infi... |
| inf3lema 9081 | Lemma for our Axiom of Inf... |
| inf3lemb 9082 | Lemma for our Axiom of Inf... |
| inf3lemc 9083 | Lemma for our Axiom of Inf... |
| inf3lemd 9084 | Lemma for our Axiom of Inf... |
| inf3lem1 9085 | Lemma for our Axiom of Inf... |
| inf3lem2 9086 | Lemma for our Axiom of Inf... |
| inf3lem3 9087 | Lemma for our Axiom of Inf... |
| inf3lem4 9088 | Lemma for our Axiom of Inf... |
| inf3lem5 9089 | Lemma for our Axiom of Inf... |
| inf3lem6 9090 | Lemma for our Axiom of Inf... |
| inf3lem7 9091 | Lemma for our Axiom of Inf... |
| inf3 9092 | Our Axiom of Infinity ~ ax... |
| infeq5i 9093 | Half of ~ infeq5 . (Contr... |
| infeq5 9094 | The statement "there exist... |
| zfinf 9096 | Axiom of Infinity expresse... |
| axinf2 9097 | A standard version of Axio... |
| zfinf2 9099 | A standard version of the ... |
| omex 9100 | The existence of omega (th... |
| axinf 9101 | The first version of the A... |
| inf5 9102 | The statement "there exist... |
| omelon 9103 | Omega is an ordinal number... |
| dfom3 9104 | The class of natural numbe... |
| elom3 9105 | A simplification of ~ elom... |
| dfom4 9106 | A simplification of ~ df-o... |
| dfom5 9107 | ` _om ` is the smallest li... |
| oancom 9108 | Ordinal addition is not co... |
| isfinite 9109 | A set is finite iff it is ... |
| fict 9110 | A finite set is countable ... |
| nnsdom 9111 | A natural number is strict... |
| omenps 9112 | Omega is equinumerous to a... |
| omensuc 9113 | The set of natural numbers... |
| infdifsn 9114 | Removing a singleton from ... |
| infdiffi 9115 | Removing a finite set from... |
| unbnn3 9116 | Any unbounded subset of na... |
| noinfep 9117 | Using the Axiom of Regular... |
| cantnffval 9120 | The value of the Cantor no... |
| cantnfdm 9121 | The domain of the Cantor n... |
| cantnfvalf 9122 | Lemma for ~ cantnf . The ... |
| cantnfs 9123 | Elementhood in the set of ... |
| cantnfcl 9124 | Basic properties of the or... |
| cantnfval 9125 | The value of the Cantor no... |
| cantnfval2 9126 | Alternate expression for t... |
| cantnfsuc 9127 | The value of the recursive... |
| cantnfle 9128 | A lower bound on the ` CNF... |
| cantnflt 9129 | An upper bound on the part... |
| cantnflt2 9130 | An upper bound on the ` CN... |
| cantnff 9131 | The ` CNF ` function is a ... |
| cantnf0 9132 | The value of the zero func... |
| cantnfrescl 9133 | A function is finitely sup... |
| cantnfres 9134 | The ` CNF ` function respe... |
| cantnfp1lem1 9135 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem2 9136 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1lem3 9137 | Lemma for ~ cantnfp1 . (C... |
| cantnfp1 9138 | If ` F ` is created by add... |
| oemapso 9139 | The relation ` T ` is a st... |
| oemapval 9140 | Value of the relation ` T ... |
| oemapvali 9141 | If ` F < G ` , then there ... |
| cantnflem1a 9142 | Lemma for ~ cantnf . (Con... |
| cantnflem1b 9143 | Lemma for ~ cantnf . (Con... |
| cantnflem1c 9144 | Lemma for ~ cantnf . (Con... |
| cantnflem1d 9145 | Lemma for ~ cantnf . (Con... |
| cantnflem1 9146 | Lemma for ~ cantnf . This... |
| cantnflem2 9147 | Lemma for ~ cantnf . (Con... |
| cantnflem3 9148 | Lemma for ~ cantnf . Here... |
| cantnflem4 9149 | Lemma for ~ cantnf . Comp... |
| cantnf 9150 | The Cantor Normal Form the... |
| oemapwe 9151 | The lexicographic order on... |
| cantnffval2 9152 | An alternate definition of... |
| cantnff1o 9153 | Simplify the isomorphism o... |
| wemapwe 9154 | Construct lexicographic or... |
| oef1o 9155 | A bijection of the base se... |
| cnfcomlem 9156 | Lemma for ~ cnfcom . (Con... |
| cnfcom 9157 | Any ordinal ` B ` is equin... |
| cnfcom2lem 9158 | Lemma for ~ cnfcom2 . (Co... |
| cnfcom2 9159 | Any nonzero ordinal ` B ` ... |
| cnfcom3lem 9160 | Lemma for ~ cnfcom3 . (Co... |
| cnfcom3 9161 | Any infinite ordinal ` B `... |
| cnfcom3clem 9162 | Lemma for ~ cnfcom3c . (C... |
| cnfcom3c 9163 | Wrap the construction of ~... |
| trcl 9164 | For any set ` A ` , show t... |
| tz9.1 9165 | Every set has a transitive... |
| tz9.1c 9166 | Alternate expression for t... |
| epfrs 9167 | The strong form of the Axi... |
| zfregs 9168 | The strong form of the Axi... |
| zfregs2 9169 | Alternate strong form of t... |
| setind 9170 | Set (epsilon) induction. ... |
| setind2 9171 | Set (epsilon) induction, s... |
| tcvalg 9174 | Value of the transitive cl... |
| tcid 9175 | Defining property of the t... |
| tctr 9176 | Defining property of the t... |
| tcmin 9177 | Defining property of the t... |
| tc2 9178 | A variant of the definitio... |
| tcsni 9179 | The transitive closure of ... |
| tcss 9180 | The transitive closure fun... |
| tcel 9181 | The transitive closure fun... |
| tcidm 9182 | The transitive closure fun... |
| tc0 9183 | The transitive closure of ... |
| tc00 9184 | The transitive closure is ... |
| r1funlim 9189 | The cumulative hierarchy o... |
| r1fnon 9190 | The cumulative hierarchy o... |
| r10 9191 | Value of the cumulative hi... |
| r1sucg 9192 | Value of the cumulative hi... |
| r1suc 9193 | Value of the cumulative hi... |
| r1limg 9194 | Value of the cumulative hi... |
| r1lim 9195 | Value of the cumulative hi... |
| r1fin 9196 | The first ` _om ` levels o... |
| r1sdom 9197 | Each stage in the cumulati... |
| r111 9198 | The cumulative hierarchy i... |
| r1tr 9199 | The cumulative hierarchy o... |
| r1tr2 9200 | The union of a cumulative ... |
| r1ordg 9201 | Ordering relation for the ... |
| r1ord3g 9202 | Ordering relation for the ... |
| r1ord 9203 | Ordering relation for the ... |
| r1ord2 9204 | Ordering relation for the ... |
| r1ord3 9205 | Ordering relation for the ... |
| r1sssuc 9206 | The value of the cumulativ... |
| r1pwss 9207 | Each set of the cumulative... |
| r1sscl 9208 | Each set of the cumulative... |
| r1val1 9209 | The value of the cumulativ... |
| tz9.12lem1 9210 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem2 9211 | Lemma for ~ tz9.12 . (Con... |
| tz9.12lem3 9212 | Lemma for ~ tz9.12 . (Con... |
| tz9.12 9213 | A set is well-founded if a... |
| tz9.13 9214 | Every set is well-founded,... |
| tz9.13g 9215 | Every set is well-founded,... |
| rankwflemb 9216 | Two ways of saying a set i... |
| rankf 9217 | The domain and range of th... |
| rankon 9218 | The rank of a set is an or... |
| r1elwf 9219 | Any member of the cumulati... |
| rankvalb 9220 | Value of the rank function... |
| rankr1ai 9221 | One direction of ~ rankr1a... |
| rankvaln 9222 | Value of the rank function... |
| rankidb 9223 | Identity law for the rank ... |
| rankdmr1 9224 | A rank is a member of the ... |
| rankr1ag 9225 | A version of ~ rankr1a tha... |
| rankr1bg 9226 | A relationship between ran... |
| r1rankidb 9227 | Any set is a subset of the... |
| r1elssi 9228 | The range of the ` R1 ` fu... |
| r1elss 9229 | The range of the ` R1 ` fu... |
| pwwf 9230 | A power set is well-founde... |
| sswf 9231 | A subset of a well-founded... |
| snwf 9232 | A singleton is well-founde... |
| unwf 9233 | A binary union is well-fou... |
| prwf 9234 | An unordered pair is well-... |
| opwf 9235 | An ordered pair is well-fo... |
| unir1 9236 | The cumulative hierarchy o... |
| jech9.3 9237 | Every set belongs to some ... |
| rankwflem 9238 | Every set is well-founded,... |
| rankval 9239 | Value of the rank function... |
| rankvalg 9240 | Value of the rank function... |
| rankval2 9241 | Value of an alternate defi... |
| uniwf 9242 | A union is well-founded if... |
| rankr1clem 9243 | Lemma for ~ rankr1c . (Co... |
| rankr1c 9244 | A relationship between the... |
| rankidn 9245 | A relationship between the... |
| rankpwi 9246 | The rank of a power set. ... |
| rankelb 9247 | The membership relation is... |
| wfelirr 9248 | A well-founded set is not ... |
| rankval3b 9249 | The value of the rank func... |
| ranksnb 9250 | The rank of a singleton. ... |
| rankonidlem 9251 | Lemma for ~ rankonid . (C... |
| rankonid 9252 | The rank of an ordinal num... |
| onwf 9253 | The ordinals are all well-... |
| onssr1 9254 | Initial segments of the or... |
| rankr1g 9255 | A relationship between the... |
| rankid 9256 | Identity law for the rank ... |
| rankr1 9257 | A relationship between the... |
| ssrankr1 9258 | A relationship between an ... |
| rankr1a 9259 | A relationship between ran... |
| r1val2 9260 | The value of the cumulativ... |
| r1val3 9261 | The value of the cumulativ... |
| rankel 9262 | The membership relation is... |
| rankval3 9263 | The value of the rank func... |
| bndrank 9264 | Any class whose elements h... |
| unbndrank 9265 | The elements of a proper c... |
| rankpw 9266 | The rank of a power set. ... |
| ranklim 9267 | The rank of a set belongs ... |
| r1pw 9268 | A stronger property of ` R... |
| r1pwALT 9269 | Alternate shorter proof of... |
| r1pwcl 9270 | The cumulative hierarchy o... |
| rankssb 9271 | The subset relation is inh... |
| rankss 9272 | The subset relation is inh... |
| rankunb 9273 | The rank of the union of t... |
| rankprb 9274 | The rank of an unordered p... |
| rankopb 9275 | The rank of an ordered pai... |
| rankuni2b 9276 | The value of the rank func... |
| ranksn 9277 | The rank of a singleton. ... |
| rankuni2 9278 | The rank of a union. Part... |
| rankun 9279 | The rank of the union of t... |
| rankpr 9280 | The rank of an unordered p... |
| rankop 9281 | The rank of an ordered pai... |
| r1rankid 9282 | Any set is a subset of the... |
| rankeq0b 9283 | A set is empty iff its ran... |
| rankeq0 9284 | A set is empty iff its ran... |
| rankr1id 9285 | The rank of the hierarchy ... |
| rankuni 9286 | The rank of a union. Part... |
| rankr1b 9287 | A relationship between ran... |
| ranksuc 9288 | The rank of a successor. ... |
| rankuniss 9289 | Upper bound of the rank of... |
| rankval4 9290 | The rank of a set is the s... |
| rankbnd 9291 | The rank of a set is bound... |
| rankbnd2 9292 | The rank of a set is bound... |
| rankc1 9293 | A relationship that can be... |
| rankc2 9294 | A relationship that can be... |
| rankelun 9295 | Rank membership is inherit... |
| rankelpr 9296 | Rank membership is inherit... |
| rankelop 9297 | Rank membership is inherit... |
| rankxpl 9298 | A lower bound on the rank ... |
| rankxpu 9299 | An upper bound on the rank... |
| rankfu 9300 | An upper bound on the rank... |
| rankmapu 9301 | An upper bound on the rank... |
| rankxplim 9302 | The rank of a Cartesian pr... |
| rankxplim2 9303 | If the rank of a Cartesian... |
| rankxplim3 9304 | The rank of a Cartesian pr... |
| rankxpsuc 9305 | The rank of a Cartesian pr... |
| tcwf 9306 | The transitive closure fun... |
| tcrank 9307 | This theorem expresses two... |
| scottex 9308 | Scott's trick collects all... |
| scott0 9309 | Scott's trick collects all... |
| scottexs 9310 | Theorem scheme version of ... |
| scott0s 9311 | Theorem scheme version of ... |
| cplem1 9312 | Lemma for the Collection P... |
| cplem2 9313 | Lemma for the Collection P... |
| cp 9314 | Collection Principle. Thi... |
| bnd 9315 | A very strong generalizati... |
| bnd2 9316 | A variant of the Boundedne... |
| kardex 9317 | The collection of all sets... |
| karden 9318 | If we allow the Axiom of R... |
| htalem 9319 | Lemma for defining an emul... |
| hta 9320 | A ZFC emulation of Hilbert... |
| djueq12 9327 | Equality theorem for disjo... |
| djueq1 9328 | Equality theorem for disjo... |
| djueq2 9329 | Equality theorem for disjo... |
| nfdju 9330 | Bound-variable hypothesis ... |
| djuex 9331 | The disjoint union of sets... |
| djuexb 9332 | The disjoint union of two ... |
| djulcl 9333 | Left closure of disjoint u... |
| djurcl 9334 | Right closure of disjoint ... |
| djulf1o 9335 | The left injection functio... |
| djurf1o 9336 | The right injection functi... |
| inlresf 9337 | The left injection restric... |
| inlresf1 9338 | The left injection restric... |
| inrresf 9339 | The right injection restri... |
| inrresf1 9340 | The right injection restri... |
| djuin 9341 | The images of any classes ... |
| djur 9342 | A member of a disjoint uni... |
| djuss 9343 | A disjoint union is a subc... |
| djuunxp 9344 | The union of a disjoint un... |
| djuexALT 9345 | Alternate proof of ~ djuex... |
| eldju1st 9346 | The first component of an ... |
| eldju2ndl 9347 | The second component of an... |
| eldju2ndr 9348 | The second component of an... |
| djuun 9349 | The disjoint union of two ... |
| 1stinl 9350 | The first component of the... |
| 2ndinl 9351 | The second component of th... |
| 1stinr 9352 | The first component of the... |
| 2ndinr 9353 | The second component of th... |
| updjudhf 9354 | The mapping of an element ... |
| updjudhcoinlf 9355 | The composition of the map... |
| updjudhcoinrg 9356 | The composition of the map... |
| updjud 9357 | Universal property of the ... |
| cardf2 9366 | The cardinality function i... |
| cardon 9367 | The cardinal number of a s... |
| isnum2 9368 | A way to express well-orde... |
| isnumi 9369 | A set equinumerous to an o... |
| ennum 9370 | Equinumerous sets are equi... |
| finnum 9371 | Every finite set is numera... |
| onenon 9372 | Every ordinal number is nu... |
| tskwe 9373 | A Tarski set is well-order... |
| xpnum 9374 | The cartesian product of n... |
| cardval3 9375 | An alternate definition of... |
| cardid2 9376 | Any numerable set is equin... |
| isnum3 9377 | A set is numerable iff it ... |
| oncardval 9378 | The value of the cardinal ... |
| oncardid 9379 | Any ordinal number is equi... |
| cardonle 9380 | The cardinal of an ordinal... |
| card0 9381 | The cardinality of the emp... |
| cardidm 9382 | The cardinality function i... |
| oncard 9383 | A set is a cardinal number... |
| ficardom 9384 | The cardinal number of a f... |
| ficardid 9385 | A finite set is equinumero... |
| cardnn 9386 | The cardinality of a natur... |
| cardnueq0 9387 | The empty set is the only ... |
| cardne 9388 | No member of a cardinal nu... |
| carden2a 9389 | If two sets have equal non... |
| carden2b 9390 | If two sets are equinumero... |
| card1 9391 | A set has cardinality one ... |
| cardsn 9392 | A singleton has cardinalit... |
| carddomi2 9393 | Two sets have the dominanc... |
| sdomsdomcardi 9394 | A set strictly dominates i... |
| cardlim 9395 | An infinite cardinal is a ... |
| cardsdomelir 9396 | A cardinal strictly domina... |
| cardsdomel 9397 | A cardinal strictly domina... |
| iscard 9398 | Two ways to express the pr... |
| iscard2 9399 | Two ways to express the pr... |
| carddom2 9400 | Two numerable sets have th... |
| harcard 9401 | The class of ordinal numbe... |
| cardprclem 9402 | Lemma for ~ cardprc . (Co... |
| cardprc 9403 | The class of all cardinal ... |
| carduni 9404 | The union of a set of card... |
| cardiun 9405 | The indexed union of a set... |
| cardennn 9406 | If ` A ` is equinumerous t... |
| cardsucinf 9407 | The cardinality of the suc... |
| cardsucnn 9408 | The cardinality of the suc... |
| cardom 9409 | The set of natural numbers... |
| carden2 9410 | Two numerable sets are equ... |
| cardsdom2 9411 | A numerable set is strictl... |
| domtri2 9412 | Trichotomy of dominance fo... |
| nnsdomel 9413 | Strict dominance and eleme... |
| cardval2 9414 | An alternate version of th... |
| isinffi 9415 | An infinite set contains s... |
| fidomtri 9416 | Trichotomy of dominance wi... |
| fidomtri2 9417 | Trichotomy of dominance wi... |
| harsdom 9418 | The Hartogs number of a we... |
| onsdom 9419 | Any well-orderable set is ... |
| harval2 9420 | An alternate expression fo... |
| cardmin2 9421 | The smallest ordinal that ... |
| pm54.43lem 9422 | In Theorem *54.43 of [Whit... |
| pm54.43 9423 | Theorem *54.43 of [Whitehe... |
| pr2nelem 9424 | Lemma for ~ pr2ne . (Cont... |
| pr2ne 9425 | If an unordered pair has t... |
| prdom2 9426 | An unordered pair has at m... |
| en2eqpr 9427 | Building a set with two el... |
| en2eleq 9428 | Express a set of pair card... |
| en2other2 9429 | Taking the other element t... |
| dif1card 9430 | The cardinality of a nonem... |
| leweon 9431 | Lexicographical order is a... |
| r0weon 9432 | A set-like well-ordering o... |
| infxpenlem 9433 | Lemma for ~ infxpen . (Co... |
| infxpen 9434 | Every infinite ordinal is ... |
| xpomen 9435 | The Cartesian product of o... |
| xpct 9436 | The cartesian product of t... |
| infxpidm2 9437 | The Cartesian product of a... |
| infxpenc 9438 | A canonical version of ~ i... |
| infxpenc2lem1 9439 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem2 9440 | Lemma for ~ infxpenc2 . (... |
| infxpenc2lem3 9441 | Lemma for ~ infxpenc2 . (... |
| infxpenc2 9442 | Existence form of ~ infxpe... |
| iunmapdisj 9443 | The union ` U_ n e. C ( A ... |
| fseqenlem1 9444 | Lemma for ~ fseqen . (Con... |
| fseqenlem2 9445 | Lemma for ~ fseqen . (Con... |
| fseqdom 9446 | One half of ~ fseqen . (C... |
| fseqen 9447 | A set that is equinumerous... |
| infpwfidom 9448 | The collection of finite s... |
| dfac8alem 9449 | Lemma for ~ dfac8a . If t... |
| dfac8a 9450 | Numeration theorem: every ... |
| dfac8b 9451 | The well-ordering theorem:... |
| dfac8clem 9452 | Lemma for ~ dfac8c . (Con... |
| dfac8c 9453 | If the union of a set is w... |
| ac10ct 9454 | A proof of the well-orderi... |
| ween 9455 | A set is numerable iff it ... |
| ac5num 9456 | A version of ~ ac5b with t... |
| ondomen 9457 | If a set is dominated by a... |
| numdom 9458 | A set dominated by a numer... |
| ssnum 9459 | A subset of a numerable se... |
| onssnum 9460 | All subsets of the ordinal... |
| indcardi 9461 | Indirect strong induction ... |
| acnrcl 9462 | Reverse closure for the ch... |
| acneq 9463 | Equality theorem for the c... |
| isacn 9464 | The property of being a ch... |
| acni 9465 | The property of being a ch... |
| acni2 9466 | The property of being a ch... |
| acni3 9467 | The property of being a ch... |
| acnlem 9468 | Construct a mapping satisf... |
| numacn 9469 | A well-orderable set has c... |
| finacn 9470 | Every set has finite choic... |
| acndom 9471 | A set with long choice seq... |
| acnnum 9472 | A set ` X ` which has choi... |
| acnen 9473 | The class of choice sets o... |
| acndom2 9474 | A set smaller than one wit... |
| acnen2 9475 | The class of sets with cho... |
| fodomacn 9476 | A version of ~ fodom that ... |
| fodomnum 9477 | A version of ~ fodom that ... |
| fonum 9478 | A surjection maps numerabl... |
| numwdom 9479 | A surjection maps numerabl... |
| fodomfi2 9480 | Onto functions define domi... |
| wdomfil 9481 | Weak dominance agrees with... |
| infpwfien 9482 | Any infinite well-orderabl... |
| inffien 9483 | The set of finite intersec... |
| wdomnumr 9484 | Weak dominance agrees with... |
| alephfnon 9485 | The aleph function is a fu... |
| aleph0 9486 | The first infinite cardina... |
| alephlim 9487 | Value of the aleph functio... |
| alephsuc 9488 | Value of the aleph functio... |
| alephon 9489 | An aleph is an ordinal num... |
| alephcard 9490 | Every aleph is a cardinal ... |
| alephnbtwn 9491 | No cardinal can be sandwic... |
| alephnbtwn2 9492 | No set has equinumerosity ... |
| alephordilem1 9493 | Lemma for ~ alephordi . (... |
| alephordi 9494 | Strict ordering property o... |
| alephord 9495 | Ordering property of the a... |
| alephord2 9496 | Ordering property of the a... |
| alephord2i 9497 | Ordering property of the a... |
| alephord3 9498 | Ordering property of the a... |
| alephsucdom 9499 | A set dominated by an alep... |
| alephsuc2 9500 | An alternate representatio... |
| alephdom 9501 | Relationship between inclu... |
| alephgeom 9502 | Every aleph is greater tha... |
| alephislim 9503 | Every aleph is a limit ord... |
| aleph11 9504 | The aleph function is one-... |
| alephf1 9505 | The aleph function is a on... |
| alephsdom 9506 | If an ordinal is smaller t... |
| alephdom2 9507 | A dominated initial ordina... |
| alephle 9508 | The argument of the aleph ... |
| cardaleph 9509 | Given any transfinite card... |
| cardalephex 9510 | Every transfinite cardinal... |
| infenaleph 9511 | An infinite numerable set ... |
| isinfcard 9512 | Two ways to express the pr... |
| iscard3 9513 | Two ways to express the pr... |
| cardnum 9514 | Two ways to express the cl... |
| alephinit 9515 | An infinite initial ordina... |
| carduniima 9516 | The union of the image of ... |
| cardinfima 9517 | If a mapping to cardinals ... |
| alephiso 9518 | Aleph is an order isomorph... |
| alephprc 9519 | The class of all transfini... |
| alephsson 9520 | The class of transfinite c... |
| unialeph 9521 | The union of the class of ... |
| alephsmo 9522 | The aleph function is stri... |
| alephf1ALT 9523 | Alternate proof of ~ aleph... |
| alephfplem1 9524 | Lemma for ~ alephfp . (Co... |
| alephfplem2 9525 | Lemma for ~ alephfp . (Co... |
| alephfplem3 9526 | Lemma for ~ alephfp . (Co... |
| alephfplem4 9527 | Lemma for ~ alephfp . (Co... |
| alephfp 9528 | The aleph function has a f... |
| alephfp2 9529 | The aleph function has at ... |
| alephval3 9530 | An alternate way to expres... |
| alephsucpw2 9531 | The power set of an aleph ... |
| mappwen 9532 | Power rule for cardinal ar... |
| finnisoeu 9533 | A finite totally ordered s... |
| iunfictbso 9534 | Countability of a countabl... |
| aceq1 9537 | Equivalence of two version... |
| aceq0 9538 | Equivalence of two version... |
| aceq2 9539 | Equivalence of two version... |
| aceq3lem 9540 | Lemma for ~ dfac3 . (Cont... |
| dfac3 9541 | Equivalence of two version... |
| dfac4 9542 | Equivalence of two version... |
| dfac5lem1 9543 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem2 9544 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem3 9545 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem4 9546 | Lemma for ~ dfac5 . (Cont... |
| dfac5lem5 9547 | Lemma for ~ dfac5 . (Cont... |
| dfac5 9548 | Equivalence of two version... |
| dfac2a 9549 | Our Axiom of Choice (in th... |
| dfac2b 9550 | Axiom of Choice (first for... |
| dfac2 9551 | Axiom of Choice (first for... |
| dfac7 9552 | Equivalence of the Axiom o... |
| dfac0 9553 | Equivalence of two version... |
| dfac1 9554 | Equivalence of two version... |
| dfac8 9555 | A proof of the equivalency... |
| dfac9 9556 | Equivalence of the axiom o... |
| dfac10 9557 | Axiom of Choice equivalent... |
| dfac10c 9558 | Axiom of Choice equivalent... |
| dfac10b 9559 | Axiom of Choice equivalent... |
| acacni 9560 | A choice equivalent: every... |
| dfacacn 9561 | A choice equivalent: every... |
| dfac13 9562 | The axiom of choice holds ... |
| dfac12lem1 9563 | Lemma for ~ dfac12 . (Con... |
| dfac12lem2 9564 | Lemma for ~ dfac12 . (Con... |
| dfac12lem3 9565 | Lemma for ~ dfac12 . (Con... |
| dfac12r 9566 | The axiom of choice holds ... |
| dfac12k 9567 | Equivalence of ~ dfac12 an... |
| dfac12a 9568 | The axiom of choice holds ... |
| dfac12 9569 | The axiom of choice holds ... |
| kmlem1 9570 | Lemma for 5-quantifier AC ... |
| kmlem2 9571 | Lemma for 5-quantifier AC ... |
| kmlem3 9572 | Lemma for 5-quantifier AC ... |
| kmlem4 9573 | Lemma for 5-quantifier AC ... |
| kmlem5 9574 | Lemma for 5-quantifier AC ... |
| kmlem6 9575 | Lemma for 5-quantifier AC ... |
| kmlem7 9576 | Lemma for 5-quantifier AC ... |
| kmlem8 9577 | Lemma for 5-quantifier AC ... |
| kmlem9 9578 | Lemma for 5-quantifier AC ... |
| kmlem10 9579 | Lemma for 5-quantifier AC ... |
| kmlem11 9580 | Lemma for 5-quantifier AC ... |
| kmlem12 9581 | Lemma for 5-quantifier AC ... |
| kmlem13 9582 | Lemma for 5-quantifier AC ... |
| kmlem14 9583 | Lemma for 5-quantifier AC ... |
| kmlem15 9584 | Lemma for 5-quantifier AC ... |
| kmlem16 9585 | Lemma for 5-quantifier AC ... |
| dfackm 9586 | Equivalence of the Axiom o... |
| undjudom 9587 | Cardinal addition dominate... |
| endjudisj 9588 | Equinumerosity of a disjoi... |
| djuen 9589 | Disjoint unions of equinum... |
| djuenun 9590 | Disjoint union is equinume... |
| dju1en 9591 | Cardinal addition with car... |
| dju1dif 9592 | Adding and subtracting one... |
| dju1p1e2 9593 | 1+1=2 for cardinal number ... |
| dju1p1e2ALT 9594 | Alternate proof of ~ dju1p... |
| dju0en 9595 | Cardinal addition with car... |
| xp2dju 9596 | Two times a cardinal numbe... |
| djucomen 9597 | Commutative law for cardin... |
| djuassen 9598 | Associative law for cardin... |
| xpdjuen 9599 | Cardinal multiplication di... |
| mapdjuen 9600 | Sum of exponents law for c... |
| pwdjuen 9601 | Sum of exponents law for c... |
| djudom1 9602 | Ordering law for cardinal ... |
| djudom2 9603 | Ordering law for cardinal ... |
| djudoml 9604 | A set is dominated by its ... |
| djuxpdom 9605 | Cartesian product dominate... |
| djufi 9606 | The disjoint union of two ... |
| cdainflem 9607 | Any partition of omega int... |
| djuinf 9608 | A set is infinite iff the ... |
| infdju1 9609 | An infinite set is equinum... |
| pwdju1 9610 | The sum of a powerset with... |
| pwdjuidm 9611 | If the natural numbers inj... |
| djulepw 9612 | If ` A ` is idempotent und... |
| onadju 9613 | The cardinal and ordinal s... |
| cardadju 9614 | The cardinal sum is equinu... |
| djunum 9615 | The disjoint union of two ... |
| unnum 9616 | The union of two numerable... |
| nnadju 9617 | The cardinal and ordinal s... |
| ficardun 9618 | The cardinality of the uni... |
| ficardun2 9619 | The cardinality of the uni... |
| pwsdompw 9620 | Lemma for ~ domtriom . Th... |
| unctb 9621 | The union of two countable... |
| infdjuabs 9622 | Absorption law for additio... |
| infunabs 9623 | An infinite set is equinum... |
| infdju 9624 | The sum of two cardinal nu... |
| infdif 9625 | The cardinality of an infi... |
| infdif2 9626 | Cardinality ordering for a... |
| infxpdom 9627 | Dominance law for multipli... |
| infxpabs 9628 | Absorption law for multipl... |
| infunsdom1 9629 | The union of two sets that... |
| infunsdom 9630 | The union of two sets that... |
| infxp 9631 | Absorption law for multipl... |
| pwdjudom 9632 | A property of dominance ov... |
| infpss 9633 | Every infinite set has an ... |
| infmap2 9634 | An exponentiation law for ... |
| ackbij2lem1 9635 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem1 9636 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem2 9637 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem3 9638 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem4 9639 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem5 9640 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem6 9641 | Lemma for ~ ackbij2 . (Co... |
| ackbij1lem7 9642 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem8 9643 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem9 9644 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem10 9645 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem11 9646 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem12 9647 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem13 9648 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem14 9649 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem15 9650 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem16 9651 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem17 9652 | Lemma for ~ ackbij1 . (Co... |
| ackbij1lem18 9653 | Lemma for ~ ackbij1 . (Co... |
| ackbij1 9654 | The Ackermann bijection, p... |
| ackbij1b 9655 | The Ackermann bijection, p... |
| ackbij2lem2 9656 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem3 9657 | Lemma for ~ ackbij2 . (Co... |
| ackbij2lem4 9658 | Lemma for ~ ackbij2 . (Co... |
| ackbij2 9659 | The Ackermann bijection, p... |
| r1om 9660 | The set of hereditarily fi... |
| fictb 9661 | A set is countable iff its... |
| cflem 9662 | A lemma used to simplify c... |
| cfval 9663 | Value of the cofinality fu... |
| cff 9664 | Cofinality is a function o... |
| cfub 9665 | An upper bound on cofinali... |
| cflm 9666 | Value of the cofinality fu... |
| cf0 9667 | Value of the cofinality fu... |
| cardcf 9668 | Cofinality is a cardinal n... |
| cflecard 9669 | Cofinality is bounded by t... |
| cfle 9670 | Cofinality is bounded by i... |
| cfon 9671 | The cofinality of any set ... |
| cfeq0 9672 | Only the ordinal zero has ... |
| cfsuc 9673 | Value of the cofinality fu... |
| cff1 9674 | There is always a map from... |
| cfflb 9675 | If there is a cofinal map ... |
| cfval2 9676 | Another expression for the... |
| coflim 9677 | A simpler expression for t... |
| cflim3 9678 | Another expression for the... |
| cflim2 9679 | The cofinality function is... |
| cfom 9680 | Value of the cofinality fu... |
| cfss 9681 | There is a cofinal subset ... |
| cfslb 9682 | Any cofinal subset of ` A ... |
| cfslbn 9683 | Any subset of ` A ` smalle... |
| cfslb2n 9684 | Any small collection of sm... |
| cofsmo 9685 | Any cofinal map implies th... |
| cfsmolem 9686 | Lemma for ~ cfsmo . (Cont... |
| cfsmo 9687 | The map in ~ cff1 can be a... |
| cfcoflem 9688 | Lemma for ~ cfcof , showin... |
| coftr 9689 | If there is a cofinal map ... |
| cfcof 9690 | If there is a cofinal map ... |
| cfidm 9691 | The cofinality function is... |
| alephsing 9692 | The cofinality of a limit ... |
| sornom 9693 | The range of a single-step... |
| isfin1a 9708 | Definition of a Ia-finite ... |
| fin1ai 9709 | Property of a Ia-finite se... |
| isfin2 9710 | Definition of a II-finite ... |
| fin2i 9711 | Property of a II-finite se... |
| isfin3 9712 | Definition of a III-finite... |
| isfin4 9713 | Definition of a IV-finite ... |
| fin4i 9714 | Infer that a set is IV-inf... |
| isfin5 9715 | Definition of a V-finite s... |
| isfin6 9716 | Definition of a VI-finite ... |
| isfin7 9717 | Definition of a VII-finite... |
| sdom2en01 9718 | A set with less than two e... |
| infpssrlem1 9719 | Lemma for ~ infpssr . (Co... |
| infpssrlem2 9720 | Lemma for ~ infpssr . (Co... |
| infpssrlem3 9721 | Lemma for ~ infpssr . (Co... |
| infpssrlem4 9722 | Lemma for ~ infpssr . (Co... |
| infpssrlem5 9723 | Lemma for ~ infpssr . (Co... |
| infpssr 9724 | Dedekind infinity implies ... |
| fin4en1 9725 | Dedekind finite is a cardi... |
| ssfin4 9726 | Dedekind finite sets have ... |
| domfin4 9727 | A set dominated by a Dedek... |
| ominf4 9728 | ` _om ` is Dedekind infini... |
| infpssALT 9729 | Alternate proof of ~ infps... |
| isfin4-2 9730 | Alternate definition of IV... |
| isfin4p1 9731 | Alternate definition of IV... |
| fin23lem7 9732 | Lemma for ~ isfin2-2 . Th... |
| fin23lem11 9733 | Lemma for ~ isfin2-2 . (C... |
| fin2i2 9734 | A II-finite set contains m... |
| isfin2-2 9735 | ` Fin2 ` expressed in term... |
| ssfin2 9736 | A subset of a II-finite se... |
| enfin2i 9737 | II-finiteness is a cardina... |
| fin23lem24 9738 | Lemma for ~ fin23 . In a ... |
| fincssdom 9739 | In a chain of finite sets,... |
| fin23lem25 9740 | Lemma for ~ fin23 . In a ... |
| fin23lem26 9741 | Lemma for ~ fin23lem22 . ... |
| fin23lem23 9742 | Lemma for ~ fin23lem22 . ... |
| fin23lem22 9743 | Lemma for ~ fin23 but coul... |
| fin23lem27 9744 | The mapping constructed in... |
| isfin3ds 9745 | Property of a III-finite s... |
| ssfin3ds 9746 | A subset of a III-finite s... |
| fin23lem12 9747 | The beginning of the proof... |
| fin23lem13 9748 | Lemma for ~ fin23 . Each ... |
| fin23lem14 9749 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem15 9750 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem16 9751 | Lemma for ~ fin23 . ` U ` ... |
| fin23lem19 9752 | Lemma for ~ fin23 . The f... |
| fin23lem20 9753 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem17 9754 | Lemma for ~ fin23 . By ? ... |
| fin23lem21 9755 | Lemma for ~ fin23 . ` X ` ... |
| fin23lem28 9756 | Lemma for ~ fin23 . The r... |
| fin23lem29 9757 | Lemma for ~ fin23 . The r... |
| fin23lem30 9758 | Lemma for ~ fin23 . The r... |
| fin23lem31 9759 | Lemma for ~ fin23 . The r... |
| fin23lem32 9760 | Lemma for ~ fin23 . Wrap ... |
| fin23lem33 9761 | Lemma for ~ fin23 . Disch... |
| fin23lem34 9762 | Lemma for ~ fin23 . Estab... |
| fin23lem35 9763 | Lemma for ~ fin23 . Stric... |
| fin23lem36 9764 | Lemma for ~ fin23 . Weak ... |
| fin23lem38 9765 | Lemma for ~ fin23 . The c... |
| fin23lem39 9766 | Lemma for ~ fin23 . Thus,... |
| fin23lem40 9767 | Lemma for ~ fin23 . ` Fin2... |
| fin23lem41 9768 | Lemma for ~ fin23 . A set... |
| isf32lem1 9769 | Lemma for ~ isfin3-2 . De... |
| isf32lem2 9770 | Lemma for ~ isfin3-2 . No... |
| isf32lem3 9771 | Lemma for ~ isfin3-2 . Be... |
| isf32lem4 9772 | Lemma for ~ isfin3-2 . Be... |
| isf32lem5 9773 | Lemma for ~ isfin3-2 . Th... |
| isf32lem6 9774 | Lemma for ~ isfin3-2 . Ea... |
| isf32lem7 9775 | Lemma for ~ isfin3-2 . Di... |
| isf32lem8 9776 | Lemma for ~ isfin3-2 . K ... |
| isf32lem9 9777 | Lemma for ~ isfin3-2 . Co... |
| isf32lem10 9778 | Lemma for isfin3-2 . Writ... |
| isf32lem11 9779 | Lemma for ~ isfin3-2 . Re... |
| isf32lem12 9780 | Lemma for ~ isfin3-2 . (C... |
| isfin32i 9781 | One half of ~ isfin3-2 . ... |
| isf33lem 9782 | Lemma for ~ isfin3-3 . (C... |
| isfin3-2 9783 | Weakly Dedekind-infinite s... |
| isfin3-3 9784 | Weakly Dedekind-infinite s... |
| fin33i 9785 | Inference from ~ isfin3-3 ... |
| compsscnvlem 9786 | Lemma for ~ compsscnv . (... |
| compsscnv 9787 | Complementation on a power... |
| isf34lem1 9788 | Lemma for ~ isfin3-4 . (C... |
| isf34lem2 9789 | Lemma for ~ isfin3-4 . (C... |
| compssiso 9790 | Complementation is an anti... |
| isf34lem3 9791 | Lemma for ~ isfin3-4 . (C... |
| compss 9792 | Express image under of the... |
| isf34lem4 9793 | Lemma for ~ isfin3-4 . (C... |
| isf34lem5 9794 | Lemma for ~ isfin3-4 . (C... |
| isf34lem7 9795 | Lemma for ~ isfin3-4 . (C... |
| isf34lem6 9796 | Lemma for ~ isfin3-4 . (C... |
| fin34i 9797 | Inference from ~ isfin3-4 ... |
| isfin3-4 9798 | Weakly Dedekind-infinite s... |
| fin11a 9799 | Every I-finite set is Ia-f... |
| enfin1ai 9800 | Ia-finiteness is a cardina... |
| isfin1-2 9801 | A set is finite in the usu... |
| isfin1-3 9802 | A set is I-finite iff ever... |
| isfin1-4 9803 | A set is I-finite iff ever... |
| dffin1-5 9804 | Compact quantifier-free ve... |
| fin23 9805 | Every II-finite set (every... |
| fin34 9806 | Every III-finite set is IV... |
| isfin5-2 9807 | Alternate definition of V-... |
| fin45 9808 | Every IV-finite set is V-f... |
| fin56 9809 | Every V-finite set is VI-f... |
| fin17 9810 | Every I-finite set is VII-... |
| fin67 9811 | Every VI-finite set is VII... |
| isfin7-2 9812 | A set is VII-finite iff it... |
| fin71num 9813 | A well-orderable set is VI... |
| dffin7-2 9814 | Class form of ~ isfin7-2 .... |
| dfacfin7 9815 | Axiom of Choice equivalent... |
| fin1a2lem1 9816 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem2 9817 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem3 9818 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem4 9819 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem5 9820 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem6 9821 | Lemma for ~ fin1a2 . Esta... |
| fin1a2lem7 9822 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem8 9823 | Lemma for ~ fin1a2 . Spli... |
| fin1a2lem9 9824 | Lemma for ~ fin1a2 . In a... |
| fin1a2lem10 9825 | Lemma for ~ fin1a2 . A no... |
| fin1a2lem11 9826 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem12 9827 | Lemma for ~ fin1a2 . (Con... |
| fin1a2lem13 9828 | Lemma for ~ fin1a2 . (Con... |
| fin12 9829 | Weak theorem which skips I... |
| fin1a2s 9830 | An II-infinite set can hav... |
| fin1a2 9831 | Every Ia-finite set is II-... |
| itunifval 9832 | Function value of iterated... |
| itunifn 9833 | Functionality of the itera... |
| ituni0 9834 | A zero-fold iterated union... |
| itunisuc 9835 | Successor iterated union. ... |
| itunitc1 9836 | Each union iterate is a me... |
| itunitc 9837 | The union of all union ite... |
| ituniiun 9838 | Unwrap an iterated union f... |
| hsmexlem7 9839 | Lemma for ~ hsmex . Prope... |
| hsmexlem8 9840 | Lemma for ~ hsmex . Prope... |
| hsmexlem9 9841 | Lemma for ~ hsmex . Prope... |
| hsmexlem1 9842 | Lemma for ~ hsmex . Bound... |
| hsmexlem2 9843 | Lemma for ~ hsmex . Bound... |
| hsmexlem3 9844 | Lemma for ~ hsmex . Clear... |
| hsmexlem4 9845 | Lemma for ~ hsmex . The c... |
| hsmexlem5 9846 | Lemma for ~ hsmex . Combi... |
| hsmexlem6 9847 | Lemma for ~ hsmex . (Cont... |
| hsmex 9848 | The collection of heredita... |
| hsmex2 9849 | The set of hereditary size... |
| hsmex3 9850 | The set of hereditary size... |
| axcc2lem 9852 | Lemma for ~ axcc2 . (Cont... |
| axcc2 9853 | A possibly more useful ver... |
| axcc3 9854 | A possibly more useful ver... |
| axcc4 9855 | A version of ~ axcc3 that ... |
| acncc 9856 | An ~ ax-cc equivalent: eve... |
| axcc4dom 9857 | Relax the constraint on ~ ... |
| domtriomlem 9858 | Lemma for ~ domtriom . (C... |
| domtriom 9859 | Trichotomy of equinumerosi... |
| fin41 9860 | Under countable choice, th... |
| dominf 9861 | A nonempty set that is a s... |
| dcomex 9863 | The Axiom of Dependent Cho... |
| axdc2lem 9864 | Lemma for ~ axdc2 . We co... |
| axdc2 9865 | An apparent strengthening ... |
| axdc3lem 9866 | The class ` S ` of finite ... |
| axdc3lem2 9867 | Lemma for ~ axdc3 . We ha... |
| axdc3lem3 9868 | Simple substitution lemma ... |
| axdc3lem4 9869 | Lemma for ~ axdc3 . We ha... |
| axdc3 9870 | Dependent Choice. Axiom D... |
| axdc4lem 9871 | Lemma for ~ axdc4 . (Cont... |
| axdc4 9872 | A more general version of ... |
| axcclem 9873 | Lemma for ~ axcc . (Contr... |
| axcc 9874 | Although CC can be proven ... |
| zfac 9876 | Axiom of Choice expressed ... |
| ac2 9877 | Axiom of Choice equivalent... |
| ac3 9878 | Axiom of Choice using abbr... |
| axac3 9880 | This theorem asserts that ... |
| ackm 9881 | A remarkable equivalent to... |
| axac2 9882 | Derive ~ ax-ac2 from ~ ax-... |
| axac 9883 | Derive ~ ax-ac from ~ ax-a... |
| axaci 9884 | Apply a choice equivalent.... |
| cardeqv 9885 | All sets are well-orderabl... |
| numth3 9886 | All sets are well-orderabl... |
| numth2 9887 | Numeration theorem: any se... |
| numth 9888 | Numeration theorem: every ... |
| ac7 9889 | An Axiom of Choice equival... |
| ac7g 9890 | An Axiom of Choice equival... |
| ac4 9891 | Equivalent of Axiom of Cho... |
| ac4c 9892 | Equivalent of Axiom of Cho... |
| ac5 9893 | An Axiom of Choice equival... |
| ac5b 9894 | Equivalent of Axiom of Cho... |
| ac6num 9895 | A version of ~ ac6 which t... |
| ac6 9896 | Equivalent of Axiom of Cho... |
| ac6c4 9897 | Equivalent of Axiom of Cho... |
| ac6c5 9898 | Equivalent of Axiom of Cho... |
| ac9 9899 | An Axiom of Choice equival... |
| ac6s 9900 | Equivalent of Axiom of Cho... |
| ac6n 9901 | Equivalent of Axiom of Cho... |
| ac6s2 9902 | Generalization of the Axio... |
| ac6s3 9903 | Generalization of the Axio... |
| ac6sg 9904 | ~ ac6s with sethood as ant... |
| ac6sf 9905 | Version of ~ ac6 with boun... |
| ac6s4 9906 | Generalization of the Axio... |
| ac6s5 9907 | Generalization of the Axio... |
| ac8 9908 | An Axiom of Choice equival... |
| ac9s 9909 | An Axiom of Choice equival... |
| numthcor 9910 | Any set is strictly domina... |
| weth 9911 | Well-ordering theorem: any... |
| zorn2lem1 9912 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem2 9913 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem3 9914 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem4 9915 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem5 9916 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem6 9917 | Lemma for ~ zorn2 . (Cont... |
| zorn2lem7 9918 | Lemma for ~ zorn2 . (Cont... |
| zorn2g 9919 | Zorn's Lemma of [Monk1] p.... |
| zorng 9920 | Zorn's Lemma. If the unio... |
| zornn0g 9921 | Variant of Zorn's lemma ~ ... |
| zorn2 9922 | Zorn's Lemma of [Monk1] p.... |
| zorn 9923 | Zorn's Lemma. If the unio... |
| zornn0 9924 | Variant of Zorn's lemma ~ ... |
| ttukeylem1 9925 | Lemma for ~ ttukey . Expa... |
| ttukeylem2 9926 | Lemma for ~ ttukey . A pr... |
| ttukeylem3 9927 | Lemma for ~ ttukey . (Con... |
| ttukeylem4 9928 | Lemma for ~ ttukey . (Con... |
| ttukeylem5 9929 | Lemma for ~ ttukey . The ... |
| ttukeylem6 9930 | Lemma for ~ ttukey . (Con... |
| ttukeylem7 9931 | Lemma for ~ ttukey . (Con... |
| ttukey2g 9932 | The Teichmüller-Tukey... |
| ttukeyg 9933 | The Teichmüller-Tukey... |
| ttukey 9934 | The Teichmüller-Tukey... |
| axdclem 9935 | Lemma for ~ axdc . (Contr... |
| axdclem2 9936 | Lemma for ~ axdc . Using ... |
| axdc 9937 | This theorem derives ~ ax-... |
| fodom 9938 | An onto function implies d... |
| fodomg 9939 | An onto function implies d... |
| dmct 9940 | The domain of a countable ... |
| rnct 9941 | The range of a countable s... |
| fodomb 9942 | Equivalence of an onto map... |
| wdomac 9943 | When assuming AC, weak and... |
| brdom3 9944 | Equivalence to a dominance... |
| brdom5 9945 | An equivalence to a domina... |
| brdom4 9946 | An equivalence to a domina... |
| brdom7disj 9947 | An equivalence to a domina... |
| brdom6disj 9948 | An equivalence to a domina... |
| fin71ac 9949 | Once we allow AC, the "str... |
| imadomg 9950 | An image of a function und... |
| fimact 9951 | The image by a function of... |
| fnrndomg 9952 | The range of a function is... |
| fnct 9953 | If the domain of a functio... |
| mptct 9954 | A countable mapping set is... |
| iunfo 9955 | Existence of an onto funct... |
| iundom2g 9956 | An upper bound for the car... |
| iundomg 9957 | An upper bound for the car... |
| iundom 9958 | An upper bound for the car... |
| unidom 9959 | An upper bound for the car... |
| uniimadom 9960 | An upper bound for the car... |
| uniimadomf 9961 | An upper bound for the car... |
| cardval 9962 | The value of the cardinal ... |
| cardid 9963 | Any set is equinumerous to... |
| cardidg 9964 | Any set is equinumerous to... |
| cardidd 9965 | Any set is equinumerous to... |
| cardf 9966 | The cardinality function i... |
| carden 9967 | Two sets are equinumerous ... |
| cardeq0 9968 | Only the empty set has car... |
| unsnen 9969 | Equinumerosity of a set wi... |
| carddom 9970 | Two sets have the dominanc... |
| cardsdom 9971 | Two sets have the strict d... |
| domtri 9972 | Trichotomy law for dominan... |
| entric 9973 | Trichotomy of equinumerosi... |
| entri2 9974 | Trichotomy of dominance an... |
| entri3 9975 | Trichotomy of dominance. ... |
| sdomsdomcard 9976 | A set strictly dominates i... |
| canth3 9977 | Cantor's theorem in terms ... |
| infxpidm 9978 | The Cartesian product of a... |
| ondomon 9979 | The collection of ordinal ... |
| cardmin 9980 | The smallest ordinal that ... |
| ficard 9981 | A set is finite iff its ca... |
| infinf 9982 | Equivalence between two in... |
| unirnfdomd 9983 | The union of the range of ... |
| konigthlem 9984 | Lemma for ~ konigth . (Co... |
| konigth 9985 | Konig's Theorem. If ` m (... |
| alephsucpw 9986 | The power set of an aleph ... |
| aleph1 9987 | The set exponentiation of ... |
| alephval2 9988 | An alternate way to expres... |
| dominfac 9989 | A nonempty set that is a s... |
| iunctb 9990 | The countable union of cou... |
| unictb 9991 | The countable union of cou... |
| infmap 9992 | An exponentiation law for ... |
| alephadd 9993 | The sum of two alephs is t... |
| alephmul 9994 | The product of two alephs ... |
| alephexp1 9995 | An exponentiation law for ... |
| alephsuc3 9996 | An alternate representatio... |
| alephexp2 9997 | An expression equinumerous... |
| alephreg 9998 | A successor aleph is regul... |
| pwcfsdom 9999 | A corollary of Konig's The... |
| cfpwsdom 10000 | A corollary of Konig's The... |
| alephom 10001 | From ~ canth2 , we know th... |
| smobeth 10002 | The beth function is stric... |
| nd1 10003 | A lemma for proving condit... |
| nd2 10004 | A lemma for proving condit... |
| nd3 10005 | A lemma for proving condit... |
| nd4 10006 | A lemma for proving condit... |
| axextnd 10007 | A version of the Axiom of ... |
| axrepndlem1 10008 | Lemma for the Axiom of Rep... |
| axrepndlem2 10009 | Lemma for the Axiom of Rep... |
| axrepnd 10010 | A version of the Axiom of ... |
| axunndlem1 10011 | Lemma for the Axiom of Uni... |
| axunnd 10012 | A version of the Axiom of ... |
| axpowndlem1 10013 | Lemma for the Axiom of Pow... |
| axpowndlem2 10014 | Lemma for the Axiom of Pow... |
| axpowndlem3 10015 | Lemma for the Axiom of Pow... |
| axpowndlem4 10016 | Lemma for the Axiom of Pow... |
| axpownd 10017 | A version of the Axiom of ... |
| axregndlem1 10018 | Lemma for the Axiom of Reg... |
| axregndlem2 10019 | Lemma for the Axiom of Reg... |
| axregnd 10020 | A version of the Axiom of ... |
| axinfndlem1 10021 | Lemma for the Axiom of Inf... |
| axinfnd 10022 | A version of the Axiom of ... |
| axacndlem1 10023 | Lemma for the Axiom of Cho... |
| axacndlem2 10024 | Lemma for the Axiom of Cho... |
| axacndlem3 10025 | Lemma for the Axiom of Cho... |
| axacndlem4 10026 | Lemma for the Axiom of Cho... |
| axacndlem5 10027 | Lemma for the Axiom of Cho... |
| axacnd 10028 | A version of the Axiom of ... |
| zfcndext 10029 | Axiom of Extensionality ~ ... |
| zfcndrep 10030 | Axiom of Replacement ~ ax-... |
| zfcndun 10031 | Axiom of Union ~ ax-un , r... |
| zfcndpow 10032 | Axiom of Power Sets ~ ax-p... |
| zfcndreg 10033 | Axiom of Regularity ~ ax-r... |
| zfcndinf 10034 | Axiom of Infinity ~ ax-inf... |
| zfcndac 10035 | Axiom of Choice ~ ax-ac , ... |
| elgch 10038 | Elementhood in the collect... |
| fingch 10039 | A finite set is a GCH-set.... |
| gchi 10040 | The only GCH-sets which ha... |
| gchen1 10041 | If ` A <_ B < ~P A ` , and... |
| gchen2 10042 | If ` A < B <_ ~P A ` , and... |
| gchor 10043 | If ` A <_ B <_ ~P A ` , an... |
| engch 10044 | The property of being a GC... |
| gchdomtri 10045 | Under certain conditions, ... |
| fpwwe2cbv 10046 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem1 10047 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem2 10048 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem3 10049 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem5 10050 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem6 10051 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem7 10052 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem8 10053 | Lemma for ~ fpwwe2 . Show... |
| fpwwe2lem9 10054 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem10 10055 | Lemma for ~ fpwwe2 . Give... |
| fpwwe2lem11 10056 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem12 10057 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2lem13 10058 | Lemma for ~ fpwwe2 . (Con... |
| fpwwe2 10059 | Given any function ` F ` f... |
| fpwwecbv 10060 | Lemma for ~ fpwwe . (Cont... |
| fpwwelem 10061 | Lemma for ~ fpwwe . (Cont... |
| fpwwe 10062 | Given any function ` F ` f... |
| canth4 10063 | An "effective" form of Can... |
| canthnumlem 10064 | Lemma for ~ canthnum . (C... |
| canthnum 10065 | The set of well-orderable ... |
| canthwelem 10066 | Lemma for ~ canthwe . (Co... |
| canthwe 10067 | The set of well-orders of ... |
| canthp1lem1 10068 | Lemma for ~ canthp1 . (Co... |
| canthp1lem2 10069 | Lemma for ~ canthp1 . (Co... |
| canthp1 10070 | A slightly stronger form o... |
| finngch 10071 | The exclusion of finite se... |
| gchdju1 10072 | An infinite GCH-set is ide... |
| gchinf 10073 | An infinite GCH-set is Ded... |
| pwfseqlem1 10074 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem2 10075 | Lemma for ~ pwfseq . (Con... |
| pwfseqlem3 10076 | Lemma for ~ pwfseq . Usin... |
| pwfseqlem4a 10077 | Lemma for ~ pwfseqlem4 . ... |
| pwfseqlem4 10078 | Lemma for ~ pwfseq . Deri... |
| pwfseqlem5 10079 | Lemma for ~ pwfseq . Alth... |
| pwfseq 10080 | The powerset of a Dedekind... |
| pwxpndom2 10081 | The powerset of a Dedekind... |
| pwxpndom 10082 | The powerset of a Dedekind... |
| pwdjundom 10083 | The powerset of a Dedekind... |
| gchdjuidm 10084 | An infinite GCH-set is ide... |
| gchxpidm 10085 | An infinite GCH-set is ide... |
| gchpwdom 10086 | A relationship between dom... |
| gchaleph 10087 | If ` ( aleph `` A ) ` is a... |
| gchaleph2 10088 | If ` ( aleph `` A ) ` and ... |
| hargch 10089 | If ` A + ~~ ~P A ` , then ... |
| alephgch 10090 | If ` ( aleph `` suc A ) ` ... |
| gch2 10091 | It is sufficient to requir... |
| gch3 10092 | An equivalent formulation ... |
| gch-kn 10093 | The equivalence of two ver... |
| gchaclem 10094 | Lemma for ~ gchac (obsolet... |
| gchhar 10095 | A "local" form of ~ gchac ... |
| gchacg 10096 | A "local" form of ~ gchac ... |
| gchac 10097 | The Generalized Continuum ... |
| elwina 10102 | Conditions of weak inacces... |
| elina 10103 | Conditions of strong inacc... |
| winaon 10104 | A weakly inaccessible card... |
| inawinalem 10105 | Lemma for ~ inawina . (Co... |
| inawina 10106 | Every strongly inaccessibl... |
| omina 10107 | ` _om ` is a strongly inac... |
| winacard 10108 | A weakly inaccessible card... |
| winainflem 10109 | A weakly inaccessible card... |
| winainf 10110 | A weakly inaccessible card... |
| winalim 10111 | A weakly inaccessible card... |
| winalim2 10112 | A nontrivial weakly inacce... |
| winafp 10113 | A nontrivial weakly inacce... |
| winafpi 10114 | This theorem, which states... |
| gchina 10115 | Assuming the GCH, weakly a... |
| iswun 10120 | Properties of a weak unive... |
| wuntr 10121 | A weak universe is transit... |
| wununi 10122 | A weak universe is closed ... |
| wunpw 10123 | A weak universe is closed ... |
| wunelss 10124 | The elements of a weak uni... |
| wunpr 10125 | A weak universe is closed ... |
| wunun 10126 | A weak universe is closed ... |
| wuntp 10127 | A weak universe is closed ... |
| wunss 10128 | A weak universe is closed ... |
| wunin 10129 | A weak universe is closed ... |
| wundif 10130 | A weak universe is closed ... |
| wunint 10131 | A weak universe is closed ... |
| wunsn 10132 | A weak universe is closed ... |
| wunsuc 10133 | A weak universe is closed ... |
| wun0 10134 | A weak universe contains t... |
| wunr1om 10135 | A weak universe is infinit... |
| wunom 10136 | A weak universe contains a... |
| wunfi 10137 | A weak universe contains a... |
| wunop 10138 | A weak universe is closed ... |
| wunot 10139 | A weak universe is closed ... |
| wunxp 10140 | A weak universe is closed ... |
| wunpm 10141 | A weak universe is closed ... |
| wunmap 10142 | A weak universe is closed ... |
| wunf 10143 | A weak universe is closed ... |
| wundm 10144 | A weak universe is closed ... |
| wunrn 10145 | A weak universe is closed ... |
| wuncnv 10146 | A weak universe is closed ... |
| wunres 10147 | A weak universe is closed ... |
| wunfv 10148 | A weak universe is closed ... |
| wunco 10149 | A weak universe is closed ... |
| wuntpos 10150 | A weak universe is closed ... |
| intwun 10151 | The intersection of a coll... |
| r1limwun 10152 | Each limit stage in the cu... |
| r1wunlim 10153 | The weak universes in the ... |
| wunex2 10154 | Construct a weak universe ... |
| wunex 10155 | Construct a weak universe ... |
| uniwun 10156 | Every set is contained in ... |
| wunex3 10157 | Construct a weak universe ... |
| wuncval 10158 | Value of the weak universe... |
| wuncid 10159 | The weak universe closure ... |
| wunccl 10160 | The weak universe closure ... |
| wuncss 10161 | The weak universe closure ... |
| wuncidm 10162 | The weak universe closure ... |
| wuncval2 10163 | Our earlier expression for... |
| eltskg 10166 | Properties of a Tarski cla... |
| eltsk2g 10167 | Properties of a Tarski cla... |
| tskpwss 10168 | First axiom of a Tarski cl... |
| tskpw 10169 | Second axiom of a Tarski c... |
| tsken 10170 | Third axiom of a Tarski cl... |
| 0tsk 10171 | The empty set is a (transi... |
| tsksdom 10172 | An element of a Tarski cla... |
| tskssel 10173 | A part of a Tarski class s... |
| tskss 10174 | The subsets of an element ... |
| tskin 10175 | The intersection of two el... |
| tsksn 10176 | A singleton of an element ... |
| tsktrss 10177 | A transitive element of a ... |
| tsksuc 10178 | If an element of a Tarski ... |
| tsk0 10179 | A nonempty Tarski class co... |
| tsk1 10180 | One is an element of a non... |
| tsk2 10181 | Two is an element of a non... |
| 2domtsk 10182 | If a Tarski class is not e... |
| tskr1om 10183 | A nonempty Tarski class is... |
| tskr1om2 10184 | A nonempty Tarski class co... |
| tskinf 10185 | A nonempty Tarski class is... |
| tskpr 10186 | If ` A ` and ` B ` are mem... |
| tskop 10187 | If ` A ` and ` B ` are mem... |
| tskxpss 10188 | A Cartesian product of two... |
| tskwe2 10189 | A Tarski class is well-ord... |
| inttsk 10190 | The intersection of a coll... |
| inar1 10191 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omALT 10192 | Alternate proof of ~ r1om ... |
| rankcf 10193 | Any set must be at least a... |
| inatsk 10194 | ` ( R1 `` A ) ` for ` A ` ... |
| r1omtsk 10195 | The set of hereditarily fi... |
| tskord 10196 | A Tarski class contains al... |
| tskcard 10197 | An even more direct relati... |
| r1tskina 10198 | There is a direct relation... |
| tskuni 10199 | The union of an element of... |
| tskwun 10200 | A nonempty transitive Tars... |
| tskint 10201 | The intersection of an ele... |
| tskun 10202 | The union of two elements ... |
| tskxp 10203 | The Cartesian product of t... |
| tskmap 10204 | Set exponentiation is an e... |
| tskurn 10205 | A transitive Tarski class ... |
| elgrug 10208 | Properties of a Grothendie... |
| grutr 10209 | A Grothendieck universe is... |
| gruelss 10210 | A Grothendieck universe is... |
| grupw 10211 | A Grothendieck universe co... |
| gruss 10212 | Any subset of an element o... |
| grupr 10213 | A Grothendieck universe co... |
| gruurn 10214 | A Grothendieck universe co... |
| gruiun 10215 | If ` B ( x ) ` is a family... |
| gruuni 10216 | A Grothendieck universe co... |
| grurn 10217 | A Grothendieck universe co... |
| gruima 10218 | A Grothendieck universe co... |
| gruel 10219 | Any element of an element ... |
| grusn 10220 | A Grothendieck universe co... |
| gruop 10221 | A Grothendieck universe co... |
| gruun 10222 | A Grothendieck universe co... |
| gruxp 10223 | A Grothendieck universe co... |
| grumap 10224 | A Grothendieck universe co... |
| gruixp 10225 | A Grothendieck universe co... |
| gruiin 10226 | A Grothendieck universe co... |
| gruf 10227 | A Grothendieck universe co... |
| gruen 10228 | A Grothendieck universe co... |
| gruwun 10229 | A nonempty Grothendieck un... |
| intgru 10230 | The intersection of a fami... |
| ingru 10231 | The intersection of a univ... |
| wfgru 10232 | The wellfounded part of a ... |
| grudomon 10233 | Each ordinal that is compa... |
| gruina 10234 | If a Grothendieck universe... |
| grur1a 10235 | A characterization of Grot... |
| grur1 10236 | A characterization of Grot... |
| grutsk1 10237 | Grothendieck universes are... |
| grutsk 10238 | Grothendieck universes are... |
| axgroth5 10240 | The Tarski-Grothendieck ax... |
| axgroth2 10241 | Alternate version of the T... |
| grothpw 10242 | Derive the Axiom of Power ... |
| grothpwex 10243 | Derive the Axiom of Power ... |
| axgroth6 10244 | The Tarski-Grothendieck ax... |
| grothomex 10245 | The Tarski-Grothendieck Ax... |
| grothac 10246 | The Tarski-Grothendieck Ax... |
| axgroth3 10247 | Alternate version of the T... |
| axgroth4 10248 | Alternate version of the T... |
| grothprimlem 10249 | Lemma for ~ grothprim . E... |
| grothprim 10250 | The Tarski-Grothendieck Ax... |
| grothtsk 10251 | The Tarski-Grothendieck Ax... |
| inaprc 10252 | An equivalent to the Tarsk... |
| tskmval 10255 | Value of our tarski map. ... |
| tskmid 10256 | The set ` A ` is an elemen... |
| tskmcl 10257 | A Tarski class that contai... |
| sstskm 10258 | Being a part of ` ( tarski... |
| eltskm 10259 | Belonging to ` ( tarskiMap... |
| elni 10292 | Membership in the class of... |
| elni2 10293 | Membership in the class of... |
| pinn 10294 | A positive integer is a na... |
| pion 10295 | A positive integer is an o... |
| piord 10296 | A positive integer is ordi... |
| niex 10297 | The class of positive inte... |
| 0npi 10298 | The empty set is not a pos... |
| 1pi 10299 | Ordinal 'one' is a positiv... |
| addpiord 10300 | Positive integer addition ... |
| mulpiord 10301 | Positive integer multiplic... |
| mulidpi 10302 | 1 is an identity element f... |
| ltpiord 10303 | Positive integer 'less tha... |
| ltsopi 10304 | Positive integer 'less tha... |
| ltrelpi 10305 | Positive integer 'less tha... |
| dmaddpi 10306 | Domain of addition on posi... |
| dmmulpi 10307 | Domain of multiplication o... |
| addclpi 10308 | Closure of addition of pos... |
| mulclpi 10309 | Closure of multiplication ... |
| addcompi 10310 | Addition of positive integ... |
| addasspi 10311 | Addition of positive integ... |
| mulcompi 10312 | Multiplication of positive... |
| mulasspi 10313 | Multiplication of positive... |
| distrpi 10314 | Multiplication of positive... |
| addcanpi 10315 | Addition cancellation law ... |
| mulcanpi 10316 | Multiplication cancellatio... |
| addnidpi 10317 | There is no identity eleme... |
| ltexpi 10318 | Ordering on positive integ... |
| ltapi 10319 | Ordering property of addit... |
| ltmpi 10320 | Ordering property of multi... |
| 1lt2pi 10321 | One is less than two (one ... |
| nlt1pi 10322 | No positive integer is les... |
| indpi 10323 | Principle of Finite Induct... |
| enqbreq 10335 | Equivalence relation for p... |
| enqbreq2 10336 | Equivalence relation for p... |
| enqer 10337 | The equivalence relation f... |
| enqex 10338 | The equivalence relation f... |
| nqex 10339 | The class of positive frac... |
| 0nnq 10340 | The empty set is not a pos... |
| elpqn 10341 | Each positive fraction is ... |
| ltrelnq 10342 | Positive fraction 'less th... |
| pinq 10343 | The representatives of pos... |
| 1nq 10344 | The positive fraction 'one... |
| nqereu 10345 | There is a unique element ... |
| nqerf 10346 | Corollary of ~ nqereu : th... |
| nqercl 10347 | Corollary of ~ nqereu : cl... |
| nqerrel 10348 | Any member of ` ( N. X. N.... |
| nqerid 10349 | Corollary of ~ nqereu : th... |
| enqeq 10350 | Corollary of ~ nqereu : if... |
| nqereq 10351 | The function ` /Q ` acts a... |
| addpipq2 10352 | Addition of positive fract... |
| addpipq 10353 | Addition of positive fract... |
| addpqnq 10354 | Addition of positive fract... |
| mulpipq2 10355 | Multiplication of positive... |
| mulpipq 10356 | Multiplication of positive... |
| mulpqnq 10357 | Multiplication of positive... |
| ordpipq 10358 | Ordering of positive fract... |
| ordpinq 10359 | Ordering of positive fract... |
| addpqf 10360 | Closure of addition on pos... |
| addclnq 10361 | Closure of addition on pos... |
| mulpqf 10362 | Closure of multiplication ... |
| mulclnq 10363 | Closure of multiplication ... |
| addnqf 10364 | Domain of addition on posi... |
| mulnqf 10365 | Domain of multiplication o... |
| addcompq 10366 | Addition of positive fract... |
| addcomnq 10367 | Addition of positive fract... |
| mulcompq 10368 | Multiplication of positive... |
| mulcomnq 10369 | Multiplication of positive... |
| adderpqlem 10370 | Lemma for ~ adderpq . (Co... |
| mulerpqlem 10371 | Lemma for ~ mulerpq . (Co... |
| adderpq 10372 | Addition is compatible wit... |
| mulerpq 10373 | Multiplication is compatib... |
| addassnq 10374 | Addition of positive fract... |
| mulassnq 10375 | Multiplication of positive... |
| mulcanenq 10376 | Lemma for distributive law... |
| distrnq 10377 | Multiplication of positive... |
| 1nqenq 10378 | The equivalence class of r... |
| mulidnq 10379 | Multiplication identity el... |
| recmulnq 10380 | Relationship between recip... |
| recidnq 10381 | A positive fraction times ... |
| recclnq 10382 | Closure law for positive f... |
| recrecnq 10383 | Reciprocal of reciprocal o... |
| dmrecnq 10384 | Domain of reciprocal on po... |
| ltsonq 10385 | 'Less than' is a strict or... |
| lterpq 10386 | Compatibility of ordering ... |
| ltanq 10387 | Ordering property of addit... |
| ltmnq 10388 | Ordering property of multi... |
| 1lt2nq 10389 | One is less than two (one ... |
| ltaddnq 10390 | The sum of two fractions i... |
| ltexnq 10391 | Ordering on positive fract... |
| halfnq 10392 | One-half of any positive f... |
| nsmallnq 10393 | The is no smallest positiv... |
| ltbtwnnq 10394 | There exists a number betw... |
| ltrnq 10395 | Ordering property of recip... |
| archnq 10396 | For any fraction, there is... |
| npex 10402 | The class of positive real... |
| elnp 10403 | Membership in positive rea... |
| elnpi 10404 | Membership in positive rea... |
| prn0 10405 | A positive real is not emp... |
| prpssnq 10406 | A positive real is a subse... |
| elprnq 10407 | A positive real is a set o... |
| 0npr 10408 | The empty set is not a pos... |
| prcdnq 10409 | A positive real is closed ... |
| prub 10410 | A positive fraction not in... |
| prnmax 10411 | A positive real has no lar... |
| npomex 10412 | A simplifying observation,... |
| prnmadd 10413 | A positive real has no lar... |
| ltrelpr 10414 | Positive real 'less than' ... |
| genpv 10415 | Value of general operation... |
| genpelv 10416 | Membership in value of gen... |
| genpprecl 10417 | Pre-closure law for genera... |
| genpdm 10418 | Domain of general operatio... |
| genpn0 10419 | The result of an operation... |
| genpss 10420 | The result of an operation... |
| genpnnp 10421 | The result of an operation... |
| genpcd 10422 | Downward closure of an ope... |
| genpnmax 10423 | An operation on positive r... |
| genpcl 10424 | Closure of an operation on... |
| genpass 10425 | Associativity of an operat... |
| plpv 10426 | Value of addition on posit... |
| mpv 10427 | Value of multiplication on... |
| dmplp 10428 | Domain of addition on posi... |
| dmmp 10429 | Domain of multiplication o... |
| nqpr 10430 | The canonical embedding of... |
| 1pr 10431 | The positive real number '... |
| addclprlem1 10432 | Lemma to prove downward cl... |
| addclprlem2 10433 | Lemma to prove downward cl... |
| addclpr 10434 | Closure of addition on pos... |
| mulclprlem 10435 | Lemma to prove downward cl... |
| mulclpr 10436 | Closure of multiplication ... |
| addcompr 10437 | Addition of positive reals... |
| addasspr 10438 | Addition of positive reals... |
| mulcompr 10439 | Multiplication of positive... |
| mulasspr 10440 | Multiplication of positive... |
| distrlem1pr 10441 | Lemma for distributive law... |
| distrlem4pr 10442 | Lemma for distributive law... |
| distrlem5pr 10443 | Lemma for distributive law... |
| distrpr 10444 | Multiplication of positive... |
| 1idpr 10445 | 1 is an identity element f... |
| ltprord 10446 | Positive real 'less than' ... |
| psslinpr 10447 | Proper subset is a linear ... |
| ltsopr 10448 | Positive real 'less than' ... |
| prlem934 10449 | Lemma 9-3.4 of [Gleason] p... |
| ltaddpr 10450 | The sum of two positive re... |
| ltaddpr2 10451 | The sum of two positive re... |
| ltexprlem1 10452 | Lemma for Proposition 9-3.... |
| ltexprlem2 10453 | Lemma for Proposition 9-3.... |
| ltexprlem3 10454 | Lemma for Proposition 9-3.... |
| ltexprlem4 10455 | Lemma for Proposition 9-3.... |
| ltexprlem5 10456 | Lemma for Proposition 9-3.... |
| ltexprlem6 10457 | Lemma for Proposition 9-3.... |
| ltexprlem7 10458 | Lemma for Proposition 9-3.... |
| ltexpri 10459 | Proposition 9-3.5(iv) of [... |
| ltaprlem 10460 | Lemma for Proposition 9-3.... |
| ltapr 10461 | Ordering property of addit... |
| addcanpr 10462 | Addition cancellation law ... |
| prlem936 10463 | Lemma 9-3.6 of [Gleason] p... |
| reclem2pr 10464 | Lemma for Proposition 9-3.... |
| reclem3pr 10465 | Lemma for Proposition 9-3.... |
| reclem4pr 10466 | Lemma for Proposition 9-3.... |
| recexpr 10467 | The reciprocal of a positi... |
| suplem1pr 10468 | The union of a nonempty, b... |
| suplem2pr 10469 | The union of a set of posi... |
| supexpr 10470 | The union of a nonempty, b... |
| enrer 10479 | The equivalence relation f... |
| nrex1 10480 | The class of signed reals ... |
| enrbreq 10481 | Equivalence relation for s... |
| enreceq 10482 | Equivalence class equality... |
| enrex 10483 | The equivalence relation f... |
| ltrelsr 10484 | Signed real 'less than' is... |
| addcmpblnr 10485 | Lemma showing compatibilit... |
| mulcmpblnrlem 10486 | Lemma used in lemma showin... |
| mulcmpblnr 10487 | Lemma showing compatibilit... |
| prsrlem1 10488 | Decomposing signed reals i... |
| addsrmo 10489 | There is at most one resul... |
| mulsrmo 10490 | There is at most one resul... |
| addsrpr 10491 | Addition of signed reals i... |
| mulsrpr 10492 | Multiplication of signed r... |
| ltsrpr 10493 | Ordering of signed reals i... |
| gt0srpr 10494 | Greater than zero in terms... |
| 0nsr 10495 | The empty set is not a sig... |
| 0r 10496 | The constant ` 0R ` is a s... |
| 1sr 10497 | The constant ` 1R ` is a s... |
| m1r 10498 | The constant ` -1R ` is a ... |
| addclsr 10499 | Closure of addition on sig... |
| mulclsr 10500 | Closure of multiplication ... |
| dmaddsr 10501 | Domain of addition on sign... |
| dmmulsr 10502 | Domain of multiplication o... |
| addcomsr 10503 | Addition of signed reals i... |
| addasssr 10504 | Addition of signed reals i... |
| mulcomsr 10505 | Multiplication of signed r... |
| mulasssr 10506 | Multiplication of signed r... |
| distrsr 10507 | Multiplication of signed r... |
| m1p1sr 10508 | Minus one plus one is zero... |
| m1m1sr 10509 | Minus one times minus one ... |
| ltsosr 10510 | Signed real 'less than' is... |
| 0lt1sr 10511 | 0 is less than 1 for signe... |
| 1ne0sr 10512 | 1 and 0 are distinct for s... |
| 0idsr 10513 | The signed real number 0 i... |
| 1idsr 10514 | 1 is an identity element f... |
| 00sr 10515 | A signed real times 0 is 0... |
| ltasr 10516 | Ordering property of addit... |
| pn0sr 10517 | A signed real plus its neg... |
| negexsr 10518 | Existence of negative sign... |
| recexsrlem 10519 | The reciprocal of a positi... |
| addgt0sr 10520 | The sum of two positive si... |
| mulgt0sr 10521 | The product of two positiv... |
| sqgt0sr 10522 | The square of a nonzero si... |
| recexsr 10523 | The reciprocal of a nonzer... |
| mappsrpr 10524 | Mapping from positive sign... |
| ltpsrpr 10525 | Mapping of order from posi... |
| map2psrpr 10526 | Equivalence for positive s... |
| supsrlem 10527 | Lemma for supremum theorem... |
| supsr 10528 | A nonempty, bounded set of... |
| opelcn 10545 | Ordered pair membership in... |
| opelreal 10546 | Ordered pair membership in... |
| elreal 10547 | Membership in class of rea... |
| elreal2 10548 | Ordered pair membership in... |
| 0ncn 10549 | The empty set is not a com... |
| ltrelre 10550 | 'Less than' is a relation ... |
| addcnsr 10551 | Addition of complex number... |
| mulcnsr 10552 | Multiplication of complex ... |
| eqresr 10553 | Equality of real numbers i... |
| addresr 10554 | Addition of real numbers i... |
| mulresr 10555 | Multiplication of real num... |
| ltresr 10556 | Ordering of real subset of... |
| ltresr2 10557 | Ordering of real subset of... |
| dfcnqs 10558 | Technical trick to permit ... |
| addcnsrec 10559 | Technical trick to permit ... |
| mulcnsrec 10560 | Technical trick to permit ... |
| axaddf 10561 | Addition is an operation o... |
| axmulf 10562 | Multiplication is an opera... |
| axcnex 10563 | The complex numbers form a... |
| axresscn 10564 | The real numbers are a sub... |
| ax1cn 10565 | 1 is a complex number. Ax... |
| axicn 10566 | ` _i ` is a complex number... |
| axaddcl 10567 | Closure law for addition o... |
| axaddrcl 10568 | Closure law for addition i... |
| axmulcl 10569 | Closure law for multiplica... |
| axmulrcl 10570 | Closure law for multiplica... |
| axmulcom 10571 | Multiplication of complex ... |
| axaddass 10572 | Addition of complex number... |
| axmulass 10573 | Multiplication of complex ... |
| axdistr 10574 | Distributive law for compl... |
| axi2m1 10575 | i-squared equals -1 (expre... |
| ax1ne0 10576 | 1 and 0 are distinct. Axi... |
| ax1rid 10577 | ` 1 ` is an identity eleme... |
| axrnegex 10578 | Existence of negative of r... |
| axrrecex 10579 | Existence of reciprocal of... |
| axcnre 10580 | A complex number can be ex... |
| axpre-lttri 10581 | Ordering on reals satisfie... |
| axpre-lttrn 10582 | Ordering on reals is trans... |
| axpre-ltadd 10583 | Ordering property of addit... |
| axpre-mulgt0 10584 | The product of two positiv... |
| axpre-sup 10585 | A nonempty, bounded-above ... |
| wuncn 10586 | A weak universe containing... |
| cnex 10612 | Alias for ~ ax-cnex . See... |
| addcl 10613 | Alias for ~ ax-addcl , for... |
| readdcl 10614 | Alias for ~ ax-addrcl , fo... |
| mulcl 10615 | Alias for ~ ax-mulcl , for... |
| remulcl 10616 | Alias for ~ ax-mulrcl , fo... |
| mulcom 10617 | Alias for ~ ax-mulcom , fo... |
| addass 10618 | Alias for ~ ax-addass , fo... |
| mulass 10619 | Alias for ~ ax-mulass , fo... |
| adddi 10620 | Alias for ~ ax-distr , for... |
| recn 10621 | A real number is a complex... |
| reex 10622 | The real numbers form a se... |
| reelprrecn 10623 | Reals are a subset of the ... |
| cnelprrecn 10624 | Complex numbers are a subs... |
| elimne0 10625 | Hypothesis for weak deduct... |
| adddir 10626 | Distributive law for compl... |
| 0cn 10627 | Zero is a complex number. ... |
| 0cnd 10628 | Zero is a complex number, ... |
| c0ex 10629 | Zero is a set. (Contribut... |
| 1cnd 10630 | One is a complex number, d... |
| 1ex 10631 | One is a set. (Contribute... |
| cnre 10632 | Alias for ~ ax-cnre , for ... |
| mulid1 10633 | The number 1 is an identit... |
| mulid2 10634 | Identity law for multiplic... |
| 1re 10635 | The number 1 is real. Thi... |
| 1red 10636 | The number 1 is real, dedu... |
| 0re 10637 | The number 0 is real. Rem... |
| 0red 10638 | The number 0 is real, dedu... |
| mulid1i 10639 | Identity law for multiplic... |
| mulid2i 10640 | Identity law for multiplic... |
| addcli 10641 | Closure law for addition. ... |
| mulcli 10642 | Closure law for multiplica... |
| mulcomi 10643 | Commutative law for multip... |
| mulcomli 10644 | Commutative law for multip... |
| addassi 10645 | Associative law for additi... |
| mulassi 10646 | Associative law for multip... |
| adddii 10647 | Distributive law (left-dis... |
| adddiri 10648 | Distributive law (right-di... |
| recni 10649 | A real number is a complex... |
| readdcli 10650 | Closure law for addition o... |
| remulcli 10651 | Closure law for multiplica... |
| mulid1d 10652 | Identity law for multiplic... |
| mulid2d 10653 | Identity law for multiplic... |
| addcld 10654 | Closure law for addition. ... |
| mulcld 10655 | Closure law for multiplica... |
| mulcomd 10656 | Commutative law for multip... |
| addassd 10657 | Associative law for additi... |
| mulassd 10658 | Associative law for multip... |
| adddid 10659 | Distributive law (left-dis... |
| adddird 10660 | Distributive law (right-di... |
| adddirp1d 10661 | Distributive law, plus 1 v... |
| joinlmuladdmuld 10662 | Join AB+CB into (A+C) on L... |
| recnd 10663 | Deduction from real number... |
| readdcld 10664 | Closure law for addition o... |
| remulcld 10665 | Closure law for multiplica... |
| pnfnre 10676 | Plus infinity is not a rea... |
| pnfnre2 10677 | Plus infinity is not a rea... |
| mnfnre 10678 | Minus infinity is not a re... |
| ressxr 10679 | The standard reals are a s... |
| rexpssxrxp 10680 | The Cartesian product of s... |
| rexr 10681 | A standard real is an exte... |
| 0xr 10682 | Zero is an extended real. ... |
| renepnf 10683 | No (finite) real equals pl... |
| renemnf 10684 | No real equals minus infin... |
| rexrd 10685 | A standard real is an exte... |
| renepnfd 10686 | No (finite) real equals pl... |
| renemnfd 10687 | No real equals minus infin... |
| pnfex 10688 | Plus infinity exists. (Co... |
| pnfxr 10689 | Plus infinity belongs to t... |
| pnfnemnf 10690 | Plus and minus infinity ar... |
| mnfnepnf 10691 | Minus and plus infinity ar... |
| mnfxr 10692 | Minus infinity belongs to ... |
| rexri 10693 | A standard real is an exte... |
| 1xr 10694 | ` 1 ` is an extended real ... |
| renfdisj 10695 | The reals and the infiniti... |
| ltrelxr 10696 | "Less than" is a relation ... |
| ltrel 10697 | "Less than" is a relation.... |
| lerelxr 10698 | "Less than or equal to" is... |
| lerel 10699 | "Less than or equal to" is... |
| xrlenlt 10700 | "Less than or equal to" ex... |
| xrlenltd 10701 | "Less than or equal to" ex... |
| xrltnle 10702 | "Less than" expressed in t... |
| xrnltled 10703 | "Not less than" implies "l... |
| ssxr 10704 | The three (non-exclusive) ... |
| ltxrlt 10705 | The standard less-than ` <... |
| axlttri 10706 | Ordering on reals satisfie... |
| axlttrn 10707 | Ordering on reals is trans... |
| axltadd 10708 | Ordering property of addit... |
| axmulgt0 10709 | The product of two positiv... |
| axsup 10710 | A nonempty, bounded-above ... |
| lttr 10711 | Alias for ~ axlttrn , for ... |
| mulgt0 10712 | The product of two positiv... |
| lenlt 10713 | 'Less than or equal to' ex... |
| ltnle 10714 | 'Less than' expressed in t... |
| ltso 10715 | 'Less than' is a strict or... |
| gtso 10716 | 'Greater than' is a strict... |
| lttri2 10717 | Consequence of trichotomy.... |
| lttri3 10718 | Trichotomy law for 'less t... |
| lttri4 10719 | Trichotomy law for 'less t... |
| letri3 10720 | Trichotomy law. (Contribu... |
| leloe 10721 | 'Less than or equal to' ex... |
| eqlelt 10722 | Equality in terms of 'less... |
| ltle 10723 | 'Less than' implies 'less ... |
| leltne 10724 | 'Less than or equal to' im... |
| lelttr 10725 | Transitive law. (Contribu... |
| ltletr 10726 | Transitive law. (Contribu... |
| ltleletr 10727 | Transitive law, weaker for... |
| letr 10728 | Transitive law. (Contribu... |
| ltnr 10729 | 'Less than' is irreflexive... |
| leid 10730 | 'Less than or equal to' is... |
| ltne 10731 | 'Less than' implies not eq... |
| ltnsym 10732 | 'Less than' is not symmetr... |
| ltnsym2 10733 | 'Less than' is antisymmetr... |
| letric 10734 | Trichotomy law. (Contribu... |
| ltlen 10735 | 'Less than' expressed in t... |
| eqle 10736 | Equality implies 'less tha... |
| eqled 10737 | Equality implies 'less tha... |
| ltadd2 10738 | Addition to both sides of ... |
| ne0gt0 10739 | A nonzero nonnegative numb... |
| lecasei 10740 | Ordering elimination by ca... |
| lelttric 10741 | Trichotomy law. (Contribu... |
| ltlecasei 10742 | Ordering elimination by ca... |
| ltnri 10743 | 'Less than' is irreflexive... |
| eqlei 10744 | Equality implies 'less tha... |
| eqlei2 10745 | Equality implies 'less tha... |
| gtneii 10746 | 'Less than' implies not eq... |
| ltneii 10747 | 'Greater than' implies not... |
| lttri2i 10748 | Consequence of trichotomy.... |
| lttri3i 10749 | Consequence of trichotomy.... |
| letri3i 10750 | Consequence of trichotomy.... |
| leloei 10751 | 'Less than or equal to' in... |
| ltleni 10752 | 'Less than' expressed in t... |
| ltnsymi 10753 | 'Less than' is not symmetr... |
| lenlti 10754 | 'Less than or equal to' in... |
| ltnlei 10755 | 'Less than' in terms of 'l... |
| ltlei 10756 | 'Less than' implies 'less ... |
| ltleii 10757 | 'Less than' implies 'less ... |
| ltnei 10758 | 'Less than' implies not eq... |
| letrii 10759 | Trichotomy law for 'less t... |
| lttri 10760 | 'Less than' is transitive.... |
| lelttri 10761 | 'Less than or equal to', '... |
| ltletri 10762 | 'Less than', 'less than or... |
| letri 10763 | 'Less than or equal to' is... |
| le2tri3i 10764 | Extended trichotomy law fo... |
| ltadd2i 10765 | Addition to both sides of ... |
| mulgt0i 10766 | The product of two positiv... |
| mulgt0ii 10767 | The product of two positiv... |
| ltnrd 10768 | 'Less than' is irreflexive... |
| gtned 10769 | 'Less than' implies not eq... |
| ltned 10770 | 'Greater than' implies not... |
| ne0gt0d 10771 | A nonzero nonnegative numb... |
| lttrid 10772 | Ordering on reals satisfie... |
| lttri2d 10773 | Consequence of trichotomy.... |
| lttri3d 10774 | Consequence of trichotomy.... |
| lttri4d 10775 | Trichotomy law for 'less t... |
| letri3d 10776 | Consequence of trichotomy.... |
| leloed 10777 | 'Less than or equal to' in... |
| eqleltd 10778 | Equality in terms of 'less... |
| ltlend 10779 | 'Less than' expressed in t... |
| lenltd 10780 | 'Less than or equal to' in... |
| ltnled 10781 | 'Less than' in terms of 'l... |
| ltled 10782 | 'Less than' implies 'less ... |
| ltnsymd 10783 | 'Less than' implies 'less ... |
| nltled 10784 | 'Not less than ' implies '... |
| lensymd 10785 | 'Less than or equal to' im... |
| letrid 10786 | Trichotomy law for 'less t... |
| leltned 10787 | 'Less than or equal to' im... |
| leneltd 10788 | 'Less than or equal to' an... |
| mulgt0d 10789 | The product of two positiv... |
| ltadd2d 10790 | Addition to both sides of ... |
| letrd 10791 | Transitive law deduction f... |
| lelttrd 10792 | Transitive law deduction f... |
| ltadd2dd 10793 | Addition to both sides of ... |
| ltletrd 10794 | Transitive law deduction f... |
| lttrd 10795 | Transitive law deduction f... |
| lelttrdi 10796 | If a number is less than a... |
| dedekind 10797 | The Dedekind cut theorem. ... |
| dedekindle 10798 | The Dedekind cut theorem, ... |
| mul12 10799 | Commutative/associative la... |
| mul32 10800 | Commutative/associative la... |
| mul31 10801 | Commutative/associative la... |
| mul4 10802 | Rearrangement of 4 factors... |
| mul4r 10803 | Rearrangement of 4 factors... |
| muladd11 10804 | A simple product of sums e... |
| 1p1times 10805 | Two times a number. (Cont... |
| peano2cn 10806 | A theorem for complex numb... |
| peano2re 10807 | A theorem for reals analog... |
| readdcan 10808 | Cancellation law for addit... |
| 00id 10809 | ` 0 ` is its own additive ... |
| mul02lem1 10810 | Lemma for ~ mul02 . If an... |
| mul02lem2 10811 | Lemma for ~ mul02 . Zero ... |
| mul02 10812 | Multiplication by ` 0 ` . ... |
| mul01 10813 | Multiplication by ` 0 ` . ... |
| addid1 10814 | ` 0 ` is an additive ident... |
| cnegex 10815 | Existence of the negative ... |
| cnegex2 10816 | Existence of a left invers... |
| addid2 10817 | ` 0 ` is a left identity f... |
| addcan 10818 | Cancellation law for addit... |
| addcan2 10819 | Cancellation law for addit... |
| addcom 10820 | Addition commutes. This u... |
| addid1i 10821 | ` 0 ` is an additive ident... |
| addid2i 10822 | ` 0 ` is a left identity f... |
| mul02i 10823 | Multiplication by 0. Theo... |
| mul01i 10824 | Multiplication by ` 0 ` . ... |
| addcomi 10825 | Addition commutes. Based ... |
| addcomli 10826 | Addition commutes. (Contr... |
| addcani 10827 | Cancellation law for addit... |
| addcan2i 10828 | Cancellation law for addit... |
| mul12i 10829 | Commutative/associative la... |
| mul32i 10830 | Commutative/associative la... |
| mul4i 10831 | Rearrangement of 4 factors... |
| mul02d 10832 | Multiplication by 0. Theo... |
| mul01d 10833 | Multiplication by ` 0 ` . ... |
| addid1d 10834 | ` 0 ` is an additive ident... |
| addid2d 10835 | ` 0 ` is a left identity f... |
| addcomd 10836 | Addition commutes. Based ... |
| addcand 10837 | Cancellation law for addit... |
| addcan2d 10838 | Cancellation law for addit... |
| addcanad 10839 | Cancelling a term on the l... |
| addcan2ad 10840 | Cancelling a term on the r... |
| addneintrd 10841 | Introducing a term on the ... |
| addneintr2d 10842 | Introducing a term on the ... |
| mul12d 10843 | Commutative/associative la... |
| mul32d 10844 | Commutative/associative la... |
| mul31d 10845 | Commutative/associative la... |
| mul4d 10846 | Rearrangement of 4 factors... |
| muladd11r 10847 | A simple product of sums e... |
| comraddd 10848 | Commute RHS addition, in d... |
| ltaddneg 10849 | Adding a negative number t... |
| ltaddnegr 10850 | Adding a negative number t... |
| add12 10851 | Commutative/associative la... |
| add32 10852 | Commutative/associative la... |
| add32r 10853 | Commutative/associative la... |
| add4 10854 | Rearrangement of 4 terms i... |
| add42 10855 | Rearrangement of 4 terms i... |
| add12i 10856 | Commutative/associative la... |
| add32i 10857 | Commutative/associative la... |
| add4i 10858 | Rearrangement of 4 terms i... |
| add42i 10859 | Rearrangement of 4 terms i... |
| add12d 10860 | Commutative/associative la... |
| add32d 10861 | Commutative/associative la... |
| add4d 10862 | Rearrangement of 4 terms i... |
| add42d 10863 | Rearrangement of 4 terms i... |
| 0cnALT 10868 | Alternate proof of ~ 0cn w... |
| 0cnALT2 10869 | Alternate proof of ~ 0cnAL... |
| negeu 10870 | Existential uniqueness of ... |
| subval 10871 | Value of subtraction, whic... |
| negeq 10872 | Equality theorem for negat... |
| negeqi 10873 | Equality inference for neg... |
| negeqd 10874 | Equality deduction for neg... |
| nfnegd 10875 | Deduction version of ~ nfn... |
| nfneg 10876 | Bound-variable hypothesis ... |
| csbnegg 10877 | Move class substitution in... |
| negex 10878 | A negative is a set. (Con... |
| subcl 10879 | Closure law for subtractio... |
| negcl 10880 | Closure law for negative. ... |
| negicn 10881 | ` -u _i ` is a complex num... |
| subf 10882 | Subtraction is an operatio... |
| subadd 10883 | Relationship between subtr... |
| subadd2 10884 | Relationship between subtr... |
| subsub23 10885 | Swap subtrahend and result... |
| pncan 10886 | Cancellation law for subtr... |
| pncan2 10887 | Cancellation law for subtr... |
| pncan3 10888 | Subtraction and addition o... |
| npcan 10889 | Cancellation law for subtr... |
| addsubass 10890 | Associative-type law for a... |
| addsub 10891 | Law for addition and subtr... |
| subadd23 10892 | Commutative/associative la... |
| addsub12 10893 | Commutative/associative la... |
| 2addsub 10894 | Law for subtraction and ad... |
| addsubeq4 10895 | Relation between sums and ... |
| pncan3oi 10896 | Subtraction and addition o... |
| mvrraddi 10897 | Move RHS right addition to... |
| mvlladdi 10898 | Move LHS left addition to ... |
| subid 10899 | Subtraction of a number fr... |
| subid1 10900 | Identity law for subtracti... |
| npncan 10901 | Cancellation law for subtr... |
| nppcan 10902 | Cancellation law for subtr... |
| nnpcan 10903 | Cancellation law for subtr... |
| nppcan3 10904 | Cancellation law for subtr... |
| subcan2 10905 | Cancellation law for subtr... |
| subeq0 10906 | If the difference between ... |
| npncan2 10907 | Cancellation law for subtr... |
| subsub2 10908 | Law for double subtraction... |
| nncan 10909 | Cancellation law for subtr... |
| subsub 10910 | Law for double subtraction... |
| nppcan2 10911 | Cancellation law for subtr... |
| subsub3 10912 | Law for double subtraction... |
| subsub4 10913 | Law for double subtraction... |
| sub32 10914 | Swap the second and third ... |
| nnncan 10915 | Cancellation law for subtr... |
| nnncan1 10916 | Cancellation law for subtr... |
| nnncan2 10917 | Cancellation law for subtr... |
| npncan3 10918 | Cancellation law for subtr... |
| pnpcan 10919 | Cancellation law for mixed... |
| pnpcan2 10920 | Cancellation law for mixed... |
| pnncan 10921 | Cancellation law for mixed... |
| ppncan 10922 | Cancellation law for mixed... |
| addsub4 10923 | Rearrangement of 4 terms i... |
| subadd4 10924 | Rearrangement of 4 terms i... |
| sub4 10925 | Rearrangement of 4 terms i... |
| neg0 10926 | Minus 0 equals 0. (Contri... |
| negid 10927 | Addition of a number and i... |
| negsub 10928 | Relationship between subtr... |
| subneg 10929 | Relationship between subtr... |
| negneg 10930 | A number is equal to the n... |
| neg11 10931 | Negative is one-to-one. (... |
| negcon1 10932 | Negative contraposition la... |
| negcon2 10933 | Negative contraposition la... |
| negeq0 10934 | A number is zero iff its n... |
| subcan 10935 | Cancellation law for subtr... |
| negsubdi 10936 | Distribution of negative o... |
| negdi 10937 | Distribution of negative o... |
| negdi2 10938 | Distribution of negative o... |
| negsubdi2 10939 | Distribution of negative o... |
| neg2sub 10940 | Relationship between subtr... |
| renegcli 10941 | Closure law for negative o... |
| resubcli 10942 | Closure law for subtractio... |
| renegcl 10943 | Closure law for negative o... |
| resubcl 10944 | Closure law for subtractio... |
| negreb 10945 | The negative of a real is ... |
| peano2cnm 10946 | "Reverse" second Peano pos... |
| peano2rem 10947 | "Reverse" second Peano pos... |
| negcli 10948 | Closure law for negative. ... |
| negidi 10949 | Addition of a number and i... |
| negnegi 10950 | A number is equal to the n... |
| subidi 10951 | Subtraction of a number fr... |
| subid1i 10952 | Identity law for subtracti... |
| negne0bi 10953 | A number is nonzero iff it... |
| negrebi 10954 | The negative of a real is ... |
| negne0i 10955 | The negative of a nonzero ... |
| subcli 10956 | Closure law for subtractio... |
| pncan3i 10957 | Subtraction and addition o... |
| negsubi 10958 | Relationship between subtr... |
| subnegi 10959 | Relationship between subtr... |
| subeq0i 10960 | If the difference between ... |
| neg11i 10961 | Negative is one-to-one. (... |
| negcon1i 10962 | Negative contraposition la... |
| negcon2i 10963 | Negative contraposition la... |
| negdii 10964 | Distribution of negative o... |
| negsubdii 10965 | Distribution of negative o... |
| negsubdi2i 10966 | Distribution of negative o... |
| subaddi 10967 | Relationship between subtr... |
| subadd2i 10968 | Relationship between subtr... |
| subaddrii 10969 | Relationship between subtr... |
| subsub23i 10970 | Swap subtrahend and result... |
| addsubassi 10971 | Associative-type law for s... |
| addsubi 10972 | Law for subtraction and ad... |
| subcani 10973 | Cancellation law for subtr... |
| subcan2i 10974 | Cancellation law for subtr... |
| pnncani 10975 | Cancellation law for mixed... |
| addsub4i 10976 | Rearrangement of 4 terms i... |
| 0reALT 10977 | Alternate proof of ~ 0re .... |
| negcld 10978 | Closure law for negative. ... |
| subidd 10979 | Subtraction of a number fr... |
| subid1d 10980 | Identity law for subtracti... |
| negidd 10981 | Addition of a number and i... |
| negnegd 10982 | A number is equal to the n... |
| negeq0d 10983 | A number is zero iff its n... |
| negne0bd 10984 | A number is nonzero iff it... |
| negcon1d 10985 | Contraposition law for una... |
| negcon1ad 10986 | Contraposition law for una... |
| neg11ad 10987 | The negatives of two compl... |
| negned 10988 | If two complex numbers are... |
| negne0d 10989 | The negative of a nonzero ... |
| negrebd 10990 | The negative of a real is ... |
| subcld 10991 | Closure law for subtractio... |
| pncand 10992 | Cancellation law for subtr... |
| pncan2d 10993 | Cancellation law for subtr... |
| pncan3d 10994 | Subtraction and addition o... |
| npcand 10995 | Cancellation law for subtr... |
| nncand 10996 | Cancellation law for subtr... |
| negsubd 10997 | Relationship between subtr... |
| subnegd 10998 | Relationship between subtr... |
| subeq0d 10999 | If the difference between ... |
| subne0d 11000 | Two unequal numbers have n... |
| subeq0ad 11001 | The difference of two comp... |
| subne0ad 11002 | If the difference of two c... |
| neg11d 11003 | If the difference between ... |
| negdid 11004 | Distribution of negative o... |
| negdi2d 11005 | Distribution of negative o... |
| negsubdid 11006 | Distribution of negative o... |
| negsubdi2d 11007 | Distribution of negative o... |
| neg2subd 11008 | Relationship between subtr... |
| subaddd 11009 | Relationship between subtr... |
| subadd2d 11010 | Relationship between subtr... |
| addsubassd 11011 | Associative-type law for s... |
| addsubd 11012 | Law for subtraction and ad... |
| subadd23d 11013 | Commutative/associative la... |
| addsub12d 11014 | Commutative/associative la... |
| npncand 11015 | Cancellation law for subtr... |
| nppcand 11016 | Cancellation law for subtr... |
| nppcan2d 11017 | Cancellation law for subtr... |
| nppcan3d 11018 | Cancellation law for subtr... |
| subsubd 11019 | Law for double subtraction... |
| subsub2d 11020 | Law for double subtraction... |
| subsub3d 11021 | Law for double subtraction... |
| subsub4d 11022 | Law for double subtraction... |
| sub32d 11023 | Swap the second and third ... |
| nnncand 11024 | Cancellation law for subtr... |
| nnncan1d 11025 | Cancellation law for subtr... |
| nnncan2d 11026 | Cancellation law for subtr... |
| npncan3d 11027 | Cancellation law for subtr... |
| pnpcand 11028 | Cancellation law for mixed... |
| pnpcan2d 11029 | Cancellation law for mixed... |
| pnncand 11030 | Cancellation law for mixed... |
| ppncand 11031 | Cancellation law for mixed... |
| subcand 11032 | Cancellation law for subtr... |
| subcan2d 11033 | Cancellation law for subtr... |
| subcanad 11034 | Cancellation law for subtr... |
| subneintrd 11035 | Introducing subtraction on... |
| subcan2ad 11036 | Cancellation law for subtr... |
| subneintr2d 11037 | Introducing subtraction on... |
| addsub4d 11038 | Rearrangement of 4 terms i... |
| subadd4d 11039 | Rearrangement of 4 terms i... |
| sub4d 11040 | Rearrangement of 4 terms i... |
| 2addsubd 11041 | Law for subtraction and ad... |
| addsubeq4d 11042 | Relation between sums and ... |
| subeqxfrd 11043 | Transfer two terms of a su... |
| mvlraddd 11044 | Move LHS right addition to... |
| mvlladdd 11045 | Move LHS left addition to ... |
| mvrraddd 11046 | Move RHS right addition to... |
| mvrladdd 11047 | Move RHS left addition to ... |
| assraddsubd 11048 | Associate RHS addition-sub... |
| subaddeqd 11049 | Transfer two terms of a su... |
| addlsub 11050 | Left-subtraction: Subtrac... |
| addrsub 11051 | Right-subtraction: Subtra... |
| subexsub 11052 | A subtraction law: Exchan... |
| addid0 11053 | If adding a number to a an... |
| addn0nid 11054 | Adding a nonzero number to... |
| pnpncand 11055 | Addition/subtraction cance... |
| subeqrev 11056 | Reverse the order of subtr... |
| addeq0 11057 | Two complex numbers add up... |
| pncan1 11058 | Cancellation law for addit... |
| npcan1 11059 | Cancellation law for subtr... |
| subeq0bd 11060 | If two complex numbers are... |
| renegcld 11061 | Closure law for negative o... |
| resubcld 11062 | Closure law for subtractio... |
| negn0 11063 | The image under negation o... |
| negf1o 11064 | Negation is an isomorphism... |
| kcnktkm1cn 11065 | k times k minus 1 is a com... |
| muladd 11066 | Product of two sums. (Con... |
| subdi 11067 | Distribution of multiplica... |
| subdir 11068 | Distribution of multiplica... |
| ine0 11069 | The imaginary unit ` _i ` ... |
| mulneg1 11070 | Product with negative is n... |
| mulneg2 11071 | The product with a negativ... |
| mulneg12 11072 | Swap the negative sign in ... |
| mul2neg 11073 | Product of two negatives. ... |
| submul2 11074 | Convert a subtraction to a... |
| mulm1 11075 | Product with minus one is ... |
| addneg1mul 11076 | Addition with product with... |
| mulsub 11077 | Product of two differences... |
| mulsub2 11078 | Swap the order of subtract... |
| mulm1i 11079 | Product with minus one is ... |
| mulneg1i 11080 | Product with negative is n... |
| mulneg2i 11081 | Product with negative is n... |
| mul2negi 11082 | Product of two negatives. ... |
| subdii 11083 | Distribution of multiplica... |
| subdiri 11084 | Distribution of multiplica... |
| muladdi 11085 | Product of two sums. (Con... |
| mulm1d 11086 | Product with minus one is ... |
| mulneg1d 11087 | Product with negative is n... |
| mulneg2d 11088 | Product with negative is n... |
| mul2negd 11089 | Product of two negatives. ... |
| subdid 11090 | Distribution of multiplica... |
| subdird 11091 | Distribution of multiplica... |
| muladdd 11092 | Product of two sums. (Con... |
| mulsubd 11093 | Product of two differences... |
| muls1d 11094 | Multiplication by one minu... |
| mulsubfacd 11095 | Multiplication followed by... |
| addmulsub 11096 | The product of a sum and a... |
| subaddmulsub 11097 | The difference with a prod... |
| mulsubaddmulsub 11098 | A special difference of a ... |
| gt0ne0 11099 | Positive implies nonzero. ... |
| lt0ne0 11100 | A number which is less tha... |
| ltadd1 11101 | Addition to both sides of ... |
| leadd1 11102 | Addition to both sides of ... |
| leadd2 11103 | Addition to both sides of ... |
| ltsubadd 11104 | 'Less than' relationship b... |
| ltsubadd2 11105 | 'Less than' relationship b... |
| lesubadd 11106 | 'Less than or equal to' re... |
| lesubadd2 11107 | 'Less than or equal to' re... |
| ltaddsub 11108 | 'Less than' relationship b... |
| ltaddsub2 11109 | 'Less than' relationship b... |
| leaddsub 11110 | 'Less than or equal to' re... |
| leaddsub2 11111 | 'Less than or equal to' re... |
| suble 11112 | Swap subtrahends in an ine... |
| lesub 11113 | Swap subtrahends in an ine... |
| ltsub23 11114 | 'Less than' relationship b... |
| ltsub13 11115 | 'Less than' relationship b... |
| le2add 11116 | Adding both sides of two '... |
| ltleadd 11117 | Adding both sides of two o... |
| leltadd 11118 | Adding both sides of two o... |
| lt2add 11119 | Adding both sides of two '... |
| addgt0 11120 | The sum of 2 positive numb... |
| addgegt0 11121 | The sum of nonnegative and... |
| addgtge0 11122 | The sum of nonnegative and... |
| addge0 11123 | The sum of 2 nonnegative n... |
| ltaddpos 11124 | Adding a positive number t... |
| ltaddpos2 11125 | Adding a positive number t... |
| ltsubpos 11126 | Subtracting a positive num... |
| posdif 11127 | Comparison of two numbers ... |
| lesub1 11128 | Subtraction from both side... |
| lesub2 11129 | Subtraction of both sides ... |
| ltsub1 11130 | Subtraction from both side... |
| ltsub2 11131 | Subtraction of both sides ... |
| lt2sub 11132 | Subtracting both sides of ... |
| le2sub 11133 | Subtracting both sides of ... |
| ltneg 11134 | Negative of both sides of ... |
| ltnegcon1 11135 | Contraposition of negative... |
| ltnegcon2 11136 | Contraposition of negative... |
| leneg 11137 | Negative of both sides of ... |
| lenegcon1 11138 | Contraposition of negative... |
| lenegcon2 11139 | Contraposition of negative... |
| lt0neg1 11140 | Comparison of a number and... |
| lt0neg2 11141 | Comparison of a number and... |
| le0neg1 11142 | Comparison of a number and... |
| le0neg2 11143 | Comparison of a number and... |
| addge01 11144 | A number is less than or e... |
| addge02 11145 | A number is less than or e... |
| add20 11146 | Two nonnegative numbers ar... |
| subge0 11147 | Nonnegative subtraction. ... |
| suble0 11148 | Nonpositive subtraction. ... |
| leaddle0 11149 | The sum of a real number a... |
| subge02 11150 | Nonnegative subtraction. ... |
| lesub0 11151 | Lemma to show a nonnegativ... |
| mulge0 11152 | The product of two nonnega... |
| mullt0 11153 | The product of two negativ... |
| msqgt0 11154 | A nonzero square is positi... |
| msqge0 11155 | A square is nonnegative. ... |
| 0lt1 11156 | 0 is less than 1. Theorem... |
| 0le1 11157 | 0 is less than or equal to... |
| relin01 11158 | An interval law for less t... |
| ltordlem 11159 | Lemma for ~ ltord1 . (Con... |
| ltord1 11160 | Infer an ordering relation... |
| leord1 11161 | Infer an ordering relation... |
| eqord1 11162 | A strictly increasing real... |
| ltord2 11163 | Infer an ordering relation... |
| leord2 11164 | Infer an ordering relation... |
| eqord2 11165 | A strictly decreasing real... |
| wloglei 11166 | Form of ~ wlogle where bot... |
| wlogle 11167 | If the predicate ` ch ( x ... |
| leidi 11168 | 'Less than or equal to' is... |
| gt0ne0i 11169 | Positive means nonzero (us... |
| gt0ne0ii 11170 | Positive implies nonzero. ... |
| msqgt0i 11171 | A nonzero square is positi... |
| msqge0i 11172 | A square is nonnegative. ... |
| addgt0i 11173 | Addition of 2 positive num... |
| addge0i 11174 | Addition of 2 nonnegative ... |
| addgegt0i 11175 | Addition of nonnegative an... |
| addgt0ii 11176 | Addition of 2 positive num... |
| add20i 11177 | Two nonnegative numbers ar... |
| ltnegi 11178 | Negative of both sides of ... |
| lenegi 11179 | Negative of both sides of ... |
| ltnegcon2i 11180 | Contraposition of negative... |
| mulge0i 11181 | The product of two nonnega... |
| lesub0i 11182 | Lemma to show a nonnegativ... |
| ltaddposi 11183 | Adding a positive number t... |
| posdifi 11184 | Comparison of two numbers ... |
| ltnegcon1i 11185 | Contraposition of negative... |
| lenegcon1i 11186 | Contraposition of negative... |
| subge0i 11187 | Nonnegative subtraction. ... |
| ltadd1i 11188 | Addition to both sides of ... |
| leadd1i 11189 | Addition to both sides of ... |
| leadd2i 11190 | Addition to both sides of ... |
| ltsubaddi 11191 | 'Less than' relationship b... |
| lesubaddi 11192 | 'Less than or equal to' re... |
| ltsubadd2i 11193 | 'Less than' relationship b... |
| lesubadd2i 11194 | 'Less than or equal to' re... |
| ltaddsubi 11195 | 'Less than' relationship b... |
| lt2addi 11196 | Adding both side of two in... |
| le2addi 11197 | Adding both side of two in... |
| gt0ne0d 11198 | Positive implies nonzero. ... |
| lt0ne0d 11199 | Something less than zero i... |
| leidd 11200 | 'Less than or equal to' is... |
| msqgt0d 11201 | A nonzero square is positi... |
| msqge0d 11202 | A square is nonnegative. ... |
| lt0neg1d 11203 | Comparison of a number and... |
| lt0neg2d 11204 | Comparison of a number and... |
| le0neg1d 11205 | Comparison of a number and... |
| le0neg2d 11206 | Comparison of a number and... |
| addgegt0d 11207 | Addition of nonnegative an... |
| addgtge0d 11208 | Addition of positive and n... |
| addgt0d 11209 | Addition of 2 positive num... |
| addge0d 11210 | Addition of 2 nonnegative ... |
| mulge0d 11211 | The product of two nonnega... |
| ltnegd 11212 | Negative of both sides of ... |
| lenegd 11213 | Negative of both sides of ... |
| ltnegcon1d 11214 | Contraposition of negative... |
| ltnegcon2d 11215 | Contraposition of negative... |
| lenegcon1d 11216 | Contraposition of negative... |
| lenegcon2d 11217 | Contraposition of negative... |
| ltaddposd 11218 | Adding a positive number t... |
| ltaddpos2d 11219 | Adding a positive number t... |
| ltsubposd 11220 | Subtracting a positive num... |
| posdifd 11221 | Comparison of two numbers ... |
| addge01d 11222 | A number is less than or e... |
| addge02d 11223 | A number is less than or e... |
| subge0d 11224 | Nonnegative subtraction. ... |
| suble0d 11225 | Nonpositive subtraction. ... |
| subge02d 11226 | Nonnegative subtraction. ... |
| ltadd1d 11227 | Addition to both sides of ... |
| leadd1d 11228 | Addition to both sides of ... |
| leadd2d 11229 | Addition to both sides of ... |
| ltsubaddd 11230 | 'Less than' relationship b... |
| lesubaddd 11231 | 'Less than or equal to' re... |
| ltsubadd2d 11232 | 'Less than' relationship b... |
| lesubadd2d 11233 | 'Less than or equal to' re... |
| ltaddsubd 11234 | 'Less than' relationship b... |
| ltaddsub2d 11235 | 'Less than' relationship b... |
| leaddsub2d 11236 | 'Less than or equal to' re... |
| subled 11237 | Swap subtrahends in an ine... |
| lesubd 11238 | Swap subtrahends in an ine... |
| ltsub23d 11239 | 'Less than' relationship b... |
| ltsub13d 11240 | 'Less than' relationship b... |
| lesub1d 11241 | Subtraction from both side... |
| lesub2d 11242 | Subtraction of both sides ... |
| ltsub1d 11243 | Subtraction from both side... |
| ltsub2d 11244 | Subtraction of both sides ... |
| ltadd1dd 11245 | Addition to both sides of ... |
| ltsub1dd 11246 | Subtraction from both side... |
| ltsub2dd 11247 | Subtraction of both sides ... |
| leadd1dd 11248 | Addition to both sides of ... |
| leadd2dd 11249 | Addition to both sides of ... |
| lesub1dd 11250 | Subtraction from both side... |
| lesub2dd 11251 | Subtraction of both sides ... |
| lesub3d 11252 | The result of subtracting ... |
| le2addd 11253 | Adding both side of two in... |
| le2subd 11254 | Subtracting both sides of ... |
| ltleaddd 11255 | Adding both sides of two o... |
| leltaddd 11256 | Adding both sides of two o... |
| lt2addd 11257 | Adding both side of two in... |
| lt2subd 11258 | Subtracting both sides of ... |
| possumd 11259 | Condition for a positive s... |
| sublt0d 11260 | When a subtraction gives a... |
| ltaddsublt 11261 | Addition and subtraction o... |
| 1le1 11262 | One is less than or equal ... |
| ixi 11263 | ` _i ` times itself is min... |
| recextlem1 11264 | Lemma for ~ recex . (Cont... |
| recextlem2 11265 | Lemma for ~ recex . (Cont... |
| recex 11266 | Existence of reciprocal of... |
| mulcand 11267 | Cancellation law for multi... |
| mulcan2d 11268 | Cancellation law for multi... |
| mulcanad 11269 | Cancellation of a nonzero ... |
| mulcan2ad 11270 | Cancellation of a nonzero ... |
| mulcan 11271 | Cancellation law for multi... |
| mulcan2 11272 | Cancellation law for multi... |
| mulcani 11273 | Cancellation law for multi... |
| mul0or 11274 | If a product is zero, one ... |
| mulne0b 11275 | The product of two nonzero... |
| mulne0 11276 | The product of two nonzero... |
| mulne0i 11277 | The product of two nonzero... |
| muleqadd 11278 | Property of numbers whose ... |
| receu 11279 | Existential uniqueness of ... |
| mulnzcnopr 11280 | Multiplication maps nonzer... |
| msq0i 11281 | A number is zero iff its s... |
| mul0ori 11282 | If a product is zero, one ... |
| msq0d 11283 | A number is zero iff its s... |
| mul0ord 11284 | If a product is zero, one ... |
| mulne0bd 11285 | The product of two nonzero... |
| mulne0d 11286 | The product of two nonzero... |
| mulcan1g 11287 | A generalized form of the ... |
| mulcan2g 11288 | A generalized form of the ... |
| mulne0bad 11289 | A factor of a nonzero comp... |
| mulne0bbd 11290 | A factor of a nonzero comp... |
| 1div0 11293 | You can't divide by zero, ... |
| divval 11294 | Value of division: if ` A ... |
| divmul 11295 | Relationship between divis... |
| divmul2 11296 | Relationship between divis... |
| divmul3 11297 | Relationship between divis... |
| divcl 11298 | Closure law for division. ... |
| reccl 11299 | Closure law for reciprocal... |
| divcan2 11300 | A cancellation law for div... |
| divcan1 11301 | A cancellation law for div... |
| diveq0 11302 | A ratio is zero iff the nu... |
| divne0b 11303 | The ratio of nonzero numbe... |
| divne0 11304 | The ratio of nonzero numbe... |
| recne0 11305 | The reciprocal of a nonzer... |
| recid 11306 | Multiplication of a number... |
| recid2 11307 | Multiplication of a number... |
| divrec 11308 | Relationship between divis... |
| divrec2 11309 | Relationship between divis... |
| divass 11310 | An associative law for div... |
| div23 11311 | A commutative/associative ... |
| div32 11312 | A commutative/associative ... |
| div13 11313 | A commutative/associative ... |
| div12 11314 | A commutative/associative ... |
| divmulass 11315 | An associative law for div... |
| divmulasscom 11316 | An associative/commutative... |
| divdir 11317 | Distribution of division o... |
| divcan3 11318 | A cancellation law for div... |
| divcan4 11319 | A cancellation law for div... |
| div11 11320 | One-to-one relationship fo... |
| divid 11321 | A number divided by itself... |
| div0 11322 | Division into zero is zero... |
| div1 11323 | A number divided by 1 is i... |
| 1div1e1 11324 | 1 divided by 1 is 1. (Con... |
| diveq1 11325 | Equality in terms of unit ... |
| divneg 11326 | Move negative sign inside ... |
| muldivdir 11327 | Distribution of division o... |
| divsubdir 11328 | Distribution of division o... |
| subdivcomb1 11329 | Bring a term in a subtract... |
| subdivcomb2 11330 | Bring a term in a subtract... |
| recrec 11331 | A number is equal to the r... |
| rec11 11332 | Reciprocal is one-to-one. ... |
| rec11r 11333 | Mutual reciprocals. (Cont... |
| divmuldiv 11334 | Multiplication of two rati... |
| divdivdiv 11335 | Division of two ratios. T... |
| divcan5 11336 | Cancellation of common fac... |
| divmul13 11337 | Swap the denominators in t... |
| divmul24 11338 | Swap the numerators in the... |
| divmuleq 11339 | Cross-multiply in an equal... |
| recdiv 11340 | The reciprocal of a ratio.... |
| divcan6 11341 | Cancellation of inverted f... |
| divdiv32 11342 | Swap denominators in a div... |
| divcan7 11343 | Cancel equal divisors in a... |
| dmdcan 11344 | Cancellation law for divis... |
| divdiv1 11345 | Division into a fraction. ... |
| divdiv2 11346 | Division by a fraction. (... |
| recdiv2 11347 | Division into a reciprocal... |
| ddcan 11348 | Cancellation in a double d... |
| divadddiv 11349 | Addition of two ratios. T... |
| divsubdiv 11350 | Subtraction of two ratios.... |
| conjmul 11351 | Two numbers whose reciproc... |
| rereccl 11352 | Closure law for reciprocal... |
| redivcl 11353 | Closure law for division o... |
| eqneg 11354 | A number equal to its nega... |
| eqnegd 11355 | A complex number equals it... |
| eqnegad 11356 | If a complex number equals... |
| div2neg 11357 | Quotient of two negatives.... |
| divneg2 11358 | Move negative sign inside ... |
| recclzi 11359 | Closure law for reciprocal... |
| recne0zi 11360 | The reciprocal of a nonzer... |
| recidzi 11361 | Multiplication of a number... |
| div1i 11362 | A number divided by 1 is i... |
| eqnegi 11363 | A number equal to its nega... |
| reccli 11364 | Closure law for reciprocal... |
| recidi 11365 | Multiplication of a number... |
| recreci 11366 | A number is equal to the r... |
| dividi 11367 | A number divided by itself... |
| div0i 11368 | Division into zero is zero... |
| divclzi 11369 | Closure law for division. ... |
| divcan1zi 11370 | A cancellation law for div... |
| divcan2zi 11371 | A cancellation law for div... |
| divreczi 11372 | Relationship between divis... |
| divcan3zi 11373 | A cancellation law for div... |
| divcan4zi 11374 | A cancellation law for div... |
| rec11i 11375 | Reciprocal is one-to-one. ... |
| divcli 11376 | Closure law for division. ... |
| divcan2i 11377 | A cancellation law for div... |
| divcan1i 11378 | A cancellation law for div... |
| divreci 11379 | Relationship between divis... |
| divcan3i 11380 | A cancellation law for div... |
| divcan4i 11381 | A cancellation law for div... |
| divne0i 11382 | The ratio of nonzero numbe... |
| rec11ii 11383 | Reciprocal is one-to-one. ... |
| divasszi 11384 | An associative law for div... |
| divmulzi 11385 | Relationship between divis... |
| divdirzi 11386 | Distribution of division o... |
| divdiv23zi 11387 | Swap denominators in a div... |
| divmuli 11388 | Relationship between divis... |
| divdiv32i 11389 | Swap denominators in a div... |
| divassi 11390 | An associative law for div... |
| divdiri 11391 | Distribution of division o... |
| div23i 11392 | A commutative/associative ... |
| div11i 11393 | One-to-one relationship fo... |
| divmuldivi 11394 | Multiplication of two rati... |
| divmul13i 11395 | Swap denominators of two r... |
| divadddivi 11396 | Addition of two ratios. T... |
| divdivdivi 11397 | Division of two ratios. T... |
| rerecclzi 11398 | Closure law for reciprocal... |
| rereccli 11399 | Closure law for reciprocal... |
| redivclzi 11400 | Closure law for division o... |
| redivcli 11401 | Closure law for division o... |
| div1d 11402 | A number divided by 1 is i... |
| reccld 11403 | Closure law for reciprocal... |
| recne0d 11404 | The reciprocal of a nonzer... |
| recidd 11405 | Multiplication of a number... |
| recid2d 11406 | Multiplication of a number... |
| recrecd 11407 | A number is equal to the r... |
| dividd 11408 | A number divided by itself... |
| div0d 11409 | Division into zero is zero... |
| divcld 11410 | Closure law for division. ... |
| divcan1d 11411 | A cancellation law for div... |
| divcan2d 11412 | A cancellation law for div... |
| divrecd 11413 | Relationship between divis... |
| divrec2d 11414 | Relationship between divis... |
| divcan3d 11415 | A cancellation law for div... |
| divcan4d 11416 | A cancellation law for div... |
| diveq0d 11417 | A ratio is zero iff the nu... |
| diveq1d 11418 | Equality in terms of unit ... |
| diveq1ad 11419 | The quotient of two comple... |
| diveq0ad 11420 | A fraction of complex numb... |
| divne1d 11421 | If two complex numbers are... |
| divne0bd 11422 | A ratio is zero iff the nu... |
| divnegd 11423 | Move negative sign inside ... |
| divneg2d 11424 | Move negative sign inside ... |
| div2negd 11425 | Quotient of two negatives.... |
| divne0d 11426 | The ratio of nonzero numbe... |
| recdivd 11427 | The reciprocal of a ratio.... |
| recdiv2d 11428 | Division into a reciprocal... |
| divcan6d 11429 | Cancellation of inverted f... |
| ddcand 11430 | Cancellation in a double d... |
| rec11d 11431 | Reciprocal is one-to-one. ... |
| divmuld 11432 | Relationship between divis... |
| div32d 11433 | A commutative/associative ... |
| div13d 11434 | A commutative/associative ... |
| divdiv32d 11435 | Swap denominators in a div... |
| divcan5d 11436 | Cancellation of common fac... |
| divcan5rd 11437 | Cancellation of common fac... |
| divcan7d 11438 | Cancel equal divisors in a... |
| dmdcand 11439 | Cancellation law for divis... |
| dmdcan2d 11440 | Cancellation law for divis... |
| divdiv1d 11441 | Division into a fraction. ... |
| divdiv2d 11442 | Division by a fraction. (... |
| divmul2d 11443 | Relationship between divis... |
| divmul3d 11444 | Relationship between divis... |
| divassd 11445 | An associative law for div... |
| div12d 11446 | A commutative/associative ... |
| div23d 11447 | A commutative/associative ... |
| divdird 11448 | Distribution of division o... |
| divsubdird 11449 | Distribution of division o... |
| div11d 11450 | One-to-one relationship fo... |
| divmuldivd 11451 | Multiplication of two rati... |
| divmul13d 11452 | Swap denominators of two r... |
| divmul24d 11453 | Swap the numerators in the... |
| divadddivd 11454 | Addition of two ratios. T... |
| divsubdivd 11455 | Subtraction of two ratios.... |
| divmuleqd 11456 | Cross-multiply in an equal... |
| divdivdivd 11457 | Division of two ratios. T... |
| diveq1bd 11458 | If two complex numbers are... |
| div2sub 11459 | Swap the order of subtract... |
| div2subd 11460 | Swap subtrahend and minuen... |
| rereccld 11461 | Closure law for reciprocal... |
| redivcld 11462 | Closure law for division o... |
| subrec 11463 | Subtraction of reciprocals... |
| subreci 11464 | Subtraction of reciprocals... |
| subrecd 11465 | Subtraction of reciprocals... |
| mvllmuld 11466 | Move LHS left multiplicati... |
| mvllmuli 11467 | Move LHS left multiplicati... |
| ldiv 11468 | Left-division. (Contribut... |
| rdiv 11469 | Right-division. (Contribu... |
| mdiv 11470 | A division law. (Contribu... |
| lineq 11471 | Solution of a (scalar) lin... |
| elimgt0 11472 | Hypothesis for weak deduct... |
| elimge0 11473 | Hypothesis for weak deduct... |
| ltp1 11474 | A number is less than itse... |
| lep1 11475 | A number is less than or e... |
| ltm1 11476 | A number minus 1 is less t... |
| lem1 11477 | A number minus 1 is less t... |
| letrp1 11478 | A transitive property of '... |
| p1le 11479 | A transitive property of p... |
| recgt0 11480 | The reciprocal of a positi... |
| prodgt0 11481 | Infer that a multiplicand ... |
| prodgt02 11482 | Infer that a multiplier is... |
| ltmul1a 11483 | Lemma for ~ ltmul1 . Mult... |
| ltmul1 11484 | Multiplication of both sid... |
| ltmul2 11485 | Multiplication of both sid... |
| lemul1 11486 | Multiplication of both sid... |
| lemul2 11487 | Multiplication of both sid... |
| lemul1a 11488 | Multiplication of both sid... |
| lemul2a 11489 | Multiplication of both sid... |
| ltmul12a 11490 | Comparison of product of t... |
| lemul12b 11491 | Comparison of product of t... |
| lemul12a 11492 | Comparison of product of t... |
| mulgt1 11493 | The product of two numbers... |
| ltmulgt11 11494 | Multiplication by a number... |
| ltmulgt12 11495 | Multiplication by a number... |
| lemulge11 11496 | Multiplication by a number... |
| lemulge12 11497 | Multiplication by a number... |
| ltdiv1 11498 | Division of both sides of ... |
| lediv1 11499 | Division of both sides of ... |
| gt0div 11500 | Division of a positive num... |
| ge0div 11501 | Division of a nonnegative ... |
| divgt0 11502 | The ratio of two positive ... |
| divge0 11503 | The ratio of nonnegative a... |
| mulge0b 11504 | A condition for multiplica... |
| mulle0b 11505 | A condition for multiplica... |
| mulsuble0b 11506 | A condition for multiplica... |
| ltmuldiv 11507 | 'Less than' relationship b... |
| ltmuldiv2 11508 | 'Less than' relationship b... |
| ltdivmul 11509 | 'Less than' relationship b... |
| ledivmul 11510 | 'Less than or equal to' re... |
| ltdivmul2 11511 | 'Less than' relationship b... |
| lt2mul2div 11512 | 'Less than' relationship b... |
| ledivmul2 11513 | 'Less than or equal to' re... |
| lemuldiv 11514 | 'Less than or equal' relat... |
| lemuldiv2 11515 | 'Less than or equal' relat... |
| ltrec 11516 | The reciprocal of both sid... |
| lerec 11517 | The reciprocal of both sid... |
| lt2msq1 11518 | Lemma for ~ lt2msq . (Con... |
| lt2msq 11519 | Two nonnegative numbers co... |
| ltdiv2 11520 | Division of a positive num... |
| ltrec1 11521 | Reciprocal swap in a 'less... |
| lerec2 11522 | Reciprocal swap in a 'less... |
| ledivdiv 11523 | Invert ratios of positive ... |
| lediv2 11524 | Division of a positive num... |
| ltdiv23 11525 | Swap denominator with othe... |
| lediv23 11526 | Swap denominator with othe... |
| lediv12a 11527 | Comparison of ratio of two... |
| lediv2a 11528 | Division of both sides of ... |
| reclt1 11529 | The reciprocal of a positi... |
| recgt1 11530 | The reciprocal of a positi... |
| recgt1i 11531 | The reciprocal of a number... |
| recp1lt1 11532 | Construct a number less th... |
| recreclt 11533 | Given a positive number ` ... |
| le2msq 11534 | The square function on non... |
| msq11 11535 | The square of a nonnegativ... |
| ledivp1 11536 | "Less than or equal to" an... |
| squeeze0 11537 | If a nonnegative number is... |
| ltp1i 11538 | A number is less than itse... |
| recgt0i 11539 | The reciprocal of a positi... |
| recgt0ii 11540 | The reciprocal of a positi... |
| prodgt0i 11541 | Infer that a multiplicand ... |
| divgt0i 11542 | The ratio of two positive ... |
| divge0i 11543 | The ratio of nonnegative a... |
| ltreci 11544 | The reciprocal of both sid... |
| lereci 11545 | The reciprocal of both sid... |
| lt2msqi 11546 | The square function on non... |
| le2msqi 11547 | The square function on non... |
| msq11i 11548 | The square of a nonnegativ... |
| divgt0i2i 11549 | The ratio of two positive ... |
| ltrecii 11550 | The reciprocal of both sid... |
| divgt0ii 11551 | The ratio of two positive ... |
| ltmul1i 11552 | Multiplication of both sid... |
| ltdiv1i 11553 | Division of both sides of ... |
| ltmuldivi 11554 | 'Less than' relationship b... |
| ltmul2i 11555 | Multiplication of both sid... |
| lemul1i 11556 | Multiplication of both sid... |
| lemul2i 11557 | Multiplication of both sid... |
| ltdiv23i 11558 | Swap denominator with othe... |
| ledivp1i 11559 | "Less than or equal to" an... |
| ltdivp1i 11560 | Less-than and division rel... |
| ltdiv23ii 11561 | Swap denominator with othe... |
| ltmul1ii 11562 | Multiplication of both sid... |
| ltdiv1ii 11563 | Division of both sides of ... |
| ltp1d 11564 | A number is less than itse... |
| lep1d 11565 | A number is less than or e... |
| ltm1d 11566 | A number minus 1 is less t... |
| lem1d 11567 | A number minus 1 is less t... |
| recgt0d 11568 | The reciprocal of a positi... |
| divgt0d 11569 | The ratio of two positive ... |
| mulgt1d 11570 | The product of two numbers... |
| lemulge11d 11571 | Multiplication by a number... |
| lemulge12d 11572 | Multiplication by a number... |
| lemul1ad 11573 | Multiplication of both sid... |
| lemul2ad 11574 | Multiplication of both sid... |
| ltmul12ad 11575 | Comparison of product of t... |
| lemul12ad 11576 | Comparison of product of t... |
| lemul12bd 11577 | Comparison of product of t... |
| fimaxre 11578 | A finite set of real numbe... |
| fimaxreOLD 11579 | Obsolete version of ~ fima... |
| fimaxre2 11580 | A nonempty finite set of r... |
| fimaxre3 11581 | A nonempty finite set of r... |
| fiminre 11582 | A nonempty finite set of r... |
| negfi 11583 | The negation of a finite s... |
| fiminreOLD 11584 | Obsolete version of ~ fimi... |
| lbreu 11585 | If a set of reals contains... |
| lbcl 11586 | If a set of reals contains... |
| lble 11587 | If a set of reals contains... |
| lbinf 11588 | If a set of reals contains... |
| lbinfcl 11589 | If a set of reals contains... |
| lbinfle 11590 | If a set of reals contains... |
| sup2 11591 | A nonempty, bounded-above ... |
| sup3 11592 | A version of the completen... |
| infm3lem 11593 | Lemma for ~ infm3 . (Cont... |
| infm3 11594 | The completeness axiom for... |
| suprcl 11595 | Closure of supremum of a n... |
| suprub 11596 | A member of a nonempty bou... |
| suprubd 11597 | Natural deduction form of ... |
| suprcld 11598 | Natural deduction form of ... |
| suprlub 11599 | The supremum of a nonempty... |
| suprnub 11600 | An upper bound is not less... |
| suprleub 11601 | The supremum of a nonempty... |
| supaddc 11602 | The supremum function dist... |
| supadd 11603 | The supremum function dist... |
| supmul1 11604 | The supremum function dist... |
| supmullem1 11605 | Lemma for ~ supmul . (Con... |
| supmullem2 11606 | Lemma for ~ supmul . (Con... |
| supmul 11607 | The supremum function dist... |
| sup3ii 11608 | A version of the completen... |
| suprclii 11609 | Closure of supremum of a n... |
| suprubii 11610 | A member of a nonempty bou... |
| suprlubii 11611 | The supremum of a nonempty... |
| suprnubii 11612 | An upper bound is not less... |
| suprleubii 11613 | The supremum of a nonempty... |
| riotaneg 11614 | The negative of the unique... |
| negiso 11615 | Negation is an order anti-... |
| dfinfre 11616 | The infimum of a set of re... |
| infrecl 11617 | Closure of infimum of a no... |
| infrenegsup 11618 | The infimum of a set of re... |
| infregelb 11619 | Any lower bound of a nonem... |
| infrelb 11620 | If a nonempty set of real ... |
| supfirege 11621 | The supremum of a finite s... |
| inelr 11622 | The imaginary unit ` _i ` ... |
| rimul 11623 | A real number times the im... |
| cru 11624 | The representation of comp... |
| crne0 11625 | The real representation of... |
| creur 11626 | The real part of a complex... |
| creui 11627 | The imaginary part of a co... |
| cju 11628 | The complex conjugate of a... |
| ofsubeq0 11629 | Function analogue of ~ sub... |
| ofnegsub 11630 | Function analogue of ~ neg... |
| ofsubge0 11631 | Function analogue of ~ sub... |
| nnexALT 11634 | Alternate proof of ~ nnex ... |
| peano5nni 11635 | Peano's inductive postulat... |
| nnssre 11636 | The positive integers are ... |
| nnsscn 11637 | The positive integers are ... |
| nnex 11638 | The set of positive intege... |
| nnre 11639 | A positive integer is a re... |
| nncn 11640 | A positive integer is a co... |
| nnrei 11641 | A positive integer is a re... |
| nncni 11642 | A positive integer is a co... |
| 1nn 11643 | Peano postulate: 1 is a po... |
| peano2nn 11644 | Peano postulate: a success... |
| dfnn2 11645 | Alternate definition of th... |
| dfnn3 11646 | Alternate definition of th... |
| nnred 11647 | A positive integer is a re... |
| nncnd 11648 | A positive integer is a co... |
| peano2nnd 11649 | Peano postulate: a success... |
| nnind 11650 | Principle of Mathematical ... |
| nnindALT 11651 | Principle of Mathematical ... |
| nn1m1nn 11652 | Every positive integer is ... |
| nn1suc 11653 | If a statement holds for 1... |
| nnaddcl 11654 | Closure of addition of pos... |
| nnmulcl 11655 | Closure of multiplication ... |
| nnmulcli 11656 | Closure of multiplication ... |
| nnmtmip 11657 | "Minus times minus is plus... |
| nn2ge 11658 | There exists a positive in... |
| nnge1 11659 | A positive integer is one ... |
| nngt1ne1 11660 | A positive integer is grea... |
| nnle1eq1 11661 | A positive integer is less... |
| nngt0 11662 | A positive integer is posi... |
| nnnlt1 11663 | A positive integer is not ... |
| nnnle0 11664 | A positive integer is not ... |
| nnne0 11665 | A positive integer is nonz... |
| nnneneg 11666 | No positive integer is equ... |
| 0nnn 11667 | Zero is not a positive int... |
| 0nnnALT 11668 | Alternate proof of ~ 0nnn ... |
| nnne0ALT 11669 | Alternate version of ~ nnn... |
| nngt0i 11670 | A positive integer is posi... |
| nnne0i 11671 | A positive integer is nonz... |
| nndivre 11672 | The quotient of a real and... |
| nnrecre 11673 | The reciprocal of a positi... |
| nnrecgt0 11674 | The reciprocal of a positi... |
| nnsub 11675 | Subtraction of positive in... |
| nnsubi 11676 | Subtraction of positive in... |
| nndiv 11677 | Two ways to express " ` A ... |
| nndivtr 11678 | Transitive property of div... |
| nnge1d 11679 | A positive integer is one ... |
| nngt0d 11680 | A positive integer is posi... |
| nnne0d 11681 | A positive integer is nonz... |
| nnrecred 11682 | The reciprocal of a positi... |
| nnaddcld 11683 | Closure of addition of pos... |
| nnmulcld 11684 | Closure of multiplication ... |
| nndivred 11685 | A positive integer is one ... |
| 0ne1 11702 | Zero is different from one... |
| 1m1e0 11703 | One minus one equals zero.... |
| 2nn 11704 | 2 is a positive integer. ... |
| 2re 11705 | The number 2 is real. (Co... |
| 2cn 11706 | The number 2 is a complex ... |
| 2cnALT 11707 | Alternate proof of ~ 2cn .... |
| 2ex 11708 | The number 2 is a set. (C... |
| 2cnd 11709 | The number 2 is a complex ... |
| 3nn 11710 | 3 is a positive integer. ... |
| 3re 11711 | The number 3 is real. (Co... |
| 3cn 11712 | The number 3 is a complex ... |
| 3ex 11713 | The number 3 is a set. (C... |
| 4nn 11714 | 4 is a positive integer. ... |
| 4re 11715 | The number 4 is real. (Co... |
| 4cn 11716 | The number 4 is a complex ... |
| 5nn 11717 | 5 is a positive integer. ... |
| 5re 11718 | The number 5 is real. (Co... |
| 5cn 11719 | The number 5 is a complex ... |
| 6nn 11720 | 6 is a positive integer. ... |
| 6re 11721 | The number 6 is real. (Co... |
| 6cn 11722 | The number 6 is a complex ... |
| 7nn 11723 | 7 is a positive integer. ... |
| 7re 11724 | The number 7 is real. (Co... |
| 7cn 11725 | The number 7 is a complex ... |
| 8nn 11726 | 8 is a positive integer. ... |
| 8re 11727 | The number 8 is real. (Co... |
| 8cn 11728 | The number 8 is a complex ... |
| 9nn 11729 | 9 is a positive integer. ... |
| 9re 11730 | The number 9 is real. (Co... |
| 9cn 11731 | The number 9 is a complex ... |
| 0le0 11732 | Zero is nonnegative. (Con... |
| 0le2 11733 | The number 0 is less than ... |
| 2pos 11734 | The number 2 is positive. ... |
| 2ne0 11735 | The number 2 is nonzero. ... |
| 3pos 11736 | The number 3 is positive. ... |
| 3ne0 11737 | The number 3 is nonzero. ... |
| 4pos 11738 | The number 4 is positive. ... |
| 4ne0 11739 | The number 4 is nonzero. ... |
| 5pos 11740 | The number 5 is positive. ... |
| 6pos 11741 | The number 6 is positive. ... |
| 7pos 11742 | The number 7 is positive. ... |
| 8pos 11743 | The number 8 is positive. ... |
| 9pos 11744 | The number 9 is positive. ... |
| neg1cn 11745 | -1 is a complex number. (... |
| neg1rr 11746 | -1 is a real number. (Con... |
| neg1ne0 11747 | -1 is nonzero. (Contribut... |
| neg1lt0 11748 | -1 is less than 0. (Contr... |
| negneg1e1 11749 | ` -u -u 1 ` is 1. (Contri... |
| 1pneg1e0 11750 | ` 1 + -u 1 ` is 0. (Contr... |
| 0m0e0 11751 | 0 minus 0 equals 0. (Cont... |
| 1m0e1 11752 | 1 - 0 = 1. (Contributed b... |
| 0p1e1 11753 | 0 + 1 = 1. (Contributed b... |
| fv0p1e1 11754 | Function value at ` N + 1 ... |
| 1p0e1 11755 | 1 + 0 = 1. (Contributed b... |
| 1p1e2 11756 | 1 + 1 = 2. (Contributed b... |
| 2m1e1 11757 | 2 - 1 = 1. The result is ... |
| 1e2m1 11758 | 1 = 2 - 1. (Contributed b... |
| 3m1e2 11759 | 3 - 1 = 2. (Contributed b... |
| 4m1e3 11760 | 4 - 1 = 3. (Contributed b... |
| 5m1e4 11761 | 5 - 1 = 4. (Contributed b... |
| 6m1e5 11762 | 6 - 1 = 5. (Contributed b... |
| 7m1e6 11763 | 7 - 1 = 6. (Contributed b... |
| 8m1e7 11764 | 8 - 1 = 7. (Contributed b... |
| 9m1e8 11765 | 9 - 1 = 8. (Contributed b... |
| 2p2e4 11766 | Two plus two equals four. ... |
| 2times 11767 | Two times a number. (Cont... |
| times2 11768 | A number times 2. (Contri... |
| 2timesi 11769 | Two times a number. (Cont... |
| times2i 11770 | A number times 2. (Contri... |
| 2txmxeqx 11771 | Two times a complex number... |
| 2div2e1 11772 | 2 divided by 2 is 1. (Con... |
| 2p1e3 11773 | 2 + 1 = 3. (Contributed b... |
| 1p2e3 11774 | 1 + 2 = 3. For a shorter ... |
| 1p2e3ALT 11775 | Alternate proof of ~ 1p2e3... |
| 3p1e4 11776 | 3 + 1 = 4. (Contributed b... |
| 4p1e5 11777 | 4 + 1 = 5. (Contributed b... |
| 5p1e6 11778 | 5 + 1 = 6. (Contributed b... |
| 6p1e7 11779 | 6 + 1 = 7. (Contributed b... |
| 7p1e8 11780 | 7 + 1 = 8. (Contributed b... |
| 8p1e9 11781 | 8 + 1 = 9. (Contributed b... |
| 3p2e5 11782 | 3 + 2 = 5. (Contributed b... |
| 3p3e6 11783 | 3 + 3 = 6. (Contributed b... |
| 4p2e6 11784 | 4 + 2 = 6. (Contributed b... |
| 4p3e7 11785 | 4 + 3 = 7. (Contributed b... |
| 4p4e8 11786 | 4 + 4 = 8. (Contributed b... |
| 5p2e7 11787 | 5 + 2 = 7. (Contributed b... |
| 5p3e8 11788 | 5 + 3 = 8. (Contributed b... |
| 5p4e9 11789 | 5 + 4 = 9. (Contributed b... |
| 6p2e8 11790 | 6 + 2 = 8. (Contributed b... |
| 6p3e9 11791 | 6 + 3 = 9. (Contributed b... |
| 7p2e9 11792 | 7 + 2 = 9. (Contributed b... |
| 1t1e1 11793 | 1 times 1 equals 1. (Cont... |
| 2t1e2 11794 | 2 times 1 equals 2. (Cont... |
| 2t2e4 11795 | 2 times 2 equals 4. (Cont... |
| 3t1e3 11796 | 3 times 1 equals 3. (Cont... |
| 3t2e6 11797 | 3 times 2 equals 6. (Cont... |
| 3t3e9 11798 | 3 times 3 equals 9. (Cont... |
| 4t2e8 11799 | 4 times 2 equals 8. (Cont... |
| 2t0e0 11800 | 2 times 0 equals 0. (Cont... |
| 4d2e2 11801 | One half of four is two. ... |
| 1lt2 11802 | 1 is less than 2. (Contri... |
| 2lt3 11803 | 2 is less than 3. (Contri... |
| 1lt3 11804 | 1 is less than 3. (Contri... |
| 3lt4 11805 | 3 is less than 4. (Contri... |
| 2lt4 11806 | 2 is less than 4. (Contri... |
| 1lt4 11807 | 1 is less than 4. (Contri... |
| 4lt5 11808 | 4 is less than 5. (Contri... |
| 3lt5 11809 | 3 is less than 5. (Contri... |
| 2lt5 11810 | 2 is less than 5. (Contri... |
| 1lt5 11811 | 1 is less than 5. (Contri... |
| 5lt6 11812 | 5 is less than 6. (Contri... |
| 4lt6 11813 | 4 is less than 6. (Contri... |
| 3lt6 11814 | 3 is less than 6. (Contri... |
| 2lt6 11815 | 2 is less than 6. (Contri... |
| 1lt6 11816 | 1 is less than 6. (Contri... |
| 6lt7 11817 | 6 is less than 7. (Contri... |
| 5lt7 11818 | 5 is less than 7. (Contri... |
| 4lt7 11819 | 4 is less than 7. (Contri... |
| 3lt7 11820 | 3 is less than 7. (Contri... |
| 2lt7 11821 | 2 is less than 7. (Contri... |
| 1lt7 11822 | 1 is less than 7. (Contri... |
| 7lt8 11823 | 7 is less than 8. (Contri... |
| 6lt8 11824 | 6 is less than 8. (Contri... |
| 5lt8 11825 | 5 is less than 8. (Contri... |
| 4lt8 11826 | 4 is less than 8. (Contri... |
| 3lt8 11827 | 3 is less than 8. (Contri... |
| 2lt8 11828 | 2 is less than 8. (Contri... |
| 1lt8 11829 | 1 is less than 8. (Contri... |
| 8lt9 11830 | 8 is less than 9. (Contri... |
| 7lt9 11831 | 7 is less than 9. (Contri... |
| 6lt9 11832 | 6 is less than 9. (Contri... |
| 5lt9 11833 | 5 is less than 9. (Contri... |
| 4lt9 11834 | 4 is less than 9. (Contri... |
| 3lt9 11835 | 3 is less than 9. (Contri... |
| 2lt9 11836 | 2 is less than 9. (Contri... |
| 1lt9 11837 | 1 is less than 9. (Contri... |
| 0ne2 11838 | 0 is not equal to 2. (Con... |
| 1ne2 11839 | 1 is not equal to 2. (Con... |
| 1le2 11840 | 1 is less than or equal to... |
| 2cnne0 11841 | 2 is a nonzero complex num... |
| 2rene0 11842 | 2 is a nonzero real number... |
| 1le3 11843 | 1 is less than or equal to... |
| neg1mulneg1e1 11844 | ` -u 1 x. -u 1 ` is 1. (C... |
| halfre 11845 | One-half is real. (Contri... |
| halfcn 11846 | One-half is a complex numb... |
| halfgt0 11847 | One-half is greater than z... |
| halfge0 11848 | One-half is not negative. ... |
| halflt1 11849 | One-half is less than one.... |
| 1mhlfehlf 11850 | Prove that 1 - 1/2 = 1/2. ... |
| 8th4div3 11851 | An eighth of four thirds i... |
| halfpm6th 11852 | One half plus or minus one... |
| it0e0 11853 | i times 0 equals 0. (Cont... |
| 2mulicn 11854 | ` ( 2 x. _i ) e. CC ` . (... |
| 2muline0 11855 | ` ( 2 x. _i ) =/= 0 ` . (... |
| halfcl 11856 | Closure of half of a numbe... |
| rehalfcl 11857 | Real closure of half. (Co... |
| half0 11858 | Half of a number is zero i... |
| 2halves 11859 | Two halves make a whole. ... |
| halfpos2 11860 | A number is positive iff i... |
| halfpos 11861 | A positive number is great... |
| halfnneg2 11862 | A number is nonnegative if... |
| halfaddsubcl 11863 | Closure of half-sum and ha... |
| halfaddsub 11864 | Sum and difference of half... |
| subhalfhalf 11865 | Subtracting the half of a ... |
| lt2halves 11866 | A sum is less than the who... |
| addltmul 11867 | Sum is less than product f... |
| nominpos 11868 | There is no smallest posit... |
| avglt1 11869 | Ordering property for aver... |
| avglt2 11870 | Ordering property for aver... |
| avgle1 11871 | Ordering property for aver... |
| avgle2 11872 | Ordering property for aver... |
| avgle 11873 | The average of two numbers... |
| 2timesd 11874 | Two times a number. (Cont... |
| times2d 11875 | A number times 2. (Contri... |
| halfcld 11876 | Closure of half of a numbe... |
| 2halvesd 11877 | Two halves make a whole. ... |
| rehalfcld 11878 | Real closure of half. (Co... |
| lt2halvesd 11879 | A sum is less than the who... |
| rehalfcli 11880 | Half a real number is real... |
| lt2addmuld 11881 | If two real numbers are le... |
| add1p1 11882 | Adding two times 1 to a nu... |
| sub1m1 11883 | Subtracting two times 1 fr... |
| cnm2m1cnm3 11884 | Subtracting 2 and afterwar... |
| xp1d2m1eqxm1d2 11885 | A complex number increased... |
| div4p1lem1div2 11886 | An integer greater than 5,... |
| nnunb 11887 | The set of positive intege... |
| arch 11888 | Archimedean property of re... |
| nnrecl 11889 | There exists a positive in... |
| bndndx 11890 | A bounded real sequence ` ... |
| elnn0 11893 | Nonnegative integers expre... |
| nnssnn0 11894 | Positive naturals are a su... |
| nn0ssre 11895 | Nonnegative integers are a... |
| nn0sscn 11896 | Nonnegative integers are a... |
| nn0ex 11897 | The set of nonnegative int... |
| nnnn0 11898 | A positive integer is a no... |
| nnnn0i 11899 | A positive integer is a no... |
| nn0re 11900 | A nonnegative integer is a... |
| nn0cn 11901 | A nonnegative integer is a... |
| nn0rei 11902 | A nonnegative integer is a... |
| nn0cni 11903 | A nonnegative integer is a... |
| dfn2 11904 | The set of positive intege... |
| elnnne0 11905 | The positive integer prope... |
| 0nn0 11906 | 0 is a nonnegative integer... |
| 1nn0 11907 | 1 is a nonnegative integer... |
| 2nn0 11908 | 2 is a nonnegative integer... |
| 3nn0 11909 | 3 is a nonnegative integer... |
| 4nn0 11910 | 4 is a nonnegative integer... |
| 5nn0 11911 | 5 is a nonnegative integer... |
| 6nn0 11912 | 6 is a nonnegative integer... |
| 7nn0 11913 | 7 is a nonnegative integer... |
| 8nn0 11914 | 8 is a nonnegative integer... |
| 9nn0 11915 | 9 is a nonnegative integer... |
| nn0ge0 11916 | A nonnegative integer is g... |
| nn0nlt0 11917 | A nonnegative integer is n... |
| nn0ge0i 11918 | Nonnegative integers are n... |
| nn0le0eq0 11919 | A nonnegative integer is l... |
| nn0p1gt0 11920 | A nonnegative integer incr... |
| nnnn0addcl 11921 | A positive integer plus a ... |
| nn0nnaddcl 11922 | A nonnegative integer plus... |
| 0mnnnnn0 11923 | The result of subtracting ... |
| un0addcl 11924 | If ` S ` is closed under a... |
| un0mulcl 11925 | If ` S ` is closed under m... |
| nn0addcl 11926 | Closure of addition of non... |
| nn0mulcl 11927 | Closure of multiplication ... |
| nn0addcli 11928 | Closure of addition of non... |
| nn0mulcli 11929 | Closure of multiplication ... |
| nn0p1nn 11930 | A nonnegative integer plus... |
| peano2nn0 11931 | Second Peano postulate for... |
| nnm1nn0 11932 | A positive integer minus 1... |
| elnn0nn 11933 | The nonnegative integer pr... |
| elnnnn0 11934 | The positive integer prope... |
| elnnnn0b 11935 | The positive integer prope... |
| elnnnn0c 11936 | The positive integer prope... |
| nn0addge1 11937 | A number is less than or e... |
| nn0addge2 11938 | A number is less than or e... |
| nn0addge1i 11939 | A number is less than or e... |
| nn0addge2i 11940 | A number is less than or e... |
| nn0sub 11941 | Subtraction of nonnegative... |
| ltsubnn0 11942 | Subtracting a nonnegative ... |
| nn0negleid 11943 | A nonnegative integer is g... |
| difgtsumgt 11944 | If the difference of a rea... |
| nn0le2xi 11945 | A nonnegative integer is l... |
| nn0lele2xi 11946 | 'Less than or equal to' im... |
| frnnn0supp 11947 | Two ways to write the supp... |
| frnnn0fsupp 11948 | A function on ` NN0 ` is f... |
| nnnn0d 11949 | A positive integer is a no... |
| nn0red 11950 | A nonnegative integer is a... |
| nn0cnd 11951 | A nonnegative integer is a... |
| nn0ge0d 11952 | A nonnegative integer is g... |
| nn0addcld 11953 | Closure of addition of non... |
| nn0mulcld 11954 | Closure of multiplication ... |
| nn0readdcl 11955 | Closure law for addition o... |
| nn0n0n1ge2 11956 | A nonnegative integer whic... |
| nn0n0n1ge2b 11957 | A nonnegative integer is n... |
| nn0ge2m1nn 11958 | If a nonnegative integer i... |
| nn0ge2m1nn0 11959 | If a nonnegative integer i... |
| nn0nndivcl 11960 | Closure law for dividing o... |
| elxnn0 11963 | An extended nonnegative in... |
| nn0ssxnn0 11964 | The standard nonnegative i... |
| nn0xnn0 11965 | A standard nonnegative int... |
| xnn0xr 11966 | An extended nonnegative in... |
| 0xnn0 11967 | Zero is an extended nonneg... |
| pnf0xnn0 11968 | Positive infinity is an ex... |
| nn0nepnf 11969 | No standard nonnegative in... |
| nn0xnn0d 11970 | A standard nonnegative int... |
| nn0nepnfd 11971 | No standard nonnegative in... |
| xnn0nemnf 11972 | No extended nonnegative in... |
| xnn0xrnemnf 11973 | The extended nonnegative i... |
| xnn0nnn0pnf 11974 | An extended nonnegative in... |
| elz 11977 | Membership in the set of i... |
| nnnegz 11978 | The negative of a positive... |
| zre 11979 | An integer is a real. (Co... |
| zcn 11980 | An integer is a complex nu... |
| zrei 11981 | An integer is a real numbe... |
| zssre 11982 | The integers are a subset ... |
| zsscn 11983 | The integers are a subset ... |
| zex 11984 | The set of integers exists... |
| elnnz 11985 | Positive integer property ... |
| 0z 11986 | Zero is an integer. (Cont... |
| 0zd 11987 | Zero is an integer, deduct... |
| elnn0z 11988 | Nonnegative integer proper... |
| elznn0nn 11989 | Integer property expressed... |
| elznn0 11990 | Integer property expressed... |
| elznn 11991 | Integer property expressed... |
| zle0orge1 11992 | There is no integer in the... |
| elz2 11993 | Membership in the set of i... |
| dfz2 11994 | Alternative definition of ... |
| zexALT 11995 | Alternate proof of ~ zex .... |
| nnssz 11996 | Positive integers are a su... |
| nn0ssz 11997 | Nonnegative integers are a... |
| nnz 11998 | A positive integer is an i... |
| nn0z 11999 | A nonnegative integer is a... |
| nnzi 12000 | A positive integer is an i... |
| nn0zi 12001 | A nonnegative integer is a... |
| elnnz1 12002 | Positive integer property ... |
| znnnlt1 12003 | An integer is not a positi... |
| nnzrab 12004 | Positive integers expresse... |
| nn0zrab 12005 | Nonnegative integers expre... |
| 1z 12006 | One is an integer. (Contr... |
| 1zzd 12007 | One is an integer, deducti... |
| 2z 12008 | 2 is an integer. (Contrib... |
| 3z 12009 | 3 is an integer. (Contrib... |
| 4z 12010 | 4 is an integer. (Contrib... |
| znegcl 12011 | Closure law for negative i... |
| neg1z 12012 | -1 is an integer. (Contri... |
| znegclb 12013 | A complex number is an int... |
| nn0negz 12014 | The negative of a nonnegat... |
| nn0negzi 12015 | The negative of a nonnegat... |
| zaddcl 12016 | Closure of addition of int... |
| peano2z 12017 | Second Peano postulate gen... |
| zsubcl 12018 | Closure of subtraction of ... |
| peano2zm 12019 | "Reverse" second Peano pos... |
| zletr 12020 | Transitive law of ordering... |
| zrevaddcl 12021 | Reverse closure law for ad... |
| znnsub 12022 | The positive difference of... |
| znn0sub 12023 | The nonnegative difference... |
| nzadd 12024 | The sum of a real number n... |
| zmulcl 12025 | Closure of multiplication ... |
| zltp1le 12026 | Integer ordering relation.... |
| zleltp1 12027 | Integer ordering relation.... |
| zlem1lt 12028 | Integer ordering relation.... |
| zltlem1 12029 | Integer ordering relation.... |
| zgt0ge1 12030 | An integer greater than ` ... |
| nnleltp1 12031 | Positive integer ordering ... |
| nnltp1le 12032 | Positive integer ordering ... |
| nnaddm1cl 12033 | Closure of addition of pos... |
| nn0ltp1le 12034 | Nonnegative integer orderi... |
| nn0leltp1 12035 | Nonnegative integer orderi... |
| nn0ltlem1 12036 | Nonnegative integer orderi... |
| nn0sub2 12037 | Subtraction of nonnegative... |
| nn0lt10b 12038 | A nonnegative integer less... |
| nn0lt2 12039 | A nonnegative integer less... |
| nn0le2is012 12040 | A nonnegative integer whic... |
| nn0lem1lt 12041 | Nonnegative integer orderi... |
| nnlem1lt 12042 | Positive integer ordering ... |
| nnltlem1 12043 | Positive integer ordering ... |
| nnm1ge0 12044 | A positive integer decreas... |
| nn0ge0div 12045 | Division of a nonnegative ... |
| zdiv 12046 | Two ways to express " ` M ... |
| zdivadd 12047 | Property of divisibility: ... |
| zdivmul 12048 | Property of divisibility: ... |
| zextle 12049 | An extensionality-like pro... |
| zextlt 12050 | An extensionality-like pro... |
| recnz 12051 | The reciprocal of a number... |
| btwnnz 12052 | A number between an intege... |
| gtndiv 12053 | A larger number does not d... |
| halfnz 12054 | One-half is not an integer... |
| 3halfnz 12055 | Three halves is not an int... |
| suprzcl 12056 | The supremum of a bounded-... |
| prime 12057 | Two ways to express " ` A ... |
| msqznn 12058 | The square of a nonzero in... |
| zneo 12059 | No even integer equals an ... |
| nneo 12060 | A positive integer is even... |
| nneoi 12061 | A positive integer is even... |
| zeo 12062 | An integer is even or odd.... |
| zeo2 12063 | An integer is even or odd ... |
| peano2uz2 12064 | Second Peano postulate for... |
| peano5uzi 12065 | Peano's inductive postulat... |
| peano5uzti 12066 | Peano's inductive postulat... |
| dfuzi 12067 | An expression for the uppe... |
| uzind 12068 | Induction on the upper int... |
| uzind2 12069 | Induction on the upper int... |
| uzind3 12070 | Induction on the upper int... |
| nn0ind 12071 | Principle of Mathematical ... |
| nn0indALT 12072 | Principle of Mathematical ... |
| nn0indd 12073 | Principle of Mathematical ... |
| fzind 12074 | Induction on the integers ... |
| fnn0ind 12075 | Induction on the integers ... |
| nn0ind-raph 12076 | Principle of Mathematical ... |
| zindd 12077 | Principle of Mathematical ... |
| btwnz 12078 | Any real number can be san... |
| nn0zd 12079 | A positive integer is an i... |
| nnzd 12080 | A nonnegative integer is a... |
| zred 12081 | An integer is a real numbe... |
| zcnd 12082 | An integer is a complex nu... |
| znegcld 12083 | Closure law for negative i... |
| peano2zd 12084 | Deduction from second Pean... |
| zaddcld 12085 | Closure of addition of int... |
| zsubcld 12086 | Closure of subtraction of ... |
| zmulcld 12087 | Closure of multiplication ... |
| znnn0nn 12088 | The negative of a negative... |
| zadd2cl 12089 | Increasing an integer by 2... |
| zriotaneg 12090 | The negative of the unique... |
| suprfinzcl 12091 | The supremum of a nonempty... |
| 9p1e10 12094 | 9 + 1 = 10. (Contributed ... |
| dfdec10 12095 | Version of the definition ... |
| decex 12096 | A decimal number is a set.... |
| deceq1 12097 | Equality theorem for the d... |
| deceq2 12098 | Equality theorem for the d... |
| deceq1i 12099 | Equality theorem for the d... |
| deceq2i 12100 | Equality theorem for the d... |
| deceq12i 12101 | Equality theorem for the d... |
| numnncl 12102 | Closure for a numeral (wit... |
| num0u 12103 | Add a zero in the units pl... |
| num0h 12104 | Add a zero in the higher p... |
| numcl 12105 | Closure for a decimal inte... |
| numsuc 12106 | The successor of a decimal... |
| deccl 12107 | Closure for a numeral. (C... |
| 10nn 12108 | 10 is a positive integer. ... |
| 10pos 12109 | The number 10 is positive.... |
| 10nn0 12110 | 10 is a nonnegative intege... |
| 10re 12111 | The number 10 is real. (C... |
| decnncl 12112 | Closure for a numeral. (C... |
| dec0u 12113 | Add a zero in the units pl... |
| dec0h 12114 | Add a zero in the higher p... |
| numnncl2 12115 | Closure for a decimal inte... |
| decnncl2 12116 | Closure for a decimal inte... |
| numlt 12117 | Comparing two decimal inte... |
| numltc 12118 | Comparing two decimal inte... |
| le9lt10 12119 | A "decimal digit" (i.e. a ... |
| declt 12120 | Comparing two decimal inte... |
| decltc 12121 | Comparing two decimal inte... |
| declth 12122 | Comparing two decimal inte... |
| decsuc 12123 | The successor of a decimal... |
| 3declth 12124 | Comparing two decimal inte... |
| 3decltc 12125 | Comparing two decimal inte... |
| decle 12126 | Comparing two decimal inte... |
| decleh 12127 | Comparing two decimal inte... |
| declei 12128 | Comparing a digit to a dec... |
| numlti 12129 | Comparing a digit to a dec... |
| declti 12130 | Comparing a digit to a dec... |
| decltdi 12131 | Comparing a digit to a dec... |
| numsucc 12132 | The successor of a decimal... |
| decsucc 12133 | The successor of a decimal... |
| 1e0p1 12134 | The successor of zero. (C... |
| dec10p 12135 | Ten plus an integer. (Con... |
| numma 12136 | Perform a multiply-add of ... |
| nummac 12137 | Perform a multiply-add of ... |
| numma2c 12138 | Perform a multiply-add of ... |
| numadd 12139 | Add two decimal integers `... |
| numaddc 12140 | Add two decimal integers `... |
| nummul1c 12141 | The product of a decimal i... |
| nummul2c 12142 | The product of a decimal i... |
| decma 12143 | Perform a multiply-add of ... |
| decmac 12144 | Perform a multiply-add of ... |
| decma2c 12145 | Perform a multiply-add of ... |
| decadd 12146 | Add two numerals ` M ` and... |
| decaddc 12147 | Add two numerals ` M ` and... |
| decaddc2 12148 | Add two numerals ` M ` and... |
| decrmanc 12149 | Perform a multiply-add of ... |
| decrmac 12150 | Perform a multiply-add of ... |
| decaddm10 12151 | The sum of two multiples o... |
| decaddi 12152 | Add two numerals ` M ` and... |
| decaddci 12153 | Add two numerals ` M ` and... |
| decaddci2 12154 | Add two numerals ` M ` and... |
| decsubi 12155 | Difference between a numer... |
| decmul1 12156 | The product of a numeral w... |
| decmul1c 12157 | The product of a numeral w... |
| decmul2c 12158 | The product of a numeral w... |
| decmulnc 12159 | The product of a numeral w... |
| 11multnc 12160 | The product of 11 (as nume... |
| decmul10add 12161 | A multiplication of a numb... |
| 6p5lem 12162 | Lemma for ~ 6p5e11 and rel... |
| 5p5e10 12163 | 5 + 5 = 10. (Contributed ... |
| 6p4e10 12164 | 6 + 4 = 10. (Contributed ... |
| 6p5e11 12165 | 6 + 5 = 11. (Contributed ... |
| 6p6e12 12166 | 6 + 6 = 12. (Contributed ... |
| 7p3e10 12167 | 7 + 3 = 10. (Contributed ... |
| 7p4e11 12168 | 7 + 4 = 11. (Contributed ... |
| 7p5e12 12169 | 7 + 5 = 12. (Contributed ... |
| 7p6e13 12170 | 7 + 6 = 13. (Contributed ... |
| 7p7e14 12171 | 7 + 7 = 14. (Contributed ... |
| 8p2e10 12172 | 8 + 2 = 10. (Contributed ... |
| 8p3e11 12173 | 8 + 3 = 11. (Contributed ... |
| 8p4e12 12174 | 8 + 4 = 12. (Contributed ... |
| 8p5e13 12175 | 8 + 5 = 13. (Contributed ... |
| 8p6e14 12176 | 8 + 6 = 14. (Contributed ... |
| 8p7e15 12177 | 8 + 7 = 15. (Contributed ... |
| 8p8e16 12178 | 8 + 8 = 16. (Contributed ... |
| 9p2e11 12179 | 9 + 2 = 11. (Contributed ... |
| 9p3e12 12180 | 9 + 3 = 12. (Contributed ... |
| 9p4e13 12181 | 9 + 4 = 13. (Contributed ... |
| 9p5e14 12182 | 9 + 5 = 14. (Contributed ... |
| 9p6e15 12183 | 9 + 6 = 15. (Contributed ... |
| 9p7e16 12184 | 9 + 7 = 16. (Contributed ... |
| 9p8e17 12185 | 9 + 8 = 17. (Contributed ... |
| 9p9e18 12186 | 9 + 9 = 18. (Contributed ... |
| 10p10e20 12187 | 10 + 10 = 20. (Contribute... |
| 10m1e9 12188 | 10 - 1 = 9. (Contributed ... |
| 4t3lem 12189 | Lemma for ~ 4t3e12 and rel... |
| 4t3e12 12190 | 4 times 3 equals 12. (Con... |
| 4t4e16 12191 | 4 times 4 equals 16. (Con... |
| 5t2e10 12192 | 5 times 2 equals 10. (Con... |
| 5t3e15 12193 | 5 times 3 equals 15. (Con... |
| 5t4e20 12194 | 5 times 4 equals 20. (Con... |
| 5t5e25 12195 | 5 times 5 equals 25. (Con... |
| 6t2e12 12196 | 6 times 2 equals 12. (Con... |
| 6t3e18 12197 | 6 times 3 equals 18. (Con... |
| 6t4e24 12198 | 6 times 4 equals 24. (Con... |
| 6t5e30 12199 | 6 times 5 equals 30. (Con... |
| 6t6e36 12200 | 6 times 6 equals 36. (Con... |
| 7t2e14 12201 | 7 times 2 equals 14. (Con... |
| 7t3e21 12202 | 7 times 3 equals 21. (Con... |
| 7t4e28 12203 | 7 times 4 equals 28. (Con... |
| 7t5e35 12204 | 7 times 5 equals 35. (Con... |
| 7t6e42 12205 | 7 times 6 equals 42. (Con... |
| 7t7e49 12206 | 7 times 7 equals 49. (Con... |
| 8t2e16 12207 | 8 times 2 equals 16. (Con... |
| 8t3e24 12208 | 8 times 3 equals 24. (Con... |
| 8t4e32 12209 | 8 times 4 equals 32. (Con... |
| 8t5e40 12210 | 8 times 5 equals 40. (Con... |
| 8t6e48 12211 | 8 times 6 equals 48. (Con... |
| 8t7e56 12212 | 8 times 7 equals 56. (Con... |
| 8t8e64 12213 | 8 times 8 equals 64. (Con... |
| 9t2e18 12214 | 9 times 2 equals 18. (Con... |
| 9t3e27 12215 | 9 times 3 equals 27. (Con... |
| 9t4e36 12216 | 9 times 4 equals 36. (Con... |
| 9t5e45 12217 | 9 times 5 equals 45. (Con... |
| 9t6e54 12218 | 9 times 6 equals 54. (Con... |
| 9t7e63 12219 | 9 times 7 equals 63. (Con... |
| 9t8e72 12220 | 9 times 8 equals 72. (Con... |
| 9t9e81 12221 | 9 times 9 equals 81. (Con... |
| 9t11e99 12222 | 9 times 11 equals 99. (Co... |
| 9lt10 12223 | 9 is less than 10. (Contr... |
| 8lt10 12224 | 8 is less than 10. (Contr... |
| 7lt10 12225 | 7 is less than 10. (Contr... |
| 6lt10 12226 | 6 is less than 10. (Contr... |
| 5lt10 12227 | 5 is less than 10. (Contr... |
| 4lt10 12228 | 4 is less than 10. (Contr... |
| 3lt10 12229 | 3 is less than 10. (Contr... |
| 2lt10 12230 | 2 is less than 10. (Contr... |
| 1lt10 12231 | 1 is less than 10. (Contr... |
| decbin0 12232 | Decompose base 4 into base... |
| decbin2 12233 | Decompose base 4 into base... |
| decbin3 12234 | Decompose base 4 into base... |
| halfthird 12235 | Half minus a third. (Cont... |
| 5recm6rec 12236 | One fifth minus one sixth.... |
| uzval 12239 | The value of the upper int... |
| uzf 12240 | The domain and range of th... |
| eluz1 12241 | Membership in the upper se... |
| eluzel2 12242 | Implication of membership ... |
| eluz2 12243 | Membership in an upper set... |
| eluzmn 12244 | Membership in an earlier u... |
| eluz1i 12245 | Membership in an upper set... |
| eluzuzle 12246 | An integer in an upper set... |
| eluzelz 12247 | A member of an upper set o... |
| eluzelre 12248 | A member of an upper set o... |
| eluzelcn 12249 | A member of an upper set o... |
| eluzle 12250 | Implication of membership ... |
| eluz 12251 | Membership in an upper set... |
| uzid 12252 | Membership of the least me... |
| uzidd 12253 | Membership of the least me... |
| uzn0 12254 | The upper integers are all... |
| uztrn 12255 | Transitive law for sets of... |
| uztrn2 12256 | Transitive law for sets of... |
| uzneg 12257 | Contraposition law for upp... |
| uzssz 12258 | An upper set of integers i... |
| uzss 12259 | Subset relationship for tw... |
| uztric 12260 | Totality of the ordering r... |
| uz11 12261 | The upper integers functio... |
| eluzp1m1 12262 | Membership in the next upp... |
| eluzp1l 12263 | Strict ordering implied by... |
| eluzp1p1 12264 | Membership in the next upp... |
| eluzaddi 12265 | Membership in a later uppe... |
| eluzsubi 12266 | Membership in an earlier u... |
| eluzadd 12267 | Membership in a later uppe... |
| eluzsub 12268 | Membership in an earlier u... |
| subeluzsub 12269 | Membership of a difference... |
| uzm1 12270 | Choices for an element of ... |
| uznn0sub 12271 | The nonnegative difference... |
| uzin 12272 | Intersection of two upper ... |
| uzp1 12273 | Choices for an element of ... |
| nn0uz 12274 | Nonnegative integers expre... |
| nnuz 12275 | Positive integers expresse... |
| elnnuz 12276 | A positive integer express... |
| elnn0uz 12277 | A nonnegative integer expr... |
| eluz2nn 12278 | An integer greater than or... |
| eluz4eluz2 12279 | An integer greater than or... |
| eluz4nn 12280 | An integer greater than or... |
| eluzge2nn0 12281 | If an integer is greater t... |
| eluz2n0 12282 | An integer greater than or... |
| uzuzle23 12283 | An integer in the upper se... |
| eluzge3nn 12284 | If an integer is greater t... |
| uz3m2nn 12285 | An integer greater than or... |
| 1eluzge0 12286 | 1 is an integer greater th... |
| 2eluzge0 12287 | 2 is an integer greater th... |
| 2eluzge1 12288 | 2 is an integer greater th... |
| uznnssnn 12289 | The upper integers startin... |
| raluz 12290 | Restricted universal quant... |
| raluz2 12291 | Restricted universal quant... |
| rexuz 12292 | Restricted existential qua... |
| rexuz2 12293 | Restricted existential qua... |
| 2rexuz 12294 | Double existential quantif... |
| peano2uz 12295 | Second Peano postulate for... |
| peano2uzs 12296 | Second Peano postulate for... |
| peano2uzr 12297 | Reversed second Peano axio... |
| uzaddcl 12298 | Addition closure law for a... |
| nn0pzuz 12299 | The sum of a nonnegative i... |
| uzind4 12300 | Induction on the upper set... |
| uzind4ALT 12301 | Induction on the upper set... |
| uzind4s 12302 | Induction on the upper set... |
| uzind4s2 12303 | Induction on the upper set... |
| uzind4i 12304 | Induction on the upper int... |
| uzwo 12305 | Well-ordering principle: a... |
| uzwo2 12306 | Well-ordering principle: a... |
| nnwo 12307 | Well-ordering principle: a... |
| nnwof 12308 | Well-ordering principle: a... |
| nnwos 12309 | Well-ordering principle: a... |
| indstr 12310 | Strong Mathematical Induct... |
| eluznn0 12311 | Membership in a nonnegativ... |
| eluznn 12312 | Membership in a positive u... |
| eluz2b1 12313 | Two ways to say "an intege... |
| eluz2gt1 12314 | An integer greater than or... |
| eluz2b2 12315 | Two ways to say "an intege... |
| eluz2b3 12316 | Two ways to say "an intege... |
| uz2m1nn 12317 | One less than an integer g... |
| 1nuz2 12318 | 1 is not in ` ( ZZ>= `` 2 ... |
| elnn1uz2 12319 | A positive integer is eith... |
| uz2mulcl 12320 | Closure of multiplication ... |
| indstr2 12321 | Strong Mathematical Induct... |
| uzinfi 12322 | Extract the lower bound of... |
| nninf 12323 | The infimum of the set of ... |
| nn0inf 12324 | The infimum of the set of ... |
| infssuzle 12325 | The infimum of a subset of... |
| infssuzcl 12326 | The infimum of a subset of... |
| ublbneg 12327 | The image under negation o... |
| eqreznegel 12328 | Two ways to express the im... |
| supminf 12329 | The supremum of a bounded-... |
| lbzbi 12330 | If a set of reals is bound... |
| zsupss 12331 | Any nonempty bounded subse... |
| suprzcl2 12332 | The supremum of a bounded-... |
| suprzub 12333 | The supremum of a bounded-... |
| uzsupss 12334 | Any bounded subset of an u... |
| nn01to3 12335 | A (nonnegative) integer be... |
| nn0ge2m1nnALT 12336 | Alternate proof of ~ nn0ge... |
| uzwo3 12337 | Well-ordering principle: a... |
| zmin 12338 | There is a unique smallest... |
| zmax 12339 | There is a unique largest ... |
| zbtwnre 12340 | There is a unique integer ... |
| rebtwnz 12341 | There is a unique greatest... |
| elq 12344 | Membership in the set of r... |
| qmulz 12345 | If ` A ` is rational, then... |
| znq 12346 | The ratio of an integer an... |
| qre 12347 | A rational number is a rea... |
| zq 12348 | An integer is a rational n... |
| zssq 12349 | The integers are a subset ... |
| nn0ssq 12350 | The nonnegative integers a... |
| nnssq 12351 | The positive integers are ... |
| qssre 12352 | The rationals are a subset... |
| qsscn 12353 | The rationals are a subset... |
| qex 12354 | The set of rational number... |
| nnq 12355 | A positive integer is rati... |
| qcn 12356 | A rational number is a com... |
| qexALT 12357 | Alternate proof of ~ qex .... |
| qaddcl 12358 | Closure of addition of rat... |
| qnegcl 12359 | Closure law for the negati... |
| qmulcl 12360 | Closure of multiplication ... |
| qsubcl 12361 | Closure of subtraction of ... |
| qreccl 12362 | Closure of reciprocal of r... |
| qdivcl 12363 | Closure of division of rat... |
| qrevaddcl 12364 | Reverse closure law for ad... |
| nnrecq 12365 | The reciprocal of a positi... |
| irradd 12366 | The sum of an irrational n... |
| irrmul 12367 | The product of an irration... |
| elpq 12368 | A positive rational is the... |
| elpqb 12369 | A class is a positive rati... |
| rpnnen1lem2 12370 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem1 12371 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem3 12372 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem4 12373 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem5 12374 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1lem6 12375 | Lemma for ~ rpnnen1 . (Co... |
| rpnnen1 12376 | One half of ~ rpnnen , whe... |
| reexALT 12377 | Alternate proof of ~ reex ... |
| cnref1o 12378 | There is a natural one-to-... |
| cnexALT 12379 | The set of complex numbers... |
| xrex 12380 | The set of extended reals ... |
| addex 12381 | The addition operation is ... |
| mulex 12382 | The multiplication operati... |
| elrp 12385 | Membership in the set of p... |
| elrpii 12386 | Membership in the set of p... |
| 1rp 12387 | 1 is a positive real. (Co... |
| 2rp 12388 | 2 is a positive real. (Co... |
| 3rp 12389 | 3 is a positive real. (Co... |
| rpssre 12390 | The positive reals are a s... |
| rpre 12391 | A positive real is a real.... |
| rpxr 12392 | A positive real is an exte... |
| rpcn 12393 | A positive real is a compl... |
| nnrp 12394 | A positive integer is a po... |
| rpgt0 12395 | A positive real is greater... |
| rpge0 12396 | A positive real is greater... |
| rpregt0 12397 | A positive real is a posit... |
| rprege0 12398 | A positive real is a nonne... |
| rpne0 12399 | A positive real is nonzero... |
| rprene0 12400 | A positive real is a nonze... |
| rpcnne0 12401 | A positive real is a nonze... |
| rpcndif0 12402 | A positive real number is ... |
| ralrp 12403 | Quantification over positi... |
| rexrp 12404 | Quantification over positi... |
| rpaddcl 12405 | Closure law for addition o... |
| rpmulcl 12406 | Closure law for multiplica... |
| rpmtmip 12407 | "Minus times minus is plus... |
| rpdivcl 12408 | Closure law for division o... |
| rpreccl 12409 | Closure law for reciprocat... |
| rphalfcl 12410 | Closure law for half of a ... |
| rpgecl 12411 | A number greater than or e... |
| rphalflt 12412 | Half of a positive real is... |
| rerpdivcl 12413 | Closure law for division o... |
| ge0p1rp 12414 | A nonnegative number plus ... |
| rpneg 12415 | Either a nonzero real or i... |
| negelrp 12416 | Elementhood of a negation ... |
| negelrpd 12417 | The negation of a negative... |
| 0nrp 12418 | Zero is not a positive rea... |
| ltsubrp 12419 | Subtracting a positive rea... |
| ltaddrp 12420 | Adding a positive number t... |
| difrp 12421 | Two ways to say one number... |
| elrpd 12422 | Membership in the set of p... |
| nnrpd 12423 | A positive integer is a po... |
| zgt1rpn0n1 12424 | An integer greater than 1 ... |
| rpred 12425 | A positive real is a real.... |
| rpxrd 12426 | A positive real is an exte... |
| rpcnd 12427 | A positive real is a compl... |
| rpgt0d 12428 | A positive real is greater... |
| rpge0d 12429 | A positive real is greater... |
| rpne0d 12430 | A positive real is nonzero... |
| rpregt0d 12431 | A positive real is real an... |
| rprege0d 12432 | A positive real is real an... |
| rprene0d 12433 | A positive real is a nonze... |
| rpcnne0d 12434 | A positive real is a nonze... |
| rpreccld 12435 | Closure law for reciprocat... |
| rprecred 12436 | Closure law for reciprocat... |
| rphalfcld 12437 | Closure law for half of a ... |
| reclt1d 12438 | The reciprocal of a positi... |
| recgt1d 12439 | The reciprocal of a positi... |
| rpaddcld 12440 | Closure law for addition o... |
| rpmulcld 12441 | Closure law for multiplica... |
| rpdivcld 12442 | Closure law for division o... |
| ltrecd 12443 | The reciprocal of both sid... |
| lerecd 12444 | The reciprocal of both sid... |
| ltrec1d 12445 | Reciprocal swap in a 'less... |
| lerec2d 12446 | Reciprocal swap in a 'less... |
| lediv2ad 12447 | Division of both sides of ... |
| ltdiv2d 12448 | Division of a positive num... |
| lediv2d 12449 | Division of a positive num... |
| ledivdivd 12450 | Invert ratios of positive ... |
| divge1 12451 | The ratio of a number over... |
| divlt1lt 12452 | A real number divided by a... |
| divle1le 12453 | A real number divided by a... |
| ledivge1le 12454 | If a number is less than o... |
| ge0p1rpd 12455 | A nonnegative number plus ... |
| rerpdivcld 12456 | Closure law for division o... |
| ltsubrpd 12457 | Subtracting a positive rea... |
| ltaddrpd 12458 | Adding a positive number t... |
| ltaddrp2d 12459 | Adding a positive number t... |
| ltmulgt11d 12460 | Multiplication by a number... |
| ltmulgt12d 12461 | Multiplication by a number... |
| gt0divd 12462 | Division of a positive num... |
| ge0divd 12463 | Division of a nonnegative ... |
| rpgecld 12464 | A number greater than or e... |
| divge0d 12465 | The ratio of nonnegative a... |
| ltmul1d 12466 | The ratio of nonnegative a... |
| ltmul2d 12467 | Multiplication of both sid... |
| lemul1d 12468 | Multiplication of both sid... |
| lemul2d 12469 | Multiplication of both sid... |
| ltdiv1d 12470 | Division of both sides of ... |
| lediv1d 12471 | Division of both sides of ... |
| ltmuldivd 12472 | 'Less than' relationship b... |
| ltmuldiv2d 12473 | 'Less than' relationship b... |
| lemuldivd 12474 | 'Less than or equal to' re... |
| lemuldiv2d 12475 | 'Less than or equal to' re... |
| ltdivmuld 12476 | 'Less than' relationship b... |
| ltdivmul2d 12477 | 'Less than' relationship b... |
| ledivmuld 12478 | 'Less than or equal to' re... |
| ledivmul2d 12479 | 'Less than or equal to' re... |
| ltmul1dd 12480 | The ratio of nonnegative a... |
| ltmul2dd 12481 | Multiplication of both sid... |
| ltdiv1dd 12482 | Division of both sides of ... |
| lediv1dd 12483 | Division of both sides of ... |
| lediv12ad 12484 | Comparison of ratio of two... |
| mul2lt0rlt0 12485 | If the result of a multipl... |
| mul2lt0rgt0 12486 | If the result of a multipl... |
| mul2lt0llt0 12487 | If the result of a multipl... |
| mul2lt0lgt0 12488 | If the result of a multipl... |
| mul2lt0bi 12489 | If the result of a multipl... |
| prodge0rd 12490 | Infer that a multiplicand ... |
| prodge0ld 12491 | Infer that a multiplier is... |
| ltdiv23d 12492 | Swap denominator with othe... |
| lediv23d 12493 | Swap denominator with othe... |
| lt2mul2divd 12494 | The ratio of nonnegative a... |
| nnledivrp 12495 | Division of a positive int... |
| nn0ledivnn 12496 | Division of a nonnegative ... |
| addlelt 12497 | If the sum of a real numbe... |
| ltxr 12504 | The 'less than' binary rel... |
| elxr 12505 | Membership in the set of e... |
| xrnemnf 12506 | An extended real other tha... |
| xrnepnf 12507 | An extended real other tha... |
| xrltnr 12508 | The extended real 'less th... |
| ltpnf 12509 | Any (finite) real is less ... |
| ltpnfd 12510 | Any (finite) real is less ... |
| 0ltpnf 12511 | Zero is less than plus inf... |
| mnflt 12512 | Minus infinity is less tha... |
| mnfltd 12513 | Minus infinity is less tha... |
| mnflt0 12514 | Minus infinity is less tha... |
| mnfltpnf 12515 | Minus infinity is less tha... |
| mnfltxr 12516 | Minus infinity is less tha... |
| pnfnlt 12517 | No extended real is greate... |
| nltmnf 12518 | No extended real is less t... |
| pnfge 12519 | Plus infinity is an upper ... |
| xnn0n0n1ge2b 12520 | An extended nonnegative in... |
| 0lepnf 12521 | 0 less than or equal to po... |
| xnn0ge0 12522 | An extended nonnegative in... |
| mnfle 12523 | Minus infinity is less tha... |
| xrltnsym 12524 | Ordering on the extended r... |
| xrltnsym2 12525 | 'Less than' is antisymmetr... |
| xrlttri 12526 | Ordering on the extended r... |
| xrlttr 12527 | Ordering on the extended r... |
| xrltso 12528 | 'Less than' is a strict or... |
| xrlttri2 12529 | Trichotomy law for 'less t... |
| xrlttri3 12530 | Trichotomy law for 'less t... |
| xrleloe 12531 | 'Less than or equal' expre... |
| xrleltne 12532 | 'Less than or equal to' im... |
| xrltlen 12533 | 'Less than' expressed in t... |
| dfle2 12534 | Alternative definition of ... |
| dflt2 12535 | Alternative definition of ... |
| xrltle 12536 | 'Less than' implies 'less ... |
| xrltled 12537 | 'Less than' implies 'less ... |
| xrleid 12538 | 'Less than or equal to' is... |
| xrleidd 12539 | 'Less than or equal to' is... |
| xrletri 12540 | Trichotomy law for extende... |
| xrletri3 12541 | Trichotomy law for extende... |
| xrletrid 12542 | Trichotomy law for extende... |
| xrlelttr 12543 | Transitive law for orderin... |
| xrltletr 12544 | Transitive law for orderin... |
| xrletr 12545 | Transitive law for orderin... |
| xrlttrd 12546 | Transitive law for orderin... |
| xrlelttrd 12547 | Transitive law for orderin... |
| xrltletrd 12548 | Transitive law for orderin... |
| xrletrd 12549 | Transitive law for orderin... |
| xrltne 12550 | 'Less than' implies not eq... |
| nltpnft 12551 | An extended real is not le... |
| xgepnf 12552 | An extended real which is ... |
| ngtmnft 12553 | An extended real is not gr... |
| xlemnf 12554 | An extended real which is ... |
| xrrebnd 12555 | An extended real is real i... |
| xrre 12556 | A way of proving that an e... |
| xrre2 12557 | An extended real between t... |
| xrre3 12558 | A way of proving that an e... |
| ge0gtmnf 12559 | A nonnegative extended rea... |
| ge0nemnf 12560 | A nonnegative extended rea... |
| xrrege0 12561 | A nonnegative extended rea... |
| xrmax1 12562 | An extended real is less t... |
| xrmax2 12563 | An extended real is less t... |
| xrmin1 12564 | The minimum of two extende... |
| xrmin2 12565 | The minimum of two extende... |
| xrmaxeq 12566 | The maximum of two extende... |
| xrmineq 12567 | The minimum of two extende... |
| xrmaxlt 12568 | Two ways of saying the max... |
| xrltmin 12569 | Two ways of saying an exte... |
| xrmaxle 12570 | Two ways of saying the max... |
| xrlemin 12571 | Two ways of saying a numbe... |
| max1 12572 | A number is less than or e... |
| max1ALT 12573 | A number is less than or e... |
| max2 12574 | A number is less than or e... |
| 2resupmax 12575 | The supremum of two real n... |
| min1 12576 | The minimum of two numbers... |
| min2 12577 | The minimum of two numbers... |
| maxle 12578 | Two ways of saying the max... |
| lemin 12579 | Two ways of saying a numbe... |
| maxlt 12580 | Two ways of saying the max... |
| ltmin 12581 | Two ways of saying a numbe... |
| lemaxle 12582 | A real number which is les... |
| max0sub 12583 | Decompose a real number in... |
| ifle 12584 | An if statement transforms... |
| z2ge 12585 | There exists an integer gr... |
| qbtwnre 12586 | The rational numbers are d... |
| qbtwnxr 12587 | The rational numbers are d... |
| qsqueeze 12588 | If a nonnegative real is l... |
| qextltlem 12589 | Lemma for ~ qextlt and qex... |
| qextlt 12590 | An extensionality-like pro... |
| qextle 12591 | An extensionality-like pro... |
| xralrple 12592 | Show that ` A ` is less th... |
| alrple 12593 | Show that ` A ` is less th... |
| xnegeq 12594 | Equality of two extended n... |
| xnegex 12595 | A negative extended real e... |
| xnegpnf 12596 | Minus ` +oo ` . Remark of... |
| xnegmnf 12597 | Minus ` -oo ` . Remark of... |
| rexneg 12598 | Minus a real number. Rema... |
| xneg0 12599 | The negative of zero. (Co... |
| xnegcl 12600 | Closure of extended real n... |
| xnegneg 12601 | Extended real version of ~... |
| xneg11 12602 | Extended real version of ~... |
| xltnegi 12603 | Forward direction of ~ xlt... |
| xltneg 12604 | Extended real version of ~... |
| xleneg 12605 | Extended real version of ~... |
| xlt0neg1 12606 | Extended real version of ~... |
| xlt0neg2 12607 | Extended real version of ~... |
| xle0neg1 12608 | Extended real version of ~... |
| xle0neg2 12609 | Extended real version of ~... |
| xaddval 12610 | Value of the extended real... |
| xaddf 12611 | The extended real addition... |
| xmulval 12612 | Value of the extended real... |
| xaddpnf1 12613 | Addition of positive infin... |
| xaddpnf2 12614 | Addition of positive infin... |
| xaddmnf1 12615 | Addition of negative infin... |
| xaddmnf2 12616 | Addition of negative infin... |
| pnfaddmnf 12617 | Addition of positive and n... |
| mnfaddpnf 12618 | Addition of negative and p... |
| rexadd 12619 | The extended real addition... |
| rexsub 12620 | Extended real subtraction ... |
| rexaddd 12621 | The extended real addition... |
| xnn0xaddcl 12622 | The extended nonnegative i... |
| xaddnemnf 12623 | Closure of extended real a... |
| xaddnepnf 12624 | Closure of extended real a... |
| xnegid 12625 | Extended real version of ~... |
| xaddcl 12626 | The extended real addition... |
| xaddcom 12627 | The extended real addition... |
| xaddid1 12628 | Extended real version of ~... |
| xaddid2 12629 | Extended real version of ~... |
| xaddid1d 12630 | ` 0 ` is a right identity ... |
| xnn0lem1lt 12631 | Extended nonnegative integ... |
| xnn0lenn0nn0 12632 | An extended nonnegative in... |
| xnn0le2is012 12633 | An extended nonnegative in... |
| xnn0xadd0 12634 | The sum of two extended no... |
| xnegdi 12635 | Extended real version of ~... |
| xaddass 12636 | Associativity of extended ... |
| xaddass2 12637 | Associativity of extended ... |
| xpncan 12638 | Extended real version of ~... |
| xnpcan 12639 | Extended real version of ~... |
| xleadd1a 12640 | Extended real version of ~... |
| xleadd2a 12641 | Commuted form of ~ xleadd1... |
| xleadd1 12642 | Weakened version of ~ xlea... |
| xltadd1 12643 | Extended real version of ~... |
| xltadd2 12644 | Extended real version of ~... |
| xaddge0 12645 | The sum of nonnegative ext... |
| xle2add 12646 | Extended real version of ~... |
| xlt2add 12647 | Extended real version of ~... |
| xsubge0 12648 | Extended real version of ~... |
| xposdif 12649 | Extended real version of ~... |
| xlesubadd 12650 | Under certain conditions, ... |
| xmullem 12651 | Lemma for ~ rexmul . (Con... |
| xmullem2 12652 | Lemma for ~ xmulneg1 . (C... |
| xmulcom 12653 | Extended real multiplicati... |
| xmul01 12654 | Extended real version of ~... |
| xmul02 12655 | Extended real version of ~... |
| xmulneg1 12656 | Extended real version of ~... |
| xmulneg2 12657 | Extended real version of ~... |
| rexmul 12658 | The extended real multipli... |
| xmulf 12659 | The extended real multipli... |
| xmulcl 12660 | Closure of extended real m... |
| xmulpnf1 12661 | Multiplication by plus inf... |
| xmulpnf2 12662 | Multiplication by plus inf... |
| xmulmnf1 12663 | Multiplication by minus in... |
| xmulmnf2 12664 | Multiplication by minus in... |
| xmulpnf1n 12665 | Multiplication by plus inf... |
| xmulid1 12666 | Extended real version of ~... |
| xmulid2 12667 | Extended real version of ~... |
| xmulm1 12668 | Extended real version of ~... |
| xmulasslem2 12669 | Lemma for ~ xmulass . (Co... |
| xmulgt0 12670 | Extended real version of ~... |
| xmulge0 12671 | Extended real version of ~... |
| xmulasslem 12672 | Lemma for ~ xmulass . (Co... |
| xmulasslem3 12673 | Lemma for ~ xmulass . (Co... |
| xmulass 12674 | Associativity of the exten... |
| xlemul1a 12675 | Extended real version of ~... |
| xlemul2a 12676 | Extended real version of ~... |
| xlemul1 12677 | Extended real version of ~... |
| xlemul2 12678 | Extended real version of ~... |
| xltmul1 12679 | Extended real version of ~... |
| xltmul2 12680 | Extended real version of ~... |
| xadddilem 12681 | Lemma for ~ xadddi . (Con... |
| xadddi 12682 | Distributive property for ... |
| xadddir 12683 | Commuted version of ~ xadd... |
| xadddi2 12684 | The assumption that the mu... |
| xadddi2r 12685 | Commuted version of ~ xadd... |
| x2times 12686 | Extended real version of ~... |
| xnegcld 12687 | Closure of extended real n... |
| xaddcld 12688 | The extended real addition... |
| xmulcld 12689 | Closure of extended real m... |
| xadd4d 12690 | Rearrangement of 4 terms i... |
| xnn0add4d 12691 | Rearrangement of 4 terms i... |
| xrsupexmnf 12692 | Adding minus infinity to a... |
| xrinfmexpnf 12693 | Adding plus infinity to a ... |
| xrsupsslem 12694 | Lemma for ~ xrsupss . (Co... |
| xrinfmsslem 12695 | Lemma for ~ xrinfmss . (C... |
| xrsupss 12696 | Any subset of extended rea... |
| xrinfmss 12697 | Any subset of extended rea... |
| xrinfmss2 12698 | Any subset of extended rea... |
| xrub 12699 | By quantifying only over r... |
| supxr 12700 | The supremum of a set of e... |
| supxr2 12701 | The supremum of a set of e... |
| supxrcl 12702 | The supremum of an arbitra... |
| supxrun 12703 | The supremum of the union ... |
| supxrmnf 12704 | Adding minus infinity to a... |
| supxrpnf 12705 | The supremum of a set of e... |
| supxrunb1 12706 | The supremum of an unbound... |
| supxrunb2 12707 | The supremum of an unbound... |
| supxrbnd1 12708 | The supremum of a bounded-... |
| supxrbnd2 12709 | The supremum of a bounded-... |
| xrsup0 12710 | The supremum of an empty s... |
| supxrub 12711 | A member of a set of exten... |
| supxrlub 12712 | The supremum of a set of e... |
| supxrleub 12713 | The supremum of a set of e... |
| supxrre 12714 | The real and extended real... |
| supxrbnd 12715 | The supremum of a bounded-... |
| supxrgtmnf 12716 | The supremum of a nonempty... |
| supxrre1 12717 | The supremum of a nonempty... |
| supxrre2 12718 | The supremum of a nonempty... |
| supxrss 12719 | Smaller sets of extended r... |
| infxrcl 12720 | The infimum of an arbitrar... |
| infxrlb 12721 | A member of a set of exten... |
| infxrgelb 12722 | The infimum of a set of ex... |
| infxrre 12723 | The real and extended real... |
| infxrmnf 12724 | The infinimum of a set of ... |
| xrinf0 12725 | The infimum of the empty s... |
| infxrss 12726 | Larger sets of extended re... |
| reltre 12727 | For all real numbers there... |
| rpltrp 12728 | For all positive real numb... |
| reltxrnmnf 12729 | For all extended real numb... |
| infmremnf 12730 | The infimum of the reals i... |
| infmrp1 12731 | The infimum of the positiv... |
| ixxval 12740 | Value of the interval func... |
| elixx1 12741 | Membership in an interval ... |
| ixxf 12742 | The set of intervals of ex... |
| ixxex 12743 | The set of intervals of ex... |
| ixxssxr 12744 | The set of intervals of ex... |
| elixx3g 12745 | Membership in a set of ope... |
| ixxssixx 12746 | An interval is a subset of... |
| ixxdisj 12747 | Split an interval into dis... |
| ixxun 12748 | Split an interval into two... |
| ixxin 12749 | Intersection of two interv... |
| ixxss1 12750 | Subset relationship for in... |
| ixxss2 12751 | Subset relationship for in... |
| ixxss12 12752 | Subset relationship for in... |
| ixxub 12753 | Extract the upper bound of... |
| ixxlb 12754 | Extract the lower bound of... |
| iooex 12755 | The set of open intervals ... |
| iooval 12756 | Value of the open interval... |
| ioo0 12757 | An empty open interval of ... |
| ioon0 12758 | An open interval of extend... |
| ndmioo 12759 | The open interval function... |
| iooid 12760 | An open interval with iden... |
| elioo3g 12761 | Membership in a set of ope... |
| elioore 12762 | A member of an open interv... |
| lbioo 12763 | An open interval does not ... |
| ubioo 12764 | An open interval does not ... |
| iooval2 12765 | Value of the open interval... |
| iooin 12766 | Intersection of two open i... |
| iooss1 12767 | Subset relationship for op... |
| iooss2 12768 | Subset relationship for op... |
| iocval 12769 | Value of the open-below, c... |
| icoval 12770 | Value of the closed-below,... |
| iccval 12771 | Value of the closed interv... |
| elioo1 12772 | Membership in an open inte... |
| elioo2 12773 | Membership in an open inte... |
| elioc1 12774 | Membership in an open-belo... |
| elico1 12775 | Membership in a closed-bel... |
| elicc1 12776 | Membership in a closed int... |
| iccid 12777 | A closed interval with ide... |
| ico0 12778 | An empty open interval of ... |
| ioc0 12779 | An empty open interval of ... |
| icc0 12780 | An empty closed interval o... |
| elicod 12781 | Membership in a left-close... |
| icogelb 12782 | An element of a left-close... |
| elicore 12783 | A member of a left-closed ... |
| ubioc1 12784 | The upper bound belongs to... |
| lbico1 12785 | The lower bound belongs to... |
| iccleub 12786 | An element of a closed int... |
| iccgelb 12787 | An element of a closed int... |
| elioo5 12788 | Membership in an open inte... |
| eliooxr 12789 | A nonempty open interval s... |
| eliooord 12790 | Ordering implied by a memb... |
| elioo4g 12791 | Membership in an open inte... |
| ioossre 12792 | An open interval is a set ... |
| elioc2 12793 | Membership in an open-belo... |
| elico2 12794 | Membership in a closed-bel... |
| elicc2 12795 | Membership in a closed rea... |
| elicc2i 12796 | Inference for membership i... |
| elicc4 12797 | Membership in a closed rea... |
| iccss 12798 | Condition for a closed int... |
| iccssioo 12799 | Condition for a closed int... |
| icossico 12800 | Condition for a closed-bel... |
| iccss2 12801 | Condition for a closed int... |
| iccssico 12802 | Condition for a closed int... |
| iccssioo2 12803 | Condition for a closed int... |
| iccssico2 12804 | Condition for a closed int... |
| ioomax 12805 | The open interval from min... |
| iccmax 12806 | The closed interval from m... |
| ioopos 12807 | The set of positive reals ... |
| ioorp 12808 | The set of positive reals ... |
| iooshf 12809 | Shift the arguments of the... |
| iocssre 12810 | A closed-above interval wi... |
| icossre 12811 | A closed-below interval wi... |
| iccssre 12812 | A closed real interval is ... |
| iccssxr 12813 | A closed interval is a set... |
| iocssxr 12814 | An open-below, closed-abov... |
| icossxr 12815 | A closed-below, open-above... |
| ioossicc 12816 | An open interval is a subs... |
| eliccxr 12817 | A member of a closed inter... |
| icossicc 12818 | A closed-below, open-above... |
| iocssicc 12819 | A closed-above, open-below... |
| ioossico 12820 | An open interval is a subs... |
| iocssioo 12821 | Condition for a closed int... |
| icossioo 12822 | Condition for a closed int... |
| ioossioo 12823 | Condition for an open inte... |
| iccsupr 12824 | A nonempty subset of a clo... |
| elioopnf 12825 | Membership in an unbounded... |
| elioomnf 12826 | Membership in an unbounded... |
| elicopnf 12827 | Membership in a closed unb... |
| repos 12828 | Two ways of saying that a ... |
| ioof 12829 | The set of open intervals ... |
| iccf 12830 | The set of closed interval... |
| unirnioo 12831 | The union of the range of ... |
| dfioo2 12832 | Alternate definition of th... |
| ioorebas 12833 | Open intervals are element... |
| xrge0neqmnf 12834 | A nonnegative extended rea... |
| xrge0nre 12835 | An extended real which is ... |
| elrege0 12836 | The predicate "is a nonneg... |
| nn0rp0 12837 | A nonnegative integer is a... |
| rge0ssre 12838 | Nonnegative real numbers a... |
| elxrge0 12839 | Elementhood in the set of ... |
| 0e0icopnf 12840 | 0 is a member of ` ( 0 [,)... |
| 0e0iccpnf 12841 | 0 is a member of ` ( 0 [,]... |
| ge0addcl 12842 | The nonnegative reals are ... |
| ge0mulcl 12843 | The nonnegative reals are ... |
| ge0xaddcl 12844 | The nonnegative reals are ... |
| ge0xmulcl 12845 | The nonnegative extended r... |
| lbicc2 12846 | The lower bound of a close... |
| ubicc2 12847 | The upper bound of a close... |
| elicc01 12848 | Membership in the closed r... |
| 0elunit 12849 | Zero is an element of the ... |
| 1elunit 12850 | One is an element of the c... |
| iooneg 12851 | Membership in a negated op... |
| iccneg 12852 | Membership in a negated cl... |
| icoshft 12853 | A shifted real is a member... |
| icoshftf1o 12854 | Shifting a closed-below, o... |
| icoun 12855 | The union of two adjacent ... |
| icodisj 12856 | Adjacent left-closed right... |
| ioounsn 12857 | The union of an open inter... |
| snunioo 12858 | The closure of one end of ... |
| snunico 12859 | The closure of the open en... |
| snunioc 12860 | The closure of the open en... |
| prunioo 12861 | The closure of an open rea... |
| ioodisj 12862 | If the upper bound of one ... |
| ioojoin 12863 | Join two open intervals to... |
| difreicc 12864 | The class difference of ` ... |
| iccsplit 12865 | Split a closed interval in... |
| iccshftr 12866 | Membership in a shifted in... |
| iccshftri 12867 | Membership in a shifted in... |
| iccshftl 12868 | Membership in a shifted in... |
| iccshftli 12869 | Membership in a shifted in... |
| iccdil 12870 | Membership in a dilated in... |
| iccdili 12871 | Membership in a dilated in... |
| icccntr 12872 | Membership in a contracted... |
| icccntri 12873 | Membership in a contracted... |
| divelunit 12874 | A condition for a ratio to... |
| lincmb01cmp 12875 | A linear combination of tw... |
| iccf1o 12876 | Describe a bijection from ... |
| iccen 12877 | Any nontrivial closed inte... |
| xov1plusxeqvd 12878 | A complex number ` X ` is ... |
| unitssre 12879 | ` ( 0 [,] 1 ) ` is a subse... |
| supicc 12880 | Supremum of a bounded set ... |
| supiccub 12881 | The supremum of a bounded ... |
| supicclub 12882 | The supremum of a bounded ... |
| supicclub2 12883 | The supremum of a bounded ... |
| zltaddlt1le 12884 | The sum of an integer and ... |
| xnn0xrge0 12885 | An extended nonnegative in... |
| fzval 12888 | The value of a finite set ... |
| fzval2 12889 | An alternative way of expr... |
| fzf 12890 | Establish the domain and c... |
| elfz1 12891 | Membership in a finite set... |
| elfz 12892 | Membership in a finite set... |
| elfz2 12893 | Membership in a finite set... |
| elfz5 12894 | Membership in a finite set... |
| elfz4 12895 | Membership in a finite set... |
| elfzuzb 12896 | Membership in a finite set... |
| eluzfz 12897 | Membership in a finite set... |
| elfzuz 12898 | A member of a finite set o... |
| elfzuz3 12899 | Membership in a finite set... |
| elfzel2 12900 | Membership in a finite set... |
| elfzel1 12901 | Membership in a finite set... |
| elfzelz 12902 | A member of a finite set o... |
| fzssz 12903 | A finite sequence of integ... |
| elfzle1 12904 | A member of a finite set o... |
| elfzle2 12905 | A member of a finite set o... |
| elfzuz2 12906 | Implication of membership ... |
| elfzle3 12907 | Membership in a finite set... |
| eluzfz1 12908 | Membership in a finite set... |
| eluzfz2 12909 | Membership in a finite set... |
| eluzfz2b 12910 | Membership in a finite set... |
| elfz3 12911 | Membership in a finite set... |
| elfz1eq 12912 | Membership in a finite set... |
| elfzubelfz 12913 | If there is a member in a ... |
| peano2fzr 12914 | A Peano-postulate-like the... |
| fzn0 12915 | Properties of a finite int... |
| fz0 12916 | A finite set of sequential... |
| fzn 12917 | A finite set of sequential... |
| fzen 12918 | A shifted finite set of se... |
| fz1n 12919 | A 1-based finite set of se... |
| 0nelfz1 12920 | 0 is not an element of a f... |
| 0fz1 12921 | Two ways to say a finite 1... |
| fz10 12922 | There are no integers betw... |
| uzsubsubfz 12923 | Membership of an integer g... |
| uzsubsubfz1 12924 | Membership of an integer g... |
| ige3m2fz 12925 | Membership of an integer g... |
| fzsplit2 12926 | Split a finite interval of... |
| fzsplit 12927 | Split a finite interval of... |
| fzdisj 12928 | Condition for two finite i... |
| fz01en 12929 | 0-based and 1-based finite... |
| elfznn 12930 | A member of a finite set o... |
| elfz1end 12931 | A nonempty finite range of... |
| fz1ssnn 12932 | A finite set of positive i... |
| fznn0sub 12933 | Subtraction closure for a ... |
| fzmmmeqm 12934 | Subtracting the difference... |
| fzaddel 12935 | Membership of a sum in a f... |
| fzadd2 12936 | Membership of a sum in a f... |
| fzsubel 12937 | Membership of a difference... |
| fzopth 12938 | A finite set of sequential... |
| fzass4 12939 | Two ways to express a nond... |
| fzss1 12940 | Subset relationship for fi... |
| fzss2 12941 | Subset relationship for fi... |
| fzssuz 12942 | A finite set of sequential... |
| fzsn 12943 | A finite interval of integ... |
| fzssp1 12944 | Subset relationship for fi... |
| fzssnn 12945 | Finite sets of sequential ... |
| ssfzunsnext 12946 | A subset of a finite seque... |
| ssfzunsn 12947 | A subset of a finite seque... |
| fzsuc 12948 | Join a successor to the en... |
| fzpred 12949 | Join a predecessor to the ... |
| fzpreddisj 12950 | A finite set of sequential... |
| elfzp1 12951 | Append an element to a fin... |
| fzp1ss 12952 | Subset relationship for fi... |
| fzelp1 12953 | Membership in a set of seq... |
| fzp1elp1 12954 | Add one to an element of a... |
| fznatpl1 12955 | Shift membership in a fini... |
| fzpr 12956 | A finite interval of integ... |
| fztp 12957 | A finite interval of integ... |
| fz12pr 12958 | An integer range between 1... |
| fzsuc2 12959 | Join a successor to the en... |
| fzp1disj 12960 | ` ( M ... ( N + 1 ) ) ` is... |
| fzdifsuc 12961 | Remove a successor from th... |
| fzprval 12962 | Two ways of defining the f... |
| fztpval 12963 | Two ways of defining the f... |
| fzrev 12964 | Reversal of start and end ... |
| fzrev2 12965 | Reversal of start and end ... |
| fzrev2i 12966 | Reversal of start and end ... |
| fzrev3 12967 | The "complement" of a memb... |
| fzrev3i 12968 | The "complement" of a memb... |
| fznn 12969 | Finite set of sequential i... |
| elfz1b 12970 | Membership in a 1-based fi... |
| elfz1uz 12971 | Membership in a 1-based fi... |
| elfzm11 12972 | Membership in a finite set... |
| uzsplit 12973 | Express an upper integer s... |
| uzdisj 12974 | The first ` N ` elements o... |
| fseq1p1m1 12975 | Add/remove an item to/from... |
| fseq1m1p1 12976 | Add/remove an item to/from... |
| fz1sbc 12977 | Quantification over a one-... |
| elfzp1b 12978 | An integer is a member of ... |
| elfzm1b 12979 | An integer is a member of ... |
| elfzp12 12980 | Options for membership in ... |
| fzm1 12981 | Choices for an element of ... |
| fzneuz 12982 | No finite set of sequentia... |
| fznuz 12983 | Disjointness of the upper ... |
| uznfz 12984 | Disjointness of the upper ... |
| fzp1nel 12985 | One plus the upper bound o... |
| fzrevral 12986 | Reversal of scanning order... |
| fzrevral2 12987 | Reversal of scanning order... |
| fzrevral3 12988 | Reversal of scanning order... |
| fzshftral 12989 | Shift the scanning order i... |
| ige2m1fz1 12990 | Membership of an integer g... |
| ige2m1fz 12991 | Membership in a 0-based fi... |
| elfz2nn0 12992 | Membership in a finite set... |
| fznn0 12993 | Characterization of a fini... |
| elfznn0 12994 | A member of a finite set o... |
| elfz3nn0 12995 | The upper bound of a nonem... |
| fz0ssnn0 12996 | Finite sets of sequential ... |
| fz1ssfz0 12997 | Subset relationship for fi... |
| 0elfz 12998 | 0 is an element of a finit... |
| nn0fz0 12999 | A nonnegative integer is a... |
| elfz0add 13000 | An element of a finite set... |
| fz0sn 13001 | An integer range from 0 to... |
| fz0tp 13002 | An integer range from 0 to... |
| fz0to3un2pr 13003 | An integer range from 0 to... |
| fz0to4untppr 13004 | An integer range from 0 to... |
| elfz0ubfz0 13005 | An element of a finite set... |
| elfz0fzfz0 13006 | A member of a finite set o... |
| fz0fzelfz0 13007 | If a member of a finite se... |
| fznn0sub2 13008 | Subtraction closure for a ... |
| uzsubfz0 13009 | Membership of an integer g... |
| fz0fzdiffz0 13010 | The difference of an integ... |
| elfzmlbm 13011 | Subtracting the lower boun... |
| elfzmlbp 13012 | Subtracting the lower boun... |
| fzctr 13013 | Lemma for theorems about t... |
| difelfzle 13014 | The difference of two inte... |
| difelfznle 13015 | The difference of two inte... |
| nn0split 13016 | Express the set of nonnega... |
| nn0disj 13017 | The first ` N + 1 ` elemen... |
| fz0sn0fz1 13018 | A finite set of sequential... |
| fvffz0 13019 | The function value of a fu... |
| 1fv 13020 | A function on a singleton.... |
| 4fvwrd4 13021 | The first four function va... |
| 2ffzeq 13022 | Two functions over 0-based... |
| preduz 13023 | The value of the predecess... |
| prednn 13024 | The value of the predecess... |
| prednn0 13025 | The value of the predecess... |
| predfz 13026 | Calculate the predecessor ... |
| fzof 13029 | Functionality of the half-... |
| elfzoel1 13030 | Reverse closure for half-o... |
| elfzoel2 13031 | Reverse closure for half-o... |
| elfzoelz 13032 | Reverse closure for half-o... |
| fzoval 13033 | Value of the half-open int... |
| elfzo 13034 | Membership in a half-open ... |
| elfzo2 13035 | Membership in a half-open ... |
| elfzouz 13036 | Membership in a half-open ... |
| nelfzo 13037 | An integer not being a mem... |
| fzolb 13038 | The left endpoint of a hal... |
| fzolb2 13039 | The left endpoint of a hal... |
| elfzole1 13040 | A member in a half-open in... |
| elfzolt2 13041 | A member in a half-open in... |
| elfzolt3 13042 | Membership in a half-open ... |
| elfzolt2b 13043 | A member in a half-open in... |
| elfzolt3b 13044 | Membership in a half-open ... |
| fzonel 13045 | A half-open range does not... |
| elfzouz2 13046 | The upper bound of a half-... |
| elfzofz 13047 | A half-open range is conta... |
| elfzo3 13048 | Express membership in a ha... |
| fzon0 13049 | A half-open integer interv... |
| fzossfz 13050 | A half-open range is conta... |
| fzossz 13051 | A half-open integer interv... |
| fzon 13052 | A half-open set of sequent... |
| fzo0n 13053 | A half-open range of nonne... |
| fzonlt0 13054 | A half-open integer range ... |
| fzo0 13055 | Half-open sets with equal ... |
| fzonnsub 13056 | If ` K < N ` then ` N - K ... |
| fzonnsub2 13057 | If ` M < N ` then ` N - M ... |
| fzoss1 13058 | Subset relationship for ha... |
| fzoss2 13059 | Subset relationship for ha... |
| fzossrbm1 13060 | Subset of a half-open rang... |
| fzo0ss1 13061 | Subset relationship for ha... |
| fzossnn0 13062 | A half-open integer range ... |
| fzospliti 13063 | One direction of splitting... |
| fzosplit 13064 | Split a half-open integer ... |
| fzodisj 13065 | Abutting half-open integer... |
| fzouzsplit 13066 | Split an upper integer set... |
| fzouzdisj 13067 | A half-open integer range ... |
| fzoun 13068 | A half-open integer range ... |
| fzodisjsn 13069 | A half-open integer range ... |
| prinfzo0 13070 | The intersection of a half... |
| lbfzo0 13071 | An integer is strictly gre... |
| elfzo0 13072 | Membership in a half-open ... |
| elfzo0z 13073 | Membership in a half-open ... |
| nn0p1elfzo 13074 | A nonnegative integer incr... |
| elfzo0le 13075 | A member in a half-open ra... |
| elfzonn0 13076 | A member of a half-open ra... |
| fzonmapblen 13077 | The result of subtracting ... |
| fzofzim 13078 | If a nonnegative integer i... |
| fz1fzo0m1 13079 | Translation of one between... |
| fzossnn 13080 | Half-open integer ranges s... |
| elfzo1 13081 | Membership in a half-open ... |
| fzo1fzo0n0 13082 | An integer between 1 and a... |
| fzo0n0 13083 | A half-open integer range ... |
| fzoaddel 13084 | Translate membership in a ... |
| fzo0addel 13085 | Translate membership in a ... |
| fzo0addelr 13086 | Translate membership in a ... |
| fzoaddel2 13087 | Translate membership in a ... |
| elfzoext 13088 | Membership of an integer i... |
| elincfzoext 13089 | Membership of an increased... |
| fzosubel 13090 | Translate membership in a ... |
| fzosubel2 13091 | Membership in a translated... |
| fzosubel3 13092 | Membership in a translated... |
| eluzgtdifelfzo 13093 | Membership of the differen... |
| ige2m2fzo 13094 | Membership of an integer g... |
| fzocatel 13095 | Translate membership in a ... |
| ubmelfzo 13096 | If an integer in a 1-based... |
| elfzodifsumelfzo 13097 | If an integer is in a half... |
| elfzom1elp1fzo 13098 | Membership of an integer i... |
| elfzom1elfzo 13099 | Membership in a half-open ... |
| fzval3 13100 | Expressing a closed intege... |
| fz0add1fz1 13101 | Translate membership in a ... |
| fzosn 13102 | Expressing a singleton as ... |
| elfzomin 13103 | Membership of an integer i... |
| zpnn0elfzo 13104 | Membership of an integer i... |
| zpnn0elfzo1 13105 | Membership of an integer i... |
| fzosplitsnm1 13106 | Removing a singleton from ... |
| elfzonlteqm1 13107 | If an element of a half-op... |
| fzonn0p1 13108 | A nonnegative integer is e... |
| fzossfzop1 13109 | A half-open range of nonne... |
| fzonn0p1p1 13110 | If a nonnegative integer i... |
| elfzom1p1elfzo 13111 | Increasing an element of a... |
| fzo0ssnn0 13112 | Half-open integer ranges s... |
| fzo01 13113 | Expressing the singleton o... |
| fzo12sn 13114 | A 1-based half-open intege... |
| fzo13pr 13115 | A 1-based half-open intege... |
| fzo0to2pr 13116 | A half-open integer range ... |
| fzo0to3tp 13117 | A half-open integer range ... |
| fzo0to42pr 13118 | A half-open integer range ... |
| fzo1to4tp 13119 | A half-open integer range ... |
| fzo0sn0fzo1 13120 | A half-open range of nonne... |
| elfzo0l 13121 | A member of a half-open ra... |
| fzoend 13122 | The endpoint of a half-ope... |
| fzo0end 13123 | The endpoint of a zero-bas... |
| ssfzo12 13124 | Subset relationship for ha... |
| ssfzoulel 13125 | If a half-open integer ran... |
| ssfzo12bi 13126 | Subset relationship for ha... |
| ubmelm1fzo 13127 | The result of subtracting ... |
| fzofzp1 13128 | If a point is in a half-op... |
| fzofzp1b 13129 | If a point is in a half-op... |
| elfzom1b 13130 | An integer is a member of ... |
| elfzom1elp1fzo1 13131 | Membership of a nonnegativ... |
| elfzo1elm1fzo0 13132 | Membership of a positive i... |
| elfzonelfzo 13133 | If an element of a half-op... |
| fzonfzoufzol 13134 | If an element of a half-op... |
| elfzomelpfzo 13135 | An integer increased by an... |
| elfznelfzo 13136 | A value in a finite set of... |
| elfznelfzob 13137 | A value in a finite set of... |
| peano2fzor 13138 | A Peano-postulate-like the... |
| fzosplitsn 13139 | Extending a half-open rang... |
| fzosplitpr 13140 | Extending a half-open inte... |
| fzosplitprm1 13141 | Extending a half-open inte... |
| fzosplitsni 13142 | Membership in a half-open ... |
| fzisfzounsn 13143 | A finite interval of integ... |
| elfzr 13144 | A member of a finite inter... |
| elfzlmr 13145 | A member of a finite inter... |
| elfz0lmr 13146 | A member of a finite inter... |
| fzostep1 13147 | Two possibilities for a nu... |
| fzoshftral 13148 | Shift the scanning order i... |
| fzind2 13149 | Induction on the integers ... |
| fvinim0ffz 13150 | The function values for th... |
| injresinjlem 13151 | Lemma for ~ injresinj . (... |
| injresinj 13152 | A function whose restricti... |
| subfzo0 13153 | The difference between two... |
| flval 13158 | Value of the floor (greate... |
| flcl 13159 | The floor (greatest intege... |
| reflcl 13160 | The floor (greatest intege... |
| fllelt 13161 | A basic property of the fl... |
| flcld 13162 | The floor (greatest intege... |
| flle 13163 | A basic property of the fl... |
| flltp1 13164 | A basic property of the fl... |
| fllep1 13165 | A basic property of the fl... |
| fraclt1 13166 | The fractional part of a r... |
| fracle1 13167 | The fractional part of a r... |
| fracge0 13168 | The fractional part of a r... |
| flge 13169 | The floor function value i... |
| fllt 13170 | The floor function value i... |
| flflp1 13171 | Move floor function betwee... |
| flid 13172 | An integer is its own floo... |
| flidm 13173 | The floor function is idem... |
| flidz 13174 | A real number equals its f... |
| flltnz 13175 | If A is not an integer, th... |
| flwordi 13176 | Ordering relationship for ... |
| flword2 13177 | Ordering relationship for ... |
| flval2 13178 | An alternate way to define... |
| flval3 13179 | An alternate way to define... |
| flbi 13180 | A condition equivalent to ... |
| flbi2 13181 | A condition equivalent to ... |
| adddivflid 13182 | The floor of a sum of an i... |
| ico01fl0 13183 | The floor of a real number... |
| flge0nn0 13184 | The floor of a number grea... |
| flge1nn 13185 | The floor of a number grea... |
| fldivnn0 13186 | The floor function of a di... |
| refldivcl 13187 | The floor function of a di... |
| divfl0 13188 | The floor of a fraction is... |
| fladdz 13189 | An integer can be moved in... |
| flzadd 13190 | An integer can be moved in... |
| flmulnn0 13191 | Move a nonnegative integer... |
| btwnzge0 13192 | A real bounded between an ... |
| 2tnp1ge0ge0 13193 | Two times an integer plus ... |
| flhalf 13194 | Ordering relation for the ... |
| fldivle 13195 | The floor function of a di... |
| fldivnn0le 13196 | The floor function of a di... |
| flltdivnn0lt 13197 | The floor function of a di... |
| ltdifltdiv 13198 | If the dividend of a divis... |
| fldiv4p1lem1div2 13199 | The floor of an integer eq... |
| fldiv4lem1div2uz2 13200 | The floor of an integer gr... |
| fldiv4lem1div2 13201 | The floor of a positive in... |
| ceilval 13202 | The value of the ceiling f... |
| dfceil2 13203 | Alternative definition of ... |
| ceilval2 13204 | The value of the ceiling f... |
| ceicl 13205 | The ceiling function retur... |
| ceilcl 13206 | Closure of the ceiling fun... |
| ceige 13207 | The ceiling of a real numb... |
| ceilge 13208 | The ceiling of a real numb... |
| ceim1l 13209 | One less than the ceiling ... |
| ceilm1lt 13210 | One less than the ceiling ... |
| ceile 13211 | The ceiling of a real numb... |
| ceille 13212 | The ceiling of a real numb... |
| ceilid 13213 | An integer is its own ceil... |
| ceilidz 13214 | A real number equals its c... |
| flleceil 13215 | The floor of a real number... |
| fleqceilz 13216 | A real number is an intege... |
| quoremz 13217 | Quotient and remainder of ... |
| quoremnn0 13218 | Quotient and remainder of ... |
| quoremnn0ALT 13219 | Alternate proof of ~ quore... |
| intfrac2 13220 | Decompose a real into inte... |
| intfracq 13221 | Decompose a rational numbe... |
| fldiv 13222 | Cancellation of the embedd... |
| fldiv2 13223 | Cancellation of an embedde... |
| fznnfl 13224 | Finite set of sequential i... |
| uzsup 13225 | An upper set of integers i... |
| ioopnfsup 13226 | An upper set of reals is u... |
| icopnfsup 13227 | An upper set of reals is u... |
| rpsup 13228 | The positive reals are unb... |
| resup 13229 | The real numbers are unbou... |
| xrsup 13230 | The extended real numbers ... |
| modval 13233 | The value of the modulo op... |
| modvalr 13234 | The value of the modulo op... |
| modcl 13235 | Closure law for the modulo... |
| flpmodeq 13236 | Partition of a division in... |
| modcld 13237 | Closure law for the modulo... |
| mod0 13238 | ` A mod B ` is zero iff ` ... |
| mulmod0 13239 | The product of an integer ... |
| negmod0 13240 | ` A ` is divisible by ` B ... |
| modge0 13241 | The modulo operation is no... |
| modlt 13242 | The modulo operation is le... |
| modelico 13243 | Modular reduction produces... |
| moddiffl 13244 | Value of the modulo operat... |
| moddifz 13245 | The modulo operation diffe... |
| modfrac 13246 | The fractional part of a n... |
| flmod 13247 | The floor function express... |
| intfrac 13248 | Break a number into its in... |
| zmod10 13249 | An integer modulo 1 is 0. ... |
| zmod1congr 13250 | Two arbitrary integers are... |
| modmulnn 13251 | Move a positive integer in... |
| modvalp1 13252 | The value of the modulo op... |
| zmodcl 13253 | Closure law for the modulo... |
| zmodcld 13254 | Closure law for the modulo... |
| zmodfz 13255 | An integer mod ` B ` lies ... |
| zmodfzo 13256 | An integer mod ` B ` lies ... |
| zmodfzp1 13257 | An integer mod ` B ` lies ... |
| modid 13258 | Identity law for modulo. ... |
| modid0 13259 | A positive real number mod... |
| modid2 13260 | Identity law for modulo. ... |
| zmodid2 13261 | Identity law for modulo re... |
| zmodidfzo 13262 | Identity law for modulo re... |
| zmodidfzoimp 13263 | Identity law for modulo re... |
| 0mod 13264 | Special case: 0 modulo a p... |
| 1mod 13265 | Special case: 1 modulo a r... |
| modabs 13266 | Absorption law for modulo.... |
| modabs2 13267 | Absorption law for modulo.... |
| modcyc 13268 | The modulo operation is pe... |
| modcyc2 13269 | The modulo operation is pe... |
| modadd1 13270 | Addition property of the m... |
| modaddabs 13271 | Absorption law for modulo.... |
| modaddmod 13272 | The sum of a real number m... |
| muladdmodid 13273 | The sum of a positive real... |
| mulp1mod1 13274 | The product of an integer ... |
| modmuladd 13275 | Decomposition of an intege... |
| modmuladdim 13276 | Implication of a decomposi... |
| modmuladdnn0 13277 | Implication of a decomposi... |
| negmod 13278 | The negation of a number m... |
| m1modnnsub1 13279 | Minus one modulo a positiv... |
| m1modge3gt1 13280 | Minus one modulo an intege... |
| addmodid 13281 | The sum of a positive inte... |
| addmodidr 13282 | The sum of a positive inte... |
| modadd2mod 13283 | The sum of a real number m... |
| modm1p1mod0 13284 | If a real number modulo a ... |
| modltm1p1mod 13285 | If a real number modulo a ... |
| modmul1 13286 | Multiplication property of... |
| modmul12d 13287 | Multiplication property of... |
| modnegd 13288 | Negation property of the m... |
| modadd12d 13289 | Additive property of the m... |
| modsub12d 13290 | Subtraction property of th... |
| modsubmod 13291 | The difference of a real n... |
| modsubmodmod 13292 | The difference of a real n... |
| 2txmodxeq0 13293 | Two times a positive real ... |
| 2submod 13294 | If a real number is betwee... |
| modifeq2int 13295 | If a nonnegative integer i... |
| modaddmodup 13296 | The sum of an integer modu... |
| modaddmodlo 13297 | The sum of an integer modu... |
| modmulmod 13298 | The product of a real numb... |
| modmulmodr 13299 | The product of an integer ... |
| modaddmulmod 13300 | The sum of a real number a... |
| moddi 13301 | Distribute multiplication ... |
| modsubdir 13302 | Distribute the modulo oper... |
| modeqmodmin 13303 | A real number equals the d... |
| modirr 13304 | A number modulo an irratio... |
| modfzo0difsn 13305 | For a number within a half... |
| modsumfzodifsn 13306 | The sum of a number within... |
| modlteq 13307 | Two nonnegative integers l... |
| addmodlteq 13308 | Two nonnegative integers l... |
| om2uz0i 13309 | The mapping ` G ` is a one... |
| om2uzsuci 13310 | The value of ` G ` (see ~ ... |
| om2uzuzi 13311 | The value ` G ` (see ~ om2... |
| om2uzlti 13312 | Less-than relation for ` G... |
| om2uzlt2i 13313 | The mapping ` G ` (see ~ o... |
| om2uzrani 13314 | Range of ` G ` (see ~ om2u... |
| om2uzf1oi 13315 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzisoi 13316 | ` G ` (see ~ om2uz0i ) is ... |
| om2uzoi 13317 | An alternative definition ... |
| om2uzrdg 13318 | A helper lemma for the val... |
| uzrdglem 13319 | A helper lemma for the val... |
| uzrdgfni 13320 | The recursive definition g... |
| uzrdg0i 13321 | Initial value of a recursi... |
| uzrdgsuci 13322 | Successor value of a recur... |
| ltweuz 13323 | ` < ` is a well-founded re... |
| ltwenn 13324 | Less than well-orders the ... |
| ltwefz 13325 | Less than well-orders a se... |
| uzenom 13326 | An upper integer set is de... |
| uzinf 13327 | An upper integer set is in... |
| nnnfi 13328 | The set of positive intege... |
| uzrdgxfr 13329 | Transfer the value of the ... |
| fzennn 13330 | The cardinality of a finit... |
| fzen2 13331 | The cardinality of a finit... |
| cardfz 13332 | The cardinality of a finit... |
| hashgf1o 13333 | ` G ` maps ` _om ` one-to-... |
| fzfi 13334 | A finite interval of integ... |
| fzfid 13335 | Commonly used special case... |
| fzofi 13336 | Half-open integer sets are... |
| fsequb 13337 | The values of a finite rea... |
| fsequb2 13338 | The values of a finite rea... |
| fseqsupcl 13339 | The values of a finite rea... |
| fseqsupubi 13340 | The values of a finite rea... |
| nn0ennn 13341 | The nonnegative integers a... |
| nnenom 13342 | The set of positive intege... |
| nnct 13343 | ` NN ` is countable. (Con... |
| uzindi 13344 | Indirect strong induction ... |
| axdc4uzlem 13345 | Lemma for ~ axdc4uz . (Co... |
| axdc4uz 13346 | A version of ~ axdc4 that ... |
| ssnn0fi 13347 | A subset of the nonnegativ... |
| rabssnn0fi 13348 | A subset of the nonnegativ... |
| uzsinds 13349 | Strong (or "total") induct... |
| nnsinds 13350 | Strong (or "total") induct... |
| nn0sinds 13351 | Strong (or "total") induct... |
| fsuppmapnn0fiublem 13352 | Lemma for ~ fsuppmapnn0fiu... |
| fsuppmapnn0fiub 13353 | If all functions of a fini... |
| fsuppmapnn0fiubex 13354 | If all functions of a fini... |
| fsuppmapnn0fiub0 13355 | If all functions of a fini... |
| suppssfz 13356 | Condition for a function o... |
| fsuppmapnn0ub 13357 | If a function over the non... |
| fsuppmapnn0fz 13358 | If a function over the non... |
| mptnn0fsupp 13359 | A mapping from the nonnega... |
| mptnn0fsuppd 13360 | A mapping from the nonnega... |
| mptnn0fsuppr 13361 | A finitely supported mappi... |
| f13idfv 13362 | A one-to-one function with... |
| seqex 13365 | Existence of the sequence ... |
| seqeq1 13366 | Equality theorem for the s... |
| seqeq2 13367 | Equality theorem for the s... |
| seqeq3 13368 | Equality theorem for the s... |
| seqeq1d 13369 | Equality deduction for the... |
| seqeq2d 13370 | Equality deduction for the... |
| seqeq3d 13371 | Equality deduction for the... |
| seqeq123d 13372 | Equality deduction for the... |
| nfseq 13373 | Hypothesis builder for the... |
| seqval 13374 | Value of the sequence buil... |
| seqfn 13375 | The sequence builder funct... |
| seq1 13376 | Value of the sequence buil... |
| seq1i 13377 | Value of the sequence buil... |
| seqp1 13378 | Value of the sequence buil... |
| seqexw 13379 | Weak version of ~ seqex th... |
| seqp1i 13380 | Value of the sequence buil... |
| seqm1 13381 | Value of the sequence buil... |
| seqcl2 13382 | Closure properties of the ... |
| seqf2 13383 | Range of the recursive seq... |
| seqcl 13384 | Closure properties of the ... |
| seqf 13385 | Range of the recursive seq... |
| seqfveq2 13386 | Equality of sequences. (C... |
| seqfeq2 13387 | Equality of sequences. (C... |
| seqfveq 13388 | Equality of sequences. (C... |
| seqfeq 13389 | Equality of sequences. (C... |
| seqshft2 13390 | Shifting the index set of ... |
| seqres 13391 | Restricting its characteri... |
| serf 13392 | An infinite series of comp... |
| serfre 13393 | An infinite series of real... |
| monoord 13394 | Ordering relation for a mo... |
| monoord2 13395 | Ordering relation for a mo... |
| sermono 13396 | The partial sums in an inf... |
| seqsplit 13397 | Split a sequence into two ... |
| seq1p 13398 | Removing the first term fr... |
| seqcaopr3 13399 | Lemma for ~ seqcaopr2 . (... |
| seqcaopr2 13400 | The sum of two infinite se... |
| seqcaopr 13401 | The sum of two infinite se... |
| seqf1olem2a 13402 | Lemma for ~ seqf1o . (Con... |
| seqf1olem1 13403 | Lemma for ~ seqf1o . (Con... |
| seqf1olem2 13404 | Lemma for ~ seqf1o . (Con... |
| seqf1o 13405 | Rearrange a sum via an arb... |
| seradd 13406 | The sum of two infinite se... |
| sersub 13407 | The difference of two infi... |
| seqid3 13408 | A sequence that consists e... |
| seqid 13409 | Discarding the first few t... |
| seqid2 13410 | The last few partial sums ... |
| seqhomo 13411 | Apply a homomorphism to a ... |
| seqz 13412 | If the operation ` .+ ` ha... |
| seqfeq4 13413 | Equality of series under d... |
| seqfeq3 13414 | Equality of series under d... |
| seqdistr 13415 | The distributive property ... |
| ser0 13416 | The value of the partial s... |
| ser0f 13417 | A zero-valued infinite ser... |
| serge0 13418 | A finite sum of nonnegativ... |
| serle 13419 | Comparison of partial sums... |
| ser1const 13420 | Value of the partial serie... |
| seqof 13421 | Distribute function operat... |
| seqof2 13422 | Distribute function operat... |
| expval 13425 | Value of exponentiation to... |
| expnnval 13426 | Value of exponentiation to... |
| exp0 13427 | Value of a complex number ... |
| 0exp0e1 13428 | ` 0 ^ 0 = 1 ` . This is o... |
| exp1 13429 | Value of a complex number ... |
| expp1 13430 | Value of a complex number ... |
| expneg 13431 | Value of a complex number ... |
| expneg2 13432 | Value of a complex number ... |
| expn1 13433 | A number to the negative o... |
| expcllem 13434 | Lemma for proving nonnegat... |
| expcl2lem 13435 | Lemma for proving integer ... |
| nnexpcl 13436 | Closure of exponentiation ... |
| nn0expcl 13437 | Closure of exponentiation ... |
| zexpcl 13438 | Closure of exponentiation ... |
| qexpcl 13439 | Closure of exponentiation ... |
| reexpcl 13440 | Closure of exponentiation ... |
| expcl 13441 | Closure law for nonnegativ... |
| rpexpcl 13442 | Closure law for exponentia... |
| reexpclz 13443 | Closure of exponentiation ... |
| qexpclz 13444 | Closure of exponentiation ... |
| m1expcl2 13445 | Closure of exponentiation ... |
| m1expcl 13446 | Closure of exponentiation ... |
| expclzlem 13447 | Closure law for integer ex... |
| expclz 13448 | Closure law for integer ex... |
| nn0expcli 13449 | Closure of exponentiation ... |
| nn0sqcl 13450 | The square of a nonnegativ... |
| expm1t 13451 | Exponentiation in terms of... |
| 1exp 13452 | Value of one raised to a n... |
| expeq0 13453 | Positive integer exponenti... |
| expne0 13454 | Positive integer exponenti... |
| expne0i 13455 | Nonnegative integer expone... |
| expgt0 13456 | Nonnegative integer expone... |
| expnegz 13457 | Value of a complex number ... |
| 0exp 13458 | Value of zero raised to a ... |
| expge0 13459 | Nonnegative integer expone... |
| expge1 13460 | Nonnegative integer expone... |
| expgt1 13461 | Positive integer exponenti... |
| mulexp 13462 | Positive integer exponenti... |
| mulexpz 13463 | Integer exponentiation of ... |
| exprec 13464 | Nonnegative integer expone... |
| expadd 13465 | Sum of exponents law for n... |
| expaddzlem 13466 | Lemma for ~ expaddz . (Co... |
| expaddz 13467 | Sum of exponents law for i... |
| expmul 13468 | Product of exponents law f... |
| expmulz 13469 | Product of exponents law f... |
| m1expeven 13470 | Exponentiation of negative... |
| expsub 13471 | Exponent subtraction law f... |
| expp1z 13472 | Value of a nonzero complex... |
| expm1 13473 | Value of a complex number ... |
| expdiv 13474 | Nonnegative integer expone... |
| sqval 13475 | Value of the square of a c... |
| sqneg 13476 | The square of the negative... |
| sqsubswap 13477 | Swap the order of subtract... |
| sqcl 13478 | Closure of square. (Contr... |
| sqmul 13479 | Distribution of square ove... |
| sqeq0 13480 | A number is zero iff its s... |
| sqdiv 13481 | Distribution of square ove... |
| sqdivid 13482 | The square of a nonzero nu... |
| sqne0 13483 | A number is nonzero iff it... |
| resqcl 13484 | Closure of the square of a... |
| sqgt0 13485 | The square of a nonzero re... |
| sqn0rp 13486 | The square of a nonzero re... |
| nnsqcl 13487 | The naturals are closed un... |
| zsqcl 13488 | Integers are closed under ... |
| qsqcl 13489 | The square of a rational i... |
| sq11 13490 | The square function is one... |
| nn0sq11 13491 | The square function is one... |
| lt2sq 13492 | The square function on non... |
| le2sq 13493 | The square function on non... |
| le2sq2 13494 | The square of a 'less than... |
| sqge0 13495 | A square of a real is nonn... |
| zsqcl2 13496 | The square of an integer i... |
| 0expd 13497 | Value of zero raised to a ... |
| exp0d 13498 | Value of a complex number ... |
| exp1d 13499 | Value of a complex number ... |
| expeq0d 13500 | Positive integer exponenti... |
| sqvald 13501 | Value of square. Inferenc... |
| sqcld 13502 | Closure of square. (Contr... |
| sqeq0d 13503 | A number is zero iff its s... |
| expcld 13504 | Closure law for nonnegativ... |
| expp1d 13505 | Value of a complex number ... |
| expaddd 13506 | Sum of exponents law for n... |
| expmuld 13507 | Product of exponents law f... |
| sqrecd 13508 | Square of reciprocal. (Co... |
| expclzd 13509 | Closure law for integer ex... |
| expne0d 13510 | Nonnegative integer expone... |
| expnegd 13511 | Value of a complex number ... |
| exprecd 13512 | Nonnegative integer expone... |
| expp1zd 13513 | Value of a nonzero complex... |
| expm1d 13514 | Value of a complex number ... |
| expsubd 13515 | Exponent subtraction law f... |
| sqmuld 13516 | Distribution of square ove... |
| sqdivd 13517 | Distribution of square ove... |
| expdivd 13518 | Nonnegative integer expone... |
| mulexpd 13519 | Positive integer exponenti... |
| znsqcld 13520 | The square of a nonzero in... |
| reexpcld 13521 | Closure of exponentiation ... |
| expge0d 13522 | Nonnegative integer expone... |
| expge1d 13523 | Nonnegative integer expone... |
| ltexp2a 13524 | Ordering relationship for ... |
| expmordi 13525 | Mantissa ordering relation... |
| rpexpmord 13526 | Mantissa ordering relation... |
| expcan 13527 | Cancellation law for expon... |
| ltexp2 13528 | Ordering law for exponenti... |
| leexp2 13529 | Ordering law for exponenti... |
| leexp2a 13530 | Weak ordering relationship... |
| ltexp2r 13531 | The power of a positive nu... |
| leexp2r 13532 | Weak ordering relationship... |
| leexp1a 13533 | Weak mantissa ordering rel... |
| exple1 13534 | Nonnegative integer expone... |
| expubnd 13535 | An upper bound on ` A ^ N ... |
| sumsqeq0 13536 | Two real numbers are equal... |
| sqvali 13537 | Value of square. Inferenc... |
| sqcli 13538 | Closure of square. (Contr... |
| sqeq0i 13539 | A number is zero iff its s... |
| sqrecii 13540 | Square of reciprocal. (Co... |
| sqmuli 13541 | Distribution of square ove... |
| sqdivi 13542 | Distribution of square ove... |
| resqcli 13543 | Closure of square in reals... |
| sqgt0i 13544 | The square of a nonzero re... |
| sqge0i 13545 | A square of a real is nonn... |
| lt2sqi 13546 | The square function on non... |
| le2sqi 13547 | The square function on non... |
| sq11i 13548 | The square function is one... |
| sq0 13549 | The square of 0 is 0. (Co... |
| sq0i 13550 | If a number is zero, its s... |
| sq0id 13551 | If a number is zero, its s... |
| sq1 13552 | The square of 1 is 1. (Co... |
| neg1sqe1 13553 | ` -u 1 ` squared is 1. (C... |
| sq2 13554 | The square of 2 is 4. (Co... |
| sq3 13555 | The square of 3 is 9. (Co... |
| sq4e2t8 13556 | The square of 4 is 2 times... |
| cu2 13557 | The cube of 2 is 8. (Cont... |
| irec 13558 | The reciprocal of ` _i ` .... |
| i2 13559 | ` _i ` squared. (Contribu... |
| i3 13560 | ` _i ` cubed. (Contribute... |
| i4 13561 | ` _i ` to the fourth power... |
| nnlesq 13562 | A positive integer is less... |
| iexpcyc 13563 | Taking ` _i ` to the ` K `... |
| expnass 13564 | A counterexample showing t... |
| sqlecan 13565 | Cancel one factor of a squ... |
| subsq 13566 | Factor the difference of t... |
| subsq2 13567 | Express the difference of ... |
| binom2i 13568 | The square of a binomial. ... |
| subsqi 13569 | Factor the difference of t... |
| sqeqori 13570 | The squares of two complex... |
| subsq0i 13571 | The two solutions to the d... |
| sqeqor 13572 | The squares of two complex... |
| binom2 13573 | The square of a binomial. ... |
| binom21 13574 | Special case of ~ binom2 w... |
| binom2sub 13575 | Expand the square of a sub... |
| binom2sub1 13576 | Special case of ~ binom2su... |
| binom2subi 13577 | Expand the square of a sub... |
| mulbinom2 13578 | The square of a binomial w... |
| binom3 13579 | The cube of a binomial. (... |
| sq01 13580 | If a complex number equals... |
| zesq 13581 | An integer is even iff its... |
| nnesq 13582 | A positive integer is even... |
| crreczi 13583 | Reciprocal of a complex nu... |
| bernneq 13584 | Bernoulli's inequality, du... |
| bernneq2 13585 | Variation of Bernoulli's i... |
| bernneq3 13586 | A corollary of ~ bernneq .... |
| expnbnd 13587 | Exponentiation with a mant... |
| expnlbnd 13588 | The reciprocal of exponent... |
| expnlbnd2 13589 | The reciprocal of exponent... |
| expmulnbnd 13590 | Exponentiation with a mant... |
| digit2 13591 | Two ways to express the ` ... |
| digit1 13592 | Two ways to express the ` ... |
| modexp 13593 | Exponentiation property of... |
| discr1 13594 | A nonnegative quadratic fo... |
| discr 13595 | If a quadratic polynomial ... |
| expnngt1 13596 | If an integer power with a... |
| expnngt1b 13597 | An integer power with an i... |
| sqoddm1div8 13598 | A squared odd number minus... |
| nnsqcld 13599 | The naturals are closed un... |
| nnexpcld 13600 | Closure of exponentiation ... |
| nn0expcld 13601 | Closure of exponentiation ... |
| rpexpcld 13602 | Closure law for exponentia... |
| ltexp2rd 13603 | The power of a positive nu... |
| reexpclzd 13604 | Closure of exponentiation ... |
| resqcld 13605 | Closure of square in reals... |
| sqge0d 13606 | A square of a real is nonn... |
| sqgt0d 13607 | The square of a nonzero re... |
| ltexp2d 13608 | Ordering relationship for ... |
| leexp2d 13609 | Ordering law for exponenti... |
| expcand 13610 | Ordering relationship for ... |
| leexp2ad 13611 | Ordering relationship for ... |
| leexp2rd 13612 | Ordering relationship for ... |
| lt2sqd 13613 | The square function on non... |
| le2sqd 13614 | The square function on non... |
| sq11d 13615 | The square function is one... |
| mulsubdivbinom2 13616 | The square of a binomial w... |
| muldivbinom2 13617 | The square of a binomial w... |
| sq10 13618 | The square of 10 is 100. ... |
| sq10e99m1 13619 | The square of 10 is 99 plu... |
| 3dec 13620 | A "decimal constructor" wh... |
| nn0le2msqi 13621 | The square function on non... |
| nn0opthlem1 13622 | A rather pretty lemma for ... |
| nn0opthlem2 13623 | Lemma for ~ nn0opthi . (C... |
| nn0opthi 13624 | An ordered pair theorem fo... |
| nn0opth2i 13625 | An ordered pair theorem fo... |
| nn0opth2 13626 | An ordered pair theorem fo... |
| facnn 13629 | Value of the factorial fun... |
| fac0 13630 | The factorial of 0. (Cont... |
| fac1 13631 | The factorial of 1. (Cont... |
| facp1 13632 | The factorial of a success... |
| fac2 13633 | The factorial of 2. (Cont... |
| fac3 13634 | The factorial of 3. (Cont... |
| fac4 13635 | The factorial of 4. (Cont... |
| facnn2 13636 | Value of the factorial fun... |
| faccl 13637 | Closure of the factorial f... |
| faccld 13638 | Closure of the factorial f... |
| facmapnn 13639 | The factorial function res... |
| facne0 13640 | The factorial function is ... |
| facdiv 13641 | A positive integer divides... |
| facndiv 13642 | No positive integer (great... |
| facwordi 13643 | Ordering property of facto... |
| faclbnd 13644 | A lower bound for the fact... |
| faclbnd2 13645 | A lower bound for the fact... |
| faclbnd3 13646 | A lower bound for the fact... |
| faclbnd4lem1 13647 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4lem2 13648 | Lemma for ~ faclbnd4 . Us... |
| faclbnd4lem3 13649 | Lemma for ~ faclbnd4 . Th... |
| faclbnd4lem4 13650 | Lemma for ~ faclbnd4 . Pr... |
| faclbnd4 13651 | Variant of ~ faclbnd5 prov... |
| faclbnd5 13652 | The factorial function gro... |
| faclbnd6 13653 | Geometric lower bound for ... |
| facubnd 13654 | An upper bound for the fac... |
| facavg 13655 | The product of two factori... |
| bcval 13658 | Value of the binomial coef... |
| bcval2 13659 | Value of the binomial coef... |
| bcval3 13660 | Value of the binomial coef... |
| bcval4 13661 | Value of the binomial coef... |
| bcrpcl 13662 | Closure of the binomial co... |
| bccmpl 13663 | "Complementing" its second... |
| bcn0 13664 | ` N ` choose 0 is 1. Rema... |
| bc0k 13665 | The binomial coefficient "... |
| bcnn 13666 | ` N ` choose ` N ` is 1. ... |
| bcn1 13667 | Binomial coefficient: ` N ... |
| bcnp1n 13668 | Binomial coefficient: ` N ... |
| bcm1k 13669 | The proportion of one bino... |
| bcp1n 13670 | The proportion of one bino... |
| bcp1nk 13671 | The proportion of one bino... |
| bcval5 13672 | Write out the top and bott... |
| bcn2 13673 | Binomial coefficient: ` N ... |
| bcp1m1 13674 | Compute the binomial coeff... |
| bcpasc 13675 | Pascal's rule for the bino... |
| bccl 13676 | A binomial coefficient, in... |
| bccl2 13677 | A binomial coefficient, in... |
| bcn2m1 13678 | Compute the binomial coeff... |
| bcn2p1 13679 | Compute the binomial coeff... |
| permnn 13680 | The number of permutations... |
| bcnm1 13681 | The binomial coefficent of... |
| 4bc3eq4 13682 | The value of four choose t... |
| 4bc2eq6 13683 | The value of four choose t... |
| hashkf 13686 | The finite part of the siz... |
| hashgval 13687 | The value of the ` # ` fun... |
| hashginv 13688 | ` ``' G ` maps the size fu... |
| hashinf 13689 | The value of the ` # ` fun... |
| hashbnd 13690 | If ` A ` has size bounded ... |
| hashfxnn0 13691 | The size function is a fun... |
| hashf 13692 | The size function maps all... |
| hashxnn0 13693 | The value of the hash func... |
| hashresfn 13694 | Restriction of the domain ... |
| dmhashres 13695 | Restriction of the domain ... |
| hashnn0pnf 13696 | The value of the hash func... |
| hashnnn0genn0 13697 | If the size of a set is no... |
| hashnemnf 13698 | The size of a set is never... |
| hashv01gt1 13699 | The size of a set is eithe... |
| hashfz1 13700 | The set ` ( 1 ... N ) ` ha... |
| hashen 13701 | Two finite sets have the s... |
| hasheni 13702 | Equinumerous sets have the... |
| hasheqf1o 13703 | The size of two finite set... |
| fiinfnf1o 13704 | There is no bijection betw... |
| focdmex 13705 | The codomain of an onto fu... |
| hasheqf1oi 13706 | The size of two sets is eq... |
| hashf1rn 13707 | The size of a finite set w... |
| hasheqf1od 13708 | The size of two sets is eq... |
| fz1eqb 13709 | Two possibly-empty 1-based... |
| hashcard 13710 | The size function of the c... |
| hashcl 13711 | Closure of the ` # ` funct... |
| hashxrcl 13712 | Extended real closure of t... |
| hashclb 13713 | Reverse closure of the ` #... |
| nfile 13714 | The size of any infinite s... |
| hashvnfin 13715 | A set of finite size is a ... |
| hashnfinnn0 13716 | The size of an infinite se... |
| isfinite4 13717 | A finite set is equinumero... |
| hasheq0 13718 | Two ways of saying a finit... |
| hashneq0 13719 | Two ways of saying a set i... |
| hashgt0n0 13720 | If the size of a set is gr... |
| hashnncl 13721 | Positive natural closure o... |
| hash0 13722 | The empty set has size zer... |
| hashelne0d 13723 | A set with an element has ... |
| hashsng 13724 | The size of a singleton. ... |
| hashen1 13725 | A set has size 1 if and on... |
| hash1elsn 13726 | A set of size 1 with a kno... |
| hashrabrsn 13727 | The size of a restricted c... |
| hashrabsn01 13728 | The size of a restricted c... |
| hashrabsn1 13729 | If the size of a restricte... |
| hashfn 13730 | A function is equinumerous... |
| fseq1hash 13731 | The value of the size func... |
| hashgadd 13732 | ` G ` maps ordinal additio... |
| hashgval2 13733 | A short expression for the... |
| hashdom 13734 | Dominance relation for the... |
| hashdomi 13735 | Non-strict order relation ... |
| hashsdom 13736 | Strict dominance relation ... |
| hashun 13737 | The size of the union of d... |
| hashun2 13738 | The size of the union of f... |
| hashun3 13739 | The size of the union of f... |
| hashinfxadd 13740 | The extended real addition... |
| hashunx 13741 | The size of the union of d... |
| hashge0 13742 | The cardinality of a set i... |
| hashgt0 13743 | The cardinality of a nonem... |
| hashge1 13744 | The cardinality of a nonem... |
| 1elfz0hash 13745 | 1 is an element of the fin... |
| hashnn0n0nn 13746 | If a nonnegative integer i... |
| hashunsng 13747 | The size of the union of a... |
| hashunsngx 13748 | The size of the union of a... |
| hashunsnggt 13749 | The size of a set is great... |
| hashprg 13750 | The size of an unordered p... |
| elprchashprn2 13751 | If one element of an unord... |
| hashprb 13752 | The size of an unordered p... |
| hashprdifel 13753 | The elements of an unorder... |
| prhash2ex 13754 | There is (at least) one se... |
| hashle00 13755 | If the size of a set is le... |
| hashgt0elex 13756 | If the size of a set is gr... |
| hashgt0elexb 13757 | The size of a set is great... |
| hashp1i 13758 | Size of a finite ordinal. ... |
| hash1 13759 | Size of a finite ordinal. ... |
| hash2 13760 | Size of a finite ordinal. ... |
| hash3 13761 | Size of a finite ordinal. ... |
| hash4 13762 | Size of a finite ordinal. ... |
| pr0hash2ex 13763 | There is (at least) one se... |
| hashss 13764 | The size of a subset is le... |
| prsshashgt1 13765 | The size of a superset of ... |
| hashin 13766 | The size of the intersecti... |
| hashssdif 13767 | The size of the difference... |
| hashdif 13768 | The size of the difference... |
| hashdifsn 13769 | The size of the difference... |
| hashdifpr 13770 | The size of the difference... |
| hashsn01 13771 | The size of a singleton is... |
| hashsnle1 13772 | The size of a singleton is... |
| hashsnlei 13773 | Get an upper bound on a co... |
| hash1snb 13774 | The size of a set is 1 if ... |
| euhash1 13775 | The size of a set is 1 in ... |
| hash1n0 13776 | If the size of a set is 1 ... |
| hashgt12el 13777 | In a set with more than on... |
| hashgt12el2 13778 | In a set with more than on... |
| hashgt23el 13779 | A set with more than two e... |
| hashunlei 13780 | Get an upper bound on a co... |
| hashsslei 13781 | Get an upper bound on a co... |
| hashfz 13782 | Value of the numeric cardi... |
| fzsdom2 13783 | Condition for finite range... |
| hashfzo 13784 | Cardinality of a half-open... |
| hashfzo0 13785 | Cardinality of a half-open... |
| hashfzp1 13786 | Value of the numeric cardi... |
| hashfz0 13787 | Value of the numeric cardi... |
| hashxplem 13788 | Lemma for ~ hashxp . (Con... |
| hashxp 13789 | The size of the Cartesian ... |
| hashmap 13790 | The size of the set expone... |
| hashpw 13791 | The size of the power set ... |
| hashfun 13792 | A finite set is a function... |
| hashres 13793 | The number of elements of ... |
| hashreshashfun 13794 | The number of elements of ... |
| hashimarn 13795 | The size of the image of a... |
| hashimarni 13796 | If the size of the image o... |
| resunimafz0 13797 | TODO-AV: Revise using ` F... |
| fnfz0hash 13798 | The size of a function on ... |
| ffz0hash 13799 | The size of a function on ... |
| fnfz0hashnn0 13800 | The size of a function on ... |
| ffzo0hash 13801 | The size of a function on ... |
| fnfzo0hash 13802 | The size of a function on ... |
| fnfzo0hashnn0 13803 | The value of the size func... |
| hashbclem 13804 | Lemma for ~ hashbc : induc... |
| hashbc 13805 | The binomial coefficient c... |
| hashfacen 13806 | The number of bijections b... |
| hashf1lem1 13807 | Lemma for ~ hashf1 . (Con... |
| hashf1lem2 13808 | Lemma for ~ hashf1 . (Con... |
| hashf1 13809 | The permutation number ` |... |
| hashfac 13810 | A factorial counts the num... |
| leiso 13811 | Two ways to write a strict... |
| leisorel 13812 | Version of ~ isorel for st... |
| fz1isolem 13813 | Lemma for ~ fz1iso . (Con... |
| fz1iso 13814 | Any finite ordered set has... |
| ishashinf 13815 | Any set that is not finite... |
| seqcoll 13816 | The function ` F ` contain... |
| seqcoll2 13817 | The function ` F ` contain... |
| phphashd 13818 | Corollary of the Pigeonhol... |
| phphashrd 13819 | Corollary of the Pigeonhol... |
| hashprlei 13820 | An unordered pair has at m... |
| hash2pr 13821 | A set of size two is an un... |
| hash2prde 13822 | A set of size two is an un... |
| hash2exprb 13823 | A set of size two is an un... |
| hash2prb 13824 | A set of size two is a pro... |
| prprrab 13825 | The set of proper pairs of... |
| nehash2 13826 | The cardinality of a set w... |
| hash2prd 13827 | A set of size two is an un... |
| hash2pwpr 13828 | If the size of a subset of... |
| hashle2pr 13829 | A nonempty set of size les... |
| hashle2prv 13830 | A nonempty subset of a pow... |
| pr2pwpr 13831 | The set of subsets of a pa... |
| hashge2el2dif 13832 | A set with size at least 2... |
| hashge2el2difr 13833 | A set with at least 2 diff... |
| hashge2el2difb 13834 | A set has size at least 2 ... |
| hashdmpropge2 13835 | The size of the domain of ... |
| hashtplei 13836 | An unordered triple has at... |
| hashtpg 13837 | The size of an unordered t... |
| hashge3el3dif 13838 | A set with size at least 3... |
| elss2prb 13839 | An element of the set of s... |
| hash2sspr 13840 | A subset of size two is an... |
| exprelprel 13841 | If there is an element of ... |
| hash3tr 13842 | A set of size three is an ... |
| hash1to3 13843 | If the size of a set is be... |
| fundmge2nop0 13844 | A function with a domain c... |
| fundmge2nop 13845 | A function with a domain c... |
| fun2dmnop0 13846 | A function with a domain c... |
| fun2dmnop 13847 | A function with a domain c... |
| hashdifsnp1 13848 | If the size of a set is a ... |
| fi1uzind 13849 | Properties of an ordered p... |
| brfi1uzind 13850 | Properties of a binary rel... |
| brfi1ind 13851 | Properties of a binary rel... |
| brfi1indALT 13852 | Alternate proof of ~ brfi1... |
| opfi1uzind 13853 | Properties of an ordered p... |
| opfi1ind 13854 | Properties of an ordered p... |
| iswrd 13857 | Property of being a word o... |
| wrdval 13858 | Value of the set of words ... |
| iswrdi 13859 | A zero-based sequence is a... |
| wrdf 13860 | A word is a zero-based seq... |
| iswrdb 13861 | A word over an alphabet is... |
| wrddm 13862 | The indices of a word (i.e... |
| sswrd 13863 | The set of words respects ... |
| snopiswrd 13864 | A singleton of an ordered ... |
| wrdexg 13865 | The set of words over a se... |
| wrdexgOLD 13866 | Obsolete proof of ~ wrdexg... |
| wrdexb 13867 | The set of words over a se... |
| wrdexi 13868 | The set of words over a se... |
| wrdsymbcl 13869 | A symbol within a word ove... |
| wrdfn 13870 | A word is a function with ... |
| wrdv 13871 | A word over an alphabet is... |
| wrdvOLD 13872 | Obsolete proof of ~ wrdv a... |
| wrdlndm 13873 | The length of a word is no... |
| wrdlndmOLD 13874 | Obsolete proof of ~ wrdlnd... |
| iswrdsymb 13875 | An arbitrary word is a wor... |
| wrdfin 13876 | A word is a finite set. (... |
| lencl 13877 | The length of a word is a ... |
| lennncl 13878 | The length of a nonempty w... |
| wrdffz 13879 | A word is a function from ... |
| wrdeq 13880 | Equality theorem for the s... |
| wrdeqi 13881 | Equality theorem for the s... |
| iswrddm0 13882 | A function with empty doma... |
| wrd0 13883 | The empty set is a word (t... |
| 0wrd0 13884 | The empty word is the only... |
| ffz0iswrd 13885 | A sequence with zero-based... |
| ffz0iswrdOLD 13886 | Obsolete proof of ~ ffz0is... |
| wrdsymb 13887 | A word is a word over the ... |
| nfwrd 13888 | Hypothesis builder for ` W... |
| csbwrdg 13889 | Class substitution for the... |
| wrdnval 13890 | Words of a fixed length ar... |
| wrdmap 13891 | Words as a mapping. (Cont... |
| hashwrdn 13892 | If there is only a finite ... |
| wrdnfi 13893 | If there is only a finite ... |
| wrdnfiOLD 13894 | Obsolete version of ~ wrdn... |
| wrdsymb0 13895 | A symbol at a position "ou... |
| wrdlenge1n0 13896 | A word with length at leas... |
| len0nnbi 13897 | The length of a word is a ... |
| wrdlenge2n0 13898 | A word with length at leas... |
| wrdsymb1 13899 | The first symbol of a none... |
| wrdlen1 13900 | A word of length 1 starts ... |
| fstwrdne 13901 | The first symbol of a none... |
| fstwrdne0 13902 | The first symbol of a none... |
| eqwrd 13903 | Two words are equal iff th... |
| elovmpowrd 13904 | Implications for the value... |
| elovmptnn0wrd 13905 | Implications for the value... |
| wrdred1 13906 | A word truncated by a symb... |
| wrdred1hash 13907 | The length of a word trunc... |
| lsw 13910 | Extract the last symbol of... |
| lsw0 13911 | The last symbol of an empt... |
| lsw0g 13912 | The last symbol of an empt... |
| lsw1 13913 | The last symbol of a word ... |
| lswcl 13914 | Closure of the last symbol... |
| lswlgt0cl 13915 | The last symbol of a nonem... |
| ccatfn 13918 | The concatenation operator... |
| ccatfval 13919 | Value of the concatenation... |
| ccatcl 13920 | The concatenation of two w... |
| ccatlen 13921 | The length of a concatenat... |
| ccatlenOLD 13922 | Obsolete version of ~ ccat... |
| ccat0 13923 | The concatenation of two w... |
| ccatval1 13924 | Value of a symbol in the l... |
| ccatval1OLD 13925 | Obsolete version of ~ ccat... |
| ccatval2 13926 | Value of a symbol in the r... |
| ccatval3 13927 | Value of a symbol in the r... |
| elfzelfzccat 13928 | An element of a finite set... |
| ccatvalfn 13929 | The concatenation of two w... |
| ccatsymb 13930 | The symbol at a given posi... |
| ccatfv0 13931 | The first symbol of a conc... |
| ccatval1lsw 13932 | The last symbol of the lef... |
| ccatval21sw 13933 | The first symbol of the ri... |
| ccatlid 13934 | Concatenation of a word by... |
| ccatrid 13935 | Concatenation of a word by... |
| ccatass 13936 | Associative law for concat... |
| ccatrn 13937 | The range of a concatenate... |
| ccatidid 13938 | Concatenation of the empty... |
| lswccatn0lsw 13939 | The last symbol of a word ... |
| lswccat0lsw 13940 | The last symbol of a word ... |
| ccatalpha 13941 | A concatenation of two arb... |
| ccatrcl1 13942 | Reverse closure of a conca... |
| ids1 13945 | Identity function protecti... |
| s1val 13946 | Value of a singleton word.... |
| s1rn 13947 | The range of a singleton w... |
| s1eq 13948 | Equality theorem for a sin... |
| s1eqd 13949 | Equality theorem for a sin... |
| s1cl 13950 | A singleton word is a word... |
| s1cld 13951 | A singleton word is a word... |
| s1prc 13952 | Value of a singleton word ... |
| s1cli 13953 | A singleton word is a word... |
| s1len 13954 | Length of a singleton word... |
| s1nz 13955 | A singleton word is not th... |
| s1dm 13956 | The domain of a singleton ... |
| s1dmALT 13957 | Alternate version of ~ s1d... |
| s1fv 13958 | Sole symbol of a singleton... |
| lsws1 13959 | The last symbol of a singl... |
| eqs1 13960 | A word of length 1 is a si... |
| wrdl1exs1 13961 | A word of length 1 is a si... |
| wrdl1s1 13962 | A word of length 1 is a si... |
| s111 13963 | The singleton word functio... |
| ccatws1cl 13964 | The concatenation of a wor... |
| ccatws1clv 13965 | The concatenation of a wor... |
| ccat2s1cl 13966 | The concatenation of two s... |
| ccats1alpha 13967 | A concatenation of a word ... |
| ccatws1len 13968 | The length of the concaten... |
| ccatws1lenp1b 13969 | The length of a word is ` ... |
| wrdlenccats1lenm1 13970 | The length of a word is th... |
| ccat2s1len 13971 | The length of the concaten... |
| ccat2s1lenOLD 13972 | Obsolete version of ~ ccat... |
| ccatw2s1cl 13973 | The concatenation of a wor... |
| ccatw2s1len 13974 | The length of the concaten... |
| ccats1val1 13975 | Value of a symbol in the l... |
| ccats1val1OLD 13976 | Obsolete version of ~ ccat... |
| ccats1val2 13977 | Value of the symbol concat... |
| ccat1st1st 13978 | The first symbol of a word... |
| ccat2s1p1 13979 | Extract the first of two c... |
| ccat2s1p2 13980 | Extract the second of two ... |
| ccat2s1p1OLD 13981 | Obsolete version of ~ ccat... |
| ccat2s1p2OLD 13982 | Obsolete version of ~ ccat... |
| ccatw2s1ass 13983 | Associative law for a conc... |
| ccatw2s1assOLD 13984 | Obsolete version of ~ ccat... |
| ccatws1n0 13985 | The concatenation of a wor... |
| ccatws1ls 13986 | The last symbol of the con... |
| lswccats1 13987 | The last symbol of a word ... |
| lswccats1fst 13988 | The last symbol of a nonem... |
| ccatw2s1p1 13989 | Extract the symbol of the ... |
| ccatw2s1p1OLD 13990 | Obsolete version of ~ ccat... |
| ccatw2s1p2 13991 | Extract the second of two ... |
| ccat2s1fvw 13992 | Extract a symbol of a word... |
| ccat2s1fvwOLD 13993 | Obsolete version of ~ ccat... |
| ccat2s1fst 13994 | The first symbol of the co... |
| ccat2s1fstOLD 13995 | Obsolete version of ~ ccat... |
| swrdnznd 13998 | The value of a subword ope... |
| swrdval 13999 | Value of a subword. (Cont... |
| swrd00 14000 | A zero length substring. ... |
| swrdcl 14001 | Closure of the subword ext... |
| swrdval2 14002 | Value of the subword extra... |
| swrdlen 14003 | Length of an extracted sub... |
| swrdfv 14004 | A symbol in an extracted s... |
| swrdfv0 14005 | The first symbol in an ext... |
| swrdf 14006 | A subword of a word is a f... |
| swrdvalfn 14007 | Value of the subword extra... |
| swrdrn 14008 | The range of a subword of ... |
| swrdlend 14009 | The value of the subword e... |
| swrdnd 14010 | The value of the subword e... |
| swrdnd2 14011 | Value of the subword extra... |
| swrdnnn0nd 14012 | The value of a subword ope... |
| swrdnd0 14013 | The value of a subword ope... |
| swrd0 14014 | A subword of an empty set ... |
| swrdrlen 14015 | Length of a right-anchored... |
| swrdlen2 14016 | Length of an extracted sub... |
| swrdfv2 14017 | A symbol in an extracted s... |
| swrdwrdsymb 14018 | A subword is a word over t... |
| swrdsb0eq 14019 | Two subwords with the same... |
| swrdsbslen 14020 | Two subwords with the same... |
| swrdspsleq 14021 | Two words have a common su... |
| swrds1 14022 | Extract a single symbol fr... |
| swrdlsw 14023 | Extract the last single sy... |
| ccatswrd 14024 | Joining two adjacent subwo... |
| swrdccat2 14025 | Recover the right half of ... |
| pfxnndmnd 14028 | The value of a prefix oper... |
| pfxval 14029 | Value of a prefix operatio... |
| pfx00 14030 | The zero length prefix is ... |
| pfx0 14031 | A prefix of an empty set i... |
| pfxval0 14032 | Value of a prefix operatio... |
| pfxcl 14033 | Closure of the prefix extr... |
| pfxmpt 14034 | Value of the prefix extrac... |
| pfxres 14035 | Value of the subword extra... |
| pfxf 14036 | A prefix of a word is a fu... |
| pfxfn 14037 | Value of the prefix extrac... |
| pfxfv 14038 | A symbol in a prefix of a ... |
| pfxlen 14039 | Length of a prefix. (Cont... |
| pfxid 14040 | A word is a prefix of itse... |
| pfxrn 14041 | The range of a prefix of a... |
| pfxn0 14042 | A prefix consisting of at ... |
| pfxnd 14043 | The value of a prefix oper... |
| pfxnd0 14044 | The value of a prefix oper... |
| pfxwrdsymb 14045 | A prefix of a word is a wo... |
| addlenrevpfx 14046 | The sum of the lengths of ... |
| addlenpfx 14047 | The sum of the lengths of ... |
| pfxfv0 14048 | The first symbol of a pref... |
| pfxtrcfv 14049 | A symbol in a word truncat... |
| pfxtrcfv0 14050 | The first symbol in a word... |
| pfxfvlsw 14051 | The last symbol in a nonem... |
| pfxeq 14052 | The prefixes of two words ... |
| pfxtrcfvl 14053 | The last symbol in a word ... |
| pfxsuffeqwrdeq 14054 | Two words are equal if and... |
| pfxsuff1eqwrdeq 14055 | Two (nonempty) words are e... |
| disjwrdpfx 14056 | Sets of words are disjoint... |
| ccatpfx 14057 | Concatenating a prefix wit... |
| pfxccat1 14058 | Recover the left half of a... |
| pfx1 14059 | The prefix of length one o... |
| swrdswrdlem 14060 | Lemma for ~ swrdswrd . (C... |
| swrdswrd 14061 | A subword of a subword is ... |
| pfxswrd 14062 | A prefix of a subword is a... |
| swrdpfx 14063 | A subword of a prefix is a... |
| pfxpfx 14064 | A prefix of a prefix is a ... |
| pfxpfxid 14065 | A prefix of a prefix with ... |
| pfxcctswrd 14066 | The concatenation of the p... |
| lenpfxcctswrd 14067 | The length of the concaten... |
| lenrevpfxcctswrd 14068 | The length of the concaten... |
| pfxlswccat 14069 | Reconstruct a nonempty wor... |
| ccats1pfxeq 14070 | The last symbol of a word ... |
| ccats1pfxeqrex 14071 | There exists a symbol such... |
| ccatopth 14072 | An ~ opth -like theorem fo... |
| ccatopth2 14073 | An ~ opth -like theorem fo... |
| ccatlcan 14074 | Concatenation of words is ... |
| ccatrcan 14075 | Concatenation of words is ... |
| wrdeqs1cat 14076 | Decompose a nonempty word ... |
| cats1un 14077 | Express a word with an ext... |
| wrdind 14078 | Perform induction over the... |
| wrd2ind 14079 | Perform induction over the... |
| swrdccatfn 14080 | The subword of a concatena... |
| swrdccatin1 14081 | The subword of a concatena... |
| pfxccatin12lem4 14082 | Lemma 4 for ~ pfxccatin12 ... |
| pfxccatin12lem2a 14083 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem1 14084 | Lemma 1 for ~ pfxccatin12 ... |
| swrdccatin2 14085 | The subword of a concatena... |
| pfxccatin12lem2c 14086 | Lemma for ~ pfxccatin12lem... |
| pfxccatin12lem2 14087 | Lemma 2 for ~ pfxccatin12 ... |
| pfxccatin12lem3 14088 | Lemma 3 for ~ pfxccatin12 ... |
| pfxccatin12 14089 | The subword of a concatena... |
| pfxccat3 14090 | The subword of a concatena... |
| swrdccat 14091 | The subword of a concatena... |
| pfxccatpfx1 14092 | A prefix of a concatenatio... |
| pfxccatpfx2 14093 | A prefix of a concatenatio... |
| pfxccat3a 14094 | A prefix of a concatenatio... |
| swrdccat3blem 14095 | Lemma for ~ swrdccat3b . ... |
| swrdccat3b 14096 | A suffix of a concatenatio... |
| pfxccatid 14097 | A prefix of a concatenatio... |
| ccats1pfxeqbi 14098 | A word is a prefix of a wo... |
| swrdccatin1d 14099 | The subword of a concatena... |
| swrdccatin2d 14100 | The subword of a concatena... |
| pfxccatin12d 14101 | The subword of a concatena... |
| reuccatpfxs1lem 14102 | Lemma for ~ reuccatpfxs1 .... |
| reuccatpfxs1 14103 | There is a unique word hav... |
| reuccatpfxs1v 14104 | There is a unique word hav... |
| splval 14107 | Value of the substring rep... |
| splcl 14108 | Closure of the substring r... |
| splid 14109 | Splicing a subword for the... |
| spllen 14110 | The length of a splice. (... |
| splfv1 14111 | Symbols to the left of a s... |
| splfv2a 14112 | Symbols within the replace... |
| splval2 14113 | Value of a splice, assumin... |
| revval 14116 | Value of the word reversin... |
| revcl 14117 | The reverse of a word is a... |
| revlen 14118 | The reverse of a word has ... |
| revfv 14119 | Reverse of a word at a poi... |
| rev0 14120 | The empty word is its own ... |
| revs1 14121 | Singleton words are their ... |
| revccat 14122 | Antiautomorphic property o... |
| revrev 14123 | Reversal is an involution ... |
| reps 14126 | Construct a function mappi... |
| repsundef 14127 | A function mapping a half-... |
| repsconst 14128 | Construct a function mappi... |
| repsf 14129 | The constructed function m... |
| repswsymb 14130 | The symbols of a "repeated... |
| repsw 14131 | A function mapping a half-... |
| repswlen 14132 | The length of a "repeated ... |
| repsw0 14133 | The "repeated symbol word"... |
| repsdf2 14134 | Alternative definition of ... |
| repswsymball 14135 | All the symbols of a "repe... |
| repswsymballbi 14136 | A word is a "repeated symb... |
| repswfsts 14137 | The first symbol of a none... |
| repswlsw 14138 | The last symbol of a nonem... |
| repsw1 14139 | The "repeated symbol word"... |
| repswswrd 14140 | A subword of a "repeated s... |
| repswpfx 14141 | A prefix of a repeated sym... |
| repswccat 14142 | The concatenation of two "... |
| repswrevw 14143 | The reverse of a "repeated... |
| cshfn 14146 | Perform a cyclical shift f... |
| cshword 14147 | Perform a cyclical shift f... |
| cshnz 14148 | A cyclical shift is the em... |
| 0csh0 14149 | Cyclically shifting an emp... |
| cshw0 14150 | A word cyclically shifted ... |
| cshwmodn 14151 | Cyclically shifting a word... |
| cshwsublen 14152 | Cyclically shifting a word... |
| cshwn 14153 | A word cyclically shifted ... |
| cshwcl 14154 | A cyclically shifted word ... |
| cshwlen 14155 | The length of a cyclically... |
| cshwf 14156 | A cyclically shifted word ... |
| cshwfn 14157 | A cyclically shifted word ... |
| cshwrn 14158 | The range of a cyclically ... |
| cshwidxmod 14159 | The symbol at a given inde... |
| cshwidxmodr 14160 | The symbol at a given inde... |
| cshwidx0mod 14161 | The symbol at index 0 of a... |
| cshwidx0 14162 | The symbol at index 0 of a... |
| cshwidxm1 14163 | The symbol at index ((n-N)... |
| cshwidxm 14164 | The symbol at index (n-N) ... |
| cshwidxn 14165 | The symbol at index (n-1) ... |
| cshf1 14166 | Cyclically shifting a word... |
| cshinj 14167 | If a word is injectiv (reg... |
| repswcshw 14168 | A cyclically shifted "repe... |
| 2cshw 14169 | Cyclically shifting a word... |
| 2cshwid 14170 | Cyclically shifting a word... |
| lswcshw 14171 | The last symbol of a word ... |
| 2cshwcom 14172 | Cyclically shifting a word... |
| cshwleneq 14173 | If the results of cyclical... |
| 3cshw 14174 | Cyclically shifting a word... |
| cshweqdif2 14175 | If cyclically shifting two... |
| cshweqdifid 14176 | If cyclically shifting a w... |
| cshweqrep 14177 | If cyclically shifting a w... |
| cshw1 14178 | If cyclically shifting a w... |
| cshw1repsw 14179 | If cyclically shifting a w... |
| cshwsexa 14180 | The class of (different!) ... |
| 2cshwcshw 14181 | If a word is a cyclically ... |
| scshwfzeqfzo 14182 | For a nonempty word the se... |
| cshwcshid 14183 | A cyclically shifted word ... |
| cshwcsh2id 14184 | A cyclically shifted word ... |
| cshimadifsn 14185 | The image of a cyclically ... |
| cshimadifsn0 14186 | The image of a cyclically ... |
| wrdco 14187 | Mapping a word by a functi... |
| lenco 14188 | Length of a mapped word is... |
| s1co 14189 | Mapping of a singleton wor... |
| revco 14190 | Mapping of words (i.e., a ... |
| ccatco 14191 | Mapping of words commutes ... |
| cshco 14192 | Mapping of words commutes ... |
| swrdco 14193 | Mapping of words commutes ... |
| pfxco 14194 | Mapping of words commutes ... |
| lswco 14195 | Mapping of (nonempty) word... |
| repsco 14196 | Mapping of words commutes ... |
| cats1cld 14211 | Closure of concatenation w... |
| cats1co 14212 | Closure of concatenation w... |
| cats1cli 14213 | Closure of concatenation w... |
| cats1fvn 14214 | The last symbol of a conca... |
| cats1fv 14215 | A symbol other than the la... |
| cats1len 14216 | The length of concatenatio... |
| cats1cat 14217 | Closure of concatenation w... |
| cats2cat 14218 | Closure of concatenation o... |
| s2eqd 14219 | Equality theorem for a dou... |
| s3eqd 14220 | Equality theorem for a len... |
| s4eqd 14221 | Equality theorem for a len... |
| s5eqd 14222 | Equality theorem for a len... |
| s6eqd 14223 | Equality theorem for a len... |
| s7eqd 14224 | Equality theorem for a len... |
| s8eqd 14225 | Equality theorem for a len... |
| s3eq2 14226 | Equality theorem for a len... |
| s2cld 14227 | A doubleton word is a word... |
| s3cld 14228 | A length 3 string is a wor... |
| s4cld 14229 | A length 4 string is a wor... |
| s5cld 14230 | A length 5 string is a wor... |
| s6cld 14231 | A length 6 string is a wor... |
| s7cld 14232 | A length 7 string is a wor... |
| s8cld 14233 | A length 7 string is a wor... |
| s2cl 14234 | A doubleton word is a word... |
| s3cl 14235 | A length 3 string is a wor... |
| s2cli 14236 | A doubleton word is a word... |
| s3cli 14237 | A length 3 string is a wor... |
| s4cli 14238 | A length 4 string is a wor... |
| s5cli 14239 | A length 5 string is a wor... |
| s6cli 14240 | A length 6 string is a wor... |
| s7cli 14241 | A length 7 string is a wor... |
| s8cli 14242 | A length 8 string is a wor... |
| s2fv0 14243 | Extract the first symbol f... |
| s2fv1 14244 | Extract the second symbol ... |
| s2len 14245 | The length of a doubleton ... |
| s2dm 14246 | The domain of a doubleton ... |
| s3fv0 14247 | Extract the first symbol f... |
| s3fv1 14248 | Extract the second symbol ... |
| s3fv2 14249 | Extract the third symbol f... |
| s3len 14250 | The length of a length 3 s... |
| s4fv0 14251 | Extract the first symbol f... |
| s4fv1 14252 | Extract the second symbol ... |
| s4fv2 14253 | Extract the third symbol f... |
| s4fv3 14254 | Extract the fourth symbol ... |
| s4len 14255 | The length of a length 4 s... |
| s5len 14256 | The length of a length 5 s... |
| s6len 14257 | The length of a length 6 s... |
| s7len 14258 | The length of a length 7 s... |
| s8len 14259 | The length of a length 8 s... |
| lsws2 14260 | The last symbol of a doubl... |
| lsws3 14261 | The last symbol of a 3 let... |
| lsws4 14262 | The last symbol of a 4 let... |
| s2prop 14263 | A length 2 word is an unor... |
| s2dmALT 14264 | Alternate version of ~ s2d... |
| s3tpop 14265 | A length 3 word is an unor... |
| s4prop 14266 | A length 4 word is a union... |
| s3fn 14267 | A length 3 word is a funct... |
| funcnvs1 14268 | The converse of a singleto... |
| funcnvs2 14269 | The converse of a length 2... |
| funcnvs3 14270 | The converse of a length 3... |
| funcnvs4 14271 | The converse of a length 4... |
| s2f1o 14272 | A length 2 word with mutua... |
| f1oun2prg 14273 | A union of unordered pairs... |
| s4f1o 14274 | A length 4 word with mutua... |
| s4dom 14275 | The domain of a length 4 w... |
| s2co 14276 | Mapping a doubleton word b... |
| s3co 14277 | Mapping a length 3 string ... |
| s0s1 14278 | Concatenation of fixed len... |
| s1s2 14279 | Concatenation of fixed len... |
| s1s3 14280 | Concatenation of fixed len... |
| s1s4 14281 | Concatenation of fixed len... |
| s1s5 14282 | Concatenation of fixed len... |
| s1s6 14283 | Concatenation of fixed len... |
| s1s7 14284 | Concatenation of fixed len... |
| s2s2 14285 | Concatenation of fixed len... |
| s4s2 14286 | Concatenation of fixed len... |
| s4s3 14287 | Concatenation of fixed len... |
| s4s4 14288 | Concatenation of fixed len... |
| s3s4 14289 | Concatenation of fixed len... |
| s2s5 14290 | Concatenation of fixed len... |
| s5s2 14291 | Concatenation of fixed len... |
| s2eq2s1eq 14292 | Two length 2 words are equ... |
| s2eq2seq 14293 | Two length 2 words are equ... |
| s3eqs2s1eq 14294 | Two length 3 words are equ... |
| s3eq3seq 14295 | Two length 3 words are equ... |
| swrds2 14296 | Extract two adjacent symbo... |
| swrds2m 14297 | Extract two adjacent symbo... |
| wrdlen2i 14298 | Implications of a word of ... |
| wrd2pr2op 14299 | A word of length two repre... |
| wrdlen2 14300 | A word of length two. (Co... |
| wrdlen2s2 14301 | A word of length two as do... |
| wrdl2exs2 14302 | A word of length two is a ... |
| pfx2 14303 | A prefix of length two. (... |
| wrd3tpop 14304 | A word of length three rep... |
| wrdlen3s3 14305 | A word of length three as ... |
| repsw2 14306 | The "repeated symbol word"... |
| repsw3 14307 | The "repeated symbol word"... |
| swrd2lsw 14308 | Extract the last two symbo... |
| 2swrd2eqwrdeq 14309 | Two words of length at lea... |
| ccatw2s1ccatws2 14310 | The concatenation of a wor... |
| ccatw2s1ccatws2OLD 14311 | Obsolete version of ~ ccat... |
| ccat2s1fvwALT 14312 | Alternate proof of ~ ccat2... |
| ccat2s1fvwALTOLD 14313 | Obsolete version of ~ ccat... |
| wwlktovf 14314 | Lemma 1 for ~ wrd2f1tovbij... |
| wwlktovf1 14315 | Lemma 2 for ~ wrd2f1tovbij... |
| wwlktovfo 14316 | Lemma 3 for ~ wrd2f1tovbij... |
| wwlktovf1o 14317 | Lemma 4 for ~ wrd2f1tovbij... |
| wrd2f1tovbij 14318 | There is a bijection betwe... |
| eqwrds3 14319 | A word is equal with a len... |
| wrdl3s3 14320 | A word of length 3 is a le... |
| s3sndisj 14321 | The singletons consisting ... |
| s3iunsndisj 14322 | The union of singletons co... |
| ofccat 14323 | Letterwise operations on w... |
| ofs1 14324 | Letterwise operations on a... |
| ofs2 14325 | Letterwise operations on a... |
| coss12d 14326 | Subset deduction for compo... |
| trrelssd 14327 | The composition of subclas... |
| xpcogend 14328 | The most interesting case ... |
| xpcoidgend 14329 | If two classes are not dis... |
| cotr2g 14330 | Two ways of saying that th... |
| cotr2 14331 | Two ways of saying a relat... |
| cotr3 14332 | Two ways of saying a relat... |
| coemptyd 14333 | Deduction about compositio... |
| xptrrel 14334 | The cross product is alway... |
| 0trrel 14335 | The empty class is a trans... |
| cleq1lem 14336 | Equality implies bijection... |
| cleq1 14337 | Equality of relations impl... |
| clsslem 14338 | The closure of a subclass ... |
| trcleq1 14343 | Equality of relations impl... |
| trclsslem 14344 | The transitive closure (as... |
| trcleq2lem 14345 | Equality implies bijection... |
| cvbtrcl 14346 | Change of bound variable i... |
| trcleq12lem 14347 | Equality implies bijection... |
| trclexlem 14348 | Existence of relation impl... |
| trclublem 14349 | If a relation exists then ... |
| trclubi 14350 | The Cartesian product of t... |
| trclubgi 14351 | The union with the Cartesi... |
| trclub 14352 | The Cartesian product of t... |
| trclubg 14353 | The union with the Cartesi... |
| trclfv 14354 | The transitive closure of ... |
| brintclab 14355 | Two ways to express a bina... |
| brtrclfv 14356 | Two ways of expressing the... |
| brcnvtrclfv 14357 | Two ways of expressing the... |
| brtrclfvcnv 14358 | Two ways of expressing the... |
| brcnvtrclfvcnv 14359 | Two ways of expressing the... |
| trclfvss 14360 | The transitive closure (as... |
| trclfvub 14361 | The transitive closure of ... |
| trclfvlb 14362 | The transitive closure of ... |
| trclfvcotr 14363 | The transitive closure of ... |
| trclfvlb2 14364 | The transitive closure of ... |
| trclfvlb3 14365 | The transitive closure of ... |
| cotrtrclfv 14366 | The transitive closure of ... |
| trclidm 14367 | The transitive closure of ... |
| trclun 14368 | Transitive closure of a un... |
| trclfvg 14369 | The value of the transitiv... |
| trclfvcotrg 14370 | The value of the transitiv... |
| reltrclfv 14371 | The transitive closure of ... |
| dmtrclfv 14372 | The domain of the transiti... |
| relexp0g 14375 | A relation composed zero t... |
| relexp0 14376 | A relation composed zero t... |
| relexp0d 14377 | A relation composed zero t... |
| relexpsucnnr 14378 | A reduction for relation e... |
| relexp1g 14379 | A relation composed once i... |
| dfid5 14380 | Identity relation is equal... |
| dfid6 14381 | Identity relation expresse... |
| relexpsucr 14382 | A reduction for relation e... |
| relexpsucrd 14383 | A reduction for relation e... |
| relexp1d 14384 | A relation composed once i... |
| relexpsucnnl 14385 | A reduction for relation e... |
| relexpsucl 14386 | A reduction for relation e... |
| relexpsucld 14387 | A reduction for relation e... |
| relexpcnv 14388 | Commutation of converse an... |
| relexpcnvd 14389 | Commutation of converse an... |
| relexp0rel 14390 | The exponentiation of a cl... |
| relexprelg 14391 | The exponentiation of a cl... |
| relexprel 14392 | The exponentiation of a re... |
| relexpreld 14393 | The exponentiation of a re... |
| relexpnndm 14394 | The domain of an exponenti... |
| relexpdmg 14395 | The domain of an exponenti... |
| relexpdm 14396 | The domain of an exponenti... |
| relexpdmd 14397 | The domain of an exponenti... |
| relexpnnrn 14398 | The range of an exponentia... |
| relexprng 14399 | The range of an exponentia... |
| relexprn 14400 | The range of an exponentia... |
| relexprnd 14401 | The range of an exponentia... |
| relexpfld 14402 | The field of an exponentia... |
| relexpfldd 14403 | The field of an exponentia... |
| relexpaddnn 14404 | Relation composition becom... |
| relexpuzrel 14405 | The exponentiation of a cl... |
| relexpaddg 14406 | Relation composition becom... |
| relexpaddd 14407 | Relation composition becom... |
| dfrtrclrec2 14410 | If two elements are connec... |
| rtrclreclem1 14411 | The reflexive, transitive ... |
| rtrclreclem2 14412 | The reflexive, transitive ... |
| rtrclreclem3 14413 | The reflexive, transitive ... |
| rtrclreclem4 14414 | The reflexive, transitive ... |
| dfrtrcl2 14415 | The two definitions ` t* `... |
| relexpindlem 14416 | Principle of transitive in... |
| relexpind 14417 | Principle of transitive in... |
| rtrclind 14418 | Principle of transitive in... |
| shftlem 14421 | Two ways to write a shifte... |
| shftuz 14422 | A shift of the upper integ... |
| shftfval 14423 | The value of the sequence ... |
| shftdm 14424 | Domain of a relation shift... |
| shftfib 14425 | Value of a fiber of the re... |
| shftfn 14426 | Functionality and domain o... |
| shftval 14427 | Value of a sequence shifte... |
| shftval2 14428 | Value of a sequence shifte... |
| shftval3 14429 | Value of a sequence shifte... |
| shftval4 14430 | Value of a sequence shifte... |
| shftval5 14431 | Value of a shifted sequenc... |
| shftf 14432 | Functionality of a shifted... |
| 2shfti 14433 | Composite shift operations... |
| shftidt2 14434 | Identity law for the shift... |
| shftidt 14435 | Identity law for the shift... |
| shftcan1 14436 | Cancellation law for the s... |
| shftcan2 14437 | Cancellation law for the s... |
| seqshft 14438 | Shifting the index set of ... |
| sgnval 14441 | Value of the signum functi... |
| sgn0 14442 | The signum of 0 is 0. (Co... |
| sgnp 14443 | The signum of a positive e... |
| sgnrrp 14444 | The signum of a positive r... |
| sgn1 14445 | The signum of 1 is 1. (Co... |
| sgnpnf 14446 | The signum of ` +oo ` is 1... |
| sgnn 14447 | The signum of a negative e... |
| sgnmnf 14448 | The signum of ` -oo ` is -... |
| cjval 14455 | The value of the conjugate... |
| cjth 14456 | The defining property of t... |
| cjf 14457 | Domain and codomain of the... |
| cjcl 14458 | The conjugate of a complex... |
| reval 14459 | The value of the real part... |
| imval 14460 | The value of the imaginary... |
| imre 14461 | The imaginary part of a co... |
| reim 14462 | The real part of a complex... |
| recl 14463 | The real part of a complex... |
| imcl 14464 | The imaginary part of a co... |
| ref 14465 | Domain and codomain of the... |
| imf 14466 | Domain and codomain of the... |
| crre 14467 | The real part of a complex... |
| crim 14468 | The real part of a complex... |
| replim 14469 | Reconstruct a complex numb... |
| remim 14470 | Value of the conjugate of ... |
| reim0 14471 | The imaginary part of a re... |
| reim0b 14472 | A number is real iff its i... |
| rereb 14473 | A number is real iff it eq... |
| mulre 14474 | A product with a nonzero r... |
| rere 14475 | A real number equals its r... |
| cjreb 14476 | A number is real iff it eq... |
| recj 14477 | Real part of a complex con... |
| reneg 14478 | Real part of negative. (C... |
| readd 14479 | Real part distributes over... |
| resub 14480 | Real part distributes over... |
| remullem 14481 | Lemma for ~ remul , ~ immu... |
| remul 14482 | Real part of a product. (... |
| remul2 14483 | Real part of a product. (... |
| rediv 14484 | Real part of a division. ... |
| imcj 14485 | Imaginary part of a comple... |
| imneg 14486 | The imaginary part of a ne... |
| imadd 14487 | Imaginary part distributes... |
| imsub 14488 | Imaginary part distributes... |
| immul 14489 | Imaginary part of a produc... |
| immul2 14490 | Imaginary part of a produc... |
| imdiv 14491 | Imaginary part of a divisi... |
| cjre 14492 | A real number equals its c... |
| cjcj 14493 | The conjugate of the conju... |
| cjadd 14494 | Complex conjugate distribu... |
| cjmul 14495 | Complex conjugate distribu... |
| ipcnval 14496 | Standard inner product on ... |
| cjmulrcl 14497 | A complex number times its... |
| cjmulval 14498 | A complex number times its... |
| cjmulge0 14499 | A complex number times its... |
| cjneg 14500 | Complex conjugate of negat... |
| addcj 14501 | A number plus its conjugat... |
| cjsub 14502 | Complex conjugate distribu... |
| cjexp 14503 | Complex conjugate of posit... |
| imval2 14504 | The imaginary part of a nu... |
| re0 14505 | The real part of zero. (C... |
| im0 14506 | The imaginary part of zero... |
| re1 14507 | The real part of one. (Co... |
| im1 14508 | The imaginary part of one.... |
| rei 14509 | The real part of ` _i ` . ... |
| imi 14510 | The imaginary part of ` _i... |
| cj0 14511 | The conjugate of zero. (C... |
| cji 14512 | The complex conjugate of t... |
| cjreim 14513 | The conjugate of a represe... |
| cjreim2 14514 | The conjugate of the repre... |
| cj11 14515 | Complex conjugate is a one... |
| cjne0 14516 | A number is nonzero iff it... |
| cjdiv 14517 | Complex conjugate distribu... |
| cnrecnv 14518 | The inverse to the canonic... |
| sqeqd 14519 | A deduction for showing tw... |
| recli 14520 | The real part of a complex... |
| imcli 14521 | The imaginary part of a co... |
| cjcli 14522 | Closure law for complex co... |
| replimi 14523 | Construct a complex number... |
| cjcji 14524 | The conjugate of the conju... |
| reim0bi 14525 | A number is real iff its i... |
| rerebi 14526 | A real number equals its r... |
| cjrebi 14527 | A number is real iff it eq... |
| recji 14528 | Real part of a complex con... |
| imcji 14529 | Imaginary part of a comple... |
| cjmulrcli 14530 | A complex number times its... |
| cjmulvali 14531 | A complex number times its... |
| cjmulge0i 14532 | A complex number times its... |
| renegi 14533 | Real part of negative. (C... |
| imnegi 14534 | Imaginary part of negative... |
| cjnegi 14535 | Complex conjugate of negat... |
| addcji 14536 | A number plus its conjugat... |
| readdi 14537 | Real part distributes over... |
| imaddi 14538 | Imaginary part distributes... |
| remuli 14539 | Real part of a product. (... |
| immuli 14540 | Imaginary part of a produc... |
| cjaddi 14541 | Complex conjugate distribu... |
| cjmuli 14542 | Complex conjugate distribu... |
| ipcni 14543 | Standard inner product on ... |
| cjdivi 14544 | Complex conjugate distribu... |
| crrei 14545 | The real part of a complex... |
| crimi 14546 | The imaginary part of a co... |
| recld 14547 | The real part of a complex... |
| imcld 14548 | The imaginary part of a co... |
| cjcld 14549 | Closure law for complex co... |
| replimd 14550 | Construct a complex number... |
| remimd 14551 | Value of the conjugate of ... |
| cjcjd 14552 | The conjugate of the conju... |
| reim0bd 14553 | A number is real iff its i... |
| rerebd 14554 | A real number equals its r... |
| cjrebd 14555 | A number is real iff it eq... |
| cjne0d 14556 | A number is nonzero iff it... |
| recjd 14557 | Real part of a complex con... |
| imcjd 14558 | Imaginary part of a comple... |
| cjmulrcld 14559 | A complex number times its... |
| cjmulvald 14560 | A complex number times its... |
| cjmulge0d 14561 | A complex number times its... |
| renegd 14562 | Real part of negative. (C... |
| imnegd 14563 | Imaginary part of negative... |
| cjnegd 14564 | Complex conjugate of negat... |
| addcjd 14565 | A number plus its conjugat... |
| cjexpd 14566 | Complex conjugate of posit... |
| readdd 14567 | Real part distributes over... |
| imaddd 14568 | Imaginary part distributes... |
| resubd 14569 | Real part distributes over... |
| imsubd 14570 | Imaginary part distributes... |
| remuld 14571 | Real part of a product. (... |
| immuld 14572 | Imaginary part of a produc... |
| cjaddd 14573 | Complex conjugate distribu... |
| cjmuld 14574 | Complex conjugate distribu... |
| ipcnd 14575 | Standard inner product on ... |
| cjdivd 14576 | Complex conjugate distribu... |
| rered 14577 | A real number equals its r... |
| reim0d 14578 | The imaginary part of a re... |
| cjred 14579 | A real number equals its c... |
| remul2d 14580 | Real part of a product. (... |
| immul2d 14581 | Imaginary part of a produc... |
| redivd 14582 | Real part of a division. ... |
| imdivd 14583 | Imaginary part of a divisi... |
| crred 14584 | The real part of a complex... |
| crimd 14585 | The imaginary part of a co... |
| sqrtval 14590 | Value of square root funct... |
| absval 14591 | The absolute value (modulu... |
| rennim 14592 | A real number does not lie... |
| cnpart 14593 | The specification of restr... |
| sqr0lem 14594 | Square root of zero. (Con... |
| sqrt0 14595 | Square root of zero. (Con... |
| sqrlem1 14596 | Lemma for ~ 01sqrex . (Co... |
| sqrlem2 14597 | Lemma for ~ 01sqrex . (Co... |
| sqrlem3 14598 | Lemma for ~ 01sqrex . (Co... |
| sqrlem4 14599 | Lemma for ~ 01sqrex . (Co... |
| sqrlem5 14600 | Lemma for ~ 01sqrex . (Co... |
| sqrlem6 14601 | Lemma for ~ 01sqrex . (Co... |
| sqrlem7 14602 | Lemma for ~ 01sqrex . (Co... |
| 01sqrex 14603 | Existence of a square root... |
| resqrex 14604 | Existence of a square root... |
| sqrmo 14605 | Uniqueness for the square ... |
| resqreu 14606 | Existence and uniqueness f... |
| resqrtcl 14607 | Closure of the square root... |
| resqrtthlem 14608 | Lemma for ~ resqrtth . (C... |
| resqrtth 14609 | Square root theorem over t... |
| remsqsqrt 14610 | Square of square root. (C... |
| sqrtge0 14611 | The square root function i... |
| sqrtgt0 14612 | The square root function i... |
| sqrtmul 14613 | Square root distributes ov... |
| sqrtle 14614 | Square root is monotonic. ... |
| sqrtlt 14615 | Square root is strictly mo... |
| sqrt11 14616 | The square root function i... |
| sqrt00 14617 | A square root is zero iff ... |
| rpsqrtcl 14618 | The square root of a posit... |
| sqrtdiv 14619 | Square root distributes ov... |
| sqrtneglem 14620 | The square root of a negat... |
| sqrtneg 14621 | The square root of a negat... |
| sqrtsq2 14622 | Relationship between squar... |
| sqrtsq 14623 | Square root of square. (C... |
| sqrtmsq 14624 | Square root of square. (C... |
| sqrt1 14625 | The square root of 1 is 1.... |
| sqrt4 14626 | The square root of 4 is 2.... |
| sqrt9 14627 | The square root of 9 is 3.... |
| sqrt2gt1lt2 14628 | The square root of 2 is bo... |
| sqrtm1 14629 | The imaginary unit is the ... |
| nn0sqeq1 14630 | A natural number with squa... |
| absneg 14631 | Absolute value of the oppo... |
| abscl 14632 | Real closure of absolute v... |
| abscj 14633 | The absolute value of a nu... |
| absvalsq 14634 | Square of value of absolut... |
| absvalsq2 14635 | Square of value of absolut... |
| sqabsadd 14636 | Square of absolute value o... |
| sqabssub 14637 | Square of absolute value o... |
| absval2 14638 | Value of absolute value fu... |
| abs0 14639 | The absolute value of 0. ... |
| absi 14640 | The absolute value of the ... |
| absge0 14641 | Absolute value is nonnegat... |
| absrpcl 14642 | The absolute value of a no... |
| abs00 14643 | The absolute value of a nu... |
| abs00ad 14644 | A complex number is zero i... |
| abs00bd 14645 | If a complex number is zer... |
| absreimsq 14646 | Square of the absolute val... |
| absreim 14647 | Absolute value of a number... |
| absmul 14648 | Absolute value distributes... |
| absdiv 14649 | Absolute value distributes... |
| absid 14650 | A nonnegative number is it... |
| abs1 14651 | The absolute value of one ... |
| absnid 14652 | A negative number is the n... |
| leabs 14653 | A real number is less than... |
| absor 14654 | The absolute value of a re... |
| absre 14655 | Absolute value of a real n... |
| absresq 14656 | Square of the absolute val... |
| absmod0 14657 | ` A ` is divisible by ` B ... |
| absexp 14658 | Absolute value of positive... |
| absexpz 14659 | Absolute value of integer ... |
| abssq 14660 | Square can be moved in and... |
| sqabs 14661 | The squares of two reals a... |
| absrele 14662 | The absolute value of a co... |
| absimle 14663 | The absolute value of a co... |
| max0add 14664 | The sum of the positive an... |
| absz 14665 | A real number is an intege... |
| nn0abscl 14666 | The absolute value of an i... |
| zabscl 14667 | The absolute value of an i... |
| abslt 14668 | Absolute value and 'less t... |
| absle 14669 | Absolute value and 'less t... |
| abssubne0 14670 | If the absolute value of a... |
| absdiflt 14671 | The absolute value of a di... |
| absdifle 14672 | The absolute value of a di... |
| elicc4abs 14673 | Membership in a symmetric ... |
| lenegsq 14674 | Comparison to a nonnegativ... |
| releabs 14675 | The real part of a number ... |
| recval 14676 | Reciprocal expressed with ... |
| absidm 14677 | The absolute value functio... |
| absgt0 14678 | The absolute value of a no... |
| nnabscl 14679 | The absolute value of a no... |
| abssub 14680 | Swapping order of subtract... |
| abssubge0 14681 | Absolute value of a nonneg... |
| abssuble0 14682 | Absolute value of a nonpos... |
| absmax 14683 | The maximum of two numbers... |
| abstri 14684 | Triangle inequality for ab... |
| abs3dif 14685 | Absolute value of differen... |
| abs2dif 14686 | Difference of absolute val... |
| abs2dif2 14687 | Difference of absolute val... |
| abs2difabs 14688 | Absolute value of differen... |
| abs1m 14689 | For any complex number, th... |
| recan 14690 | Cancellation law involving... |
| absf 14691 | Mapping domain and codomai... |
| abs3lem 14692 | Lemma involving absolute v... |
| abslem2 14693 | Lemma involving absolute v... |
| rddif 14694 | The difference between a r... |
| absrdbnd 14695 | Bound on the absolute valu... |
| fzomaxdiflem 14696 | Lemma for ~ fzomaxdif . (... |
| fzomaxdif 14697 | A bound on the separation ... |
| uzin2 14698 | The upper integers are clo... |
| rexanuz 14699 | Combine two different uppe... |
| rexanre 14700 | Combine two different uppe... |
| rexfiuz 14701 | Combine finitely many diff... |
| rexuz3 14702 | Restrict the base of the u... |
| rexanuz2 14703 | Combine two different uppe... |
| r19.29uz 14704 | A version of ~ 19.29 for u... |
| r19.2uz 14705 | A version of ~ r19.2z for ... |
| rexuzre 14706 | Convert an upper real quan... |
| rexico 14707 | Restrict the base of an up... |
| cau3lem 14708 | Lemma for ~ cau3 . (Contr... |
| cau3 14709 | Convert between three-quan... |
| cau4 14710 | Change the base of a Cauch... |
| caubnd2 14711 | A Cauchy sequence of compl... |
| caubnd 14712 | A Cauchy sequence of compl... |
| sqreulem 14713 | Lemma for ~ sqreu : write ... |
| sqreu 14714 | Existence and uniqueness f... |
| sqrtcl 14715 | Closure of the square root... |
| sqrtthlem 14716 | Lemma for ~ sqrtth . (Con... |
| sqrtf 14717 | Mapping domain and codomai... |
| sqrtth 14718 | Square root theorem over t... |
| sqrtrege0 14719 | The square root function m... |
| eqsqrtor 14720 | Solve an equation containi... |
| eqsqrtd 14721 | A deduction for showing th... |
| eqsqrt2d 14722 | A deduction for showing th... |
| amgm2 14723 | Arithmetic-geometric mean ... |
| sqrtthi 14724 | Square root theorem. Theo... |
| sqrtcli 14725 | The square root of a nonne... |
| sqrtgt0i 14726 | The square root of a posit... |
| sqrtmsqi 14727 | Square root of square. (C... |
| sqrtsqi 14728 | Square root of square. (C... |
| sqsqrti 14729 | Square of square root. (C... |
| sqrtge0i 14730 | The square root of a nonne... |
| absidi 14731 | A nonnegative number is it... |
| absnidi 14732 | A negative number is the n... |
| leabsi 14733 | A real number is less than... |
| absori 14734 | The absolute value of a re... |
| absrei 14735 | Absolute value of a real n... |
| sqrtpclii 14736 | The square root of a posit... |
| sqrtgt0ii 14737 | The square root of a posit... |
| sqrt11i 14738 | The square root function i... |
| sqrtmuli 14739 | Square root distributes ov... |
| sqrtmulii 14740 | Square root distributes ov... |
| sqrtmsq2i 14741 | Relationship between squar... |
| sqrtlei 14742 | Square root is monotonic. ... |
| sqrtlti 14743 | Square root is strictly mo... |
| abslti 14744 | Absolute value and 'less t... |
| abslei 14745 | Absolute value and 'less t... |
| cnsqrt00 14746 | A square root of a complex... |
| absvalsqi 14747 | Square of value of absolut... |
| absvalsq2i 14748 | Square of value of absolut... |
| abscli 14749 | Real closure of absolute v... |
| absge0i 14750 | Absolute value is nonnegat... |
| absval2i 14751 | Value of absolute value fu... |
| abs00i 14752 | The absolute value of a nu... |
| absgt0i 14753 | The absolute value of a no... |
| absnegi 14754 | Absolute value of negative... |
| abscji 14755 | The absolute value of a nu... |
| releabsi 14756 | The real part of a number ... |
| abssubi 14757 | Swapping order of subtract... |
| absmuli 14758 | Absolute value distributes... |
| sqabsaddi 14759 | Square of absolute value o... |
| sqabssubi 14760 | Square of absolute value o... |
| absdivzi 14761 | Absolute value distributes... |
| abstrii 14762 | Triangle inequality for ab... |
| abs3difi 14763 | Absolute value of differen... |
| abs3lemi 14764 | Lemma involving absolute v... |
| rpsqrtcld 14765 | The square root of a posit... |
| sqrtgt0d 14766 | The square root of a posit... |
| absnidd 14767 | A negative number is the n... |
| leabsd 14768 | A real number is less than... |
| absord 14769 | The absolute value of a re... |
| absred 14770 | Absolute value of a real n... |
| resqrtcld 14771 | The square root of a nonne... |
| sqrtmsqd 14772 | Square root of square. (C... |
| sqrtsqd 14773 | Square root of square. (C... |
| sqrtge0d 14774 | The square root of a nonne... |
| sqrtnegd 14775 | The square root of a negat... |
| absidd 14776 | A nonnegative number is it... |
| sqrtdivd 14777 | Square root distributes ov... |
| sqrtmuld 14778 | Square root distributes ov... |
| sqrtsq2d 14779 | Relationship between squar... |
| sqrtled 14780 | Square root is monotonic. ... |
| sqrtltd 14781 | Square root is strictly mo... |
| sqr11d 14782 | The square root function i... |
| absltd 14783 | Absolute value and 'less t... |
| absled 14784 | Absolute value and 'less t... |
| abssubge0d 14785 | Absolute value of a nonneg... |
| abssuble0d 14786 | Absolute value of a nonpos... |
| absdifltd 14787 | The absolute value of a di... |
| absdifled 14788 | The absolute value of a di... |
| icodiamlt 14789 | Two elements in a half-ope... |
| abscld 14790 | Real closure of absolute v... |
| sqrtcld 14791 | Closure of the square root... |
| sqrtrege0d 14792 | The real part of the squar... |
| sqsqrtd 14793 | Square root theorem. Theo... |
| msqsqrtd 14794 | Square root theorem. Theo... |
| sqr00d 14795 | A square root is zero iff ... |
| absvalsqd 14796 | Square of value of absolut... |
| absvalsq2d 14797 | Square of value of absolut... |
| absge0d 14798 | Absolute value is nonnegat... |
| absval2d 14799 | Value of absolute value fu... |
| abs00d 14800 | The absolute value of a nu... |
| absne0d 14801 | The absolute value of a nu... |
| absrpcld 14802 | The absolute value of a no... |
| absnegd 14803 | Absolute value of negative... |
| abscjd 14804 | The absolute value of a nu... |
| releabsd 14805 | The real part of a number ... |
| absexpd 14806 | Absolute value of positive... |
| abssubd 14807 | Swapping order of subtract... |
| absmuld 14808 | Absolute value distributes... |
| absdivd 14809 | Absolute value distributes... |
| abstrid 14810 | Triangle inequality for ab... |
| abs2difd 14811 | Difference of absolute val... |
| abs2dif2d 14812 | Difference of absolute val... |
| abs2difabsd 14813 | Absolute value of differen... |
| abs3difd 14814 | Absolute value of differen... |
| abs3lemd 14815 | Lemma involving absolute v... |
| reusq0 14816 | A complex number is the sq... |
| bhmafibid1cn 14817 | The Brahmagupta-Fibonacci ... |
| bhmafibid2cn 14818 | The Brahmagupta-Fibonacci ... |
| bhmafibid1 14819 | The Brahmagupta-Fibonacci ... |
| bhmafibid2 14820 | The Brahmagupta-Fibonacci ... |
| limsupgord 14823 | Ordering property of the s... |
| limsupcl 14824 | Closure of the superior li... |
| limsupval 14825 | The superior limit of an i... |
| limsupgf 14826 | Closure of the superior li... |
| limsupgval 14827 | Value of the superior limi... |
| limsupgle 14828 | The defining property of t... |
| limsuple 14829 | The defining property of t... |
| limsuplt 14830 | The defining property of t... |
| limsupval2 14831 | The superior limit, relati... |
| limsupgre 14832 | If a sequence of real numb... |
| limsupbnd1 14833 | If a sequence is eventuall... |
| limsupbnd2 14834 | If a sequence is eventuall... |
| climrel 14843 | The limit relation is a re... |
| rlimrel 14844 | The limit relation is a re... |
| clim 14845 | Express the predicate: Th... |
| rlim 14846 | Express the predicate: Th... |
| rlim2 14847 | Rewrite ~ rlim for a mappi... |
| rlim2lt 14848 | Use strictly less-than in ... |
| rlim3 14849 | Restrict the range of the ... |
| climcl 14850 | Closure of the limit of a ... |
| rlimpm 14851 | Closure of a function with... |
| rlimf 14852 | Closure of a function with... |
| rlimss 14853 | Domain closure of a functi... |
| rlimcl 14854 | Closure of the limit of a ... |
| clim2 14855 | Express the predicate: Th... |
| clim2c 14856 | Express the predicate ` F ... |
| clim0 14857 | Express the predicate ` F ... |
| clim0c 14858 | Express the predicate ` F ... |
| rlim0 14859 | Express the predicate ` B ... |
| rlim0lt 14860 | Use strictly less-than in ... |
| climi 14861 | Convergence of a sequence ... |
| climi2 14862 | Convergence of a sequence ... |
| climi0 14863 | Convergence of a sequence ... |
| rlimi 14864 | Convergence at infinity of... |
| rlimi2 14865 | Convergence at infinity of... |
| ello1 14866 | Elementhood in the set of ... |
| ello12 14867 | Elementhood in the set of ... |
| ello12r 14868 | Sufficient condition for e... |
| lo1f 14869 | An eventually upper bounde... |
| lo1dm 14870 | An eventually upper bounde... |
| lo1bdd 14871 | The defining property of a... |
| ello1mpt 14872 | Elementhood in the set of ... |
| ello1mpt2 14873 | Elementhood in the set of ... |
| ello1d 14874 | Sufficient condition for e... |
| lo1bdd2 14875 | If an eventually bounded f... |
| lo1bddrp 14876 | Refine ~ o1bdd2 to give a ... |
| elo1 14877 | Elementhood in the set of ... |
| elo12 14878 | Elementhood in the set of ... |
| elo12r 14879 | Sufficient condition for e... |
| o1f 14880 | An eventually bounded func... |
| o1dm 14881 | An eventually bounded func... |
| o1bdd 14882 | The defining property of a... |
| lo1o1 14883 | A function is eventually b... |
| lo1o12 14884 | A function is eventually b... |
| elo1mpt 14885 | Elementhood in the set of ... |
| elo1mpt2 14886 | Elementhood in the set of ... |
| elo1d 14887 | Sufficient condition for e... |
| o1lo1 14888 | A real function is eventua... |
| o1lo12 14889 | A lower bounded real funct... |
| o1lo1d 14890 | A real eventually bounded ... |
| icco1 14891 | Derive eventual boundednes... |
| o1bdd2 14892 | If an eventually bounded f... |
| o1bddrp 14893 | Refine ~ o1bdd2 to give a ... |
| climconst 14894 | An (eventually) constant s... |
| rlimconst 14895 | A constant sequence conver... |
| rlimclim1 14896 | Forward direction of ~ rli... |
| rlimclim 14897 | A sequence on an upper int... |
| climrlim2 14898 | Produce a real limit from ... |
| climconst2 14899 | A constant sequence conver... |
| climz 14900 | The zero sequence converge... |
| rlimuni 14901 | A real function whose doma... |
| rlimdm 14902 | Two ways to express that a... |
| climuni 14903 | An infinite sequence of co... |
| fclim 14904 | The limit relation is func... |
| climdm 14905 | Two ways to express that a... |
| climeu 14906 | An infinite sequence of co... |
| climreu 14907 | An infinite sequence of co... |
| climmo 14908 | An infinite sequence of co... |
| rlimres 14909 | The restriction of a funct... |
| lo1res 14910 | The restriction of an even... |
| o1res 14911 | The restriction of an even... |
| rlimres2 14912 | The restriction of a funct... |
| lo1res2 14913 | The restriction of a funct... |
| o1res2 14914 | The restriction of a funct... |
| lo1resb 14915 | The restriction of a funct... |
| rlimresb 14916 | The restriction of a funct... |
| o1resb 14917 | The restriction of a funct... |
| climeq 14918 | Two functions that are eve... |
| lo1eq 14919 | Two functions that are eve... |
| rlimeq 14920 | Two functions that are eve... |
| o1eq 14921 | Two functions that are eve... |
| climmpt 14922 | Exhibit a function ` G ` w... |
| 2clim 14923 | If two sequences converge ... |
| climmpt2 14924 | Relate an integer limit on... |
| climshftlem 14925 | A shifted function converg... |
| climres 14926 | A function restricted to u... |
| climshft 14927 | A shifted function converg... |
| serclim0 14928 | The zero series converges ... |
| rlimcld2 14929 | If ` D ` is a closed set i... |
| rlimrege0 14930 | The limit of a sequence of... |
| rlimrecl 14931 | The limit of a real sequen... |
| rlimge0 14932 | The limit of a sequence of... |
| climshft2 14933 | A shifted function converg... |
| climrecl 14934 | The limit of a convergent ... |
| climge0 14935 | A nonnegative sequence con... |
| climabs0 14936 | Convergence to zero of the... |
| o1co 14937 | Sufficient condition for t... |
| o1compt 14938 | Sufficient condition for t... |
| rlimcn1 14939 | Image of a limit under a c... |
| rlimcn1b 14940 | Image of a limit under a c... |
| rlimcn2 14941 | Image of a limit under a c... |
| climcn1 14942 | Image of a limit under a c... |
| climcn2 14943 | Image of a limit under a c... |
| addcn2 14944 | Complex number addition is... |
| subcn2 14945 | Complex number subtraction... |
| mulcn2 14946 | Complex number multiplicat... |
| reccn2 14947 | The reciprocal function is... |
| cn1lem 14948 | A sufficient condition for... |
| abscn2 14949 | The absolute value functio... |
| cjcn2 14950 | The complex conjugate func... |
| recn2 14951 | The real part function is ... |
| imcn2 14952 | The imaginary part functio... |
| climcn1lem 14953 | The limit of a continuous ... |
| climabs 14954 | Limit of the absolute valu... |
| climcj 14955 | Limit of the complex conju... |
| climre 14956 | Limit of the real part of ... |
| climim 14957 | Limit of the imaginary par... |
| rlimmptrcl 14958 | Reverse closure for a real... |
| rlimabs 14959 | Limit of the absolute valu... |
| rlimcj 14960 | Limit of the complex conju... |
| rlimre 14961 | Limit of the real part of ... |
| rlimim 14962 | Limit of the imaginary par... |
| o1of2 14963 | Show that a binary operati... |
| o1add 14964 | The sum of two eventually ... |
| o1mul 14965 | The product of two eventua... |
| o1sub 14966 | The difference of two even... |
| rlimo1 14967 | Any function with a finite... |
| rlimdmo1 14968 | A convergent function is e... |
| o1rlimmul 14969 | The product of an eventual... |
| o1const 14970 | A constant function is eve... |
| lo1const 14971 | A constant function is eve... |
| lo1mptrcl 14972 | Reverse closure for an eve... |
| o1mptrcl 14973 | Reverse closure for an eve... |
| o1add2 14974 | The sum of two eventually ... |
| o1mul2 14975 | The product of two eventua... |
| o1sub2 14976 | The product of two eventua... |
| lo1add 14977 | The sum of two eventually ... |
| lo1mul 14978 | The product of an eventual... |
| lo1mul2 14979 | The product of an eventual... |
| o1dif 14980 | If the difference of two f... |
| lo1sub 14981 | The difference of an event... |
| climadd 14982 | Limit of the sum of two co... |
| climmul 14983 | Limit of the product of tw... |
| climsub 14984 | Limit of the difference of... |
| climaddc1 14985 | Limit of a constant ` C ` ... |
| climaddc2 14986 | Limit of a constant ` C ` ... |
| climmulc2 14987 | Limit of a sequence multip... |
| climsubc1 14988 | Limit of a constant ` C ` ... |
| climsubc2 14989 | Limit of a constant ` C ` ... |
| climle 14990 | Comparison of the limits o... |
| climsqz 14991 | Convergence of a sequence ... |
| climsqz2 14992 | Convergence of a sequence ... |
| rlimadd 14993 | Limit of the sum of two co... |
| rlimsub 14994 | Limit of the difference of... |
| rlimmul 14995 | Limit of the product of tw... |
| rlimdiv 14996 | Limit of the quotient of t... |
| rlimneg 14997 | Limit of the negative of a... |
| rlimle 14998 | Comparison of the limits o... |
| rlimsqzlem 14999 | Lemma for ~ rlimsqz and ~ ... |
| rlimsqz 15000 | Convergence of a sequence ... |
| rlimsqz2 15001 | Convergence of a sequence ... |
| lo1le 15002 | Transfer eventual upper bo... |
| o1le 15003 | Transfer eventual boundedn... |
| rlimno1 15004 | A function whose inverse c... |
| clim2ser 15005 | The limit of an infinite s... |
| clim2ser2 15006 | The limit of an infinite s... |
| iserex 15007 | An infinite series converg... |
| isermulc2 15008 | Multiplication of an infin... |
| climlec2 15009 | Comparison of a constant t... |
| iserle 15010 | Comparison of the limits o... |
| iserge0 15011 | The limit of an infinite s... |
| climub 15012 | The limit of a monotonic s... |
| climserle 15013 | The partial sums of a conv... |
| isershft 15014 | Index shift of the limit o... |
| isercolllem1 15015 | Lemma for ~ isercoll . (C... |
| isercolllem2 15016 | Lemma for ~ isercoll . (C... |
| isercolllem3 15017 | Lemma for ~ isercoll . (C... |
| isercoll 15018 | Rearrange an infinite seri... |
| isercoll2 15019 | Generalize ~ isercoll so t... |
| climsup 15020 | A bounded monotonic sequen... |
| climcau 15021 | A converging sequence of c... |
| climbdd 15022 | A converging sequence of c... |
| caucvgrlem 15023 | Lemma for ~ caurcvgr . (C... |
| caurcvgr 15024 | A Cauchy sequence of real ... |
| caucvgrlem2 15025 | Lemma for ~ caucvgr . (Co... |
| caucvgr 15026 | A Cauchy sequence of compl... |
| caurcvg 15027 | A Cauchy sequence of real ... |
| caurcvg2 15028 | A Cauchy sequence of real ... |
| caucvg 15029 | A Cauchy sequence of compl... |
| caucvgb 15030 | A function is convergent i... |
| serf0 15031 | If an infinite series conv... |
| iseraltlem1 15032 | Lemma for ~ iseralt . A d... |
| iseraltlem2 15033 | Lemma for ~ iseralt . The... |
| iseraltlem3 15034 | Lemma for ~ iseralt . Fro... |
| iseralt 15035 | The alternating series tes... |
| sumex 15038 | A sum is a set. (Contribu... |
| sumeq1 15039 | Equality theorem for a sum... |
| nfsum1 15040 | Bound-variable hypothesis ... |
| nfsumw 15041 | Bound-variable hypothesis ... |
| nfsum 15042 | Bound-variable hypothesis ... |
| sumeq2w 15043 | Equality theorem for sum, ... |
| sumeq2ii 15044 | Equality theorem for sum, ... |
| sumeq2 15045 | Equality theorem for sum. ... |
| cbvsum 15046 | Change bound variable in a... |
| cbvsumv 15047 | Change bound variable in a... |
| cbvsumi 15048 | Change bound variable in a... |
| sumeq1i 15049 | Equality inference for sum... |
| sumeq2i 15050 | Equality inference for sum... |
| sumeq12i 15051 | Equality inference for sum... |
| sumeq1d 15052 | Equality deduction for sum... |
| sumeq2d 15053 | Equality deduction for sum... |
| sumeq2dv 15054 | Equality deduction for sum... |
| sumeq2sdv 15055 | Equality deduction for sum... |
| 2sumeq2dv 15056 | Equality deduction for dou... |
| sumeq12dv 15057 | Equality deduction for sum... |
| sumeq12rdv 15058 | Equality deduction for sum... |
| sum2id 15059 | The second class argument ... |
| sumfc 15060 | A lemma to facilitate conv... |
| fz1f1o 15061 | A lemma for working with f... |
| sumrblem 15062 | Lemma for ~ sumrb . (Cont... |
| fsumcvg 15063 | The sequence of partial su... |
| sumrb 15064 | Rebase the starting point ... |
| summolem3 15065 | Lemma for ~ summo . (Cont... |
| summolem2a 15066 | Lemma for ~ summo . (Cont... |
| summolem2 15067 | Lemma for ~ summo . (Cont... |
| summo 15068 | A sum has at most one limi... |
| zsum 15069 | Series sum with index set ... |
| isum 15070 | Series sum with an upper i... |
| fsum 15071 | The value of a sum over a ... |
| sum0 15072 | Any sum over the empty set... |
| sumz 15073 | Any sum of zero over a sum... |
| fsumf1o 15074 | Re-index a finite sum usin... |
| sumss 15075 | Change the index set to a ... |
| fsumss 15076 | Change the index set to a ... |
| sumss2 15077 | Change the index set of a ... |
| fsumcvg2 15078 | The sequence of partial su... |
| fsumsers 15079 | Special case of series sum... |
| fsumcvg3 15080 | A finite sum is convergent... |
| fsumser 15081 | A finite sum expressed in ... |
| fsumcl2lem 15082 | - Lemma for finite sum clo... |
| fsumcllem 15083 | - Lemma for finite sum clo... |
| fsumcl 15084 | Closure of a finite sum of... |
| fsumrecl 15085 | Closure of a finite sum of... |
| fsumzcl 15086 | Closure of a finite sum of... |
| fsumnn0cl 15087 | Closure of a finite sum of... |
| fsumrpcl 15088 | Closure of a finite sum of... |
| fsumzcl2 15089 | A finite sum with integer ... |
| fsumadd 15090 | The sum of two finite sums... |
| fsumsplit 15091 | Split a sum into two parts... |
| fsumsplitf 15092 | Split a sum into two parts... |
| sumsnf 15093 | A sum of a singleton is th... |
| fsumsplitsn 15094 | Separate out a term in a f... |
| sumsn 15095 | A sum of a singleton is th... |
| fsum1 15096 | The finite sum of ` A ( k ... |
| sumpr 15097 | A sum over a pair is the s... |
| sumtp 15098 | A sum over a triple is the... |
| sumsns 15099 | A sum of a singleton is th... |
| fsumm1 15100 | Separate out the last term... |
| fzosump1 15101 | Separate out the last term... |
| fsum1p 15102 | Separate out the first ter... |
| fsummsnunz 15103 | A finite sum all of whose ... |
| fsumsplitsnun 15104 | Separate out a term in a f... |
| fsump1 15105 | The addition of the next t... |
| isumclim 15106 | An infinite sum equals the... |
| isumclim2 15107 | A converging series conver... |
| isumclim3 15108 | The sequence of partial fi... |
| sumnul 15109 | The sum of a non-convergen... |
| isumcl 15110 | The sum of a converging in... |
| isummulc2 15111 | An infinite sum multiplied... |
| isummulc1 15112 | An infinite sum multiplied... |
| isumdivc 15113 | An infinite sum divided by... |
| isumrecl 15114 | The sum of a converging in... |
| isumge0 15115 | An infinite sum of nonnega... |
| isumadd 15116 | Addition of infinite sums.... |
| sumsplit 15117 | Split a sum into two parts... |
| fsump1i 15118 | Optimized version of ~ fsu... |
| fsum2dlem 15119 | Lemma for ~ fsum2d - induc... |
| fsum2d 15120 | Write a double sum as a su... |
| fsumxp 15121 | Combine two sums into a si... |
| fsumcnv 15122 | Transform a region of summ... |
| fsumcom2 15123 | Interchange order of summa... |
| fsumcom 15124 | Interchange order of summa... |
| fsum0diaglem 15125 | Lemma for ~ fsum0diag . (... |
| fsum0diag 15126 | Two ways to express "the s... |
| mptfzshft 15127 | 1-1 onto function in maps-... |
| fsumrev 15128 | Reversal of a finite sum. ... |
| fsumshft 15129 | Index shift of a finite su... |
| fsumshftm 15130 | Negative index shift of a ... |
| fsumrev2 15131 | Reversal of a finite sum. ... |
| fsum0diag2 15132 | Two ways to express "the s... |
| fsummulc2 15133 | A finite sum multiplied by... |
| fsummulc1 15134 | A finite sum multiplied by... |
| fsumdivc 15135 | A finite sum divided by a ... |
| fsumneg 15136 | Negation of a finite sum. ... |
| fsumsub 15137 | Split a finite sum over a ... |
| fsum2mul 15138 | Separate the nested sum of... |
| fsumconst 15139 | The sum of constant terms ... |
| fsumdifsnconst 15140 | The sum of constant terms ... |
| modfsummodslem1 15141 | Lemma 1 for ~ modfsummods ... |
| modfsummods 15142 | Induction step for ~ modfs... |
| modfsummod 15143 | A finite sum modulo a posi... |
| fsumge0 15144 | If all of the terms of a f... |
| fsumless 15145 | A shorter sum of nonnegati... |
| fsumge1 15146 | A sum of nonnegative numbe... |
| fsum00 15147 | A sum of nonnegative numbe... |
| fsumle 15148 | If all of the terms of fin... |
| fsumlt 15149 | If every term in one finit... |
| fsumabs 15150 | Generalized triangle inequ... |
| telfsumo 15151 | Sum of a telescoping serie... |
| telfsumo2 15152 | Sum of a telescoping serie... |
| telfsum 15153 | Sum of a telescoping serie... |
| telfsum2 15154 | Sum of a telescoping serie... |
| fsumparts 15155 | Summation by parts. (Cont... |
| fsumrelem 15156 | Lemma for ~ fsumre , ~ fsu... |
| fsumre 15157 | The real part of a sum. (... |
| fsumim 15158 | The imaginary part of a su... |
| fsumcj 15159 | The complex conjugate of a... |
| fsumrlim 15160 | Limit of a finite sum of c... |
| fsumo1 15161 | The finite sum of eventual... |
| o1fsum 15162 | If ` A ( k ) ` is O(1), th... |
| seqabs 15163 | Generalized triangle inequ... |
| iserabs 15164 | Generalized triangle inequ... |
| cvgcmp 15165 | A comparison test for conv... |
| cvgcmpub 15166 | An upper bound for the lim... |
| cvgcmpce 15167 | A comparison test for conv... |
| abscvgcvg 15168 | An absolutely convergent s... |
| climfsum 15169 | Limit of a finite sum of c... |
| fsumiun 15170 | Sum over a disjoint indexe... |
| hashiun 15171 | The cardinality of a disjo... |
| hash2iun 15172 | The cardinality of a neste... |
| hash2iun1dif1 15173 | The cardinality of a neste... |
| hashrabrex 15174 | The number of elements in ... |
| hashuni 15175 | The cardinality of a disjo... |
| qshash 15176 | The cardinality of a set w... |
| ackbijnn 15177 | Translate the Ackermann bi... |
| binomlem 15178 | Lemma for ~ binom (binomia... |
| binom 15179 | The binomial theorem: ` ( ... |
| binom1p 15180 | Special case of the binomi... |
| binom11 15181 | Special case of the binomi... |
| binom1dif 15182 | A summation for the differ... |
| bcxmaslem1 15183 | Lemma for ~ bcxmas . (Con... |
| bcxmas 15184 | Parallel summation (Christ... |
| incexclem 15185 | Lemma for ~ incexc . (Con... |
| incexc 15186 | The inclusion/exclusion pr... |
| incexc2 15187 | The inclusion/exclusion pr... |
| isumshft 15188 | Index shift of an infinite... |
| isumsplit 15189 | Split off the first ` N ` ... |
| isum1p 15190 | The infinite sum of a conv... |
| isumnn0nn 15191 | Sum from 0 to infinity in ... |
| isumrpcl 15192 | The infinite sum of positi... |
| isumle 15193 | Comparison of two infinite... |
| isumless 15194 | A finite sum of nonnegativ... |
| isumsup2 15195 | An infinite sum of nonnega... |
| isumsup 15196 | An infinite sum of nonnega... |
| isumltss 15197 | A partial sum of a series ... |
| climcndslem1 15198 | Lemma for ~ climcnds : bou... |
| climcndslem2 15199 | Lemma for ~ climcnds : bou... |
| climcnds 15200 | The Cauchy condensation te... |
| divrcnv 15201 | The sequence of reciprocal... |
| divcnv 15202 | The sequence of reciprocal... |
| flo1 15203 | The floor function satisfi... |
| divcnvshft 15204 | Limit of a ratio function.... |
| supcvg 15205 | Extract a sequence ` f ` i... |
| infcvgaux1i 15206 | Auxiliary theorem for appl... |
| infcvgaux2i 15207 | Auxiliary theorem for appl... |
| harmonic 15208 | The harmonic series ` H ` ... |
| arisum 15209 | Arithmetic series sum of t... |
| arisum2 15210 | Arithmetic series sum of t... |
| trireciplem 15211 | Lemma for ~ trirecip . Sh... |
| trirecip 15212 | The sum of the reciprocals... |
| expcnv 15213 | A sequence of powers of a ... |
| explecnv 15214 | A sequence of terms conver... |
| geoserg 15215 | The value of the finite ge... |
| geoser 15216 | The value of the finite ge... |
| pwdif 15217 | The difference of two numb... |
| pwm1geoser 15218 | The n-th power of a number... |
| pwm1geoserOLD 15219 | Obsolete version of ~ pwm1... |
| geolim 15220 | The partial sums in the in... |
| geolim2 15221 | The partial sums in the ge... |
| georeclim 15222 | The limit of a geometric s... |
| geo2sum 15223 | The value of the finite ge... |
| geo2sum2 15224 | The value of the finite ge... |
| geo2lim 15225 | The value of the infinite ... |
| geomulcvg 15226 | The geometric series conve... |
| geoisum 15227 | The infinite sum of ` 1 + ... |
| geoisumr 15228 | The infinite sum of recipr... |
| geoisum1 15229 | The infinite sum of ` A ^ ... |
| geoisum1c 15230 | The infinite sum of ` A x.... |
| 0.999... 15231 | The recurring decimal 0.99... |
| geoihalfsum 15232 | Prove that the infinite ge... |
| cvgrat 15233 | Ratio test for convergence... |
| mertenslem1 15234 | Lemma for ~ mertens . (Co... |
| mertenslem2 15235 | Lemma for ~ mertens . (Co... |
| mertens 15236 | Mertens' theorem. If ` A ... |
| prodf 15237 | An infinite product of com... |
| clim2prod 15238 | The limit of an infinite p... |
| clim2div 15239 | The limit of an infinite p... |
| prodfmul 15240 | The product of two infinit... |
| prodf1 15241 | The value of the partial p... |
| prodf1f 15242 | A one-valued infinite prod... |
| prodfclim1 15243 | The constant one product c... |
| prodfn0 15244 | No term of a nonzero infin... |
| prodfrec 15245 | The reciprocal of an infin... |
| prodfdiv 15246 | The quotient of two infini... |
| ntrivcvg 15247 | A non-trivially converging... |
| ntrivcvgn0 15248 | A product that converges t... |
| ntrivcvgfvn0 15249 | Any value of a product seq... |
| ntrivcvgtail 15250 | A tail of a non-trivially ... |
| ntrivcvgmullem 15251 | Lemma for ~ ntrivcvgmul . ... |
| ntrivcvgmul 15252 | The product of two non-tri... |
| prodex 15255 | A product is a set. (Cont... |
| prodeq1f 15256 | Equality theorem for a pro... |
| prodeq1 15257 | Equality theorem for a pro... |
| nfcprod1 15258 | Bound-variable hypothesis ... |
| nfcprod 15259 | Bound-variable hypothesis ... |
| prodeq2w 15260 | Equality theorem for produ... |
| prodeq2ii 15261 | Equality theorem for produ... |
| prodeq2 15262 | Equality theorem for produ... |
| cbvprod 15263 | Change bound variable in a... |
| cbvprodv 15264 | Change bound variable in a... |
| cbvprodi 15265 | Change bound variable in a... |
| prodeq1i 15266 | Equality inference for pro... |
| prodeq2i 15267 | Equality inference for pro... |
| prodeq12i 15268 | Equality inference for pro... |
| prodeq1d 15269 | Equality deduction for pro... |
| prodeq2d 15270 | Equality deduction for pro... |
| prodeq2dv 15271 | Equality deduction for pro... |
| prodeq2sdv 15272 | Equality deduction for pro... |
| 2cprodeq2dv 15273 | Equality deduction for dou... |
| prodeq12dv 15274 | Equality deduction for pro... |
| prodeq12rdv 15275 | Equality deduction for pro... |
| prod2id 15276 | The second class argument ... |
| prodrblem 15277 | Lemma for ~ prodrb . (Con... |
| fprodcvg 15278 | The sequence of partial pr... |
| prodrblem2 15279 | Lemma for ~ prodrb . (Con... |
| prodrb 15280 | Rebase the starting point ... |
| prodmolem3 15281 | Lemma for ~ prodmo . (Con... |
| prodmolem2a 15282 | Lemma for ~ prodmo . (Con... |
| prodmolem2 15283 | Lemma for ~ prodmo . (Con... |
| prodmo 15284 | A product has at most one ... |
| zprod 15285 | Series product with index ... |
| iprod 15286 | Series product with an upp... |
| zprodn0 15287 | Nonzero series product wit... |
| iprodn0 15288 | Nonzero series product wit... |
| fprod 15289 | The value of a product ove... |
| fprodntriv 15290 | A non-triviality lemma for... |
| prod0 15291 | A product over the empty s... |
| prod1 15292 | Any product of one over a ... |
| prodfc 15293 | A lemma to facilitate conv... |
| fprodf1o 15294 | Re-index a finite product ... |
| prodss 15295 | Change the index set to a ... |
| fprodss 15296 | Change the index set to a ... |
| fprodser 15297 | A finite product expressed... |
| fprodcl2lem 15298 | Finite product closure lem... |
| fprodcllem 15299 | Finite product closure lem... |
| fprodcl 15300 | Closure of a finite produc... |
| fprodrecl 15301 | Closure of a finite produc... |
| fprodzcl 15302 | Closure of a finite produc... |
| fprodnncl 15303 | Closure of a finite produc... |
| fprodrpcl 15304 | Closure of a finite produc... |
| fprodnn0cl 15305 | Closure of a finite produc... |
| fprodcllemf 15306 | Finite product closure lem... |
| fprodreclf 15307 | Closure of a finite produc... |
| fprodmul 15308 | The product of two finite ... |
| fproddiv 15309 | The quotient of two finite... |
| prodsn 15310 | A product of a singleton i... |
| fprod1 15311 | A finite product of only o... |
| prodsnf 15312 | A product of a singleton i... |
| climprod1 15313 | The limit of a product ove... |
| fprodsplit 15314 | Split a finite product int... |
| fprodm1 15315 | Separate out the last term... |
| fprod1p 15316 | Separate out the first ter... |
| fprodp1 15317 | Multiply in the last term ... |
| fprodm1s 15318 | Separate out the last term... |
| fprodp1s 15319 | Multiply in the last term ... |
| prodsns 15320 | A product of the singleton... |
| fprodfac 15321 | Factorial using product no... |
| fprodabs 15322 | The absolute value of a fi... |
| fprodeq0 15323 | Anything finite product co... |
| fprodshft 15324 | Shift the index of a finit... |
| fprodrev 15325 | Reversal of a finite produ... |
| fprodconst 15326 | The product of constant te... |
| fprodn0 15327 | A finite product of nonzer... |
| fprod2dlem 15328 | Lemma for ~ fprod2d - indu... |
| fprod2d 15329 | Write a double product as ... |
| fprodxp 15330 | Combine two products into ... |
| fprodcnv 15331 | Transform a product region... |
| fprodcom2 15332 | Interchange order of multi... |
| fprodcom 15333 | Interchange product order.... |
| fprod0diag 15334 | Two ways to express "the p... |
| fproddivf 15335 | The quotient of two finite... |
| fprodsplitf 15336 | Split a finite product int... |
| fprodsplitsn 15337 | Separate out a term in a f... |
| fprodsplit1f 15338 | Separate out a term in a f... |
| fprodn0f 15339 | A finite product of nonzer... |
| fprodclf 15340 | Closure of a finite produc... |
| fprodge0 15341 | If all the terms of a fini... |
| fprodeq0g 15342 | Any finite product contain... |
| fprodge1 15343 | If all of the terms of a f... |
| fprodle 15344 | If all the terms of two fi... |
| fprodmodd 15345 | If all factors of two fini... |
| iprodclim 15346 | An infinite product equals... |
| iprodclim2 15347 | A converging product conve... |
| iprodclim3 15348 | The sequence of partial fi... |
| iprodcl 15349 | The product of a non-trivi... |
| iprodrecl 15350 | The product of a non-trivi... |
| iprodmul 15351 | Multiplication of infinite... |
| risefacval 15356 | The value of the rising fa... |
| fallfacval 15357 | The value of the falling f... |
| risefacval2 15358 | One-based value of rising ... |
| fallfacval2 15359 | One-based value of falling... |
| fallfacval3 15360 | A product representation o... |
| risefaccllem 15361 | Lemma for rising factorial... |
| fallfaccllem 15362 | Lemma for falling factoria... |
| risefaccl 15363 | Closure law for rising fac... |
| fallfaccl 15364 | Closure law for falling fa... |
| rerisefaccl 15365 | Closure law for rising fac... |
| refallfaccl 15366 | Closure law for falling fa... |
| nnrisefaccl 15367 | Closure law for rising fac... |
| zrisefaccl 15368 | Closure law for rising fac... |
| zfallfaccl 15369 | Closure law for falling fa... |
| nn0risefaccl 15370 | Closure law for rising fac... |
| rprisefaccl 15371 | Closure law for rising fac... |
| risefallfac 15372 | A relationship between ris... |
| fallrisefac 15373 | A relationship between fal... |
| risefall0lem 15374 | Lemma for ~ risefac0 and ~... |
| risefac0 15375 | The value of the rising fa... |
| fallfac0 15376 | The value of the falling f... |
| risefacp1 15377 | The value of the rising fa... |
| fallfacp1 15378 | The value of the falling f... |
| risefacp1d 15379 | The value of the rising fa... |
| fallfacp1d 15380 | The value of the falling f... |
| risefac1 15381 | The value of rising factor... |
| fallfac1 15382 | The value of falling facto... |
| risefacfac 15383 | Relate rising factorial to... |
| fallfacfwd 15384 | The forward difference of ... |
| 0fallfac 15385 | The value of the zero fall... |
| 0risefac 15386 | The value of the zero risi... |
| binomfallfaclem1 15387 | Lemma for ~ binomfallfac .... |
| binomfallfaclem2 15388 | Lemma for ~ binomfallfac .... |
| binomfallfac 15389 | A version of the binomial ... |
| binomrisefac 15390 | A version of the binomial ... |
| fallfacval4 15391 | Represent the falling fact... |
| bcfallfac 15392 | Binomial coefficient in te... |
| fallfacfac 15393 | Relate falling factorial t... |
| bpolylem 15396 | Lemma for ~ bpolyval . (C... |
| bpolyval 15397 | The value of the Bernoulli... |
| bpoly0 15398 | The value of the Bernoulli... |
| bpoly1 15399 | The value of the Bernoulli... |
| bpolycl 15400 | Closure law for Bernoulli ... |
| bpolysum 15401 | A sum for Bernoulli polyno... |
| bpolydiflem 15402 | Lemma for ~ bpolydif . (C... |
| bpolydif 15403 | Calculate the difference b... |
| fsumkthpow 15404 | A closed-form expression f... |
| bpoly2 15405 | The Bernoulli polynomials ... |
| bpoly3 15406 | The Bernoulli polynomials ... |
| bpoly4 15407 | The Bernoulli polynomials ... |
| fsumcube 15408 | Express the sum of cubes i... |
| eftcl 15421 | Closure of a term in the s... |
| reeftcl 15422 | The terms of the series ex... |
| eftabs 15423 | The absolute value of a te... |
| eftval 15424 | The value of a term in the... |
| efcllem 15425 | Lemma for ~ efcl . The se... |
| ef0lem 15426 | The series defining the ex... |
| efval 15427 | Value of the exponential f... |
| esum 15428 | Value of Euler's constant ... |
| eff 15429 | Domain and codomain of the... |
| efcl 15430 | Closure law for the expone... |
| efval2 15431 | Value of the exponential f... |
| efcvg 15432 | The series that defines th... |
| efcvgfsum 15433 | Exponential function conve... |
| reefcl 15434 | The exponential function i... |
| reefcld 15435 | The exponential function i... |
| ere 15436 | Euler's constant ` _e ` = ... |
| ege2le3 15437 | Lemma for ~ egt2lt3 . (Co... |
| ef0 15438 | Value of the exponential f... |
| efcj 15439 | The exponential of a compl... |
| efaddlem 15440 | Lemma for ~ efadd (exponen... |
| efadd 15441 | Sum of exponents law for e... |
| fprodefsum 15442 | Move the exponential funct... |
| efcan 15443 | Cancellation law for expon... |
| efne0 15444 | The exponential of a compl... |
| efneg 15445 | The exponential of the opp... |
| eff2 15446 | The exponential function m... |
| efsub 15447 | Difference of exponents la... |
| efexp 15448 | The exponential of an inte... |
| efzval 15449 | Value of the exponential f... |
| efgt0 15450 | The exponential of a real ... |
| rpefcl 15451 | The exponential of a real ... |
| rpefcld 15452 | The exponential of a real ... |
| eftlcvg 15453 | The tail series of the exp... |
| eftlcl 15454 | Closure of the sum of an i... |
| reeftlcl 15455 | Closure of the sum of an i... |
| eftlub 15456 | An upper bound on the abso... |
| efsep 15457 | Separate out the next term... |
| effsumlt 15458 | The partial sums of the se... |
| eft0val 15459 | The value of the first ter... |
| ef4p 15460 | Separate out the first fou... |
| efgt1p2 15461 | The exponential of a posit... |
| efgt1p 15462 | The exponential of a posit... |
| efgt1 15463 | The exponential of a posit... |
| eflt 15464 | The exponential function o... |
| efle 15465 | The exponential function o... |
| reef11 15466 | The exponential function o... |
| reeff1 15467 | The exponential function m... |
| eflegeo 15468 | The exponential function o... |
| sinval 15469 | Value of the sine function... |
| cosval 15470 | Value of the cosine functi... |
| sinf 15471 | Domain and codomain of the... |
| cosf 15472 | Domain and codomain of the... |
| sincl 15473 | Closure of the sine functi... |
| coscl 15474 | Closure of the cosine func... |
| tanval 15475 | Value of the tangent funct... |
| tancl 15476 | The closure of the tangent... |
| sincld 15477 | Closure of the sine functi... |
| coscld 15478 | Closure of the cosine func... |
| tancld 15479 | Closure of the tangent fun... |
| tanval2 15480 | Express the tangent functi... |
| tanval3 15481 | Express the tangent functi... |
| resinval 15482 | The sine of a real number ... |
| recosval 15483 | The cosine of a real numbe... |
| efi4p 15484 | Separate out the first fou... |
| resin4p 15485 | Separate out the first fou... |
| recos4p 15486 | Separate out the first fou... |
| resincl 15487 | The sine of a real number ... |
| recoscl 15488 | The cosine of a real numbe... |
| retancl 15489 | The closure of the tangent... |
| resincld 15490 | Closure of the sine functi... |
| recoscld 15491 | Closure of the cosine func... |
| retancld 15492 | Closure of the tangent fun... |
| sinneg 15493 | The sine of a negative is ... |
| cosneg 15494 | The cosines of a number an... |
| tanneg 15495 | The tangent of a negative ... |
| sin0 15496 | Value of the sine function... |
| cos0 15497 | Value of the cosine functi... |
| tan0 15498 | The value of the tangent f... |
| efival 15499 | The exponential function i... |
| efmival 15500 | The exponential function i... |
| sinhval 15501 | Value of the hyperbolic si... |
| coshval 15502 | Value of the hyperbolic co... |
| resinhcl 15503 | The hyperbolic sine of a r... |
| rpcoshcl 15504 | The hyperbolic cosine of a... |
| recoshcl 15505 | The hyperbolic cosine of a... |
| retanhcl 15506 | The hyperbolic tangent of ... |
| tanhlt1 15507 | The hyperbolic tangent of ... |
| tanhbnd 15508 | The hyperbolic tangent of ... |
| efeul 15509 | Eulerian representation of... |
| efieq 15510 | The exponentials of two im... |
| sinadd 15511 | Addition formula for sine.... |
| cosadd 15512 | Addition formula for cosin... |
| tanaddlem 15513 | A useful intermediate step... |
| tanadd 15514 | Addition formula for tange... |
| sinsub 15515 | Sine of difference. (Cont... |
| cossub 15516 | Cosine of difference. (Co... |
| addsin 15517 | Sum of sines. (Contribute... |
| subsin 15518 | Difference of sines. (Con... |
| sinmul 15519 | Product of sines can be re... |
| cosmul 15520 | Product of cosines can be ... |
| addcos 15521 | Sum of cosines. (Contribu... |
| subcos 15522 | Difference of cosines. (C... |
| sincossq 15523 | Sine squared plus cosine s... |
| sin2t 15524 | Double-angle formula for s... |
| cos2t 15525 | Double-angle formula for c... |
| cos2tsin 15526 | Double-angle formula for c... |
| sinbnd 15527 | The sine of a real number ... |
| cosbnd 15528 | The cosine of a real numbe... |
| sinbnd2 15529 | The sine of a real number ... |
| cosbnd2 15530 | The cosine of a real numbe... |
| ef01bndlem 15531 | Lemma for ~ sin01bnd and ~... |
| sin01bnd 15532 | Bounds on the sine of a po... |
| cos01bnd 15533 | Bounds on the cosine of a ... |
| cos1bnd 15534 | Bounds on the cosine of 1.... |
| cos2bnd 15535 | Bounds on the cosine of 2.... |
| sinltx 15536 | The sine of a positive rea... |
| sin01gt0 15537 | The sine of a positive rea... |
| cos01gt0 15538 | The cosine of a positive r... |
| sin02gt0 15539 | The sine of a positive rea... |
| sincos1sgn 15540 | The signs of the sine and ... |
| sincos2sgn 15541 | The signs of the sine and ... |
| sin4lt0 15542 | The sine of 4 is negative.... |
| absefi 15543 | The absolute value of the ... |
| absef 15544 | The absolute value of the ... |
| absefib 15545 | A complex number is real i... |
| efieq1re 15546 | A number whose imaginary e... |
| demoivre 15547 | De Moivre's Formula. Proo... |
| demoivreALT 15548 | Alternate proof of ~ demoi... |
| eirrlem 15551 | Lemma for ~ eirr . (Contr... |
| eirr 15552 | ` _e ` is irrational. (Co... |
| egt2lt3 15553 | Euler's constant ` _e ` = ... |
| epos 15554 | Euler's constant ` _e ` is... |
| epr 15555 | Euler's constant ` _e ` is... |
| ene0 15556 | ` _e ` is not 0. (Contrib... |
| ene1 15557 | ` _e ` is not 1. (Contrib... |
| xpnnen 15558 | The Cartesian product of t... |
| znnen 15559 | The set of integers and th... |
| qnnen 15560 | The rational numbers are c... |
| rpnnen2lem1 15561 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem2 15562 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem3 15563 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem4 15564 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem5 15565 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem6 15566 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem7 15567 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem8 15568 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem9 15569 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem10 15570 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem11 15571 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2lem12 15572 | Lemma for ~ rpnnen2 . (Co... |
| rpnnen2 15573 | The other half of ~ rpnnen... |
| rpnnen 15574 | The cardinality of the con... |
| rexpen 15575 | The real numbers are equin... |
| cpnnen 15576 | The complex numbers are eq... |
| rucALT 15577 | Alternate proof of ~ ruc .... |
| ruclem1 15578 | Lemma for ~ ruc (the reals... |
| ruclem2 15579 | Lemma for ~ ruc . Orderin... |
| ruclem3 15580 | Lemma for ~ ruc . The con... |
| ruclem4 15581 | Lemma for ~ ruc . Initial... |
| ruclem6 15582 | Lemma for ~ ruc . Domain ... |
| ruclem7 15583 | Lemma for ~ ruc . Success... |
| ruclem8 15584 | Lemma for ~ ruc . The int... |
| ruclem9 15585 | Lemma for ~ ruc . The fir... |
| ruclem10 15586 | Lemma for ~ ruc . Every f... |
| ruclem11 15587 | Lemma for ~ ruc . Closure... |
| ruclem12 15588 | Lemma for ~ ruc . The sup... |
| ruclem13 15589 | Lemma for ~ ruc . There i... |
| ruc 15590 | The set of positive intege... |
| resdomq 15591 | The set of rationals is st... |
| aleph1re 15592 | There are at least aleph-o... |
| aleph1irr 15593 | There are at least aleph-o... |
| cnso 15594 | The complex numbers can be... |
| sqrt2irrlem 15595 | Lemma for ~ sqrt2irr . Th... |
| sqrt2irr 15596 | The square root of 2 is ir... |
| sqrt2re 15597 | The square root of 2 exist... |
| sqrt2irr0 15598 | The square root of 2 is an... |
| nthruc 15599 | The sequence ` NN ` , ` ZZ... |
| nthruz 15600 | The sequence ` NN ` , ` NN... |
| divides 15603 | Define the divides relatio... |
| dvdsval2 15604 | One nonzero integer divide... |
| dvdsval3 15605 | One nonzero integer divide... |
| dvdszrcl 15606 | Reverse closure for the di... |
| dvdsmod0 15607 | If a positive integer divi... |
| p1modz1 15608 | If a number greater than 1... |
| dvdsmodexp 15609 | If a positive integer divi... |
| nndivdvds 15610 | Strong form of ~ dvdsval2 ... |
| nndivides 15611 | Definition of the divides ... |
| moddvds 15612 | Two ways to say ` A == B `... |
| modm1div 15613 | A number greater than 1 di... |
| dvds0lem 15614 | A lemma to assist theorems... |
| dvds1lem 15615 | A lemma to assist theorems... |
| dvds2lem 15616 | A lemma to assist theorems... |
| iddvds 15617 | An integer divides itself.... |
| 1dvds 15618 | 1 divides any integer. Th... |
| dvds0 15619 | Any integer divides 0. Th... |
| negdvdsb 15620 | An integer divides another... |
| dvdsnegb 15621 | An integer divides another... |
| absdvdsb 15622 | An integer divides another... |
| dvdsabsb 15623 | An integer divides another... |
| 0dvds 15624 | Only 0 is divisible by 0. ... |
| dvdsmul1 15625 | An integer divides a multi... |
| dvdsmul2 15626 | An integer divides a multi... |
| iddvdsexp 15627 | An integer divides a posit... |
| muldvds1 15628 | If a product divides an in... |
| muldvds2 15629 | If a product divides an in... |
| dvdscmul 15630 | Multiplication by a consta... |
| dvdsmulc 15631 | Multiplication by a consta... |
| dvdscmulr 15632 | Cancellation law for the d... |
| dvdsmulcr 15633 | Cancellation law for the d... |
| summodnegmod 15634 | The sum of two integers mo... |
| modmulconst 15635 | Constant multiplication in... |
| dvds2ln 15636 | If an integer divides each... |
| dvds2add 15637 | If an integer divides each... |
| dvds2sub 15638 | If an integer divides each... |
| dvds2subd 15639 | Natural deduction form of ... |
| dvdstr 15640 | The divides relation is tr... |
| dvdsmultr1 15641 | If an integer divides anot... |
| dvdsmultr1d 15642 | Natural deduction form of ... |
| dvdsmultr2 15643 | If an integer divides anot... |
| ordvdsmul 15644 | If an integer divides eith... |
| dvdssub2 15645 | If an integer divides a di... |
| dvdsadd 15646 | An integer divides another... |
| dvdsaddr 15647 | An integer divides another... |
| dvdssub 15648 | An integer divides another... |
| dvdssubr 15649 | An integer divides another... |
| dvdsadd2b 15650 | Adding a multiple of the b... |
| dvdsaddre2b 15651 | Adding a multiple of the b... |
| fsumdvds 15652 | If every term in a sum is ... |
| dvdslelem 15653 | Lemma for ~ dvdsle . (Con... |
| dvdsle 15654 | The divisors of a positive... |
| dvdsleabs 15655 | The divisors of a nonzero ... |
| dvdsleabs2 15656 | Transfer divisibility to a... |
| dvdsabseq 15657 | If two integers divide eac... |
| dvdseq 15658 | If two nonnegative integer... |
| divconjdvds 15659 | If a nonzero integer ` M `... |
| dvdsdivcl 15660 | The complement of a diviso... |
| dvdsflip 15661 | An involution of the divis... |
| dvdsssfz1 15662 | The set of divisors of a n... |
| dvds1 15663 | The only nonnegative integ... |
| alzdvds 15664 | Only 0 is divisible by all... |
| dvdsext 15665 | Poset extensionality for d... |
| fzm1ndvds 15666 | No number between ` 1 ` an... |
| fzo0dvdseq 15667 | Zero is the only one of th... |
| fzocongeq 15668 | Two different elements of ... |
| addmodlteqALT 15669 | Two nonnegative integers l... |
| dvdsfac 15670 | A positive integer divides... |
| dvdsexp 15671 | A power divides a power wi... |
| dvdsmod 15672 | Any number ` K ` whose mod... |
| mulmoddvds 15673 | If an integer is divisible... |
| 3dvds 15674 | A rule for divisibility by... |
| 3dvdsdec 15675 | A decimal number is divisi... |
| 3dvds2dec 15676 | A decimal number is divisi... |
| fprodfvdvdsd 15677 | A finite product of intege... |
| fproddvdsd 15678 | A finite product of intege... |
| evenelz 15679 | An even number is an integ... |
| zeo3 15680 | An integer is even or odd.... |
| zeo4 15681 | An integer is even or odd ... |
| zeneo 15682 | No even integer equals an ... |
| odd2np1lem 15683 | Lemma for ~ odd2np1 . (Co... |
| odd2np1 15684 | An integer is odd iff it i... |
| even2n 15685 | An integer is even iff it ... |
| oddm1even 15686 | An integer is odd iff its ... |
| oddp1even 15687 | An integer is odd iff its ... |
| oexpneg 15688 | The exponential of the neg... |
| mod2eq0even 15689 | An integer is 0 modulo 2 i... |
| mod2eq1n2dvds 15690 | An integer is 1 modulo 2 i... |
| oddnn02np1 15691 | A nonnegative integer is o... |
| oddge22np1 15692 | An integer greater than on... |
| evennn02n 15693 | A nonnegative integer is e... |
| evennn2n 15694 | A positive integer is even... |
| 2tp1odd 15695 | A number which is twice an... |
| mulsucdiv2z 15696 | An integer multiplied with... |
| sqoddm1div8z 15697 | A squared odd number minus... |
| 2teven 15698 | A number which is twice an... |
| zeo5 15699 | An integer is either even ... |
| evend2 15700 | An integer is even iff its... |
| oddp1d2 15701 | An integer is odd iff its ... |
| zob 15702 | Alternate characterization... |
| oddm1d2 15703 | An integer is odd iff its ... |
| ltoddhalfle 15704 | An integer is less than ha... |
| halfleoddlt 15705 | An integer is greater than... |
| opoe 15706 | The sum of two odds is eve... |
| omoe 15707 | The difference of two odds... |
| opeo 15708 | The sum of an odd and an e... |
| omeo 15709 | The difference of an odd a... |
| z0even 15710 | 2 divides 0. That means 0... |
| n2dvds1 15711 | 2 does not divide 1. That... |
| n2dvds1OLD 15712 | Obsolete version of ~ n2dv... |
| n2dvdsm1 15713 | 2 does not divide -1. Tha... |
| z2even 15714 | 2 divides 2. That means 2... |
| n2dvds3 15715 | 2 does not divide 3. That... |
| n2dvds3OLD 15716 | Obsolete version of ~ n2dv... |
| z4even 15717 | 2 divides 4. That means 4... |
| 4dvdseven 15718 | An integer which is divisi... |
| m1expe 15719 | Exponentiation of -1 by an... |
| m1expo 15720 | Exponentiation of -1 by an... |
| m1exp1 15721 | Exponentiation of negative... |
| nn0enne 15722 | A positive integer is an e... |
| nn0ehalf 15723 | The half of an even nonneg... |
| nnehalf 15724 | The half of an even positi... |
| nn0onn 15725 | An odd nonnegative integer... |
| nn0o1gt2 15726 | An odd nonnegative integer... |
| nno 15727 | An alternate characterizat... |
| nn0o 15728 | An alternate characterizat... |
| nn0ob 15729 | Alternate characterization... |
| nn0oddm1d2 15730 | A positive integer is odd ... |
| nnoddm1d2 15731 | A positive integer is odd ... |
| sumeven 15732 | If every term in a sum is ... |
| sumodd 15733 | If every term in a sum is ... |
| evensumodd 15734 | If every term in a sum wit... |
| oddsumodd 15735 | If every term in a sum wit... |
| pwp1fsum 15736 | The n-th power of a number... |
| oddpwp1fsum 15737 | An odd power of a number i... |
| divalglem0 15738 | Lemma for ~ divalg . (Con... |
| divalglem1 15739 | Lemma for ~ divalg . (Con... |
| divalglem2 15740 | Lemma for ~ divalg . (Con... |
| divalglem4 15741 | Lemma for ~ divalg . (Con... |
| divalglem5 15742 | Lemma for ~ divalg . (Con... |
| divalglem6 15743 | Lemma for ~ divalg . (Con... |
| divalglem7 15744 | Lemma for ~ divalg . (Con... |
| divalglem8 15745 | Lemma for ~ divalg . (Con... |
| divalglem9 15746 | Lemma for ~ divalg . (Con... |
| divalglem10 15747 | Lemma for ~ divalg . (Con... |
| divalg 15748 | The division algorithm (th... |
| divalgb 15749 | Express the division algor... |
| divalg2 15750 | The division algorithm (th... |
| divalgmod 15751 | The result of the ` mod ` ... |
| divalgmodcl 15752 | The result of the ` mod ` ... |
| modremain 15753 | The result of the modulo o... |
| ndvdssub 15754 | Corollary of the division ... |
| ndvdsadd 15755 | Corollary of the division ... |
| ndvdsp1 15756 | Special case of ~ ndvdsadd... |
| ndvdsi 15757 | A quick test for non-divis... |
| flodddiv4 15758 | The floor of an odd intege... |
| fldivndvdslt 15759 | The floor of an integer di... |
| flodddiv4lt 15760 | The floor of an odd number... |
| flodddiv4t2lthalf 15761 | The floor of an odd number... |
| bitsfval 15766 | Expand the definition of t... |
| bitsval 15767 | Expand the definition of t... |
| bitsval2 15768 | Expand the definition of t... |
| bitsss 15769 | The set of bits of an inte... |
| bitsf 15770 | The ` bits ` function is a... |
| bits0 15771 | Value of the zeroth bit. ... |
| bits0e 15772 | The zeroth bit of an even ... |
| bits0o 15773 | The zeroth bit of an odd n... |
| bitsp1 15774 | The ` M + 1 ` -th bit of `... |
| bitsp1e 15775 | The ` M + 1 ` -th bit of `... |
| bitsp1o 15776 | The ` M + 1 ` -th bit of `... |
| bitsfzolem 15777 | Lemma for ~ bitsfzo . (Co... |
| bitsfzo 15778 | The bits of a number are a... |
| bitsmod 15779 | Truncating the bit sequenc... |
| bitsfi 15780 | Every number is associated... |
| bitscmp 15781 | The bit complement of ` N ... |
| 0bits 15782 | The bits of zero. (Contri... |
| m1bits 15783 | The bits of negative one. ... |
| bitsinv1lem 15784 | Lemma for ~ bitsinv1 . (C... |
| bitsinv1 15785 | There is an explicit inver... |
| bitsinv2 15786 | There is an explicit inver... |
| bitsf1ocnv 15787 | The ` bits ` function rest... |
| bitsf1o 15788 | The ` bits ` function rest... |
| bitsf1 15789 | The ` bits ` function is a... |
| 2ebits 15790 | The bits of a power of two... |
| bitsinv 15791 | The inverse of the ` bits ... |
| bitsinvp1 15792 | Recursive definition of th... |
| sadadd2lem2 15793 | The core of the proof of ~... |
| sadfval 15795 | Define the addition of two... |
| sadcf 15796 | The carry sequence is a se... |
| sadc0 15797 | The initial element of the... |
| sadcp1 15798 | The carry sequence (which ... |
| sadval 15799 | The full adder sequence is... |
| sadcaddlem 15800 | Lemma for ~ sadcadd . (Co... |
| sadcadd 15801 | Non-recursive definition o... |
| sadadd2lem 15802 | Lemma for ~ sadadd2 . (Co... |
| sadadd2 15803 | Sum of initial segments of... |
| sadadd3 15804 | Sum of initial segments of... |
| sadcl 15805 | The sum of two sequences i... |
| sadcom 15806 | The adder sequence functio... |
| saddisjlem 15807 | Lemma for ~ sadadd . (Con... |
| saddisj 15808 | The sum of disjoint sequen... |
| sadaddlem 15809 | Lemma for ~ sadadd . (Con... |
| sadadd 15810 | For sequences that corresp... |
| sadid1 15811 | The adder sequence functio... |
| sadid2 15812 | The adder sequence functio... |
| sadasslem 15813 | Lemma for ~ sadass . (Con... |
| sadass 15814 | Sequence addition is assoc... |
| sadeq 15815 | Any element of a sequence ... |
| bitsres 15816 | Restrict the bits of a num... |
| bitsuz 15817 | The bits of a number are a... |
| bitsshft 15818 | Shifting a bit sequence to... |
| smufval 15820 | The multiplication of two ... |
| smupf 15821 | The sequence of partial su... |
| smup0 15822 | The initial element of the... |
| smupp1 15823 | The initial element of the... |
| smuval 15824 | Define the addition of two... |
| smuval2 15825 | The partial sum sequence s... |
| smupvallem 15826 | If ` A ` only has elements... |
| smucl 15827 | The product of two sequenc... |
| smu01lem 15828 | Lemma for ~ smu01 and ~ sm... |
| smu01 15829 | Multiplication of a sequen... |
| smu02 15830 | Multiplication of a sequen... |
| smupval 15831 | Rewrite the elements of th... |
| smup1 15832 | Rewrite ~ smupp1 using onl... |
| smueqlem 15833 | Any element of a sequence ... |
| smueq 15834 | Any element of a sequence ... |
| smumullem 15835 | Lemma for ~ smumul . (Con... |
| smumul 15836 | For sequences that corresp... |
| gcdval 15839 | The value of the ` gcd ` o... |
| gcd0val 15840 | The value, by convention, ... |
| gcdn0val 15841 | The value of the ` gcd ` o... |
| gcdcllem1 15842 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem2 15843 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdcllem3 15844 | Lemma for ~ gcdn0cl , ~ gc... |
| gcdn0cl 15845 | Closure of the ` gcd ` ope... |
| gcddvds 15846 | The gcd of two integers di... |
| dvdslegcd 15847 | An integer which divides b... |
| nndvdslegcd 15848 | A positive integer which d... |
| gcdcl 15849 | Closure of the ` gcd ` ope... |
| gcdnncl 15850 | Closure of the ` gcd ` ope... |
| gcdcld 15851 | Closure of the ` gcd ` ope... |
| gcd2n0cl 15852 | Closure of the ` gcd ` ope... |
| zeqzmulgcd 15853 | An integer is the product ... |
| divgcdz 15854 | An integer divided by the ... |
| gcdf 15855 | Domain and codomain of the... |
| gcdcom 15856 | The ` gcd ` operator is co... |
| divgcdnn 15857 | A positive integer divided... |
| divgcdnnr 15858 | A positive integer divided... |
| gcdeq0 15859 | The gcd of two integers is... |
| gcdn0gt0 15860 | The gcd of two integers is... |
| gcd0id 15861 | The gcd of 0 and an intege... |
| gcdid0 15862 | The gcd of an integer and ... |
| nn0gcdid0 15863 | The gcd of a nonnegative i... |
| gcdneg 15864 | Negating one operand of th... |
| neggcd 15865 | Negating one operand of th... |
| gcdaddmlem 15866 | Lemma for ~ gcdaddm . (Co... |
| gcdaddm 15867 | Adding a multiple of one o... |
| gcdadd 15868 | The GCD of two numbers is ... |
| gcdid 15869 | The gcd of a number and it... |
| gcd1 15870 | The gcd of a number with 1... |
| gcdabs 15871 | The gcd of two integers is... |
| gcdabs1 15872 | ` gcd ` of the absolute va... |
| gcdabs2 15873 | ` gcd ` of the absolute va... |
| modgcd 15874 | The gcd remains unchanged ... |
| 1gcd 15875 | The GCD of one and an inte... |
| gcdmultipled 15876 | The greatest common diviso... |
| gcdmultiplez 15877 | The GCD of a multiple of a... |
| gcdmultiple 15878 | The GCD of a multiple of a... |
| dvdsgcdidd 15879 | The greatest common diviso... |
| 6gcd4e2 15880 | The greatest common diviso... |
| bezoutlem1 15881 | Lemma for ~ bezout . (Con... |
| bezoutlem2 15882 | Lemma for ~ bezout . (Con... |
| bezoutlem3 15883 | Lemma for ~ bezout . (Con... |
| bezoutlem4 15884 | Lemma for ~ bezout . (Con... |
| bezout 15885 | Bézout's identity: ... |
| dvdsgcd 15886 | An integer which divides e... |
| dvdsgcdb 15887 | Biconditional form of ~ dv... |
| dfgcd2 15888 | Alternate definition of th... |
| gcdass 15889 | Associative law for ` gcd ... |
| mulgcd 15890 | Distribute multiplication ... |
| absmulgcd 15891 | Distribute absolute value ... |
| mulgcdr 15892 | Reverse distribution law f... |
| gcddiv 15893 | Division law for GCD. (Con... |
| gcdmultipleOLD 15894 | Obsolete proof of ~ gcdmul... |
| gcdmultiplezOLD 15895 | Obsolete proof of ~ gcdmul... |
| gcdzeq 15896 | A positive integer ` A ` i... |
| gcdeq 15897 | ` A ` is equal to its gcd ... |
| dvdssqim 15898 | Unidirectional form of ~ d... |
| dvdsmulgcd 15899 | A divisibility equivalent ... |
| rpmulgcd 15900 | If ` K ` and ` M ` are rel... |
| rplpwr 15901 | If ` A ` and ` B ` are rel... |
| rppwr 15902 | If ` A ` and ` B ` are rel... |
| sqgcd 15903 | Square distributes over GC... |
| dvdssqlem 15904 | Lemma for ~ dvdssq . (Con... |
| dvdssq 15905 | Two numbers are divisible ... |
| bezoutr 15906 | Partial converse to ~ bezo... |
| bezoutr1 15907 | Converse of ~ bezout for w... |
| nn0seqcvgd 15908 | A strictly-decreasing nonn... |
| seq1st 15909 | A sequence whose iteration... |
| algr0 15910 | The value of the algorithm... |
| algrf 15911 | An algorithm is a step fun... |
| algrp1 15912 | The value of the algorithm... |
| alginv 15913 | If ` I ` is an invariant o... |
| algcvg 15914 | One way to prove that an a... |
| algcvgblem 15915 | Lemma for ~ algcvgb . (Co... |
| algcvgb 15916 | Two ways of expressing tha... |
| algcvga 15917 | The countdown function ` C... |
| algfx 15918 | If ` F ` reaches a fixed p... |
| eucalgval2 15919 | The value of the step func... |
| eucalgval 15920 | Euclid's Algorithm ~ eucal... |
| eucalgf 15921 | Domain and codomain of the... |
| eucalginv 15922 | The invariant of the step ... |
| eucalglt 15923 | The second member of the s... |
| eucalgcvga 15924 | Once Euclid's Algorithm ha... |
| eucalg 15925 | Euclid's Algorithm compute... |
| lcmval 15930 | Value of the ` lcm ` opera... |
| lcmcom 15931 | The ` lcm ` operator is co... |
| lcm0val 15932 | The value, by convention, ... |
| lcmn0val 15933 | The value of the ` lcm ` o... |
| lcmcllem 15934 | Lemma for ~ lcmn0cl and ~ ... |
| lcmn0cl 15935 | Closure of the ` lcm ` ope... |
| dvdslcm 15936 | The lcm of two integers is... |
| lcmledvds 15937 | A positive integer which b... |
| lcmeq0 15938 | The lcm of two integers is... |
| lcmcl 15939 | Closure of the ` lcm ` ope... |
| gcddvdslcm 15940 | The greatest common diviso... |
| lcmneg 15941 | Negating one operand of th... |
| neglcm 15942 | Negating one operand of th... |
| lcmabs 15943 | The lcm of two integers is... |
| lcmgcdlem 15944 | Lemma for ~ lcmgcd and ~ l... |
| lcmgcd 15945 | The product of two numbers... |
| lcmdvds 15946 | The lcm of two integers di... |
| lcmid 15947 | The lcm of an integer and ... |
| lcm1 15948 | The lcm of an integer and ... |
| lcmgcdnn 15949 | The product of two positiv... |
| lcmgcdeq 15950 | Two integers' absolute val... |
| lcmdvdsb 15951 | Biconditional form of ~ lc... |
| lcmass 15952 | Associative law for ` lcm ... |
| 3lcm2e6woprm 15953 | The least common multiple ... |
| 6lcm4e12 15954 | The least common multiple ... |
| absproddvds 15955 | The absolute value of the ... |
| absprodnn 15956 | The absolute value of the ... |
| fissn0dvds 15957 | For each finite subset of ... |
| fissn0dvdsn0 15958 | For each finite subset of ... |
| lcmfval 15959 | Value of the ` _lcm ` func... |
| lcmf0val 15960 | The value, by convention, ... |
| lcmfn0val 15961 | The value of the ` _lcm ` ... |
| lcmfnnval 15962 | The value of the ` _lcm ` ... |
| lcmfcllem 15963 | Lemma for ~ lcmfn0cl and ~... |
| lcmfn0cl 15964 | Closure of the ` _lcm ` fu... |
| lcmfpr 15965 | The value of the ` _lcm ` ... |
| lcmfcl 15966 | Closure of the ` _lcm ` fu... |
| lcmfnncl 15967 | Closure of the ` _lcm ` fu... |
| lcmfeq0b 15968 | The least common multiple ... |
| dvdslcmf 15969 | The least common multiple ... |
| lcmfledvds 15970 | A positive integer which i... |
| lcmf 15971 | Characterization of the le... |
| lcmf0 15972 | The least common multiple ... |
| lcmfsn 15973 | The least common multiple ... |
| lcmftp 15974 | The least common multiple ... |
| lcmfunsnlem1 15975 | Lemma for ~ lcmfdvds and ~... |
| lcmfunsnlem2lem1 15976 | Lemma 1 for ~ lcmfunsnlem2... |
| lcmfunsnlem2lem2 15977 | Lemma 2 for ~ lcmfunsnlem2... |
| lcmfunsnlem2 15978 | Lemma for ~ lcmfunsn and ~... |
| lcmfunsnlem 15979 | Lemma for ~ lcmfdvds and ~... |
| lcmfdvds 15980 | The least common multiple ... |
| lcmfdvdsb 15981 | Biconditional form of ~ lc... |
| lcmfunsn 15982 | The ` _lcm ` function for ... |
| lcmfun 15983 | The ` _lcm ` function for ... |
| lcmfass 15984 | Associative law for the ` ... |
| lcmf2a3a4e12 15985 | The least common multiple ... |
| lcmflefac 15986 | The least common multiple ... |
| coprmgcdb 15987 | Two positive integers are ... |
| ncoprmgcdne1b 15988 | Two positive integers are ... |
| ncoprmgcdgt1b 15989 | Two positive integers are ... |
| coprmdvds1 15990 | If two positive integers a... |
| coprmdvds 15991 | Euclid's Lemma (see ProofW... |
| coprmdvds2 15992 | If an integer is divisible... |
| mulgcddvds 15993 | One half of ~ rpmulgcd2 , ... |
| rpmulgcd2 15994 | If ` M ` is relatively pri... |
| qredeq 15995 | Two equal reduced fraction... |
| qredeu 15996 | Every rational number has ... |
| rpmul 15997 | If ` K ` is relatively pri... |
| rpdvds 15998 | If ` K ` is relatively pri... |
| coprmprod 15999 | The product of the element... |
| coprmproddvdslem 16000 | Lemma for ~ coprmproddvds ... |
| coprmproddvds 16001 | If a positive integer is d... |
| congr 16002 | Definition of congruence b... |
| divgcdcoprm0 16003 | Integers divided by gcd ar... |
| divgcdcoprmex 16004 | Integers divided by gcd ar... |
| cncongr1 16005 | One direction of the bicon... |
| cncongr2 16006 | The other direction of the... |
| cncongr 16007 | Cancellability of Congruen... |
| cncongrcoprm 16008 | Corollary 1 of Cancellabil... |
| isprm 16011 | The predicate "is a prime ... |
| prmnn 16012 | A prime number is a positi... |
| prmz 16013 | A prime number is an integ... |
| prmssnn 16014 | The prime numbers are a su... |
| prmex 16015 | The set of prime numbers e... |
| 0nprm 16016 | 0 is not a prime number. ... |
| 1nprm 16017 | 1 is not a prime number. ... |
| 1idssfct 16018 | The positive divisors of a... |
| isprm2lem 16019 | Lemma for ~ isprm2 . (Con... |
| isprm2 16020 | The predicate "is a prime ... |
| isprm3 16021 | The predicate "is a prime ... |
| isprm4 16022 | The predicate "is a prime ... |
| prmind2 16023 | A variation on ~ prmind as... |
| prmind 16024 | Perform induction over the... |
| dvdsprime 16025 | If ` M ` divides a prime, ... |
| nprm 16026 | A product of two integers ... |
| nprmi 16027 | An inference for composite... |
| dvdsnprmd 16028 | If a number is divisible b... |
| prm2orodd 16029 | A prime number is either 2... |
| 2prm 16030 | 2 is a prime number. (Con... |
| 2mulprm 16031 | A multiple of two is prime... |
| 3prm 16032 | 3 is a prime number. (Con... |
| 4nprm 16033 | 4 is not a prime number. ... |
| prmuz2 16034 | A prime number is an integ... |
| prmgt1 16035 | A prime number is an integ... |
| prmm2nn0 16036 | Subtracting 2 from a prime... |
| oddprmgt2 16037 | An odd prime is greater th... |
| oddprmge3 16038 | An odd prime is greater th... |
| ge2nprmge4 16039 | A composite integer greate... |
| sqnprm 16040 | A square is never prime. ... |
| dvdsprm 16041 | An integer greater than or... |
| exprmfct 16042 | Every integer greater than... |
| prmdvdsfz 16043 | Each integer greater than ... |
| nprmdvds1 16044 | No prime number divides 1.... |
| isprm5 16045 | One need only check prime ... |
| isprm7 16046 | One need only check prime ... |
| maxprmfct 16047 | The set of prime factors o... |
| divgcdodd 16048 | Either ` A / ( A gcd B ) `... |
| coprm 16049 | A prime number either divi... |
| prmrp 16050 | Unequal prime numbers are ... |
| euclemma 16051 | Euclid's lemma. A prime n... |
| isprm6 16052 | A number is prime iff it s... |
| prmdvdsexp 16053 | A prime divides a positive... |
| prmdvdsexpb 16054 | A prime divides a positive... |
| prmdvdsexpr 16055 | If a prime divides a nonne... |
| prmexpb 16056 | Two positive prime powers ... |
| prmfac1 16057 | The factorial of a number ... |
| rpexp 16058 | If two numbers ` A ` and `... |
| rpexp1i 16059 | Relative primality passes ... |
| rpexp12i 16060 | Relative primality passes ... |
| prmndvdsfaclt 16061 | A prime number does not di... |
| ncoprmlnprm 16062 | If two positive integers a... |
| cncongrprm 16063 | Corollary 2 of Cancellabil... |
| isevengcd2 16064 | The predicate "is an even ... |
| isoddgcd1 16065 | The predicate "is an odd n... |
| 3lcm2e6 16066 | The least common multiple ... |
| qnumval 16071 | Value of the canonical num... |
| qdenval 16072 | Value of the canonical den... |
| qnumdencl 16073 | Lemma for ~ qnumcl and ~ q... |
| qnumcl 16074 | The canonical numerator of... |
| qdencl 16075 | The canonical denominator ... |
| fnum 16076 | Canonical numerator define... |
| fden 16077 | Canonical denominator defi... |
| qnumdenbi 16078 | Two numbers are the canoni... |
| qnumdencoprm 16079 | The canonical representati... |
| qeqnumdivden 16080 | Recover a rational number ... |
| qmuldeneqnum 16081 | Multiplying a rational by ... |
| divnumden 16082 | Calculate the reduced form... |
| divdenle 16083 | Reducing a quotient never ... |
| qnumgt0 16084 | A rational is positive iff... |
| qgt0numnn 16085 | A rational is positive iff... |
| nn0gcdsq 16086 | Squaring commutes with GCD... |
| zgcdsq 16087 | ~ nn0gcdsq extended to int... |
| numdensq 16088 | Squaring a rational square... |
| numsq 16089 | Square commutes with canon... |
| densq 16090 | Square commutes with canon... |
| qden1elz 16091 | A rational is an integer i... |
| zsqrtelqelz 16092 | If an integer has a ration... |
| nonsq 16093 | Any integer strictly betwe... |
| phival 16098 | Value of the Euler ` phi `... |
| phicl2 16099 | Bounds and closure for the... |
| phicl 16100 | Closure for the value of t... |
| phibndlem 16101 | Lemma for ~ phibnd . (Con... |
| phibnd 16102 | A slightly tighter bound o... |
| phicld 16103 | Closure for the value of t... |
| phi1 16104 | Value of the Euler ` phi `... |
| dfphi2 16105 | Alternate definition of th... |
| hashdvds 16106 | The number of numbers in a... |
| phiprmpw 16107 | Value of the Euler ` phi `... |
| phiprm 16108 | Value of the Euler ` phi `... |
| crth 16109 | The Chinese Remainder Theo... |
| phimullem 16110 | Lemma for ~ phimul . (Con... |
| phimul 16111 | The Euler ` phi ` function... |
| eulerthlem1 16112 | Lemma for ~ eulerth . (Co... |
| eulerthlem2 16113 | Lemma for ~ eulerth . (Co... |
| eulerth 16114 | Euler's theorem, a general... |
| fermltl 16115 | Fermat's little theorem. ... |
| prmdiv 16116 | Show an explicit expressio... |
| prmdiveq 16117 | The modular inverse of ` A... |
| prmdivdiv 16118 | The (modular) inverse of t... |
| hashgcdlem 16119 | A correspondence between e... |
| hashgcdeq 16120 | Number of initial positive... |
| phisum 16121 | The divisor sum identity o... |
| odzval 16122 | Value of the order functio... |
| odzcllem 16123 | - Lemma for ~ odzcl , show... |
| odzcl 16124 | The order of a group eleme... |
| odzid 16125 | Any element raised to the ... |
| odzdvds 16126 | The only powers of ` A ` t... |
| odzphi 16127 | The order of any group ele... |
| modprm1div 16128 | A prime number divides an ... |
| m1dvdsndvds 16129 | If an integer minus 1 is d... |
| modprminv 16130 | Show an explicit expressio... |
| modprminveq 16131 | The modular inverse of ` A... |
| vfermltl 16132 | Variant of Fermat's little... |
| vfermltlALT 16133 | Alternate proof of ~ vferm... |
| powm2modprm 16134 | If an integer minus 1 is d... |
| reumodprminv 16135 | For any prime number and f... |
| modprm0 16136 | For two positive integers ... |
| nnnn0modprm0 16137 | For a positive integer and... |
| modprmn0modprm0 16138 | For an integer not being 0... |
| coprimeprodsq 16139 | If three numbers are copri... |
| coprimeprodsq2 16140 | If three numbers are copri... |
| oddprm 16141 | A prime not equal to ` 2 `... |
| nnoddn2prm 16142 | A prime not equal to ` 2 `... |
| oddn2prm 16143 | A prime not equal to ` 2 `... |
| nnoddn2prmb 16144 | A number is a prime number... |
| prm23lt5 16145 | A prime less than 5 is eit... |
| prm23ge5 16146 | A prime is either 2 or 3 o... |
| pythagtriplem1 16147 | Lemma for ~ pythagtrip . ... |
| pythagtriplem2 16148 | Lemma for ~ pythagtrip . ... |
| pythagtriplem3 16149 | Lemma for ~ pythagtrip . ... |
| pythagtriplem4 16150 | Lemma for ~ pythagtrip . ... |
| pythagtriplem10 16151 | Lemma for ~ pythagtrip . ... |
| pythagtriplem6 16152 | Lemma for ~ pythagtrip . ... |
| pythagtriplem7 16153 | Lemma for ~ pythagtrip . ... |
| pythagtriplem8 16154 | Lemma for ~ pythagtrip . ... |
| pythagtriplem9 16155 | Lemma for ~ pythagtrip . ... |
| pythagtriplem11 16156 | Lemma for ~ pythagtrip . ... |
| pythagtriplem12 16157 | Lemma for ~ pythagtrip . ... |
| pythagtriplem13 16158 | Lemma for ~ pythagtrip . ... |
| pythagtriplem14 16159 | Lemma for ~ pythagtrip . ... |
| pythagtriplem15 16160 | Lemma for ~ pythagtrip . ... |
| pythagtriplem16 16161 | Lemma for ~ pythagtrip . ... |
| pythagtriplem17 16162 | Lemma for ~ pythagtrip . ... |
| pythagtriplem18 16163 | Lemma for ~ pythagtrip . ... |
| pythagtriplem19 16164 | Lemma for ~ pythagtrip . ... |
| pythagtrip 16165 | Parameterize the Pythagore... |
| iserodd 16166 | Collect the odd terms in a... |
| pclem 16169 | - Lemma for the prime powe... |
| pcprecl 16170 | Closure of the prime power... |
| pcprendvds 16171 | Non-divisibility property ... |
| pcprendvds2 16172 | Non-divisibility property ... |
| pcpre1 16173 | Value of the prime power p... |
| pcpremul 16174 | Multiplicative property of... |
| pcval 16175 | The value of the prime pow... |
| pceulem 16176 | Lemma for ~ pceu . (Contr... |
| pceu 16177 | Uniqueness for the prime p... |
| pczpre 16178 | Connect the prime count pr... |
| pczcl 16179 | Closure of the prime power... |
| pccl 16180 | Closure of the prime power... |
| pccld 16181 | Closure of the prime power... |
| pcmul 16182 | Multiplication property of... |
| pcdiv 16183 | Division property of the p... |
| pcqmul 16184 | Multiplication property of... |
| pc0 16185 | The value of the prime pow... |
| pc1 16186 | Value of the prime count f... |
| pcqcl 16187 | Closure of the general pri... |
| pcqdiv 16188 | Division property of the p... |
| pcrec 16189 | Prime power of a reciproca... |
| pcexp 16190 | Prime power of an exponent... |
| pcxcl 16191 | Extended real closure of t... |
| pcge0 16192 | The prime count of an inte... |
| pczdvds 16193 | Defining property of the p... |
| pcdvds 16194 | Defining property of the p... |
| pczndvds 16195 | Defining property of the p... |
| pcndvds 16196 | Defining property of the p... |
| pczndvds2 16197 | The remainder after dividi... |
| pcndvds2 16198 | The remainder after dividi... |
| pcdvdsb 16199 | ` P ^ A ` divides ` N ` if... |
| pcelnn 16200 | There are a positive numbe... |
| pceq0 16201 | There are zero powers of a... |
| pcidlem 16202 | The prime count of a prime... |
| pcid 16203 | The prime count of a prime... |
| pcneg 16204 | The prime count of a negat... |
| pcabs 16205 | The prime count of an abso... |
| pcdvdstr 16206 | The prime count increases ... |
| pcgcd1 16207 | The prime count of a GCD i... |
| pcgcd 16208 | The prime count of a GCD i... |
| pc2dvds 16209 | A characterization of divi... |
| pc11 16210 | The prime count function, ... |
| pcz 16211 | The prime count function c... |
| pcprmpw2 16212 | Self-referential expressio... |
| pcprmpw 16213 | Self-referential expressio... |
| dvdsprmpweq 16214 | If a positive integer divi... |
| dvdsprmpweqnn 16215 | If an integer greater than... |
| dvdsprmpweqle 16216 | If a positive integer divi... |
| difsqpwdvds 16217 | If the difference of two s... |
| pcaddlem 16218 | Lemma for ~ pcadd . The o... |
| pcadd 16219 | An inequality for the prim... |
| pcadd2 16220 | The inequality of ~ pcadd ... |
| pcmptcl 16221 | Closure for the prime powe... |
| pcmpt 16222 | Construct a function with ... |
| pcmpt2 16223 | Dividing two prime count m... |
| pcmptdvds 16224 | The partial products of th... |
| pcprod 16225 | The product of the primes ... |
| sumhash 16226 | The sum of 1 over a set is... |
| fldivp1 16227 | The difference between the... |
| pcfaclem 16228 | Lemma for ~ pcfac . (Cont... |
| pcfac 16229 | Calculate the prime count ... |
| pcbc 16230 | Calculate the prime count ... |
| qexpz 16231 | If a power of a rational n... |
| expnprm 16232 | A second or higher power o... |
| oddprmdvds 16233 | Every positive integer whi... |
| prmpwdvds 16234 | A relation involving divis... |
| pockthlem 16235 | Lemma for ~ pockthg . (Co... |
| pockthg 16236 | The generalized Pocklingto... |
| pockthi 16237 | Pocklington's theorem, whi... |
| unbenlem 16238 | Lemma for ~ unben . (Cont... |
| unben 16239 | An unbounded set of positi... |
| infpnlem1 16240 | Lemma for ~ infpn . The s... |
| infpnlem2 16241 | Lemma for ~ infpn . For a... |
| infpn 16242 | There exist infinitely man... |
| infpn2 16243 | There exist infinitely man... |
| prmunb 16244 | The primes are unbounded. ... |
| prminf 16245 | There are an infinite numb... |
| prmreclem1 16246 | Lemma for ~ prmrec . Prop... |
| prmreclem2 16247 | Lemma for ~ prmrec . Ther... |
| prmreclem3 16248 | Lemma for ~ prmrec . The ... |
| prmreclem4 16249 | Lemma for ~ prmrec . Show... |
| prmreclem5 16250 | Lemma for ~ prmrec . Here... |
| prmreclem6 16251 | Lemma for ~ prmrec . If t... |
| prmrec 16252 | The sum of the reciprocals... |
| 1arithlem1 16253 | Lemma for ~ 1arith . (Con... |
| 1arithlem2 16254 | Lemma for ~ 1arith . (Con... |
| 1arithlem3 16255 | Lemma for ~ 1arith . (Con... |
| 1arithlem4 16256 | Lemma for ~ 1arith . (Con... |
| 1arith 16257 | Fundamental theorem of ari... |
| 1arith2 16258 | Fundamental theorem of ari... |
| elgz 16261 | Elementhood in the gaussia... |
| gzcn 16262 | A gaussian integer is a co... |
| zgz 16263 | An integer is a gaussian i... |
| igz 16264 | ` _i ` is a gaussian integ... |
| gznegcl 16265 | The gaussian integers are ... |
| gzcjcl 16266 | The gaussian integers are ... |
| gzaddcl 16267 | The gaussian integers are ... |
| gzmulcl 16268 | The gaussian integers are ... |
| gzreim 16269 | Construct a gaussian integ... |
| gzsubcl 16270 | The gaussian integers are ... |
| gzabssqcl 16271 | The squared norm of a gaus... |
| 4sqlem5 16272 | Lemma for ~ 4sq . (Contri... |
| 4sqlem6 16273 | Lemma for ~ 4sq . (Contri... |
| 4sqlem7 16274 | Lemma for ~ 4sq . (Contri... |
| 4sqlem8 16275 | Lemma for ~ 4sq . (Contri... |
| 4sqlem9 16276 | Lemma for ~ 4sq . (Contri... |
| 4sqlem10 16277 | Lemma for ~ 4sq . (Contri... |
| 4sqlem1 16278 | Lemma for ~ 4sq . The set... |
| 4sqlem2 16279 | Lemma for ~ 4sq . Change ... |
| 4sqlem3 16280 | Lemma for ~ 4sq . Suffici... |
| 4sqlem4a 16281 | Lemma for ~ 4sqlem4 . (Co... |
| 4sqlem4 16282 | Lemma for ~ 4sq . We can ... |
| mul4sqlem 16283 | Lemma for ~ mul4sq : algeb... |
| mul4sq 16284 | Euler's four-square identi... |
| 4sqlem11 16285 | Lemma for ~ 4sq . Use the... |
| 4sqlem12 16286 | Lemma for ~ 4sq . For any... |
| 4sqlem13 16287 | Lemma for ~ 4sq . (Contri... |
| 4sqlem14 16288 | Lemma for ~ 4sq . (Contri... |
| 4sqlem15 16289 | Lemma for ~ 4sq . (Contri... |
| 4sqlem16 16290 | Lemma for ~ 4sq . (Contri... |
| 4sqlem17 16291 | Lemma for ~ 4sq . (Contri... |
| 4sqlem18 16292 | Lemma for ~ 4sq . Inducti... |
| 4sqlem19 16293 | Lemma for ~ 4sq . The pro... |
| 4sq 16294 | Lagrange's four-square the... |
| vdwapfval 16301 | Define the arithmetic prog... |
| vdwapf 16302 | The arithmetic progression... |
| vdwapval 16303 | Value of the arithmetic pr... |
| vdwapun 16304 | Remove the first element o... |
| vdwapid1 16305 | The first element of an ar... |
| vdwap0 16306 | Value of a length-1 arithm... |
| vdwap1 16307 | Value of a length-1 arithm... |
| vdwmc 16308 | The predicate " The ` <. R... |
| vdwmc2 16309 | Expand out the definition ... |
| vdwpc 16310 | The predicate " The colori... |
| vdwlem1 16311 | Lemma for ~ vdw . (Contri... |
| vdwlem2 16312 | Lemma for ~ vdw . (Contri... |
| vdwlem3 16313 | Lemma for ~ vdw . (Contri... |
| vdwlem4 16314 | Lemma for ~ vdw . (Contri... |
| vdwlem5 16315 | Lemma for ~ vdw . (Contri... |
| vdwlem6 16316 | Lemma for ~ vdw . (Contri... |
| vdwlem7 16317 | Lemma for ~ vdw . (Contri... |
| vdwlem8 16318 | Lemma for ~ vdw . (Contri... |
| vdwlem9 16319 | Lemma for ~ vdw . (Contri... |
| vdwlem10 16320 | Lemma for ~ vdw . Set up ... |
| vdwlem11 16321 | Lemma for ~ vdw . (Contri... |
| vdwlem12 16322 | Lemma for ~ vdw . ` K = 2 ... |
| vdwlem13 16323 | Lemma for ~ vdw . Main in... |
| vdw 16324 | Van der Waerden's theorem.... |
| vdwnnlem1 16325 | Corollary of ~ vdw , and l... |
| vdwnnlem2 16326 | Lemma for ~ vdwnn . The s... |
| vdwnnlem3 16327 | Lemma for ~ vdwnn . (Cont... |
| vdwnn 16328 | Van der Waerden's theorem,... |
| ramtlecl 16330 | The set ` T ` of numbers w... |
| hashbcval 16332 | Value of the "binomial set... |
| hashbccl 16333 | The binomial set is a fini... |
| hashbcss 16334 | Subset relation for the bi... |
| hashbc0 16335 | The set of subsets of size... |
| hashbc2 16336 | The size of the binomial s... |
| 0hashbc 16337 | There are no subsets of th... |
| ramval 16338 | The value of the Ramsey nu... |
| ramcl2lem 16339 | Lemma for extended real cl... |
| ramtcl 16340 | The Ramsey number has the ... |
| ramtcl2 16341 | The Ramsey number is an in... |
| ramtub 16342 | The Ramsey number is a low... |
| ramub 16343 | The Ramsey number is a low... |
| ramub2 16344 | It is sufficient to check ... |
| rami 16345 | The defining property of a... |
| ramcl2 16346 | The Ramsey number is eithe... |
| ramxrcl 16347 | The Ramsey number is an ex... |
| ramubcl 16348 | If the Ramsey number is up... |
| ramlb 16349 | Establish a lower bound on... |
| 0ram 16350 | The Ramsey number when ` M... |
| 0ram2 16351 | The Ramsey number when ` M... |
| ram0 16352 | The Ramsey number when ` R... |
| 0ramcl 16353 | Lemma for ~ ramcl : Exist... |
| ramz2 16354 | The Ramsey number when ` F... |
| ramz 16355 | The Ramsey number when ` F... |
| ramub1lem1 16356 | Lemma for ~ ramub1 . (Con... |
| ramub1lem2 16357 | Lemma for ~ ramub1 . (Con... |
| ramub1 16358 | Inductive step for Ramsey'... |
| ramcl 16359 | Ramsey's theorem: the Rams... |
| ramsey 16360 | Ramsey's theorem with the ... |
| prmoval 16363 | Value of the primorial fun... |
| prmocl 16364 | Closure of the primorial f... |
| prmone0 16365 | The primorial function is ... |
| prmo0 16366 | The primorial of 0. (Cont... |
| prmo1 16367 | The primorial of 1. (Cont... |
| prmop1 16368 | The primorial of a success... |
| prmonn2 16369 | Value of the primorial fun... |
| prmo2 16370 | The primorial of 2. (Cont... |
| prmo3 16371 | The primorial of 3. (Cont... |
| prmdvdsprmo 16372 | The primorial of a number ... |
| prmdvdsprmop 16373 | The primorial of a number ... |
| fvprmselelfz 16374 | The value of the prime sel... |
| fvprmselgcd1 16375 | The greatest common diviso... |
| prmolefac 16376 | The primorial of a positiv... |
| prmodvdslcmf 16377 | The primorial of a nonnega... |
| prmolelcmf 16378 | The primorial of a positiv... |
| prmgaplem1 16379 | Lemma for ~ prmgap : The ... |
| prmgaplem2 16380 | Lemma for ~ prmgap : The ... |
| prmgaplcmlem1 16381 | Lemma for ~ prmgaplcm : T... |
| prmgaplcmlem2 16382 | Lemma for ~ prmgaplcm : T... |
| prmgaplem3 16383 | Lemma for ~ prmgap . (Con... |
| prmgaplem4 16384 | Lemma for ~ prmgap . (Con... |
| prmgaplem5 16385 | Lemma for ~ prmgap : for e... |
| prmgaplem6 16386 | Lemma for ~ prmgap : for e... |
| prmgaplem7 16387 | Lemma for ~ prmgap . (Con... |
| prmgaplem8 16388 | Lemma for ~ prmgap . (Con... |
| prmgap 16389 | The prime gap theorem: for... |
| prmgaplcm 16390 | Alternate proof of ~ prmga... |
| prmgapprmolem 16391 | Lemma for ~ prmgapprmo : ... |
| prmgapprmo 16392 | Alternate proof of ~ prmga... |
| dec2dvds 16393 | Divisibility by two is obv... |
| dec5dvds 16394 | Divisibility by five is ob... |
| dec5dvds2 16395 | Divisibility by five is ob... |
| dec5nprm 16396 | Divisibility by five is ob... |
| dec2nprm 16397 | Divisibility by two is obv... |
| modxai 16398 | Add exponents in a power m... |
| mod2xi 16399 | Double exponents in a powe... |
| modxp1i 16400 | Add one to an exponent in ... |
| mod2xnegi 16401 | Version of ~ mod2xi with a... |
| modsubi 16402 | Subtract from within a mod... |
| gcdi 16403 | Calculate a GCD via Euclid... |
| gcdmodi 16404 | Calculate a GCD via Euclid... |
| decexp2 16405 | Calculate a power of two. ... |
| numexp0 16406 | Calculate an integer power... |
| numexp1 16407 | Calculate an integer power... |
| numexpp1 16408 | Calculate an integer power... |
| numexp2x 16409 | Double an integer power. ... |
| decsplit0b 16410 | Split a decimal number int... |
| decsplit0 16411 | Split a decimal number int... |
| decsplit1 16412 | Split a decimal number int... |
| decsplit 16413 | Split a decimal number int... |
| karatsuba 16414 | The Karatsuba multiplicati... |
| 2exp4 16415 | Two to the fourth power is... |
| 2exp6 16416 | Two to the sixth power is ... |
| 2exp8 16417 | Two to the eighth power is... |
| 2exp16 16418 | Two to the sixteenth power... |
| 3exp3 16419 | Three to the third power i... |
| 2expltfac 16420 | The factorial grows faster... |
| cshwsidrepsw 16421 | If cyclically shifting a w... |
| cshwsidrepswmod0 16422 | If cyclically shifting a w... |
| cshwshashlem1 16423 | If cyclically shifting a w... |
| cshwshashlem2 16424 | If cyclically shifting a w... |
| cshwshashlem3 16425 | If cyclically shifting a w... |
| cshwsdisj 16426 | The singletons resulting b... |
| cshwsiun 16427 | The set of (different!) wo... |
| cshwsex 16428 | The class of (different!) ... |
| cshws0 16429 | The size of the set of (di... |
| cshwrepswhash1 16430 | The size of the set of (di... |
| cshwshashnsame 16431 | If a word (not consisting ... |
| cshwshash 16432 | If a word has a length bei... |
| prmlem0 16433 | Lemma for ~ prmlem1 and ~ ... |
| prmlem1a 16434 | A quick proof skeleton to ... |
| prmlem1 16435 | A quick proof skeleton to ... |
| 5prm 16436 | 5 is a prime number. (Con... |
| 6nprm 16437 | 6 is not a prime number. ... |
| 7prm 16438 | 7 is a prime number. (Con... |
| 8nprm 16439 | 8 is not a prime number. ... |
| 9nprm 16440 | 9 is not a prime number. ... |
| 10nprm 16441 | 10 is not a prime number. ... |
| 11prm 16442 | 11 is a prime number. (Co... |
| 13prm 16443 | 13 is a prime number. (Co... |
| 17prm 16444 | 17 is a prime number. (Co... |
| 19prm 16445 | 19 is a prime number. (Co... |
| 23prm 16446 | 23 is a prime number. (Co... |
| prmlem2 16447 | Our last proving session g... |
| 37prm 16448 | 37 is a prime number. (Co... |
| 43prm 16449 | 43 is a prime number. (Co... |
| 83prm 16450 | 83 is a prime number. (Co... |
| 139prm 16451 | 139 is a prime number. (C... |
| 163prm 16452 | 163 is a prime number. (C... |
| 317prm 16453 | 317 is a prime number. (C... |
| 631prm 16454 | 631 is a prime number. (C... |
| prmo4 16455 | The primorial of 4. (Cont... |
| prmo5 16456 | The primorial of 5. (Cont... |
| prmo6 16457 | The primorial of 6. (Cont... |
| 1259lem1 16458 | Lemma for ~ 1259prm . Cal... |
| 1259lem2 16459 | Lemma for ~ 1259prm . Cal... |
| 1259lem3 16460 | Lemma for ~ 1259prm . Cal... |
| 1259lem4 16461 | Lemma for ~ 1259prm . Cal... |
| 1259lem5 16462 | Lemma for ~ 1259prm . Cal... |
| 1259prm 16463 | 1259 is a prime number. (... |
| 2503lem1 16464 | Lemma for ~ 2503prm . Cal... |
| 2503lem2 16465 | Lemma for ~ 2503prm . Cal... |
| 2503lem3 16466 | Lemma for ~ 2503prm . Cal... |
| 2503prm 16467 | 2503 is a prime number. (... |
| 4001lem1 16468 | Lemma for ~ 4001prm . Cal... |
| 4001lem2 16469 | Lemma for ~ 4001prm . Cal... |
| 4001lem3 16470 | Lemma for ~ 4001prm . Cal... |
| 4001lem4 16471 | Lemma for ~ 4001prm . Cal... |
| 4001prm 16472 | 4001 is a prime number. (... |
| sloteq 16482 | Equality theorem for the `... |
| brstruct 16486 | The structure relation is ... |
| isstruct2 16487 | The property of being a st... |
| structex 16488 | A structure is a set. (Co... |
| structn0fun 16489 | A structure without the em... |
| isstruct 16490 | The property of being a st... |
| structcnvcnv 16491 | Two ways to express the re... |
| structfung 16492 | The converse of the conver... |
| structfun 16493 | Convert between two kinds ... |
| structfn 16494 | Convert between two kinds ... |
| slotfn 16495 | A slot is a function on se... |
| strfvnd 16496 | Deduction version of ~ str... |
| basfn 16497 | The base set extractor is ... |
| wunndx 16498 | Closure of the index extra... |
| strfvn 16499 | Value of a structure compo... |
| strfvss 16500 | A structure component extr... |
| wunstr 16501 | Closure of a structure ind... |
| ndxarg 16502 | Get the numeric argument f... |
| ndxid 16503 | A structure component extr... |
| strndxid 16504 | The value of a structure c... |
| reldmsets 16505 | The structure override ope... |
| setsvalg 16506 | Value of the structure rep... |
| setsval 16507 | Value of the structure rep... |
| setsidvald 16508 | Value of the structure rep... |
| fvsetsid 16509 | The value of the structure... |
| fsets 16510 | The structure replacement ... |
| setsdm 16511 | The domain of a structure ... |
| setsfun 16512 | A structure with replaceme... |
| setsfun0 16513 | A structure with replaceme... |
| setsn0fun 16514 | The value of the structure... |
| setsstruct2 16515 | An extensible structure wi... |
| setsexstruct2 16516 | An extensible structure wi... |
| setsstruct 16517 | An extensible structure wi... |
| wunsets 16518 | Closure of structure repla... |
| setsres 16519 | The structure replacement ... |
| setsabs 16520 | Replacing the same compone... |
| setscom 16521 | Component-setting is commu... |
| strfvd 16522 | Deduction version of ~ str... |
| strfv2d 16523 | Deduction version of ~ str... |
| strfv2 16524 | A variation on ~ strfv to ... |
| strfv 16525 | Extract a structure compon... |
| strfv3 16526 | Variant on ~ strfv for lar... |
| strssd 16527 | Deduction version of ~ str... |
| strss 16528 | Propagate component extrac... |
| str0 16529 | All components of the empt... |
| base0 16530 | The base set of the empty ... |
| strfvi 16531 | Structure slot extractors ... |
| setsid 16532 | Value of the structure rep... |
| setsnid 16533 | Value of the structure rep... |
| sbcie2s 16534 | A special version of class... |
| sbcie3s 16535 | A special version of class... |
| baseval 16536 | Value of the base set extr... |
| baseid 16537 | Utility theorem: index-ind... |
| elbasfv 16538 | Utility theorem: reverse c... |
| elbasov 16539 | Utility theorem: reverse c... |
| strov2rcl 16540 | Partial reverse closure fo... |
| basendx 16541 | Index value of the base se... |
| basendxnn 16542 | The index value of the bas... |
| basprssdmsets 16543 | The pair of the base index... |
| reldmress 16544 | The structure restriction ... |
| ressval 16545 | Value of structure restric... |
| ressid2 16546 | General behavior of trivia... |
| ressval2 16547 | Value of nontrivial struct... |
| ressbas 16548 | Base set of a structure re... |
| ressbas2 16549 | Base set of a structure re... |
| ressbasss 16550 | The base set of a restrict... |
| resslem 16551 | Other elements of a struct... |
| ress0 16552 | All restrictions of the nu... |
| ressid 16553 | Behavior of trivial restri... |
| ressinbas 16554 | Restriction only cares abo... |
| ressval3d 16555 | Value of structure restric... |
| ressress 16556 | Restriction composition la... |
| ressabs 16557 | Restriction absorption law... |
| wunress 16558 | Closure of structure restr... |
| strleun 16585 | Combine two structures int... |
| strle1 16586 | Make a structure from a si... |
| strle2 16587 | Make a structure from a pa... |
| strle3 16588 | Make a structure from a tr... |
| plusgndx 16589 | Index value of the ~ df-pl... |
| plusgid 16590 | Utility theorem: index-ind... |
| opelstrbas 16591 | The base set of a structur... |
| 1strstr 16592 | A constructed one-slot str... |
| 1strbas 16593 | The base set of a construc... |
| 1strwunbndx 16594 | A constructed one-slot str... |
| 1strwun 16595 | A constructed one-slot str... |
| 2strstr 16596 | A constructed two-slot str... |
| 2strbas 16597 | The base set of a construc... |
| 2strop 16598 | The other slot of a constr... |
| 2strstr1 16599 | A constructed two-slot str... |
| 2strbas1 16600 | The base set of a construc... |
| 2strop1 16601 | The other slot of a constr... |
| basendxnplusgndx 16602 | The slot for the base set ... |
| grpstr 16603 | A constructed group is a s... |
| grpbase 16604 | The base set of a construc... |
| grpplusg 16605 | The operation of a constru... |
| ressplusg 16606 | ` +g ` is unaffected by re... |
| grpbasex 16607 | The base of an explicitly ... |
| grpplusgx 16608 | The operation of an explic... |
| mulrndx 16609 | Index value of the ~ df-mu... |
| mulrid 16610 | Utility theorem: index-ind... |
| plusgndxnmulrndx 16611 | The slot for the group (ad... |
| basendxnmulrndx 16612 | The slot for the base set ... |
| rngstr 16613 | A constructed ring is a st... |
| rngbase 16614 | The base set of a construc... |
| rngplusg 16615 | The additive operation of ... |
| rngmulr 16616 | The multiplicative operati... |
| starvndx 16617 | Index value of the ~ df-st... |
| starvid 16618 | Utility theorem: index-ind... |
| ressmulr 16619 | ` .r ` is unaffected by re... |
| ressstarv 16620 | ` *r ` is unaffected by re... |
| srngstr 16621 | A constructed star ring is... |
| srngbase 16622 | The base set of a construc... |
| srngplusg 16623 | The addition operation of ... |
| srngmulr 16624 | The multiplication operati... |
| srnginvl 16625 | The involution function of... |
| scandx 16626 | Index value of the ~ df-sc... |
| scaid 16627 | Utility theorem: index-ind... |
| vscandx 16628 | Index value of the ~ df-vs... |
| vscaid 16629 | Utility theorem: index-ind... |
| lmodstr 16630 | A constructed left module ... |
| lmodbase 16631 | The base set of a construc... |
| lmodplusg 16632 | The additive operation of ... |
| lmodsca 16633 | The set of scalars of a co... |
| lmodvsca 16634 | The scalar product operati... |
| ipndx 16635 | Index value of the ~ df-ip... |
| ipid 16636 | Utility theorem: index-ind... |
| ipsstr 16637 | Lemma to shorten proofs of... |
| ipsbase 16638 | The base set of a construc... |
| ipsaddg 16639 | The additive operation of ... |
| ipsmulr 16640 | The multiplicative operati... |
| ipssca 16641 | The set of scalars of a co... |
| ipsvsca 16642 | The scalar product operati... |
| ipsip 16643 | The multiplicative operati... |
| resssca 16644 | ` Scalar ` is unaffected b... |
| ressvsca 16645 | ` .s ` is unaffected by re... |
| ressip 16646 | The inner product is unaff... |
| phlstr 16647 | A constructed pre-Hilbert ... |
| phlbase 16648 | The base set of a construc... |
| phlplusg 16649 | The additive operation of ... |
| phlsca 16650 | The ring of scalars of a c... |
| phlvsca 16651 | The scalar product operati... |
| phlip 16652 | The inner product (Hermiti... |
| tsetndx 16653 | Index value of the ~ df-ts... |
| tsetid 16654 | Utility theorem: index-ind... |
| topgrpstr 16655 | A constructed topological ... |
| topgrpbas 16656 | The base set of a construc... |
| topgrpplusg 16657 | The additive operation of ... |
| topgrptset 16658 | The topology of a construc... |
| resstset 16659 | ` TopSet ` is unaffected b... |
| plendx 16660 | Index value of the ~ df-pl... |
| pleid 16661 | Utility theorem: self-refe... |
| otpsstr 16662 | Functionality of a topolog... |
| otpsbas 16663 | The base set of a topologi... |
| otpstset 16664 | The open sets of a topolog... |
| otpsle 16665 | The order of a topological... |
| ressle 16666 | ` le ` is unaffected by re... |
| ocndx 16667 | Index value of the ~ df-oc... |
| ocid 16668 | Utility theorem: index-ind... |
| dsndx 16669 | Index value of the ~ df-ds... |
| dsid 16670 | Utility theorem: index-ind... |
| unifndx 16671 | Index value of the ~ df-un... |
| unifid 16672 | Utility theorem: index-ind... |
| odrngstr 16673 | Functionality of an ordere... |
| odrngbas 16674 | The base set of an ordered... |
| odrngplusg 16675 | The addition operation of ... |
| odrngmulr 16676 | The multiplication operati... |
| odrngtset 16677 | The open sets of an ordere... |
| odrngle 16678 | The order of an ordered me... |
| odrngds 16679 | The metric of an ordered m... |
| ressds 16680 | ` dist ` is unaffected by ... |
| homndx 16681 | Index value of the ~ df-ho... |
| homid 16682 | Utility theorem: index-ind... |
| ccondx 16683 | Index value of the ~ df-cc... |
| ccoid 16684 | Utility theorem: index-ind... |
| resshom 16685 | ` Hom ` is unaffected by r... |
| ressco 16686 | ` comp ` is unaffected by ... |
| slotsbhcdif 16687 | The slots ` Base ` , ` Hom... |
| restfn 16692 | The subspace topology oper... |
| topnfn 16693 | The topology extractor fun... |
| restval 16694 | The subspace topology indu... |
| elrest 16695 | The predicate "is an open ... |
| elrestr 16696 | Sufficient condition for b... |
| 0rest 16697 | Value of the structure res... |
| restid2 16698 | The subspace topology over... |
| restsspw 16699 | The subspace topology is a... |
| firest 16700 | The finite intersections o... |
| restid 16701 | The subspace topology of t... |
| topnval 16702 | Value of the topology extr... |
| topnid 16703 | Value of the topology extr... |
| topnpropd 16704 | The topology extractor fun... |
| reldmprds 16716 | The structure product is a... |
| prdsbasex 16718 | Lemma for structure produc... |
| imasvalstr 16719 | Structure product value is... |
| prdsvalstr 16720 | Structure product value is... |
| prdsvallem 16721 | Lemma for ~ prdsbas and si... |
| prdsval 16722 | Value of the structure pro... |
| prdssca 16723 | Scalar ring of a structure... |
| prdsbas 16724 | Base set of a structure pr... |
| prdsplusg 16725 | Addition in a structure pr... |
| prdsmulr 16726 | Multiplication in a struct... |
| prdsvsca 16727 | Scalar multiplication in a... |
| prdsip 16728 | Inner product in a structu... |
| prdsle 16729 | Structure product weak ord... |
| prdsless 16730 | Closure of the order relat... |
| prdsds 16731 | Structure product distance... |
| prdsdsfn 16732 | Structure product distance... |
| prdstset 16733 | Structure product topology... |
| prdshom 16734 | Structure product hom-sets... |
| prdsco 16735 | Structure product composit... |
| prdsbas2 16736 | The base set of a structur... |
| prdsbasmpt 16737 | A constructed tuple is a p... |
| prdsbasfn 16738 | Points in the structure pr... |
| prdsbasprj 16739 | Each point in a structure ... |
| prdsplusgval 16740 | Value of a componentwise s... |
| prdsplusgfval 16741 | Value of a structure produ... |
| prdsmulrval 16742 | Value of a componentwise r... |
| prdsmulrfval 16743 | Value of a structure produ... |
| prdsleval 16744 | Value of the product order... |
| prdsdsval 16745 | Value of the metric in a s... |
| prdsvscaval 16746 | Scalar multiplication in a... |
| prdsvscafval 16747 | Scalar multiplication of a... |
| prdsbas3 16748 | The base set of an indexed... |
| prdsbasmpt2 16749 | A constructed tuple is a p... |
| prdsbascl 16750 | An element of the base has... |
| prdsdsval2 16751 | Value of the metric in a s... |
| prdsdsval3 16752 | Value of the metric in a s... |
| pwsval 16753 | Value of a structure power... |
| pwsbas 16754 | Base set of a structure po... |
| pwselbasb 16755 | Membership in the base set... |
| pwselbas 16756 | An element of a structure ... |
| pwsplusgval 16757 | Value of addition in a str... |
| pwsmulrval 16758 | Value of multiplication in... |
| pwsle 16759 | Ordering in a structure po... |
| pwsleval 16760 | Ordering in a structure po... |
| pwsvscafval 16761 | Scalar multiplication in a... |
| pwsvscaval 16762 | Scalar multiplication of a... |
| pwssca 16763 | The ring of scalars of a s... |
| pwsdiagel 16764 | Membership of diagonal ele... |
| pwssnf1o 16765 | Triviality of singleton po... |
| imasval 16778 | Value of an image structur... |
| imasbas 16779 | The base set of an image s... |
| imasds 16780 | The distance function of a... |
| imasdsfn 16781 | The distance function is a... |
| imasdsval 16782 | The distance function of a... |
| imasdsval2 16783 | The distance function of a... |
| imasplusg 16784 | The group operation in an ... |
| imasmulr 16785 | The ring multiplication in... |
| imassca 16786 | The scalar field of an ima... |
| imasvsca 16787 | The scalar multiplication ... |
| imasip 16788 | The inner product of an im... |
| imastset 16789 | The topology of an image s... |
| imasle 16790 | The ordering of an image s... |
| f1ocpbllem 16791 | Lemma for ~ f1ocpbl . (Co... |
| f1ocpbl 16792 | An injection is compatible... |
| f1ovscpbl 16793 | An injection is compatible... |
| f1olecpbl 16794 | An injection is compatible... |
| imasaddfnlem 16795 | The image structure operat... |
| imasaddvallem 16796 | The operation of an image ... |
| imasaddflem 16797 | The image set operations a... |
| imasaddfn 16798 | The image structure's grou... |
| imasaddval 16799 | The value of an image stru... |
| imasaddf 16800 | The image structure's grou... |
| imasmulfn 16801 | The image structure's ring... |
| imasmulval 16802 | The value of an image stru... |
| imasmulf 16803 | The image structure's ring... |
| imasvscafn 16804 | The image structure's scal... |
| imasvscaval 16805 | The value of an image stru... |
| imasvscaf 16806 | The image structure's scal... |
| imasless 16807 | The order relation defined... |
| imasleval 16808 | The value of the image str... |
| qusval 16809 | Value of a quotient struct... |
| quslem 16810 | The function in ~ qusval i... |
| qusin 16811 | Restrict the equivalence r... |
| qusbas 16812 | Base set of a quotient str... |
| quss 16813 | The scalar field of a quot... |
| divsfval 16814 | Value of the function in ~... |
| ercpbllem 16815 | Lemma for ~ ercpbl . (Con... |
| ercpbl 16816 | Translate the function com... |
| erlecpbl 16817 | Translate the relation com... |
| qusaddvallem 16818 | Value of an operation defi... |
| qusaddflem 16819 | The operation of a quotien... |
| qusaddval 16820 | The base set of an image s... |
| qusaddf 16821 | The base set of an image s... |
| qusmulval 16822 | The base set of an image s... |
| qusmulf 16823 | The base set of an image s... |
| fnpr2o 16824 | Function with a domain of ... |
| fnpr2ob 16825 | Biconditional version of ~... |
| fvpr0o 16826 | The value of a function wi... |
| fvpr1o 16827 | The value of a function wi... |
| fvprif 16828 | The value of the pair func... |
| xpsfrnel 16829 | Elementhood in the target ... |
| xpsfeq 16830 | A function on ` 2o ` is de... |
| xpsfrnel2 16831 | Elementhood in the target ... |
| xpscf 16832 | Equivalent condition for t... |
| xpsfval 16833 | The value of the function ... |
| xpsff1o 16834 | The function appearing in ... |
| xpsfrn 16835 | A short expression for the... |
| xpsff1o2 16836 | The function appearing in ... |
| xpsval 16837 | Value of the binary struct... |
| xpsrnbas 16838 | The indexed structure prod... |
| xpsbas 16839 | The base set of the binary... |
| xpsaddlem 16840 | Lemma for ~ xpsadd and ~ x... |
| xpsadd 16841 | Value of the addition oper... |
| xpsmul 16842 | Value of the multiplicatio... |
| xpssca 16843 | Value of the scalar field ... |
| xpsvsca 16844 | Value of the scalar multip... |
| xpsless 16845 | Closure of the ordering in... |
| xpsle 16846 | Value of the ordering in a... |
| ismre 16855 | Property of being a Moore ... |
| fnmre 16856 | The Moore collection gener... |
| mresspw 16857 | A Moore collection is a su... |
| mress 16858 | A Moore-closed subset is a... |
| mre1cl 16859 | In any Moore collection th... |
| mreintcl 16860 | A nonempty collection of c... |
| mreiincl 16861 | A nonempty indexed interse... |
| mrerintcl 16862 | The relative intersection ... |
| mreriincl 16863 | The relative intersection ... |
| mreincl 16864 | Two closed sets have a clo... |
| mreuni 16865 | Since the entire base set ... |
| mreunirn 16866 | Two ways to express the no... |
| ismred 16867 | Properties that determine ... |
| ismred2 16868 | Properties that determine ... |
| mremre 16869 | The Moore collections of s... |
| submre 16870 | The subcollection of a clo... |
| mrcflem 16871 | The domain and range of th... |
| fnmrc 16872 | Moore-closure is a well-be... |
| mrcfval 16873 | Value of the function expr... |
| mrcf 16874 | The Moore closure is a fun... |
| mrcval 16875 | Evaluation of the Moore cl... |
| mrccl 16876 | The Moore closure of a set... |
| mrcsncl 16877 | The Moore closure of a sin... |
| mrcid 16878 | The closure of a closed se... |
| mrcssv 16879 | The closure of a set is a ... |
| mrcidb 16880 | A set is closed iff it is ... |
| mrcss 16881 | Closure preserves subset o... |
| mrcssid 16882 | The closure of a set is a ... |
| mrcidb2 16883 | A set is closed iff it con... |
| mrcidm 16884 | The closure operation is i... |
| mrcsscl 16885 | The closure is the minimal... |
| mrcuni 16886 | Idempotence of closure und... |
| mrcun 16887 | Idempotence of closure und... |
| mrcssvd 16888 | The Moore closure of a set... |
| mrcssd 16889 | Moore closure preserves su... |
| mrcssidd 16890 | A set is contained in its ... |
| mrcidmd 16891 | Moore closure is idempoten... |
| mressmrcd 16892 | In a Moore system, if a se... |
| submrc 16893 | In a closure system which ... |
| mrieqvlemd 16894 | In a Moore system, if ` Y ... |
| mrisval 16895 | Value of the set of indepe... |
| ismri 16896 | Criterion for a set to be ... |
| ismri2 16897 | Criterion for a subset of ... |
| ismri2d 16898 | Criterion for a subset of ... |
| ismri2dd 16899 | Definition of independence... |
| mriss 16900 | An independent set of a Mo... |
| mrissd 16901 | An independent set of a Mo... |
| ismri2dad 16902 | Consequence of a set in a ... |
| mrieqvd 16903 | In a Moore system, a set i... |
| mrieqv2d 16904 | In a Moore system, a set i... |
| mrissmrcd 16905 | In a Moore system, if an i... |
| mrissmrid 16906 | In a Moore system, subsets... |
| mreexd 16907 | In a Moore system, the clo... |
| mreexmrid 16908 | In a Moore system whose cl... |
| mreexexlemd 16909 | This lemma is used to gene... |
| mreexexlem2d 16910 | Used in ~ mreexexlem4d to ... |
| mreexexlem3d 16911 | Base case of the induction... |
| mreexexlem4d 16912 | Induction step of the indu... |
| mreexexd 16913 | Exchange-type theorem. In... |
| mreexdomd 16914 | In a Moore system whose cl... |
| mreexfidimd 16915 | In a Moore system whose cl... |
| isacs 16916 | A set is an algebraic clos... |
| acsmre 16917 | Algebraic closure systems ... |
| isacs2 16918 | In the definition of an al... |
| acsfiel 16919 | A set is closed in an alge... |
| acsfiel2 16920 | A set is closed in an alge... |
| acsmred 16921 | An algebraic closure syste... |
| isacs1i 16922 | A closure system determine... |
| mreacs 16923 | Algebraicity is a composab... |
| acsfn 16924 | Algebraicity of a conditio... |
| acsfn0 16925 | Algebraicity of a point cl... |
| acsfn1 16926 | Algebraicity of a one-argu... |
| acsfn1c 16927 | Algebraicity of a one-argu... |
| acsfn2 16928 | Algebraicity of a two-argu... |
| iscat 16937 | The predicate "is a catego... |
| iscatd 16938 | Properties that determine ... |
| catidex 16939 | Each object in a category ... |
| catideu 16940 | Each object in a category ... |
| cidfval 16941 | Each object in a category ... |
| cidval 16942 | Each object in a category ... |
| cidffn 16943 | The identity arrow constru... |
| cidfn 16944 | The identity arrow operato... |
| catidd 16945 | Deduce the identity arrow ... |
| iscatd2 16946 | Version of ~ iscatd with a... |
| catidcl 16947 | Each object in a category ... |
| catlid 16948 | Left identity property of ... |
| catrid 16949 | Right identity property of... |
| catcocl 16950 | Closure of a composition a... |
| catass 16951 | Associativity of compositi... |
| 0catg 16952 | Any structure with an empt... |
| 0cat 16953 | The empty set is a categor... |
| homffval 16954 | Value of the functionalize... |
| fnhomeqhomf 16955 | If the Hom-set operation i... |
| homfval 16956 | Value of the functionalize... |
| homffn 16957 | The functionalized Hom-set... |
| homfeq 16958 | Condition for two categori... |
| homfeqd 16959 | If two structures have the... |
| homfeqbas 16960 | Deduce equality of base se... |
| homfeqval 16961 | Value of the functionalize... |
| comfffval 16962 | Value of the functionalize... |
| comffval 16963 | Value of the functionalize... |
| comfval 16964 | Value of the functionalize... |
| comfffval2 16965 | Value of the functionalize... |
| comffval2 16966 | Value of the functionalize... |
| comfval2 16967 | Value of the functionalize... |
| comfffn 16968 | The functionalized composi... |
| comffn 16969 | The functionalized composi... |
| comfeq 16970 | Condition for two categori... |
| comfeqd 16971 | Condition for two categori... |
| comfeqval 16972 | Equality of two compositio... |
| catpropd 16973 | Two structures with the sa... |
| cidpropd 16974 | Two structures with the sa... |
| oppcval 16977 | Value of the opposite cate... |
| oppchomfval 16978 | Hom-sets of the opposite c... |
| oppchom 16979 | Hom-sets of the opposite c... |
| oppccofval 16980 | Composition in the opposit... |
| oppcco 16981 | Composition in the opposit... |
| oppcbas 16982 | Base set of an opposite ca... |
| oppccatid 16983 | Lemma for ~ oppccat . (Co... |
| oppchomf 16984 | Hom-sets of the opposite c... |
| oppcid 16985 | Identity function of an op... |
| oppccat 16986 | An opposite category is a ... |
| 2oppcbas 16987 | The double opposite catego... |
| 2oppchomf 16988 | The double opposite catego... |
| 2oppccomf 16989 | The double opposite catego... |
| oppchomfpropd 16990 | If two categories have the... |
| oppccomfpropd 16991 | If two categories have the... |
| monfval 16996 | Definition of a monomorphi... |
| ismon 16997 | Definition of a monomorphi... |
| ismon2 16998 | Write out the monomorphism... |
| monhom 16999 | A monomorphism is a morphi... |
| moni 17000 | Property of a monomorphism... |
| monpropd 17001 | If two categories have the... |
| oppcmon 17002 | A monomorphism in the oppo... |
| oppcepi 17003 | An epimorphism in the oppo... |
| isepi 17004 | Definition of an epimorphi... |
| isepi2 17005 | Write out the epimorphism ... |
| epihom 17006 | An epimorphism is a morphi... |
| epii 17007 | Property of an epimorphism... |
| sectffval 17014 | Value of the section opera... |
| sectfval 17015 | Value of the section relat... |
| sectss 17016 | The section relation is a ... |
| issect 17017 | The property " ` F ` is a ... |
| issect2 17018 | Property of being a sectio... |
| sectcan 17019 | If ` G ` is a section of `... |
| sectco 17020 | Composition of two section... |
| isofval 17021 | Function value of the func... |
| invffval 17022 | Value of the inverse relat... |
| invfval 17023 | Value of the inverse relat... |
| isinv 17024 | Value of the inverse relat... |
| invss 17025 | The inverse relation is a ... |
| invsym 17026 | The inverse relation is sy... |
| invsym2 17027 | The inverse relation is sy... |
| invfun 17028 | The inverse relation is a ... |
| isoval 17029 | The isomorphisms are the d... |
| inviso1 17030 | If ` G ` is an inverse to ... |
| inviso2 17031 | If ` G ` is an inverse to ... |
| invf 17032 | The inverse relation is a ... |
| invf1o 17033 | The inverse relation is a ... |
| invinv 17034 | The inverse of the inverse... |
| invco 17035 | The composition of two iso... |
| dfiso2 17036 | Alternate definition of an... |
| dfiso3 17037 | Alternate definition of an... |
| inveq 17038 | If there are two inverses ... |
| isofn 17039 | The function value of the ... |
| isohom 17040 | An isomorphism is a homomo... |
| isoco 17041 | The composition of two iso... |
| oppcsect 17042 | A section in the opposite ... |
| oppcsect2 17043 | A section in the opposite ... |
| oppcinv 17044 | An inverse in the opposite... |
| oppciso 17045 | An isomorphism in the oppo... |
| sectmon 17046 | If ` F ` is a section of `... |
| monsect 17047 | If ` F ` is a monomorphism... |
| sectepi 17048 | If ` F ` is a section of `... |
| episect 17049 | If ` F ` is an epimorphism... |
| sectid 17050 | The identity is a section ... |
| invid 17051 | The inverse of the identit... |
| idiso 17052 | The identity is an isomorp... |
| idinv 17053 | The inverse of the identit... |
| invisoinvl 17054 | The inverse of an isomorph... |
| invisoinvr 17055 | The inverse of an isomorph... |
| invcoisoid 17056 | The inverse of an isomorph... |
| isocoinvid 17057 | The inverse of an isomorph... |
| rcaninv 17058 | Right cancellation of an i... |
| cicfval 17061 | The set of isomorphic obje... |
| brcic 17062 | The relation "is isomorphi... |
| cic 17063 | Objects ` X ` and ` Y ` in... |
| brcici 17064 | Prove that two objects are... |
| cicref 17065 | Isomorphism is reflexive. ... |
| ciclcl 17066 | Isomorphism implies the le... |
| cicrcl 17067 | Isomorphism implies the ri... |
| cicsym 17068 | Isomorphism is symmetric. ... |
| cictr 17069 | Isomorphism is transitive.... |
| cicer 17070 | Isomorphism is an equivale... |
| sscrel 17077 | The subcategory subset rel... |
| brssc 17078 | The subcategory subset rel... |
| sscpwex 17079 | An analogue of ~ pwex for ... |
| subcrcl 17080 | Reverse closure for the su... |
| sscfn1 17081 | The subcategory subset rel... |
| sscfn2 17082 | The subcategory subset rel... |
| ssclem 17083 | Lemma for ~ ssc1 and simil... |
| isssc 17084 | Value of the subcategory s... |
| ssc1 17085 | Infer subset relation on o... |
| ssc2 17086 | Infer subset relation on m... |
| sscres 17087 | Any function restricted to... |
| sscid 17088 | The subcategory subset rel... |
| ssctr 17089 | The subcategory subset rel... |
| ssceq 17090 | The subcategory subset rel... |
| rescval 17091 | Value of the category rest... |
| rescval2 17092 | Value of the category rest... |
| rescbas 17093 | Base set of the category r... |
| reschom 17094 | Hom-sets of the category r... |
| reschomf 17095 | Hom-sets of the category r... |
| rescco 17096 | Composition in the categor... |
| rescabs 17097 | Restriction absorption law... |
| rescabs2 17098 | Restriction absorption law... |
| issubc 17099 | Elementhood in the set of ... |
| issubc2 17100 | Elementhood in the set of ... |
| 0ssc 17101 | For any category ` C ` , t... |
| 0subcat 17102 | For any category ` C ` , t... |
| catsubcat 17103 | For any category ` C ` , `... |
| subcssc 17104 | An element in the set of s... |
| subcfn 17105 | An element in the set of s... |
| subcss1 17106 | The objects of a subcatego... |
| subcss2 17107 | The morphisms of a subcate... |
| subcidcl 17108 | The identity of the origin... |
| subccocl 17109 | A subcategory is closed un... |
| subccatid 17110 | A subcategory is a categor... |
| subcid 17111 | The identity in a subcateg... |
| subccat 17112 | A subcategory is a categor... |
| issubc3 17113 | Alternate definition of a ... |
| fullsubc 17114 | The full subcategory gener... |
| fullresc 17115 | The category formed by str... |
| resscat 17116 | A category restricted to a... |
| subsubc 17117 | A subcategory of a subcate... |
| relfunc 17126 | The set of functors is a r... |
| funcrcl 17127 | Reverse closure for a func... |
| isfunc 17128 | Value of the set of functo... |
| isfuncd 17129 | Deduce that an operation i... |
| funcf1 17130 | The object part of a funct... |
| funcixp 17131 | The morphism part of a fun... |
| funcf2 17132 | The morphism part of a fun... |
| funcfn2 17133 | The morphism part of a fun... |
| funcid 17134 | A functor maps each identi... |
| funcco 17135 | A functor maps composition... |
| funcsect 17136 | The image of a section und... |
| funcinv 17137 | The image of an inverse un... |
| funciso 17138 | The image of an isomorphis... |
| funcoppc 17139 | A functor on categories yi... |
| idfuval 17140 | Value of the identity func... |
| idfu2nd 17141 | Value of the morphism part... |
| idfu2 17142 | Value of the morphism part... |
| idfu1st 17143 | Value of the object part o... |
| idfu1 17144 | Value of the object part o... |
| idfucl 17145 | The identity functor is a ... |
| cofuval 17146 | Value of the composition o... |
| cofu1st 17147 | Value of the object part o... |
| cofu1 17148 | Value of the object part o... |
| cofu2nd 17149 | Value of the morphism part... |
| cofu2 17150 | Value of the morphism part... |
| cofuval2 17151 | Value of the composition o... |
| cofucl 17152 | The composition of two fun... |
| cofuass 17153 | Functor composition is ass... |
| cofulid 17154 | The identity functor is a ... |
| cofurid 17155 | The identity functor is a ... |
| resfval 17156 | Value of the functor restr... |
| resfval2 17157 | Value of the functor restr... |
| resf1st 17158 | Value of the functor restr... |
| resf2nd 17159 | Value of the functor restr... |
| funcres 17160 | A functor restricted to a ... |
| funcres2b 17161 | Condition for a functor to... |
| funcres2 17162 | A functor into a restricte... |
| wunfunc 17163 | A weak universe is closed ... |
| funcpropd 17164 | If two categories have the... |
| funcres2c 17165 | Condition for a functor to... |
| fullfunc 17170 | A full functor is a functo... |
| fthfunc 17171 | A faithful functor is a fu... |
| relfull 17172 | The set of full functors i... |
| relfth 17173 | The set of faithful functo... |
| isfull 17174 | Value of the set of full f... |
| isfull2 17175 | Equivalent condition for a... |
| fullfo 17176 | The morphism map of a full... |
| fulli 17177 | The morphism map of a full... |
| isfth 17178 | Value of the set of faithf... |
| isfth2 17179 | Equivalent condition for a... |
| isffth2 17180 | A fully faithful functor i... |
| fthf1 17181 | The morphism map of a fait... |
| fthi 17182 | The morphism map of a fait... |
| ffthf1o 17183 | The morphism map of a full... |
| fullpropd 17184 | If two categories have the... |
| fthpropd 17185 | If two categories have the... |
| fulloppc 17186 | The opposite functor of a ... |
| fthoppc 17187 | The opposite functor of a ... |
| ffthoppc 17188 | The opposite functor of a ... |
| fthsect 17189 | A faithful functor reflect... |
| fthinv 17190 | A faithful functor reflect... |
| fthmon 17191 | A faithful functor reflect... |
| fthepi 17192 | A faithful functor reflect... |
| ffthiso 17193 | A fully faithful functor r... |
| fthres2b 17194 | Condition for a faithful f... |
| fthres2c 17195 | Condition for a faithful f... |
| fthres2 17196 | A faithful functor into a ... |
| idffth 17197 | The identity functor is a ... |
| cofull 17198 | The composition of two ful... |
| cofth 17199 | The composition of two fai... |
| coffth 17200 | The composition of two ful... |
| rescfth 17201 | The inclusion functor from... |
| ressffth 17202 | The inclusion functor from... |
| fullres2c 17203 | Condition for a full funct... |
| ffthres2c 17204 | Condition for a fully fait... |
| fnfuc 17209 | The ` FuncCat ` operation ... |
| natfval 17210 | Value of the function givi... |
| isnat 17211 | Property of being a natura... |
| isnat2 17212 | Property of being a natura... |
| natffn 17213 | The natural transformation... |
| natrcl 17214 | Reverse closure for a natu... |
| nat1st2nd 17215 | Rewrite the natural transf... |
| natixp 17216 | A natural transformation i... |
| natcl 17217 | A component of a natural t... |
| natfn 17218 | A natural transformation i... |
| nati 17219 | Naturality property of a n... |
| wunnat 17220 | A weak universe is closed ... |
| catstr 17221 | A category structure is a ... |
| fucval 17222 | Value of the functor categ... |
| fuccofval 17223 | Value of the functor categ... |
| fucbas 17224 | The objects of the functor... |
| fuchom 17225 | The morphisms in the funct... |
| fucco 17226 | Value of the composition o... |
| fuccoval 17227 | Value of the functor categ... |
| fuccocl 17228 | The composition of two nat... |
| fucidcl 17229 | The identity natural trans... |
| fuclid 17230 | Left identity of natural t... |
| fucrid 17231 | Right identity of natural ... |
| fucass 17232 | Associativity of natural t... |
| fuccatid 17233 | The functor category is a ... |
| fuccat 17234 | The functor category is a ... |
| fucid 17235 | The identity morphism in t... |
| fucsect 17236 | Two natural transformation... |
| fucinv 17237 | Two natural transformation... |
| invfuc 17238 | If ` V ( x ) ` is an inver... |
| fuciso 17239 | A natural transformation i... |
| natpropd 17240 | If two categories have the... |
| fucpropd 17241 | If two categories have the... |
| initorcl 17248 | Reverse closure for an ini... |
| termorcl 17249 | Reverse closure for a term... |
| zeroorcl 17250 | Reverse closure for a zero... |
| initoval 17251 | The value of the initial o... |
| termoval 17252 | The value of the terminal ... |
| zerooval 17253 | The value of the zero obje... |
| isinito 17254 | The predicate "is an initi... |
| istermo 17255 | The predicate "is a termin... |
| iszeroo 17256 | The predicate "is a zero o... |
| isinitoi 17257 | Implication of a class bei... |
| istermoi 17258 | Implication of a class bei... |
| initoid 17259 | For an initial object, the... |
| termoid 17260 | For a terminal object, the... |
| initoo 17261 | An initial object is an ob... |
| termoo 17262 | A terminal object is an ob... |
| iszeroi 17263 | Implication of a class bei... |
| 2initoinv 17264 | Morphisms between two init... |
| initoeu1 17265 | Initial objects are essent... |
| initoeu1w 17266 | Initial objects are essent... |
| initoeu2lem0 17267 | Lemma 0 for ~ initoeu2 . ... |
| initoeu2lem1 17268 | Lemma 1 for ~ initoeu2 . ... |
| initoeu2lem2 17269 | Lemma 2 for ~ initoeu2 . ... |
| initoeu2 17270 | Initial objects are essent... |
| 2termoinv 17271 | Morphisms between two term... |
| termoeu1 17272 | Terminal objects are essen... |
| termoeu1w 17273 | Terminal objects are essen... |
| homarcl 17282 | Reverse closure for an arr... |
| homafval 17283 | Value of the disjointified... |
| homaf 17284 | Functionality of the disjo... |
| homaval 17285 | Value of the disjointified... |
| elhoma 17286 | Value of the disjointified... |
| elhomai 17287 | Produce an arrow from a mo... |
| elhomai2 17288 | Produce an arrow from a mo... |
| homarcl2 17289 | Reverse closure for the do... |
| homarel 17290 | An arrow is an ordered pai... |
| homa1 17291 | The first component of an ... |
| homahom2 17292 | The second component of an... |
| homahom 17293 | The second component of an... |
| homadm 17294 | The domain of an arrow wit... |
| homacd 17295 | The codomain of an arrow w... |
| homadmcd 17296 | Decompose an arrow into do... |
| arwval 17297 | The set of arrows is the u... |
| arwrcl 17298 | The first component of an ... |
| arwhoma 17299 | An arrow is contained in t... |
| homarw 17300 | A hom-set is a subset of t... |
| arwdm 17301 | The domain of an arrow is ... |
| arwcd 17302 | The codomain of an arrow i... |
| dmaf 17303 | The domain function is a f... |
| cdaf 17304 | The codomain function is a... |
| arwhom 17305 | The second component of an... |
| arwdmcd 17306 | Decompose an arrow into do... |
| idafval 17311 | Value of the identity arro... |
| idaval 17312 | Value of the identity arro... |
| ida2 17313 | Morphism part of the ident... |
| idahom 17314 | Domain and codomain of the... |
| idadm 17315 | Domain of the identity arr... |
| idacd 17316 | Codomain of the identity a... |
| idaf 17317 | The identity arrow functio... |
| coafval 17318 | The value of the compositi... |
| eldmcoa 17319 | A pair ` <. G , F >. ` is ... |
| dmcoass 17320 | The domain of composition ... |
| homdmcoa 17321 | If ` F : X --> Y ` and ` G... |
| coaval 17322 | Value of composition for c... |
| coa2 17323 | The morphism part of arrow... |
| coahom 17324 | The composition of two com... |
| coapm 17325 | Composition of arrows is a... |
| arwlid 17326 | Left identity of a categor... |
| arwrid 17327 | Right identity of a catego... |
| arwass 17328 | Associativity of compositi... |
| setcval 17331 | Value of the category of s... |
| setcbas 17332 | Set of objects of the cate... |
| setchomfval 17333 | Set of arrows of the categ... |
| setchom 17334 | Set of arrows of the categ... |
| elsetchom 17335 | A morphism of sets is a fu... |
| setccofval 17336 | Composition in the categor... |
| setcco 17337 | Composition in the categor... |
| setccatid 17338 | Lemma for ~ setccat . (Co... |
| setccat 17339 | The category of sets is a ... |
| setcid 17340 | The identity arrow in the ... |
| setcmon 17341 | A monomorphism of sets is ... |
| setcepi 17342 | An epimorphism of sets is ... |
| setcsect 17343 | A section in the category ... |
| setcinv 17344 | An inverse in the category... |
| setciso 17345 | An isomorphism in the cate... |
| resssetc 17346 | The restriction of the cat... |
| funcsetcres2 17347 | A functor into a smaller c... |
| catcval 17350 | Value of the category of c... |
| catcbas 17351 | Set of objects of the cate... |
| catchomfval 17352 | Set of arrows of the categ... |
| catchom 17353 | Set of arrows of the categ... |
| catccofval 17354 | Composition in the categor... |
| catcco 17355 | Composition in the categor... |
| catccatid 17356 | Lemma for ~ catccat . (Co... |
| catcid 17357 | The identity arrow in the ... |
| catccat 17358 | The category of categories... |
| resscatc 17359 | The restriction of the cat... |
| catcisolem 17360 | Lemma for ~ catciso . (Co... |
| catciso 17361 | A functor is an isomorphis... |
| catcoppccl 17362 | The category of categories... |
| catcfuccl 17363 | The category of categories... |
| fncnvimaeqv 17364 | The inverse images of the ... |
| bascnvimaeqv 17365 | The inverse image of the u... |
| estrcval 17368 | Value of the category of e... |
| estrcbas 17369 | Set of objects of the cate... |
| estrchomfval 17370 | Set of morphisms ("arrows"... |
| estrchom 17371 | The morphisms between exte... |
| elestrchom 17372 | A morphism between extensi... |
| estrccofval 17373 | Composition in the categor... |
| estrcco 17374 | Composition in the categor... |
| estrcbasbas 17375 | An element of the base set... |
| estrccatid 17376 | Lemma for ~ estrccat . (C... |
| estrccat 17377 | The category of extensible... |
| estrcid 17378 | The identity arrow in the ... |
| estrchomfn 17379 | The Hom-set operation in t... |
| estrchomfeqhom 17380 | The functionalized Hom-set... |
| estrreslem1 17381 | Lemma 1 for ~ estrres . (... |
| estrreslem2 17382 | Lemma 2 for ~ estrres . (... |
| estrres 17383 | Any restriction of a categ... |
| funcestrcsetclem1 17384 | Lemma 1 for ~ funcestrcset... |
| funcestrcsetclem2 17385 | Lemma 2 for ~ funcestrcset... |
| funcestrcsetclem3 17386 | Lemma 3 for ~ funcestrcset... |
| funcestrcsetclem4 17387 | Lemma 4 for ~ funcestrcset... |
| funcestrcsetclem5 17388 | Lemma 5 for ~ funcestrcset... |
| funcestrcsetclem6 17389 | Lemma 6 for ~ funcestrcset... |
| funcestrcsetclem7 17390 | Lemma 7 for ~ funcestrcset... |
| funcestrcsetclem8 17391 | Lemma 8 for ~ funcestrcset... |
| funcestrcsetclem9 17392 | Lemma 9 for ~ funcestrcset... |
| funcestrcsetc 17393 | The "natural forgetful fun... |
| fthestrcsetc 17394 | The "natural forgetful fun... |
| fullestrcsetc 17395 | The "natural forgetful fun... |
| equivestrcsetc 17396 | The "natural forgetful fun... |
| setc1strwun 17397 | A constructed one-slot str... |
| funcsetcestrclem1 17398 | Lemma 1 for ~ funcsetcestr... |
| funcsetcestrclem2 17399 | Lemma 2 for ~ funcsetcestr... |
| funcsetcestrclem3 17400 | Lemma 3 for ~ funcsetcestr... |
| embedsetcestrclem 17401 | Lemma for ~ embedsetcestrc... |
| funcsetcestrclem4 17402 | Lemma 4 for ~ funcsetcestr... |
| funcsetcestrclem5 17403 | Lemma 5 for ~ funcsetcestr... |
| funcsetcestrclem6 17404 | Lemma 6 for ~ funcsetcestr... |
| funcsetcestrclem7 17405 | Lemma 7 for ~ funcsetcestr... |
| funcsetcestrclem8 17406 | Lemma 8 for ~ funcsetcestr... |
| funcsetcestrclem9 17407 | Lemma 9 for ~ funcsetcestr... |
| funcsetcestrc 17408 | The "embedding functor" fr... |
| fthsetcestrc 17409 | The "embedding functor" fr... |
| fullsetcestrc 17410 | The "embedding functor" fr... |
| embedsetcestrc 17411 | The "embedding functor" fr... |
| fnxpc 17420 | The binary product of cate... |
| xpcval 17421 | Value of the binary produc... |
| xpcbas 17422 | Set of objects of the bina... |
| xpchomfval 17423 | Set of morphisms of the bi... |
| xpchom 17424 | Set of morphisms of the bi... |
| relxpchom 17425 | A hom-set in the binary pr... |
| xpccofval 17426 | Value of composition in th... |
| xpcco 17427 | Value of composition in th... |
| xpcco1st 17428 | Value of composition in th... |
| xpcco2nd 17429 | Value of composition in th... |
| xpchom2 17430 | Value of the set of morphi... |
| xpcco2 17431 | Value of composition in th... |
| xpccatid 17432 | The product of two categor... |
| xpcid 17433 | The identity morphism in t... |
| xpccat 17434 | The product of two categor... |
| 1stfval 17435 | Value of the first project... |
| 1stf1 17436 | Value of the first project... |
| 1stf2 17437 | Value of the first project... |
| 2ndfval 17438 | Value of the first project... |
| 2ndf1 17439 | Value of the first project... |
| 2ndf2 17440 | Value of the first project... |
| 1stfcl 17441 | The first projection funct... |
| 2ndfcl 17442 | The second projection func... |
| prfval 17443 | Value of the pairing funct... |
| prf1 17444 | Value of the pairing funct... |
| prf2fval 17445 | Value of the pairing funct... |
| prf2 17446 | Value of the pairing funct... |
| prfcl 17447 | The pairing of functors ` ... |
| prf1st 17448 | Cancellation of pairing wi... |
| prf2nd 17449 | Cancellation of pairing wi... |
| 1st2ndprf 17450 | Break a functor into a pro... |
| catcxpccl 17451 | The category of categories... |
| xpcpropd 17452 | If two categories have the... |
| evlfval 17461 | Value of the evaluation fu... |
| evlf2 17462 | Value of the evaluation fu... |
| evlf2val 17463 | Value of the evaluation na... |
| evlf1 17464 | Value of the evaluation fu... |
| evlfcllem 17465 | Lemma for ~ evlfcl . (Con... |
| evlfcl 17466 | The evaluation functor is ... |
| curfval 17467 | Value of the curry functor... |
| curf1fval 17468 | Value of the object part o... |
| curf1 17469 | Value of the object part o... |
| curf11 17470 | Value of the double evalua... |
| curf12 17471 | The partially evaluated cu... |
| curf1cl 17472 | The partially evaluated cu... |
| curf2 17473 | Value of the curry functor... |
| curf2val 17474 | Value of a component of th... |
| curf2cl 17475 | The curry functor at a mor... |
| curfcl 17476 | The curry functor of a fun... |
| curfpropd 17477 | If two categories have the... |
| uncfval 17478 | Value of the uncurry funct... |
| uncfcl 17479 | The uncurry operation take... |
| uncf1 17480 | Value of the uncurry funct... |
| uncf2 17481 | Value of the uncurry funct... |
| curfuncf 17482 | Cancellation of curry with... |
| uncfcurf 17483 | Cancellation of uncurry wi... |
| diagval 17484 | Define the diagonal functo... |
| diagcl 17485 | The diagonal functor is a ... |
| diag1cl 17486 | The constant functor of ` ... |
| diag11 17487 | Value of the constant func... |
| diag12 17488 | Value of the constant func... |
| diag2 17489 | Value of the diagonal func... |
| diag2cl 17490 | The diagonal functor at a ... |
| curf2ndf 17491 | As shown in ~ diagval , th... |
| hofval 17496 | Value of the Hom functor, ... |
| hof1fval 17497 | The object part of the Hom... |
| hof1 17498 | The object part of the Hom... |
| hof2fval 17499 | The morphism part of the H... |
| hof2val 17500 | The morphism part of the H... |
| hof2 17501 | The morphism part of the H... |
| hofcllem 17502 | Lemma for ~ hofcl . (Cont... |
| hofcl 17503 | Closure of the Hom functor... |
| oppchofcl 17504 | Closure of the opposite Ho... |
| yonval 17505 | Value of the Yoneda embedd... |
| yoncl 17506 | The Yoneda embedding is a ... |
| yon1cl 17507 | The Yoneda embedding at an... |
| yon11 17508 | Value of the Yoneda embedd... |
| yon12 17509 | Value of the Yoneda embedd... |
| yon2 17510 | Value of the Yoneda embedd... |
| hofpropd 17511 | If two categories have the... |
| yonpropd 17512 | If two categories have the... |
| oppcyon 17513 | Value of the opposite Yone... |
| oyoncl 17514 | The opposite Yoneda embedd... |
| oyon1cl 17515 | The opposite Yoneda embedd... |
| yonedalem1 17516 | Lemma for ~ yoneda . (Con... |
| yonedalem21 17517 | Lemma for ~ yoneda . (Con... |
| yonedalem3a 17518 | Lemma for ~ yoneda . (Con... |
| yonedalem4a 17519 | Lemma for ~ yoneda . (Con... |
| yonedalem4b 17520 | Lemma for ~ yoneda . (Con... |
| yonedalem4c 17521 | Lemma for ~ yoneda . (Con... |
| yonedalem22 17522 | Lemma for ~ yoneda . (Con... |
| yonedalem3b 17523 | Lemma for ~ yoneda . (Con... |
| yonedalem3 17524 | Lemma for ~ yoneda . (Con... |
| yonedainv 17525 | The Yoneda Lemma with expl... |
| yonffthlem 17526 | Lemma for ~ yonffth . (Co... |
| yoneda 17527 | The Yoneda Lemma. There i... |
| yonffth 17528 | The Yoneda Lemma. The Yon... |
| yoniso 17529 | If the codomain is recover... |
| isprs 17534 | Property of being a preord... |
| prslem 17535 | Lemma for ~ prsref and ~ p... |
| prsref 17536 | "Less than or equal to" is... |
| prstr 17537 | "Less than or equal to" is... |
| isdrs 17538 | Property of being a direct... |
| drsdir 17539 | Direction of a directed se... |
| drsprs 17540 | A directed set is a proset... |
| drsbn0 17541 | The base of a directed set... |
| drsdirfi 17542 | Any _finite_ number of ele... |
| isdrs2 17543 | Directed sets may be defin... |
| ispos 17551 | The predicate "is a poset.... |
| ispos2 17552 | A poset is an antisymmetri... |
| posprs 17553 | A poset is a proset. (Con... |
| posi 17554 | Lemma for poset properties... |
| posref 17555 | A poset ordering is reflex... |
| posasymb 17556 | A poset ordering is asymme... |
| postr 17557 | A poset ordering is transi... |
| 0pos 17558 | Technical lemma to simplif... |
| isposd 17559 | Properties that determine ... |
| isposi 17560 | Properties that determine ... |
| isposix 17561 | Properties that determine ... |
| pltfval 17563 | Value of the less-than rel... |
| pltval 17564 | Less-than relation. ( ~ d... |
| pltle 17565 | "Less than" implies "less ... |
| pltne 17566 | The "less than" relation i... |
| pltirr 17567 | The "less than" relation i... |
| pleval2i 17568 | One direction of ~ pleval2... |
| pleval2 17569 | "Less than or equal to" in... |
| pltnle 17570 | "Less than" implies not co... |
| pltval3 17571 | Alternate expression for t... |
| pltnlt 17572 | The less-than relation imp... |
| pltn2lp 17573 | The less-than relation has... |
| plttr 17574 | The less-than relation is ... |
| pltletr 17575 | Transitive law for chained... |
| plelttr 17576 | Transitive law for chained... |
| pospo 17577 | Write a poset structure in... |
| lubfval 17582 | Value of the least upper b... |
| lubdm 17583 | Domain of the least upper ... |
| lubfun 17584 | The LUB is a function. (C... |
| lubeldm 17585 | Member of the domain of th... |
| lubelss 17586 | A member of the domain of ... |
| lubeu 17587 | Unique existence proper of... |
| lubval 17588 | Value of the least upper b... |
| lubcl 17589 | The least upper bound func... |
| lubprop 17590 | Properties of greatest low... |
| luble 17591 | The greatest lower bound i... |
| lublecllem 17592 | Lemma for ~ lublecl and ~ ... |
| lublecl 17593 | The set of all elements le... |
| lubid 17594 | The LUB of elements less t... |
| glbfval 17595 | Value of the greatest lowe... |
| glbdm 17596 | Domain of the greatest low... |
| glbfun 17597 | The GLB is a function. (C... |
| glbeldm 17598 | Member of the domain of th... |
| glbelss 17599 | A member of the domain of ... |
| glbeu 17600 | Unique existence proper of... |
| glbval 17601 | Value of the greatest lowe... |
| glbcl 17602 | The least upper bound func... |
| glbprop 17603 | Properties of greatest low... |
| glble 17604 | The greatest lower bound i... |
| joinfval 17605 | Value of join function for... |
| joinfval2 17606 | Value of join function for... |
| joindm 17607 | Domain of join function fo... |
| joindef 17608 | Two ways to say that a joi... |
| joinval 17609 | Join value. Since both si... |
| joincl 17610 | Closure of join of element... |
| joindmss 17611 | Subset property of domain ... |
| joinval2lem 17612 | Lemma for ~ joinval2 and ~... |
| joinval2 17613 | Value of join for a poset ... |
| joineu 17614 | Uniqueness of join of elem... |
| joinlem 17615 | Lemma for join properties.... |
| lejoin1 17616 | A join's first argument is... |
| lejoin2 17617 | A join's second argument i... |
| joinle 17618 | A join is less than or equ... |
| meetfval 17619 | Value of meet function for... |
| meetfval2 17620 | Value of meet function for... |
| meetdm 17621 | Domain of meet function fo... |
| meetdef 17622 | Two ways to say that a mee... |
| meetval 17623 | Meet value. Since both si... |
| meetcl 17624 | Closure of meet of element... |
| meetdmss 17625 | Subset property of domain ... |
| meetval2lem 17626 | Lemma for ~ meetval2 and ~... |
| meetval2 17627 | Value of meet for a poset ... |
| meeteu 17628 | Uniqueness of meet of elem... |
| meetlem 17629 | Lemma for meet properties.... |
| lemeet1 17630 | A meet's first argument is... |
| lemeet2 17631 | A meet's second argument i... |
| meetle 17632 | A meet is less than or equ... |
| joincomALT 17633 | The join of a poset commut... |
| joincom 17634 | The join of a poset commut... |
| meetcomALT 17635 | The meet of a poset commut... |
| meetcom 17636 | The meet of a poset commut... |
| istos 17639 | The predicate "is a toset.... |
| tosso 17640 | Write the totally ordered ... |
| p0val 17645 | Value of poset zero. (Con... |
| p1val 17646 | Value of poset zero. (Con... |
| p0le 17647 | Any element is less than o... |
| ple1 17648 | Any element is less than o... |
| islat 17651 | The predicate "is a lattic... |
| latcl2 17652 | The join and meet of any t... |
| latlem 17653 | Lemma for lattice properti... |
| latpos 17654 | A lattice is a poset. (Co... |
| latjcl 17655 | Closure of join operation ... |
| latmcl 17656 | Closure of meet operation ... |
| latref 17657 | A lattice ordering is refl... |
| latasymb 17658 | A lattice ordering is asym... |
| latasym 17659 | A lattice ordering is asym... |
| lattr 17660 | A lattice ordering is tran... |
| latasymd 17661 | Deduce equality from latti... |
| lattrd 17662 | A lattice ordering is tran... |
| latjcom 17663 | The join of a lattice comm... |
| latlej1 17664 | A join's first argument is... |
| latlej2 17665 | A join's second argument i... |
| latjle12 17666 | A join is less than or equ... |
| latleeqj1 17667 | "Less than or equal to" in... |
| latleeqj2 17668 | "Less than or equal to" in... |
| latjlej1 17669 | Add join to both sides of ... |
| latjlej2 17670 | Add join to both sides of ... |
| latjlej12 17671 | Add join to both sides of ... |
| latnlej 17672 | An idiom to express that a... |
| latnlej1l 17673 | An idiom to express that a... |
| latnlej1r 17674 | An idiom to express that a... |
| latnlej2 17675 | An idiom to express that a... |
| latnlej2l 17676 | An idiom to express that a... |
| latnlej2r 17677 | An idiom to express that a... |
| latjidm 17678 | Lattice join is idempotent... |
| latmcom 17679 | The join of a lattice comm... |
| latmle1 17680 | A meet is less than or equ... |
| latmle2 17681 | A meet is less than or equ... |
| latlem12 17682 | An element is less than or... |
| latleeqm1 17683 | "Less than or equal to" in... |
| latleeqm2 17684 | "Less than or equal to" in... |
| latmlem1 17685 | Add meet to both sides of ... |
| latmlem2 17686 | Add meet to both sides of ... |
| latmlem12 17687 | Add join to both sides of ... |
| latnlemlt 17688 | Negation of "less than or ... |
| latnle 17689 | Equivalent expressions for... |
| latmidm 17690 | Lattice join is idempotent... |
| latabs1 17691 | Lattice absorption law. F... |
| latabs2 17692 | Lattice absorption law. F... |
| latledi 17693 | An ortholattice is distrib... |
| latmlej11 17694 | Ordering of a meet and joi... |
| latmlej12 17695 | Ordering of a meet and joi... |
| latmlej21 17696 | Ordering of a meet and joi... |
| latmlej22 17697 | Ordering of a meet and joi... |
| lubsn 17698 | The least upper bound of a... |
| latjass 17699 | Lattice join is associativ... |
| latj12 17700 | Swap 1st and 2nd members o... |
| latj32 17701 | Swap 2nd and 3rd members o... |
| latj13 17702 | Swap 1st and 3rd members o... |
| latj31 17703 | Swap 2nd and 3rd members o... |
| latjrot 17704 | Rotate lattice join of 3 c... |
| latj4 17705 | Rearrangement of lattice j... |
| latj4rot 17706 | Rotate lattice join of 4 c... |
| latjjdi 17707 | Lattice join distributes o... |
| latjjdir 17708 | Lattice join distributes o... |
| mod1ile 17709 | The weak direction of the ... |
| mod2ile 17710 | The weak direction of the ... |
| isclat 17713 | The predicate "is a comple... |
| clatpos 17714 | A complete lattice is a po... |
| clatlem 17715 | Lemma for properties of a ... |
| clatlubcl 17716 | Any subset of the base set... |
| clatlubcl2 17717 | Any subset of the base set... |
| clatglbcl 17718 | Any subset of the base set... |
| clatglbcl2 17719 | Any subset of the base set... |
| clatl 17720 | A complete lattice is a la... |
| isglbd 17721 | Properties that determine ... |
| lublem 17722 | Lemma for the least upper ... |
| lubub 17723 | The LUB of a complete latt... |
| lubl 17724 | The LUB of a complete latt... |
| lubss 17725 | Subset law for least upper... |
| lubel 17726 | An element of a set is les... |
| lubun 17727 | The LUB of a union. (Cont... |
| clatglb 17728 | Properties of greatest low... |
| clatglble 17729 | The greatest lower bound i... |
| clatleglb 17730 | Two ways of expressing "le... |
| clatglbss 17731 | Subset law for greatest lo... |
| oduval 17734 | Value of an order dual str... |
| oduleval 17735 | Value of the less-equal re... |
| oduleg 17736 | Truth of the less-equal re... |
| odubas 17737 | Base set of an order dual ... |
| pospropd 17738 | Posethood is determined on... |
| odupos 17739 | Being a poset is a self-du... |
| oduposb 17740 | Being a poset is a self-du... |
| meet0 17741 | Lemma for ~ odujoin . (Co... |
| join0 17742 | Lemma for ~ odumeet . (Co... |
| oduglb 17743 | Greatest lower bounds in a... |
| odumeet 17744 | Meets in a dual order are ... |
| odulub 17745 | Least upper bounds in a du... |
| odujoin 17746 | Joins in a dual order are ... |
| odulatb 17747 | Being a lattice is self-du... |
| oduclatb 17748 | Being a complete lattice i... |
| odulat 17749 | Being a lattice is self-du... |
| poslubmo 17750 | Least upper bounds in a po... |
| posglbmo 17751 | Greatest lower bounds in a... |
| poslubd 17752 | Properties which determine... |
| poslubdg 17753 | Properties which determine... |
| posglbd 17754 | Properties which determine... |
| ipostr 17757 | The structure of ~ df-ipo ... |
| ipoval 17758 | Value of the inclusion pos... |
| ipobas 17759 | Base set of the inclusion ... |
| ipolerval 17760 | Relation of the inclusion ... |
| ipotset 17761 | Topology of the inclusion ... |
| ipole 17762 | Weak order condition of th... |
| ipolt 17763 | Strict order condition of ... |
| ipopos 17764 | The inclusion poset on a f... |
| isipodrs 17765 | Condition for a family of ... |
| ipodrscl 17766 | Direction by inclusion as ... |
| ipodrsfi 17767 | Finite upper bound propert... |
| fpwipodrs 17768 | The finite subsets of any ... |
| ipodrsima 17769 | The monotone image of a di... |
| isacs3lem 17770 | An algebraic closure syste... |
| acsdrsel 17771 | An algebraic closure syste... |
| isacs4lem 17772 | In a closure system in whi... |
| isacs5lem 17773 | If closure commutes with d... |
| acsdrscl 17774 | In an algebraic closure sy... |
| acsficl 17775 | A closure in an algebraic ... |
| isacs5 17776 | A closure system is algebr... |
| isacs4 17777 | A closure system is algebr... |
| isacs3 17778 | A closure system is algebr... |
| acsficld 17779 | In an algebraic closure sy... |
| acsficl2d 17780 | In an algebraic closure sy... |
| acsfiindd 17781 | In an algebraic closure sy... |
| acsmapd 17782 | In an algebraic closure sy... |
| acsmap2d 17783 | In an algebraic closure sy... |
| acsinfd 17784 | In an algebraic closure sy... |
| acsdomd 17785 | In an algebraic closure sy... |
| acsinfdimd 17786 | In an algebraic closure sy... |
| acsexdimd 17787 | In an algebraic closure sy... |
| mrelatglb 17788 | Greatest lower bounds in a... |
| mrelatglb0 17789 | The empty intersection in ... |
| mrelatlub 17790 | Least upper bounds in a Mo... |
| mreclatBAD 17791 | A Moore space is a complet... |
| latmass 17792 | Lattice meet is associativ... |
| latdisdlem 17793 | Lemma for ~ latdisd . (Co... |
| latdisd 17794 | In a lattice, joins distri... |
| isdlat 17797 | Property of being a distri... |
| dlatmjdi 17798 | In a distributive lattice,... |
| dlatl 17799 | A distributive lattice is ... |
| odudlatb 17800 | The dual of a distributive... |
| dlatjmdi 17801 | In a distributive lattice,... |
| isps 17806 | The predicate "is a poset"... |
| psrel 17807 | A poset is a relation. (C... |
| psref2 17808 | A poset is antisymmetric a... |
| pstr2 17809 | A poset is transitive. (C... |
| pslem 17810 | Lemma for ~ psref and othe... |
| psdmrn 17811 | The domain and range of a ... |
| psref 17812 | A poset is reflexive. (Co... |
| psrn 17813 | The range of a poset equal... |
| psasym 17814 | A poset is antisymmetric. ... |
| pstr 17815 | A poset is transitive. (C... |
| cnvps 17816 | The converse of a poset is... |
| cnvpsb 17817 | The converse of a poset is... |
| psss 17818 | Any subset of a partially ... |
| psssdm2 17819 | Field of a subposet. (Con... |
| psssdm 17820 | Field of a subposet. (Con... |
| istsr 17821 | The predicate is a toset. ... |
| istsr2 17822 | The predicate is a toset. ... |
| tsrlin 17823 | A toset is a linear order.... |
| tsrlemax 17824 | Two ways of saying a numbe... |
| tsrps 17825 | A toset is a poset. (Cont... |
| cnvtsr 17826 | The converse of a toset is... |
| tsrss 17827 | Any subset of a totally or... |
| ledm 17828 | The domain of ` <_ ` is ` ... |
| lern 17829 | The range of ` <_ ` is ` R... |
| lefld 17830 | The field of the 'less or ... |
| letsr 17831 | The "less than or equal to... |
| isdir 17836 | A condition for a relation... |
| reldir 17837 | A direction is a relation.... |
| dirdm 17838 | A direction's domain is eq... |
| dirref 17839 | A direction is reflexive. ... |
| dirtr 17840 | A direction is transitive.... |
| dirge 17841 | For any two elements of a ... |
| tsrdir 17842 | A totally ordered set is a... |
| ismgm 17847 | The predicate "is a magma"... |
| ismgmn0 17848 | The predicate "is a magma"... |
| mgmcl 17849 | Closure of the operation o... |
| isnmgm 17850 | A condition for a structur... |
| mgmsscl 17851 | If the base set of a magma... |
| plusffval 17852 | The group addition operati... |
| plusfval 17853 | The group addition operati... |
| plusfeq 17854 | If the addition operation ... |
| plusffn 17855 | The group addition operati... |
| mgmplusf 17856 | The group addition functio... |
| issstrmgm 17857 | Characterize a substructur... |
| intopsn 17858 | The internal operation for... |
| mgmb1mgm1 17859 | The only magma with a base... |
| mgm0 17860 | Any set with an empty base... |
| mgm0b 17861 | The structure with an empt... |
| mgm1 17862 | The structure with one ele... |
| opifismgm 17863 | A structure with a group a... |
| mgmidmo 17864 | A two-sided identity eleme... |
| grpidval 17865 | The value of the identity ... |
| grpidpropd 17866 | If two structures have the... |
| fn0g 17867 | The group zero extractor i... |
| 0g0 17868 | The identity element funct... |
| ismgmid 17869 | The identity element of a ... |
| mgmidcl 17870 | The identity element of a ... |
| mgmlrid 17871 | The identity element of a ... |
| ismgmid2 17872 | Show that a given element ... |
| lidrideqd 17873 | If there is a left and rig... |
| lidrididd 17874 | If there is a left and rig... |
| grpidd 17875 | Deduce the identity elemen... |
| mgmidsssn0 17876 | Property of the set of ide... |
| grprinvlem 17877 | Lemma for ~ grprinvd . (C... |
| grprinvd 17878 | Deduce right inverse from ... |
| grpridd 17879 | Deduce right identity from... |
| gsumvalx 17880 | Expand out the substitutio... |
| gsumval 17881 | Expand out the substitutio... |
| gsumpropd 17882 | The group sum depends only... |
| gsumpropd2lem 17883 | Lemma for ~ gsumpropd2 . ... |
| gsumpropd2 17884 | A stronger version of ~ gs... |
| gsummgmpropd 17885 | A stronger version of ~ gs... |
| gsumress 17886 | The group sum in a substru... |
| gsumval1 17887 | Value of the group sum ope... |
| gsum0 17888 | Value of the empty group s... |
| gsumval2a 17889 | Value of the group sum ope... |
| gsumval2 17890 | Value of the group sum ope... |
| gsumsplit1r 17891 | Splitting off the rightmos... |
| gsumprval 17892 | Value of the group sum ope... |
| gsumpr12val 17893 | Value of the group sum ope... |
| issgrp 17896 | The predicate "is a semigr... |
| issgrpv 17897 | The predicate "is a semigr... |
| issgrpn0 17898 | The predicate "is a semigr... |
| isnsgrp 17899 | A condition for a structur... |
| sgrpmgm 17900 | A semigroup is a magma. (... |
| sgrpass 17901 | A semigroup operation is a... |
| sgrp0 17902 | Any set with an empty base... |
| sgrp0b 17903 | The structure with an empt... |
| sgrp1 17904 | The structure with one ele... |
| ismnddef 17907 | The predicate "is a monoid... |
| ismnd 17908 | The predicate "is a monoid... |
| isnmnd 17909 | A condition for a structur... |
| sgrpidmnd 17910 | A semigroup with an identi... |
| mndsgrp 17911 | A monoid is a semigroup. ... |
| mndmgm 17912 | A monoid is a magma. (Con... |
| mndcl 17913 | Closure of the operation o... |
| mndass 17914 | A monoid operation is asso... |
| mndid 17915 | A monoid has a two-sided i... |
| mndideu 17916 | The two-sided identity ele... |
| mnd32g 17917 | Commutative/associative la... |
| mnd12g 17918 | Commutative/associative la... |
| mnd4g 17919 | Commutative/associative la... |
| mndidcl 17920 | The identity element of a ... |
| mndbn0 17921 | The base set of a monoid i... |
| hashfinmndnn 17922 | A finite monoid has positi... |
| mndplusf 17923 | The group addition operati... |
| mndlrid 17924 | A monoid's identity elemen... |
| mndlid 17925 | The identity element of a ... |
| mndrid 17926 | The identity element of a ... |
| ismndd 17927 | Deduce a monoid from its p... |
| mndpfo 17928 | The addition operation of ... |
| mndfo 17929 | The addition operation of ... |
| mndpropd 17930 | If two structures have the... |
| mndprop 17931 | If two structures have the... |
| issubmnd 17932 | Characterize a submonoid b... |
| ress0g 17933 | ` 0g ` is unaffected by re... |
| submnd0 17934 | The zero of a submonoid is... |
| mndinvmod 17935 | Uniqueness of an inverse e... |
| prdsplusgcl 17936 | Structure product pointwis... |
| prdsidlem 17937 | Characterization of identi... |
| prdsmndd 17938 | The product of a family of... |
| prds0g 17939 | Zero in a product of monoi... |
| pwsmnd 17940 | The structure power of a m... |
| pws0g 17941 | Zero in a structure power ... |
| imasmnd2 17942 | The image structure of a m... |
| imasmnd 17943 | The image structure of a m... |
| imasmndf1 17944 | The image of a monoid unde... |
| xpsmnd 17945 | The binary product of mono... |
| mnd1 17946 | The (smallest) structure r... |
| mnd1id 17947 | The singleton element of a... |
| ismhm 17952 | Property of a monoid homom... |
| mhmrcl1 17953 | Reverse closure of a monoi... |
| mhmrcl2 17954 | Reverse closure of a monoi... |
| mhmf 17955 | A monoid homomorphism is a... |
| mhmpropd 17956 | Monoid homomorphism depend... |
| mhmlin 17957 | A monoid homomorphism comm... |
| mhm0 17958 | A monoid homomorphism pres... |
| idmhm 17959 | The identity homomorphism ... |
| mhmf1o 17960 | A monoid homomorphism is b... |
| submrcl 17961 | Reverse closure for submon... |
| issubm 17962 | Expand definition of a sub... |
| issubm2 17963 | Submonoids are subsets tha... |
| issubmndb 17964 | The submonoid predicate. ... |
| issubmd 17965 | Deduction for proving a su... |
| mndissubm 17966 | If the base set of a monoi... |
| resmndismnd 17967 | If the base set of a monoi... |
| submss 17968 | Submonoids are subsets of ... |
| submid 17969 | Every monoid is trivially ... |
| subm0cl 17970 | Submonoids contain zero. ... |
| submcl 17971 | Submonoids are closed unde... |
| submmnd 17972 | Submonoids are themselves ... |
| submbas 17973 | The base set of a submonoi... |
| subm0 17974 | Submonoids have the same i... |
| subsubm 17975 | A submonoid of a submonoid... |
| 0subm 17976 | The zero submonoid of an a... |
| insubm 17977 | The intersection of two su... |
| 0mhm 17978 | The constant zero linear f... |
| resmhm 17979 | Restriction of a monoid ho... |
| resmhm2 17980 | One direction of ~ resmhm2... |
| resmhm2b 17981 | Restriction of the codomai... |
| mhmco 17982 | The composition of monoid ... |
| mhmima 17983 | The homomorphic image of a... |
| mhmeql 17984 | The equalizer of two monoi... |
| submacs 17985 | Submonoids are an algebrai... |
| mndind 17986 | Induction in a monoid. In... |
| prdspjmhm 17987 | A projection from a produc... |
| pwspjmhm 17988 | A projection from a struct... |
| pwsdiagmhm 17989 | Diagonal monoid homomorphi... |
| pwsco1mhm 17990 | Right composition with a f... |
| pwsco2mhm 17991 | Left composition with a mo... |
| gsumvallem2 17992 | Lemma for properties of th... |
| gsumsubm 17993 | Evaluate a group sum in a ... |
| gsumz 17994 | Value of a group sum over ... |
| gsumwsubmcl 17995 | Closure of the composite i... |
| gsumws1 17996 | A singleton composite reco... |
| gsumwcl 17997 | Closure of the composite o... |
| gsumsgrpccat 17998 | Homomorphic property of no... |
| gsumccatOLD 17999 | Obsolete version of ~ gsum... |
| gsumccat 18000 | Homomorphic property of co... |
| gsumws2 18001 | Valuation of a pair in a m... |
| gsumccatsn 18002 | Homomorphic property of co... |
| gsumspl 18003 | The primary purpose of the... |
| gsumwmhm 18004 | Behavior of homomorphisms ... |
| gsumwspan 18005 | The submonoid generated by... |
| frmdval 18010 | Value of the free monoid c... |
| frmdbas 18011 | The base set of a free mon... |
| frmdelbas 18012 | An element of the base set... |
| frmdplusg 18013 | The monoid operation of a ... |
| frmdadd 18014 | Value of the monoid operat... |
| vrmdfval 18015 | The canonical injection fr... |
| vrmdval 18016 | The value of the generatin... |
| vrmdf 18017 | The mapping from the index... |
| frmdmnd 18018 | A free monoid is a monoid.... |
| frmd0 18019 | The identity of the free m... |
| frmdsssubm 18020 | The set of words taking va... |
| frmdgsum 18021 | Any word in a free monoid ... |
| frmdss2 18022 | A subset of generators is ... |
| frmdup1 18023 | Any assignment of the gene... |
| frmdup2 18024 | The evaluation map has the... |
| frmdup3lem 18025 | Lemma for ~ frmdup3 . (Co... |
| frmdup3 18026 | Universal property of the ... |
| efmnd 18029 | The monoid of endofunction... |
| efmndbas 18030 | The base set of the monoid... |
| efmndbasabf 18031 | The base set of the monoid... |
| elefmndbas 18032 | Two ways of saying a funct... |
| elefmndbas2 18033 | Two ways of saying a funct... |
| efmndbasf 18034 | Elements in the monoid of ... |
| efmndhash 18035 | The monoid of endofunction... |
| efmndbasfi 18036 | The monoid of endofunction... |
| efmndfv 18037 | The function value of an e... |
| efmndtset 18038 | The topology of the monoid... |
| efmndplusg 18039 | The group operation of a m... |
| efmndov 18040 | The value of the group ope... |
| efmndcl 18041 | The group operation of the... |
| efmndtopn 18042 | The topology of the monoid... |
| symggrplem 18043 | Lemma for ~ symggrp and ~ ... |
| efmndmgm 18044 | The monoid of endofunction... |
| efmndsgrp 18045 | The monoid of endofunction... |
| ielefmnd 18046 | The identity function rest... |
| efmndid 18047 | The identity function rest... |
| efmndmnd 18048 | The monoid of endofunction... |
| efmnd0nmnd 18049 | Even the monoid of endofun... |
| efmndbas0 18050 | The base set of the monoid... |
| efmnd1hash 18051 | The monoid of endofunction... |
| efmnd1bas 18052 | The monoid of endofunction... |
| efmnd2hash 18053 | The monoid of endofunction... |
| submefmnd 18054 | If the base set of a monoi... |
| sursubmefmnd 18055 | The set of surjective endo... |
| injsubmefmnd 18056 | The set of injective endof... |
| idressubmefmnd 18057 | The singleton containing o... |
| idresefmnd 18058 | The structure with the sin... |
| smndex1ibas 18059 | The modulo function ` I ` ... |
| smndex1iidm 18060 | The modulo function ` I ` ... |
| smndex1gbas 18061 | The constant functions ` (... |
| smndex1gid 18062 | The composition of a const... |
| smndex1igid 18063 | The composition of the mod... |
| smndex1basss 18064 | The modulo function ` I ` ... |
| smndex1bas 18065 | The base set of the monoid... |
| smndex1mgm 18066 | The monoid of endofunction... |
| smndex1sgrp 18067 | The monoid of endofunction... |
| smndex1mndlem 18068 | Lemma for ~ smndex1mnd and... |
| smndex1mnd 18069 | The monoid of endofunction... |
| smndex1id 18070 | The modulo function ` I ` ... |
| smndex1n0mnd 18071 | The identity of the monoid... |
| nsmndex1 18072 | The base set ` B ` of the ... |
| smndex2dbas 18073 | The doubling function ` D ... |
| smndex2dnrinv 18074 | The doubling function ` D ... |
| smndex2hbas 18075 | The halving functions ` H ... |
| smndex2dlinvh 18076 | The halving functions ` H ... |
| mgm2nsgrplem1 18077 | Lemma 1 for ~ mgm2nsgrp : ... |
| mgm2nsgrplem2 18078 | Lemma 2 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem3 18079 | Lemma 3 for ~ mgm2nsgrp . ... |
| mgm2nsgrplem4 18080 | Lemma 4 for ~ mgm2nsgrp : ... |
| mgm2nsgrp 18081 | A small magma (with two el... |
| sgrp2nmndlem1 18082 | Lemma 1 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem2 18083 | Lemma 2 for ~ sgrp2nmnd . ... |
| sgrp2nmndlem3 18084 | Lemma 3 for ~ sgrp2nmnd . ... |
| sgrp2rid2 18085 | A small semigroup (with tw... |
| sgrp2rid2ex 18086 | A small semigroup (with tw... |
| sgrp2nmndlem4 18087 | Lemma 4 for ~ sgrp2nmnd : ... |
| sgrp2nmndlem5 18088 | Lemma 5 for ~ sgrp2nmnd : ... |
| sgrp2nmnd 18089 | A small semigroup (with tw... |
| mgmnsgrpex 18090 | There is a magma which is ... |
| sgrpnmndex 18091 | There is a semigroup which... |
| sgrpssmgm 18092 | The class of all semigroup... |
| mndsssgrp 18093 | The class of all monoids i... |
| pwmndgplus 18094 | The operation of the monoi... |
| pwmndid 18095 | The identity of the monoid... |
| pwmnd 18096 | The power set of a class `... |
| isgrp 18103 | The predicate "is a group.... |
| grpmnd 18104 | A group is a monoid. (Con... |
| grpcl 18105 | Closure of the operation o... |
| grpass 18106 | A group operation is assoc... |
| grpinvex 18107 | Every member of a group ha... |
| grpideu 18108 | The two-sided identity ele... |
| grpplusf 18109 | The group addition operati... |
| grpplusfo 18110 | The group addition operati... |
| resgrpplusfrn 18111 | The underlying set of a gr... |
| grppropd 18112 | If two structures have the... |
| grpprop 18113 | If two structures have the... |
| grppropstr 18114 | Generalize a specific 2-el... |
| grpss 18115 | Show that a structure exte... |
| isgrpd2e 18116 | Deduce a group from its pr... |
| isgrpd2 18117 | Deduce a group from its pr... |
| isgrpde 18118 | Deduce a group from its pr... |
| isgrpd 18119 | Deduce a group from its pr... |
| isgrpi 18120 | Properties that determine ... |
| grpsgrp 18121 | A group is a semigroup. (... |
| dfgrp2 18122 | Alternate definition of a ... |
| dfgrp2e 18123 | Alternate definition of a ... |
| isgrpix 18124 | Properties that determine ... |
| grpidcl 18125 | The identity element of a ... |
| grpbn0 18126 | The base set of a group is... |
| grplid 18127 | The identity element of a ... |
| grprid 18128 | The identity element of a ... |
| grpn0 18129 | A group is not empty. (Co... |
| hashfingrpnn 18130 | A finite group has positiv... |
| grprcan 18131 | Right cancellation law for... |
| grpinveu 18132 | The left inverse element o... |
| grpid 18133 | Two ways of saying that an... |
| isgrpid2 18134 | Properties showing that an... |
| grpidd2 18135 | Deduce the identity elemen... |
| grpinvfval 18136 | The inverse function of a ... |
| grpinvfvalALT 18137 | Shorter proof of ~ grpinvf... |
| grpinvval 18138 | The inverse of a group ele... |
| grpinvfn 18139 | Functionality of the group... |
| grpinvfvi 18140 | The group inverse function... |
| grpsubfval 18141 | Group subtraction (divisio... |
| grpsubfvalALT 18142 | Shorter proof of ~ grpsubf... |
| grpsubval 18143 | Group subtraction (divisio... |
| grpinvf 18144 | The group inversion operat... |
| grpinvcl 18145 | A group element's inverse ... |
| grplinv 18146 | The left inverse of a grou... |
| grprinv 18147 | The right inverse of a gro... |
| grpinvid1 18148 | The inverse of a group ele... |
| grpinvid2 18149 | The inverse of a group ele... |
| isgrpinv 18150 | Properties showing that a ... |
| grplrinv 18151 | In a group, every member h... |
| grpidinv2 18152 | A group's properties using... |
| grpidinv 18153 | A group has a left and rig... |
| grpinvid 18154 | The inverse of the identit... |
| grplcan 18155 | Left cancellation law for ... |
| grpasscan1 18156 | An associative cancellatio... |
| grpasscan2 18157 | An associative cancellatio... |
| grpidrcan 18158 | If right adding an element... |
| grpidlcan 18159 | If left adding an element ... |
| grpinvinv 18160 | Double inverse law for gro... |
| grpinvcnv 18161 | The group inverse is its o... |
| grpinv11 18162 | The group inverse is one-t... |
| grpinvf1o 18163 | The group inverse is a one... |
| grpinvnz 18164 | The inverse of a nonzero g... |
| grpinvnzcl 18165 | The inverse of a nonzero g... |
| grpsubinv 18166 | Subtraction of an inverse.... |
| grplmulf1o 18167 | Left multiplication by a g... |
| grpinvpropd 18168 | If two structures have the... |
| grpidssd 18169 | If the base set of a group... |
| grpinvssd 18170 | If the base set of a group... |
| grpinvadd 18171 | The inverse of the group o... |
| grpsubf 18172 | Functionality of group sub... |
| grpsubcl 18173 | Closure of group subtracti... |
| grpsubrcan 18174 | Right cancellation law for... |
| grpinvsub 18175 | Inverse of a group subtrac... |
| grpinvval2 18176 | A ~ df-neg -like equation ... |
| grpsubid 18177 | Subtraction of a group ele... |
| grpsubid1 18178 | Subtraction of the identit... |
| grpsubeq0 18179 | If the difference between ... |
| grpsubadd0sub 18180 | Subtraction expressed as a... |
| grpsubadd 18181 | Relationship between group... |
| grpsubsub 18182 | Double group subtraction. ... |
| grpaddsubass 18183 | Associative-type law for g... |
| grppncan 18184 | Cancellation law for subtr... |
| grpnpcan 18185 | Cancellation law for subtr... |
| grpsubsub4 18186 | Double group subtraction (... |
| grppnpcan2 18187 | Cancellation law for mixed... |
| grpnpncan 18188 | Cancellation law for group... |
| grpnpncan0 18189 | Cancellation law for group... |
| grpnnncan2 18190 | Cancellation law for group... |
| dfgrp3lem 18191 | Lemma for ~ dfgrp3 . (Con... |
| dfgrp3 18192 | Alternate definition of a ... |
| dfgrp3e 18193 | Alternate definition of a ... |
| grplactfval 18194 | The left group action of e... |
| grplactval 18195 | The value of the left grou... |
| grplactcnv 18196 | The left group action of e... |
| grplactf1o 18197 | The left group action of e... |
| grpsubpropd 18198 | Weak property deduction fo... |
| grpsubpropd2 18199 | Strong property deduction ... |
| grp1 18200 | The (smallest) structure r... |
| grp1inv 18201 | The inverse function of th... |
| prdsinvlem 18202 | Characterization of invers... |
| prdsgrpd 18203 | The product of a family of... |
| prdsinvgd 18204 | Negation in a product of g... |
| pwsgrp 18205 | A structure power of a gro... |
| pwsinvg 18206 | Negation in a group power.... |
| pwssub 18207 | Subtraction in a group pow... |
| imasgrp2 18208 | The image structure of a g... |
| imasgrp 18209 | The image structure of a g... |
| imasgrpf1 18210 | The image of a group under... |
| qusgrp2 18211 | Prove that a quotient stru... |
| xpsgrp 18212 | The binary product of grou... |
| mhmlem 18213 | Lemma for ~ mhmmnd and ~ g... |
| mhmid 18214 | A surjective monoid morphi... |
| mhmmnd 18215 | The image of a monoid ` G ... |
| mhmfmhm 18216 | The function fulfilling th... |
| ghmgrp 18217 | The image of a group ` G `... |
| mulgfval 18220 | Group multiple (exponentia... |
| mulgfvalALT 18221 | Shorter proof of ~ mulgfva... |
| mulgval 18222 | Value of the group multipl... |
| mulgfn 18223 | Functionality of the group... |
| mulgfvi 18224 | The group multiple operati... |
| mulg0 18225 | Group multiple (exponentia... |
| mulgnn 18226 | Group multiple (exponentia... |
| mulgnngsum 18227 | Group multiple (exponentia... |
| mulgnn0gsum 18228 | Group multiple (exponentia... |
| mulg1 18229 | Group multiple (exponentia... |
| mulgnnp1 18230 | Group multiple (exponentia... |
| mulg2 18231 | Group multiple (exponentia... |
| mulgnegnn 18232 | Group multiple (exponentia... |
| mulgnn0p1 18233 | Group multiple (exponentia... |
| mulgnnsubcl 18234 | Closure of the group multi... |
| mulgnn0subcl 18235 | Closure of the group multi... |
| mulgsubcl 18236 | Closure of the group multi... |
| mulgnncl 18237 | Closure of the group multi... |
| mulgnn0cl 18238 | Closure of the group multi... |
| mulgcl 18239 | Closure of the group multi... |
| mulgneg 18240 | Group multiple (exponentia... |
| mulgnegneg 18241 | The inverse of a negative ... |
| mulgm1 18242 | Group multiple (exponentia... |
| mulgcld 18243 | Deduction associated with ... |
| mulgaddcomlem 18244 | Lemma for ~ mulgaddcom . ... |
| mulgaddcom 18245 | The group multiple operato... |
| mulginvcom 18246 | The group multiple operato... |
| mulginvinv 18247 | The group multiple operato... |
| mulgnn0z 18248 | A group multiple of the id... |
| mulgz 18249 | A group multiple of the id... |
| mulgnndir 18250 | Sum of group multiples, fo... |
| mulgnn0dir 18251 | Sum of group multiples, ge... |
| mulgdirlem 18252 | Lemma for ~ mulgdir . (Co... |
| mulgdir 18253 | Sum of group multiples, ge... |
| mulgp1 18254 | Group multiple (exponentia... |
| mulgneg2 18255 | Group multiple (exponentia... |
| mulgnnass 18256 | Product of group multiples... |
| mulgnn0ass 18257 | Product of group multiples... |
| mulgass 18258 | Product of group multiples... |
| mulgassr 18259 | Reversed product of group ... |
| mulgmodid 18260 | Casting out multiples of t... |
| mulgsubdir 18261 | Subtraction of a group ele... |
| mhmmulg 18262 | A homomorphism of monoids ... |
| mulgpropd 18263 | Two structures with the sa... |
| submmulgcl 18264 | Closure of the group multi... |
| submmulg 18265 | A group multiple is the sa... |
| pwsmulg 18266 | Value of a group multiple ... |
| issubg 18273 | The subgroup predicate. (... |
| subgss 18274 | A subgroup is a subset. (... |
| subgid 18275 | A group is a subgroup of i... |
| subggrp 18276 | A subgroup is a group. (C... |
| subgbas 18277 | The base of the restricted... |
| subgrcl 18278 | Reverse closure for the su... |
| subg0 18279 | A subgroup of a group must... |
| subginv 18280 | The inverse of an element ... |
| subg0cl 18281 | The group identity is an e... |
| subginvcl 18282 | The inverse of an element ... |
| subgcl 18283 | A subgroup is closed under... |
| subgsubcl 18284 | A subgroup is closed under... |
| subgsub 18285 | The subtraction of element... |
| subgmulgcl 18286 | Closure of the group multi... |
| subgmulg 18287 | A group multiple is the sa... |
| issubg2 18288 | Characterize the subgroups... |
| issubgrpd2 18289 | Prove a subgroup by closur... |
| issubgrpd 18290 | Prove a subgroup by closur... |
| issubg3 18291 | A subgroup is a symmetric ... |
| issubg4 18292 | A subgroup is a nonempty s... |
| grpissubg 18293 | If the base set of a group... |
| resgrpisgrp 18294 | If the base set of a group... |
| subgsubm 18295 | A subgroup is a submonoid.... |
| subsubg 18296 | A subgroup of a subgroup i... |
| subgint 18297 | The intersection of a none... |
| 0subg 18298 | The zero subgroup of an ar... |
| trivsubgd 18299 | The only subgroup of a tri... |
| trivsubgsnd 18300 | The only subgroup of a tri... |
| isnsg 18301 | Property of being a normal... |
| isnsg2 18302 | Weaken the condition of ~ ... |
| nsgbi 18303 | Defining property of a nor... |
| nsgsubg 18304 | A normal subgroup is a sub... |
| nsgconj 18305 | The conjugation of an elem... |
| isnsg3 18306 | A subgroup is normal iff t... |
| subgacs 18307 | Subgroups are an algebraic... |
| nsgacs 18308 | Normal subgroups form an a... |
| elnmz 18309 | Elementhood in the normali... |
| nmzbi 18310 | Defining property of the n... |
| nmzsubg 18311 | The normalizer N_G(S) of a... |
| ssnmz 18312 | A subgroup is a subset of ... |
| isnsg4 18313 | A subgroup is normal iff i... |
| nmznsg 18314 | Any subgroup is a normal s... |
| 0nsg 18315 | The zero subgroup is norma... |
| nsgid 18316 | The whole group is a norma... |
| 0idnsgd 18317 | The whole group and the ze... |
| trivnsgd 18318 | The only normal subgroup o... |
| triv1nsgd 18319 | A trivial group has exactl... |
| 1nsgtrivd 18320 | A group with exactly one n... |
| releqg 18321 | The left coset equivalence... |
| eqgfval 18322 | Value of the subgroup left... |
| eqgval 18323 | Value of the subgroup left... |
| eqger 18324 | The subgroup coset equival... |
| eqglact 18325 | A left coset can be expres... |
| eqgid 18326 | The left coset containing ... |
| eqgen 18327 | Each coset is equipotent t... |
| eqgcpbl 18328 | The subgroup coset equival... |
| qusgrp 18329 | If ` Y ` is a normal subgr... |
| quseccl 18330 | Closure of the quotient ma... |
| qusadd 18331 | Value of the group operati... |
| qus0 18332 | Value of the group identit... |
| qusinv 18333 | Value of the group inverse... |
| qussub 18334 | Value of the group subtrac... |
| lagsubg2 18335 | Lagrange's theorem for fin... |
| lagsubg 18336 | Lagrange's theorem for Gro... |
| cycsubmel 18337 | Characterization of an ele... |
| cycsubmcl 18338 | The set of nonnegative int... |
| cycsubm 18339 | The set of nonnegative int... |
| cyccom 18340 | Condition for an operation... |
| cycsubmcom 18341 | The operation of a monoid ... |
| cycsubggend 18342 | The cyclic subgroup genera... |
| cycsubgcl 18343 | The set of integer powers ... |
| cycsubgss 18344 | The cyclic subgroup genera... |
| cycsubg 18345 | The cyclic group generated... |
| cycsubgcld 18346 | The cyclic subgroup genera... |
| cycsubg2 18347 | The subgroup generated by ... |
| cycsubg2cl 18348 | Any multiple of an element... |
| reldmghm 18351 | Lemma for group homomorphi... |
| isghm 18352 | Property of being a homomo... |
| isghm3 18353 | Property of a group homomo... |
| ghmgrp1 18354 | A group homomorphism is on... |
| ghmgrp2 18355 | A group homomorphism is on... |
| ghmf 18356 | A group homomorphism is a ... |
| ghmlin 18357 | A homomorphism of groups i... |
| ghmid 18358 | A homomorphism of groups p... |
| ghminv 18359 | A homomorphism of groups p... |
| ghmsub 18360 | Linearity of subtraction t... |
| isghmd 18361 | Deduction for a group homo... |
| ghmmhm 18362 | A group homomorphism is a ... |
| ghmmhmb 18363 | Group homomorphisms and mo... |
| ghmmulg 18364 | A homomorphism of monoids ... |
| ghmrn 18365 | The range of a homomorphis... |
| 0ghm 18366 | The constant zero linear f... |
| idghm 18367 | The identity homomorphism ... |
| resghm 18368 | Restriction of a homomorph... |
| resghm2 18369 | One direction of ~ resghm2... |
| resghm2b 18370 | Restriction of the codomai... |
| ghmghmrn 18371 | A group homomorphism from ... |
| ghmco 18372 | The composition of group h... |
| ghmima 18373 | The image of a subgroup un... |
| ghmpreima 18374 | The inverse image of a sub... |
| ghmeql 18375 | The equalizer of two group... |
| ghmnsgima 18376 | The image of a normal subg... |
| ghmnsgpreima 18377 | The inverse image of a nor... |
| ghmker 18378 | The kernel of a homomorphi... |
| ghmeqker 18379 | Two source points map to t... |
| pwsdiagghm 18380 | Diagonal homomorphism into... |
| ghmf1 18381 | Two ways of saying a group... |
| ghmf1o 18382 | A bijective group homomorp... |
| conjghm 18383 | Conjugation is an automorp... |
| conjsubg 18384 | A conjugated subgroup is a... |
| conjsubgen 18385 | A conjugated subgroup is e... |
| conjnmz 18386 | A subgroup is unchanged un... |
| conjnmzb 18387 | Alternative condition for ... |
| conjnsg 18388 | A normal subgroup is uncha... |
| qusghm 18389 | If ` Y ` is a normal subgr... |
| ghmpropd 18390 | Group homomorphism depends... |
| gimfn 18395 | The group isomorphism func... |
| isgim 18396 | An isomorphism of groups i... |
| gimf1o 18397 | An isomorphism of groups i... |
| gimghm 18398 | An isomorphism of groups i... |
| isgim2 18399 | A group isomorphism is a h... |
| subggim 18400 | Behavior of subgroups unde... |
| gimcnv 18401 | The converse of a bijectiv... |
| gimco 18402 | The composition of group i... |
| brgic 18403 | The relation "is isomorphi... |
| brgici 18404 | Prove isomorphic by an exp... |
| gicref 18405 | Isomorphism is reflexive. ... |
| giclcl 18406 | Isomorphism implies the le... |
| gicrcl 18407 | Isomorphism implies the ri... |
| gicsym 18408 | Isomorphism is symmetric. ... |
| gictr 18409 | Isomorphism is transitive.... |
| gicer 18410 | Isomorphism is an equivale... |
| gicen 18411 | Isomorphic groups have equ... |
| gicsubgen 18412 | A less trivial example of ... |
| isga 18415 | The predicate "is a (left)... |
| gagrp 18416 | The left argument of a gro... |
| gaset 18417 | The right argument of a gr... |
| gagrpid 18418 | The identity of the group ... |
| gaf 18419 | The mapping of the group a... |
| gafo 18420 | A group action is onto its... |
| gaass 18421 | An "associative" property ... |
| ga0 18422 | The action of a group on t... |
| gaid 18423 | The trivial action of a gr... |
| subgga 18424 | A subgroup acts on its par... |
| gass 18425 | A subset of a group action... |
| gasubg 18426 | The restriction of a group... |
| gaid2 18427 | A group operation is a lef... |
| galcan 18428 | The action of a particular... |
| gacan 18429 | Group inverses cancel in a... |
| gapm 18430 | The action of a particular... |
| gaorb 18431 | The orbit equivalence rela... |
| gaorber 18432 | The orbit equivalence rela... |
| gastacl 18433 | The stabilizer subgroup in... |
| gastacos 18434 | Write the coset relation f... |
| orbstafun 18435 | Existence and uniqueness f... |
| orbstaval 18436 | Value of the function at a... |
| orbsta 18437 | The Orbit-Stabilizer theor... |
| orbsta2 18438 | Relation between the size ... |
| cntrval 18443 | Substitute definition of t... |
| cntzfval 18444 | First level substitution f... |
| cntzval 18445 | Definition substitution fo... |
| elcntz 18446 | Elementhood in the central... |
| cntzel 18447 | Membership in a centralize... |
| cntzsnval 18448 | Special substitution for t... |
| elcntzsn 18449 | Value of the centralizer o... |
| sscntz 18450 | A centralizer expression f... |
| cntzrcl 18451 | Reverse closure for elemen... |
| cntzssv 18452 | The centralizer is uncondi... |
| cntzi 18453 | Membership in a centralize... |
| cntrss 18454 | The center is a subset of ... |
| cntri 18455 | Defining property of the c... |
| resscntz 18456 | Centralizer in a substruct... |
| cntz2ss 18457 | Centralizers reverse the s... |
| cntzrec 18458 | Reciprocity relationship f... |
| cntziinsn 18459 | Express any centralizer as... |
| cntzsubm 18460 | Centralizers in a monoid a... |
| cntzsubg 18461 | Centralizers in a group ar... |
| cntzidss 18462 | If the elements of ` S ` c... |
| cntzmhm 18463 | Centralizers in a monoid a... |
| cntzmhm2 18464 | Centralizers in a monoid a... |
| cntrsubgnsg 18465 | A central subgroup is norm... |
| cntrnsg 18466 | The center of a group is a... |
| oppgval 18469 | Value of the opposite grou... |
| oppgplusfval 18470 | Value of the addition oper... |
| oppgplus 18471 | Value of the addition oper... |
| oppglem 18472 | Lemma for ~ oppgbas . (Co... |
| oppgbas 18473 | Base set of an opposite gr... |
| oppgtset 18474 | Topology of an opposite gr... |
| oppgtopn 18475 | Topology of an opposite gr... |
| oppgmnd 18476 | The opposite of a monoid i... |
| oppgmndb 18477 | Bidirectional form of ~ op... |
| oppgid 18478 | Zero in a monoid is a symm... |
| oppggrp 18479 | The opposite of a group is... |
| oppggrpb 18480 | Bidirectional form of ~ op... |
| oppginv 18481 | Inverses in a group are a ... |
| invoppggim 18482 | The inverse is an antiauto... |
| oppggic 18483 | Every group is (naturally)... |
| oppgsubm 18484 | Being a submonoid is a sym... |
| oppgsubg 18485 | Being a subgroup is a symm... |
| oppgcntz 18486 | A centralizer in a group i... |
| oppgcntr 18487 | The center of a group is t... |
| gsumwrev 18488 | A sum in an opposite monoi... |
| symgval 18491 | The value of the symmetric... |
| permsetex 18492 | The set of permutations of... |
| symgbas 18493 | The base set of the symmet... |
| symgbasex 18494 | The base set of the symmet... |
| elsymgbas2 18495 | Two ways of saying a funct... |
| elsymgbas 18496 | Two ways of saying a funct... |
| symgbasf1o 18497 | Elements in the symmetric ... |
| symgbasf 18498 | A permutation (element of ... |
| symgbasmap 18499 | A permutation (element of ... |
| symghash 18500 | The symmetric group on ` n... |
| symgbasfi 18501 | The symmetric group on a f... |
| symgfv 18502 | The function value of a pe... |
| symgfvne 18503 | The function values of a p... |
| symgressbas 18504 | The symmetric group on ` A... |
| symgplusg 18505 | The group operation of a s... |
| symgov 18506 | The value of the group ope... |
| symgcl 18507 | The group operation of the... |
| idresperm 18508 | The identity function rest... |
| symgmov1 18509 | For a permutation of a set... |
| symgmov2 18510 | For a permutation of a set... |
| symgbas0 18511 | The base set of the symmet... |
| symg1hash 18512 | The symmetric group on a s... |
| symg1bas 18513 | The symmetric group on a s... |
| symg2hash 18514 | The symmetric group on a (... |
| symg2bas 18515 | The symmetric group on a p... |
| 0symgefmndeq 18516 | The symmetric group on the... |
| snsymgefmndeq 18517 | The symmetric group on a s... |
| symgpssefmnd 18518 | For a set ` A ` with more ... |
| symgvalstruct 18519 | The value of the symmetric... |
| symgsubmefmnd 18520 | The symmetric group on a s... |
| symgtset 18521 | The topology of the symmet... |
| symggrp 18522 | The symmetric group on a s... |
| symgid 18523 | The group identity element... |
| symginv 18524 | The group inverse in the s... |
| symgsubmefmndALT 18525 | The symmetric group on a s... |
| galactghm 18526 | The currying of a group ac... |
| lactghmga 18527 | The converse of ~ galactgh... |
| symgtopn 18528 | The topology of the symmet... |
| symgga 18529 | The symmetric group induce... |
| pgrpsubgsymgbi 18530 | Every permutation group is... |
| pgrpsubgsymg 18531 | Every permutation group is... |
| idressubgsymg 18532 | The singleton containing o... |
| idrespermg 18533 | The structure with the sin... |
| cayleylem1 18534 | Lemma for ~ cayley . (Con... |
| cayleylem2 18535 | Lemma for ~ cayley . (Con... |
| cayley 18536 | Cayley's Theorem (construc... |
| cayleyth 18537 | Cayley's Theorem (existenc... |
| symgfix2 18538 | If a permutation does not ... |
| symgextf 18539 | The extension of a permuta... |
| symgextfv 18540 | The function value of the ... |
| symgextfve 18541 | The function value of the ... |
| symgextf1lem 18542 | Lemma for ~ symgextf1 . (... |
| symgextf1 18543 | The extension of a permuta... |
| symgextfo 18544 | The extension of a permuta... |
| symgextf1o 18545 | The extension of a permuta... |
| symgextsymg 18546 | The extension of a permuta... |
| symgextres 18547 | The restriction of the ext... |
| gsumccatsymgsn 18548 | Homomorphic property of co... |
| gsmsymgrfixlem1 18549 | Lemma 1 for ~ gsmsymgrfix ... |
| gsmsymgrfix 18550 | The composition of permuta... |
| fvcosymgeq 18551 | The values of two composit... |
| gsmsymgreqlem1 18552 | Lemma 1 for ~ gsmsymgreq .... |
| gsmsymgreqlem2 18553 | Lemma 2 for ~ gsmsymgreq .... |
| gsmsymgreq 18554 | Two combination of permuta... |
| symgfixelq 18555 | A permutation of a set fix... |
| symgfixels 18556 | The restriction of a permu... |
| symgfixelsi 18557 | The restriction of a permu... |
| symgfixf 18558 | The mapping of a permutati... |
| symgfixf1 18559 | The mapping of a permutati... |
| symgfixfolem1 18560 | Lemma 1 for ~ symgfixfo . ... |
| symgfixfo 18561 | The mapping of a permutati... |
| symgfixf1o 18562 | The mapping of a permutati... |
| f1omvdmvd 18565 | A permutation of any class... |
| f1omvdcnv 18566 | A permutation and its inve... |
| mvdco 18567 | Composing two permutations... |
| f1omvdconj 18568 | Conjugation of a permutati... |
| f1otrspeq 18569 | A transposition is charact... |
| f1omvdco2 18570 | If exactly one of two perm... |
| f1omvdco3 18571 | If a point is moved by exa... |
| pmtrfval 18572 | The function generating tr... |
| pmtrval 18573 | A generated transposition,... |
| pmtrfv 18574 | General value of mapping a... |
| pmtrprfv 18575 | In a transposition of two ... |
| pmtrprfv3 18576 | In a transposition of two ... |
| pmtrf 18577 | Functionality of a transpo... |
| pmtrmvd 18578 | A transposition moves prec... |
| pmtrrn 18579 | Transposing two points giv... |
| pmtrfrn 18580 | A transposition (as a kind... |
| pmtrffv 18581 | Mapping of a point under a... |
| pmtrrn2 18582 | For any transposition ther... |
| pmtrfinv 18583 | A transposition function i... |
| pmtrfmvdn0 18584 | A transposition moves at l... |
| pmtrff1o 18585 | A transposition function i... |
| pmtrfcnv 18586 | A transposition function i... |
| pmtrfb 18587 | An intrinsic characterizat... |
| pmtrfconj 18588 | Any conjugate of a transpo... |
| symgsssg 18589 | The symmetric group has su... |
| symgfisg 18590 | The symmetric group has a ... |
| symgtrf 18591 | Transpositions are element... |
| symggen 18592 | The span of the transposit... |
| symggen2 18593 | A finite permutation group... |
| symgtrinv 18594 | To invert a permutation re... |
| pmtr3ncomlem1 18595 | Lemma 1 for ~ pmtr3ncom . ... |
| pmtr3ncomlem2 18596 | Lemma 2 for ~ pmtr3ncom . ... |
| pmtr3ncom 18597 | Transpositions over sets w... |
| pmtrdifellem1 18598 | Lemma 1 for ~ pmtrdifel . ... |
| pmtrdifellem2 18599 | Lemma 2 for ~ pmtrdifel . ... |
| pmtrdifellem3 18600 | Lemma 3 for ~ pmtrdifel . ... |
| pmtrdifellem4 18601 | Lemma 4 for ~ pmtrdifel . ... |
| pmtrdifel 18602 | A transposition of element... |
| pmtrdifwrdellem1 18603 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem2 18604 | Lemma 2 for ~ pmtrdifwrdel... |
| pmtrdifwrdellem3 18605 | Lemma 3 for ~ pmtrdifwrdel... |
| pmtrdifwrdel2lem1 18606 | Lemma 1 for ~ pmtrdifwrdel... |
| pmtrdifwrdel 18607 | A sequence of transpositio... |
| pmtrdifwrdel2 18608 | A sequence of transpositio... |
| pmtrprfval 18609 | The transpositions on a pa... |
| pmtrprfvalrn 18610 | The range of the transposi... |
| psgnunilem1 18615 | Lemma for ~ psgnuni . Giv... |
| psgnunilem5 18616 | Lemma for ~ psgnuni . It ... |
| psgnunilem2 18617 | Lemma for ~ psgnuni . Ind... |
| psgnunilem3 18618 | Lemma for ~ psgnuni . Any... |
| psgnunilem4 18619 | Lemma for ~ psgnuni . An ... |
| m1expaddsub 18620 | Addition and subtraction o... |
| psgnuni 18621 | If the same permutation ca... |
| psgnfval 18622 | Function definition of the... |
| psgnfn 18623 | Functionality and domain o... |
| psgndmsubg 18624 | The finitary permutations ... |
| psgneldm 18625 | Property of being a finita... |
| psgneldm2 18626 | The finitary permutations ... |
| psgneldm2i 18627 | A sequence of transpositio... |
| psgneu 18628 | A finitary permutation has... |
| psgnval 18629 | Value of the permutation s... |
| psgnvali 18630 | A finitary permutation has... |
| psgnvalii 18631 | Any representation of a pe... |
| psgnpmtr 18632 | All transpositions are odd... |
| psgn0fv0 18633 | The permutation sign funct... |
| sygbasnfpfi 18634 | The class of non-fixed poi... |
| psgnfvalfi 18635 | Function definition of the... |
| psgnvalfi 18636 | Value of the permutation s... |
| psgnran 18637 | The range of the permutati... |
| gsmtrcl 18638 | The group sum of transposi... |
| psgnfitr 18639 | A permutation of a finite ... |
| psgnfieu 18640 | A permutation of a finite ... |
| pmtrsn 18641 | The value of the transposi... |
| psgnsn 18642 | The permutation sign funct... |
| psgnprfval 18643 | The permutation sign funct... |
| psgnprfval1 18644 | The permutation sign of th... |
| psgnprfval2 18645 | The permutation sign of th... |
| odfval 18654 | Value of the order functio... |
| odfvalALT 18655 | Shorter proof of ~ odfval ... |
| odval 18656 | Second substitution for th... |
| odlem1 18657 | The group element order is... |
| odcl 18658 | The order of a group eleme... |
| odf 18659 | Functionality of the group... |
| odid 18660 | Any element to the power o... |
| odlem2 18661 | Any positive annihilator o... |
| odmodnn0 18662 | Reduce the argument of a g... |
| mndodconglem 18663 | Lemma for ~ mndodcong . (... |
| mndodcong 18664 | If two multipliers are con... |
| mndodcongi 18665 | If two multipliers are con... |
| oddvdsnn0 18666 | The only multiples of ` A ... |
| odnncl 18667 | If a nonzero multiple of a... |
| odmod 18668 | Reduce the argument of a g... |
| oddvds 18669 | The only multiples of ` A ... |
| oddvdsi 18670 | Any group element is annih... |
| odcong 18671 | If two multipliers are con... |
| odeq 18672 | The ~ oddvds property uniq... |
| odval2 18673 | A non-conditional definiti... |
| odcld 18674 | The order of a group eleme... |
| odmulgid 18675 | A relationship between the... |
| odmulg2 18676 | The order of a multiple di... |
| odmulg 18677 | Relationship between the o... |
| odmulgeq 18678 | A multiple of a point of f... |
| odbezout 18679 | If ` N ` is coprime to the... |
| od1 18680 | The order of the group ide... |
| odeq1 18681 | The group identity is the ... |
| odinv 18682 | The order of the inverse o... |
| odf1 18683 | The multiples of an elemen... |
| odinf 18684 | The multiples of an elemen... |
| dfod2 18685 | An alternative definition ... |
| odcl2 18686 | The order of an element of... |
| oddvds2 18687 | The order of an element of... |
| submod 18688 | The order of an element is... |
| subgod 18689 | The order of an element is... |
| odsubdvds 18690 | The order of an element of... |
| odf1o1 18691 | An element with zero order... |
| odf1o2 18692 | An element with nonzero or... |
| odhash 18693 | An element of zero order g... |
| odhash2 18694 | If an element has nonzero ... |
| odhash3 18695 | An element which generates... |
| odngen 18696 | A cyclic subgroup of size ... |
| gexval 18697 | Value of the exponent of a... |
| gexlem1 18698 | The group element order is... |
| gexcl 18699 | The exponent of a group is... |
| gexid 18700 | Any element to the power o... |
| gexlem2 18701 | Any positive annihilator o... |
| gexdvdsi 18702 | Any group element is annih... |
| gexdvds 18703 | The only ` N ` that annihi... |
| gexdvds2 18704 | An integer divides the gro... |
| gexod 18705 | Any group element is annih... |
| gexcl3 18706 | If the order of every grou... |
| gexnnod 18707 | Every group element has fi... |
| gexcl2 18708 | The exponent of a finite g... |
| gexdvds3 18709 | The exponent of a finite g... |
| gex1 18710 | A group or monoid has expo... |
| ispgp 18711 | A group is a ` P ` -group ... |
| pgpprm 18712 | Reverse closure for the fi... |
| pgpgrp 18713 | Reverse closure for the se... |
| pgpfi1 18714 | A finite group with order ... |
| pgp0 18715 | The identity subgroup is a... |
| subgpgp 18716 | A subgroup of a p-group is... |
| sylow1lem1 18717 | Lemma for ~ sylow1 . The ... |
| sylow1lem2 18718 | Lemma for ~ sylow1 . The ... |
| sylow1lem3 18719 | Lemma for ~ sylow1 . One ... |
| sylow1lem4 18720 | Lemma for ~ sylow1 . The ... |
| sylow1lem5 18721 | Lemma for ~ sylow1 . Usin... |
| sylow1 18722 | Sylow's first theorem. If... |
| odcau 18723 | Cauchy's theorem for the o... |
| pgpfi 18724 | The converse to ~ pgpfi1 .... |
| pgpfi2 18725 | Alternate version of ~ pgp... |
| pgphash 18726 | The order of a p-group. (... |
| isslw 18727 | The property of being a Sy... |
| slwprm 18728 | Reverse closure for the fi... |
| slwsubg 18729 | A Sylow ` P ` -subgroup is... |
| slwispgp 18730 | Defining property of a Syl... |
| slwpss 18731 | A proper superset of a Syl... |
| slwpgp 18732 | A Sylow ` P ` -subgroup is... |
| pgpssslw 18733 | Every ` P ` -subgroup is c... |
| slwn0 18734 | Every finite group contain... |
| subgslw 18735 | A Sylow subgroup that is c... |
| sylow2alem1 18736 | Lemma for ~ sylow2a . An ... |
| sylow2alem2 18737 | Lemma for ~ sylow2a . All... |
| sylow2a 18738 | A named lemma of Sylow's s... |
| sylow2blem1 18739 | Lemma for ~ sylow2b . Eva... |
| sylow2blem2 18740 | Lemma for ~ sylow2b . Lef... |
| sylow2blem3 18741 | Sylow's second theorem. P... |
| sylow2b 18742 | Sylow's second theorem. A... |
| slwhash 18743 | A sylow subgroup has cardi... |
| fislw 18744 | The sylow subgroups of a f... |
| sylow2 18745 | Sylow's second theorem. S... |
| sylow3lem1 18746 | Lemma for ~ sylow3 , first... |
| sylow3lem2 18747 | Lemma for ~ sylow3 , first... |
| sylow3lem3 18748 | Lemma for ~ sylow3 , first... |
| sylow3lem4 18749 | Lemma for ~ sylow3 , first... |
| sylow3lem5 18750 | Lemma for ~ sylow3 , secon... |
| sylow3lem6 18751 | Lemma for ~ sylow3 , secon... |
| sylow3 18752 | Sylow's third theorem. Th... |
| lsmfval 18757 | The subgroup sum function ... |
| lsmvalx 18758 | Subspace sum value (for a ... |
| lsmelvalx 18759 | Subspace sum membership (f... |
| lsmelvalix 18760 | Subspace sum membership (f... |
| oppglsm 18761 | The subspace sum operation... |
| lsmssv 18762 | Subgroup sum is a subset o... |
| lsmless1x 18763 | Subset implies subgroup su... |
| lsmless2x 18764 | Subset implies subgroup su... |
| lsmub1x 18765 | Subgroup sum is an upper b... |
| lsmub2x 18766 | Subgroup sum is an upper b... |
| lsmval 18767 | Subgroup sum value (for a ... |
| lsmelval 18768 | Subgroup sum membership (f... |
| lsmelvali 18769 | Subgroup sum membership (f... |
| lsmelvalm 18770 | Subgroup sum membership an... |
| lsmelvalmi 18771 | Membership of vector subtr... |
| lsmsubm 18772 | The sum of two commuting s... |
| lsmsubg 18773 | The sum of two commuting s... |
| lsmcom2 18774 | Subgroup sum commutes. (C... |
| smndlsmidm 18775 | The direct product is idem... |
| lsmub1 18776 | Subgroup sum is an upper b... |
| lsmub2 18777 | Subgroup sum is an upper b... |
| lsmunss 18778 | Union of subgroups is a su... |
| lsmless1 18779 | Subset implies subgroup su... |
| lsmless2 18780 | Subset implies subgroup su... |
| lsmless12 18781 | Subset implies subgroup su... |
| lsmidm 18782 | Subgroup sum is idempotent... |
| lsmidmOLD 18783 | Obsolete proof of ~ lsmidm... |
| lsmlub 18784 | The least upper bound prop... |
| lsmss1 18785 | Subgroup sum with a subset... |
| lsmss1b 18786 | Subgroup sum with a subset... |
| lsmss2 18787 | Subgroup sum with a subset... |
| lsmss2b 18788 | Subgroup sum with a subset... |
| lsmass 18789 | Subgroup sum is associativ... |
| mndlsmidm 18790 | Subgroup sum is idempotent... |
| lsm01 18791 | Subgroup sum with the zero... |
| lsm02 18792 | Subgroup sum with the zero... |
| subglsm 18793 | The subgroup sum evaluated... |
| lssnle 18794 | Equivalent expressions for... |
| lsmmod 18795 | The modular law holds for ... |
| lsmmod2 18796 | Modular law dual for subgr... |
| lsmpropd 18797 | If two structures have the... |
| cntzrecd 18798 | Commute the "subgroups com... |
| lsmcntz 18799 | The "subgroups commute" pr... |
| lsmcntzr 18800 | The "subgroups commute" pr... |
| lsmdisj 18801 | Disjointness from a subgro... |
| lsmdisj2 18802 | Association of the disjoin... |
| lsmdisj3 18803 | Association of the disjoin... |
| lsmdisjr 18804 | Disjointness from a subgro... |
| lsmdisj2r 18805 | Association of the disjoin... |
| lsmdisj3r 18806 | Association of the disjoin... |
| lsmdisj2a 18807 | Association of the disjoin... |
| lsmdisj2b 18808 | Association of the disjoin... |
| lsmdisj3a 18809 | Association of the disjoin... |
| lsmdisj3b 18810 | Association of the disjoin... |
| subgdisj1 18811 | Vectors belonging to disjo... |
| subgdisj2 18812 | Vectors belonging to disjo... |
| subgdisjb 18813 | Vectors belonging to disjo... |
| pj1fval 18814 | The left projection functi... |
| pj1val 18815 | The left projection functi... |
| pj1eu 18816 | Uniqueness of a left proje... |
| pj1f 18817 | The left projection functi... |
| pj2f 18818 | The right projection funct... |
| pj1id 18819 | Any element of a direct su... |
| pj1eq 18820 | Any element of a direct su... |
| pj1lid 18821 | The left projection functi... |
| pj1rid 18822 | The left projection functi... |
| pj1ghm 18823 | The left projection functi... |
| pj1ghm2 18824 | The left projection functi... |
| lsmhash 18825 | The order of the direct pr... |
| efgmval 18832 | Value of the formal invers... |
| efgmf 18833 | The formal inverse operati... |
| efgmnvl 18834 | The inversion function on ... |
| efgrcl 18835 | Lemma for ~ efgval . (Con... |
| efglem 18836 | Lemma for ~ efgval . (Con... |
| efgval 18837 | Value of the free group co... |
| efger 18838 | Value of the free group co... |
| efgi 18839 | Value of the free group co... |
| efgi0 18840 | Value of the free group co... |
| efgi1 18841 | Value of the free group co... |
| efgtf 18842 | Value of the free group co... |
| efgtval 18843 | Value of the extension fun... |
| efgval2 18844 | Value of the free group co... |
| efgi2 18845 | Value of the free group co... |
| efgtlen 18846 | Value of the free group co... |
| efginvrel2 18847 | The inverse of the reverse... |
| efginvrel1 18848 | The inverse of the reverse... |
| efgsf 18849 | Value of the auxiliary fun... |
| efgsdm 18850 | Elementhood in the domain ... |
| efgsval 18851 | Value of the auxiliary fun... |
| efgsdmi 18852 | Property of the last link ... |
| efgsval2 18853 | Value of the auxiliary fun... |
| efgsrel 18854 | The start and end of any e... |
| efgs1 18855 | A singleton of an irreduci... |
| efgs1b 18856 | Every extension sequence e... |
| efgsp1 18857 | If ` F ` is an extension s... |
| efgsres 18858 | An initial segment of an e... |
| efgsfo 18859 | For any word, there is a s... |
| efgredlema 18860 | The reduced word that form... |
| efgredlemf 18861 | Lemma for ~ efgredleme . ... |
| efgredlemg 18862 | Lemma for ~ efgred . (Con... |
| efgredleme 18863 | Lemma for ~ efgred . (Con... |
| efgredlemd 18864 | The reduced word that form... |
| efgredlemc 18865 | The reduced word that form... |
| efgredlemb 18866 | The reduced word that form... |
| efgredlem 18867 | The reduced word that form... |
| efgred 18868 | The reduced word that form... |
| efgrelexlema 18869 | If two words ` A , B ` are... |
| efgrelexlemb 18870 | If two words ` A , B ` are... |
| efgrelex 18871 | If two words ` A , B ` are... |
| efgredeu 18872 | There is a unique reduced ... |
| efgred2 18873 | Two extension sequences ha... |
| efgcpbllema 18874 | Lemma for ~ efgrelex . De... |
| efgcpbllemb 18875 | Lemma for ~ efgrelex . Sh... |
| efgcpbl 18876 | Two extension sequences ha... |
| efgcpbl2 18877 | Two extension sequences ha... |
| frgpval 18878 | Value of the free group co... |
| frgpcpbl 18879 | Compatibility of the group... |
| frgp0 18880 | The free group is a group.... |
| frgpeccl 18881 | Closure of the quotient ma... |
| frgpgrp 18882 | The free group is a group.... |
| frgpadd 18883 | Addition in the free group... |
| frgpinv 18884 | The inverse of an element ... |
| frgpmhm 18885 | The "natural map" from wor... |
| vrgpfval 18886 | The canonical injection fr... |
| vrgpval 18887 | The value of the generatin... |
| vrgpf 18888 | The mapping from the index... |
| vrgpinv 18889 | The inverse of a generatin... |
| frgpuptf 18890 | Any assignment of the gene... |
| frgpuptinv 18891 | Any assignment of the gene... |
| frgpuplem 18892 | Any assignment of the gene... |
| frgpupf 18893 | Any assignment of the gene... |
| frgpupval 18894 | Any assignment of the gene... |
| frgpup1 18895 | Any assignment of the gene... |
| frgpup2 18896 | The evaluation map has the... |
| frgpup3lem 18897 | The evaluation map has the... |
| frgpup3 18898 | Universal property of the ... |
| 0frgp 18899 | The free group on zero gen... |
| isabl 18904 | The predicate "is an Abeli... |
| ablgrp 18905 | An Abelian group is a grou... |
| ablgrpd 18906 | An Abelian group is a grou... |
| ablcmn 18907 | An Abelian group is a comm... |
| iscmn 18908 | The predicate "is a commut... |
| isabl2 18909 | The predicate "is an Abeli... |
| cmnpropd 18910 | If two structures have the... |
| ablpropd 18911 | If two structures have the... |
| ablprop 18912 | If two structures have the... |
| iscmnd 18913 | Properties that determine ... |
| isabld 18914 | Properties that determine ... |
| isabli 18915 | Properties that determine ... |
| cmnmnd 18916 | A commutative monoid is a ... |
| cmncom 18917 | A commutative monoid is co... |
| ablcom 18918 | An Abelian group operation... |
| cmn32 18919 | Commutative/associative la... |
| cmn4 18920 | Commutative/associative la... |
| cmn12 18921 | Commutative/associative la... |
| abl32 18922 | Commutative/associative la... |
| rinvmod 18923 | Uniqueness of a right inve... |
| ablinvadd 18924 | The inverse of an Abelian ... |
| ablsub2inv 18925 | Abelian group subtraction ... |
| ablsubadd 18926 | Relationship between Abeli... |
| ablsub4 18927 | Commutative/associative su... |
| abladdsub4 18928 | Abelian group addition/sub... |
| abladdsub 18929 | Associative-type law for g... |
| ablpncan2 18930 | Cancellation law for subtr... |
| ablpncan3 18931 | A cancellation law for com... |
| ablsubsub 18932 | Law for double subtraction... |
| ablsubsub4 18933 | Law for double subtraction... |
| ablpnpcan 18934 | Cancellation law for mixed... |
| ablnncan 18935 | Cancellation law for group... |
| ablsub32 18936 | Swap the second and third ... |
| ablnnncan 18937 | Cancellation law for group... |
| ablnnncan1 18938 | Cancellation law for group... |
| ablsubsub23 18939 | Swap subtrahend and result... |
| mulgnn0di 18940 | Group multiple of a sum, f... |
| mulgdi 18941 | Group multiple of a sum. ... |
| mulgmhm 18942 | The map from ` x ` to ` n ... |
| mulgghm 18943 | The map from ` x ` to ` n ... |
| mulgsubdi 18944 | Group multiple of a differ... |
| ghmfghm 18945 | The function fulfilling th... |
| ghmcmn 18946 | The image of a commutative... |
| ghmabl 18947 | The image of an abelian gr... |
| invghm 18948 | The inversion map is a gro... |
| eqgabl 18949 | Value of the subgroup cose... |
| subgabl 18950 | A subgroup of an abelian g... |
| subcmn 18951 | A submonoid of a commutati... |
| submcmn 18952 | A submonoid of a commutati... |
| submcmn2 18953 | A submonoid is commutative... |
| cntzcmn 18954 | The centralizer of any sub... |
| cntzcmnss 18955 | Any subset in a commutativ... |
| cntrcmnd 18956 | The center of a monoid is ... |
| cntrabl 18957 | The center of a group is a... |
| cntzspan 18958 | If the generators commute,... |
| cntzcmnf 18959 | Discharge the centralizer ... |
| ghmplusg 18960 | The pointwise sum of two l... |
| ablnsg 18961 | Every subgroup of an abeli... |
| odadd1 18962 | The order of a product in ... |
| odadd2 18963 | The order of a product in ... |
| odadd 18964 | The order of a product is ... |
| gex2abl 18965 | A group with exponent 2 (o... |
| gexexlem 18966 | Lemma for ~ gexex . (Cont... |
| gexex 18967 | In an abelian group with f... |
| torsubg 18968 | The set of all elements of... |
| oddvdssubg 18969 | The set of all elements wh... |
| lsmcomx 18970 | Subgroup sum commutes (ext... |
| ablcntzd 18971 | All subgroups in an abelia... |
| lsmcom 18972 | Subgroup sum commutes. (C... |
| lsmsubg2 18973 | The sum of two subgroups i... |
| lsm4 18974 | Commutative/associative la... |
| prdscmnd 18975 | The product of a family of... |
| prdsabld 18976 | The product of a family of... |
| pwscmn 18977 | The structure power on a c... |
| pwsabl 18978 | The structure power on an ... |
| qusabl 18979 | If ` Y ` is a subgroup of ... |
| abl1 18980 | The (smallest) structure r... |
| abln0 18981 | Abelian groups (and theref... |
| cnaddablx 18982 | The complex numbers are an... |
| cnaddabl 18983 | The complex numbers are an... |
| cnaddid 18984 | The group identity element... |
| cnaddinv 18985 | Value of the group inverse... |
| zaddablx 18986 | The integers are an Abelia... |
| frgpnabllem1 18987 | Lemma for ~ frgpnabl . (C... |
| frgpnabllem2 18988 | Lemma for ~ frgpnabl . (C... |
| frgpnabl 18989 | The free group on two or m... |
| iscyg 18992 | Definition of a cyclic gro... |
| iscyggen 18993 | The property of being a cy... |
| iscyggen2 18994 | The property of being a cy... |
| iscyg2 18995 | A cyclic group is a group ... |
| cyggeninv 18996 | The inverse of a cyclic ge... |
| cyggenod 18997 | An element is the generato... |
| cyggenod2 18998 | In an infinite cyclic grou... |
| iscyg3 18999 | Definition of a cyclic gro... |
| iscygd 19000 | Definition of a cyclic gro... |
| iscygodd 19001 | Show that a group with an ... |
| cycsubmcmn 19002 | The set of nonnegative int... |
| cyggrp 19003 | A cyclic group is a group.... |
| cygabl 19004 | A cyclic group is abelian.... |
| cygablOLD 19005 | Obsolete proof of ~ cygabl... |
| cygctb 19006 | A cyclic group is countabl... |
| 0cyg 19007 | The trivial group is cycli... |
| prmcyg 19008 | A group with prime order i... |
| lt6abl 19009 | A group with fewer than ` ... |
| ghmcyg 19010 | The image of a cyclic grou... |
| cyggex2 19011 | The exponent of a cyclic g... |
| cyggex 19012 | The exponent of a finite c... |
| cyggexb 19013 | A finite abelian group is ... |
| giccyg 19014 | Cyclicity is a group prope... |
| cycsubgcyg 19015 | The cyclic subgroup genera... |
| cycsubgcyg2 19016 | The cyclic subgroup genera... |
| gsumval3a 19017 | Value of the group sum ope... |
| gsumval3eu 19018 | The group sum as defined i... |
| gsumval3lem1 19019 | Lemma 1 for ~ gsumval3 . ... |
| gsumval3lem2 19020 | Lemma 2 for ~ gsumval3 . ... |
| gsumval3 19021 | Value of the group sum ope... |
| gsumcllem 19022 | Lemma for ~ gsumcl and rel... |
| gsumzres 19023 | Extend a finite group sum ... |
| gsumzcl2 19024 | Closure of a finite group ... |
| gsumzcl 19025 | Closure of a finite group ... |
| gsumzf1o 19026 | Re-index a finite group su... |
| gsumres 19027 | Extend a finite group sum ... |
| gsumcl2 19028 | Closure of a finite group ... |
| gsumcl 19029 | Closure of a finite group ... |
| gsumf1o 19030 | Re-index a finite group su... |
| gsumreidx 19031 | Re-index a finite group su... |
| gsumzsubmcl 19032 | Closure of a group sum in ... |
| gsumsubmcl 19033 | Closure of a group sum in ... |
| gsumsubgcl 19034 | Closure of a group sum in ... |
| gsumzaddlem 19035 | The sum of two group sums.... |
| gsumzadd 19036 | The sum of two group sums.... |
| gsumadd 19037 | The sum of two group sums.... |
| gsummptfsadd 19038 | The sum of two group sums ... |
| gsummptfidmadd 19039 | The sum of two group sums ... |
| gsummptfidmadd2 19040 | The sum of two group sums ... |
| gsumzsplit 19041 | Split a group sum into two... |
| gsumsplit 19042 | Split a group sum into two... |
| gsumsplit2 19043 | Split a group sum into two... |
| gsummptfidmsplit 19044 | Split a group sum expresse... |
| gsummptfidmsplitres 19045 | Split a group sum expresse... |
| gsummptfzsplit 19046 | Split a group sum expresse... |
| gsummptfzsplitl 19047 | Split a group sum expresse... |
| gsumconst 19048 | Sum of a constant series. ... |
| gsumconstf 19049 | Sum of a constant series. ... |
| gsummptshft 19050 | Index shift of a finite gr... |
| gsumzmhm 19051 | Apply a group homomorphism... |
| gsummhm 19052 | Apply a group homomorphism... |
| gsummhm2 19053 | Apply a group homomorphism... |
| gsummptmhm 19054 | Apply a group homomorphism... |
| gsummulglem 19055 | Lemma for ~ gsummulg and ~... |
| gsummulg 19056 | Nonnegative multiple of a ... |
| gsummulgz 19057 | Integer multiple of a grou... |
| gsumzoppg 19058 | The opposite of a group su... |
| gsumzinv 19059 | Inverse of a group sum. (... |
| gsuminv 19060 | Inverse of a group sum. (... |
| gsummptfidminv 19061 | Inverse of a group sum exp... |
| gsumsub 19062 | The difference of two grou... |
| gsummptfssub 19063 | The difference of two grou... |
| gsummptfidmsub 19064 | The difference of two grou... |
| gsumsnfd 19065 | Group sum of a singleton, ... |
| gsumsnd 19066 | Group sum of a singleton, ... |
| gsumsnf 19067 | Group sum of a singleton, ... |
| gsumsn 19068 | Group sum of a singleton. ... |
| gsumpr 19069 | Group sum of a pair. (Con... |
| gsumzunsnd 19070 | Append an element to a fin... |
| gsumunsnfd 19071 | Append an element to a fin... |
| gsumunsnd 19072 | Append an element to a fin... |
| gsumunsnf 19073 | Append an element to a fin... |
| gsumunsn 19074 | Append an element to a fin... |
| gsumdifsnd 19075 | Extract a summand from a f... |
| gsumpt 19076 | Sum of a family that is no... |
| gsummptf1o 19077 | Re-index a finite group su... |
| gsummptun 19078 | Group sum of a disjoint un... |
| gsummpt1n0 19079 | If only one summand in a f... |
| gsummptif1n0 19080 | If only one summand in a f... |
| gsummptcl 19081 | Closure of a finite group ... |
| gsummptfif1o 19082 | Re-index a finite group su... |
| gsummptfzcl 19083 | Closure of a finite group ... |
| gsum2dlem1 19084 | Lemma 1 for ~ gsum2d . (C... |
| gsum2dlem2 19085 | Lemma for ~ gsum2d . (Con... |
| gsum2d 19086 | Write a sum over a two-dim... |
| gsum2d2lem 19087 | Lemma for ~ gsum2d2 : show... |
| gsum2d2 19088 | Write a group sum over a t... |
| gsumcom2 19089 | Two-dimensional commutatio... |
| gsumxp 19090 | Write a group sum over a c... |
| gsumcom 19091 | Commute the arguments of a... |
| gsumcom3 19092 | A commutative law for fini... |
| gsumcom3fi 19093 | A commutative law for fini... |
| gsumxp2 19094 | Write a group sum over a c... |
| prdsgsum 19095 | Finite commutative sums in... |
| pwsgsum 19096 | Finite commutative sums in... |
| fsfnn0gsumfsffz 19097 | Replacing a finitely suppo... |
| nn0gsumfz 19098 | Replacing a finitely suppo... |
| nn0gsumfz0 19099 | Replacing a finitely suppo... |
| gsummptnn0fz 19100 | A final group sum over a f... |
| gsummptnn0fzfv 19101 | A final group sum over a f... |
| telgsumfzslem 19102 | Lemma for ~ telgsumfzs (in... |
| telgsumfzs 19103 | Telescoping group sum rang... |
| telgsumfz 19104 | Telescoping group sum rang... |
| telgsumfz0s 19105 | Telescoping finite group s... |
| telgsumfz0 19106 | Telescoping finite group s... |
| telgsums 19107 | Telescoping finitely suppo... |
| telgsum 19108 | Telescoping finitely suppo... |
| reldmdprd 19113 | The domain of the internal... |
| dmdprd 19114 | The domain of definition o... |
| dmdprdd 19115 | Show that a given family i... |
| dprddomprc 19116 | A family of subgroups inde... |
| dprddomcld 19117 | If a family of subgroups i... |
| dprdval0prc 19118 | The internal direct produc... |
| dprdval 19119 | The value of the internal ... |
| eldprd 19120 | A class ` A ` is an intern... |
| dprdgrp 19121 | Reverse closure for the in... |
| dprdf 19122 | The function ` S ` is a fa... |
| dprdf2 19123 | The function ` S ` is a fa... |
| dprdcntz 19124 | The function ` S ` is a fa... |
| dprddisj 19125 | The function ` S ` is a fa... |
| dprdw 19126 | The property of being a fi... |
| dprdwd 19127 | A mapping being a finitely... |
| dprdff 19128 | A finitely supported funct... |
| dprdfcl 19129 | A finitely supported funct... |
| dprdffsupp 19130 | A finitely supported funct... |
| dprdfcntz 19131 | A function on the elements... |
| dprdssv 19132 | The internal direct produc... |
| dprdfid 19133 | A function mapping all but... |
| eldprdi 19134 | The domain of definition o... |
| dprdfinv 19135 | Take the inverse of a grou... |
| dprdfadd 19136 | Take the sum of group sums... |
| dprdfsub 19137 | Take the difference of gro... |
| dprdfeq0 19138 | The zero function is the o... |
| dprdf11 19139 | Two group sums over a dire... |
| dprdsubg 19140 | The internal direct produc... |
| dprdub 19141 | Each factor is a subset of... |
| dprdlub 19142 | The direct product is smal... |
| dprdspan 19143 | The direct product is the ... |
| dprdres 19144 | Restriction of a direct pr... |
| dprdss 19145 | Create a direct product by... |
| dprdz 19146 | A family consisting entire... |
| dprd0 19147 | The empty family is an int... |
| dprdf1o 19148 | Rearrange the index set of... |
| dprdf1 19149 | Rearrange the index set of... |
| subgdmdprd 19150 | A direct product in a subg... |
| subgdprd 19151 | A direct product in a subg... |
| dprdsn 19152 | A singleton family is an i... |
| dmdprdsplitlem 19153 | Lemma for ~ dmdprdsplit . ... |
| dprdcntz2 19154 | The function ` S ` is a fa... |
| dprddisj2 19155 | The function ` S ` is a fa... |
| dprd2dlem2 19156 | The direct product of a co... |
| dprd2dlem1 19157 | The direct product of a co... |
| dprd2da 19158 | The direct product of a co... |
| dprd2db 19159 | The direct product of a co... |
| dprd2d2 19160 | The direct product of a co... |
| dmdprdsplit2lem 19161 | Lemma for ~ dmdprdsplit . ... |
| dmdprdsplit2 19162 | The direct product splits ... |
| dmdprdsplit 19163 | The direct product splits ... |
| dprdsplit 19164 | The direct product is the ... |
| dmdprdpr 19165 | A singleton family is an i... |
| dprdpr 19166 | A singleton family is an i... |
| dpjlem 19167 | Lemma for theorems about d... |
| dpjcntz 19168 | The two subgroups that app... |
| dpjdisj 19169 | The two subgroups that app... |
| dpjlsm 19170 | The two subgroups that app... |
| dpjfval 19171 | Value of the direct produc... |
| dpjval 19172 | Value of the direct produc... |
| dpjf 19173 | The ` X ` -th index projec... |
| dpjidcl 19174 | The key property of projec... |
| dpjeq 19175 | Decompose a group sum into... |
| dpjid 19176 | The key property of projec... |
| dpjlid 19177 | The ` X ` -th index projec... |
| dpjrid 19178 | The ` Y ` -th index projec... |
| dpjghm 19179 | The direct product is the ... |
| dpjghm2 19180 | The direct product is the ... |
| ablfacrplem 19181 | Lemma for ~ ablfacrp2 . (... |
| ablfacrp 19182 | A finite abelian group who... |
| ablfacrp2 19183 | The factors ` K , L ` of ~... |
| ablfac1lem 19184 | Lemma for ~ ablfac1b . Sa... |
| ablfac1a 19185 | The factors of ~ ablfac1b ... |
| ablfac1b 19186 | Any abelian group is the d... |
| ablfac1c 19187 | The factors of ~ ablfac1b ... |
| ablfac1eulem 19188 | Lemma for ~ ablfac1eu . (... |
| ablfac1eu 19189 | The factorization of ~ abl... |
| pgpfac1lem1 19190 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem2 19191 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3a 19192 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem3 19193 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem4 19194 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1lem5 19195 | Lemma for ~ pgpfac1 . (Co... |
| pgpfac1 19196 | Factorization of a finite ... |
| pgpfaclem1 19197 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem2 19198 | Lemma for ~ pgpfac . (Con... |
| pgpfaclem3 19199 | Lemma for ~ pgpfac . (Con... |
| pgpfac 19200 | Full factorization of a fi... |
| ablfaclem1 19201 | Lemma for ~ ablfac . (Con... |
| ablfaclem2 19202 | Lemma for ~ ablfac . (Con... |
| ablfaclem3 19203 | Lemma for ~ ablfac . (Con... |
| ablfac 19204 | The Fundamental Theorem of... |
| ablfac2 19205 | Choose generators for each... |
| issimpg 19208 | The predicate "is a simple... |
| issimpgd 19209 | Deduce a simple group from... |
| simpggrp 19210 | A simple group is a group.... |
| simpggrpd 19211 | A simple group is a group.... |
| simpg2nsg 19212 | A simple group has two nor... |
| trivnsimpgd 19213 | Trivial groups are not sim... |
| simpgntrivd 19214 | Simple groups are nontrivi... |
| simpgnideld 19215 | A simple group contains a ... |
| simpgnsgd 19216 | The only normal subgroups ... |
| simpgnsgeqd 19217 | A normal subgroup of a sim... |
| 2nsgsimpgd 19218 | If any normal subgroup of ... |
| simpgnsgbid 19219 | A nontrivial group is simp... |
| ablsimpnosubgd 19220 | A subgroup of an abelian s... |
| ablsimpg1gend 19221 | An abelian simple group is... |
| ablsimpgcygd 19222 | An abelian simple group is... |
| ablsimpgfindlem1 19223 | Lemma for ~ ablsimpgfind .... |
| ablsimpgfindlem2 19224 | Lemma for ~ ablsimpgfind .... |
| cycsubggenodd 19225 | Relationship between the o... |
| ablsimpgfind 19226 | An abelian simple group is... |
| fincygsubgd 19227 | The subgroup referenced in... |
| fincygsubgodd 19228 | Calculate the order of a s... |
| fincygsubgodexd 19229 | A finite cyclic group has ... |
| prmgrpsimpgd 19230 | A group of prime order is ... |
| ablsimpgprmd 19231 | An abelian simple group ha... |
| ablsimpgd 19232 | An abelian group is simple... |
| fnmgp 19235 | The multiplicative group o... |
| mgpval 19236 | Value of the multiplicatio... |
| mgpplusg 19237 | Value of the group operati... |
| mgplem 19238 | Lemma for ~ mgpbas . (Con... |
| mgpbas 19239 | Base set of the multiplica... |
| mgpsca 19240 | The multiplication monoid ... |
| mgptset 19241 | Topology component of the ... |
| mgptopn 19242 | Topology of the multiplica... |
| mgpds 19243 | Distance function of the m... |
| mgpress 19244 | Subgroup commutes with the... |
| ringidval 19247 | The value of the unity ele... |
| dfur2 19248 | The multiplicative identit... |
| issrg 19251 | The predicate "is a semiri... |
| srgcmn 19252 | A semiring is a commutativ... |
| srgmnd 19253 | A semiring is a monoid. (... |
| srgmgp 19254 | A semiring is a monoid und... |
| srgi 19255 | Properties of a semiring. ... |
| srgcl 19256 | Closure of the multiplicat... |
| srgass 19257 | Associative law for the mu... |
| srgideu 19258 | The unit element of a semi... |
| srgfcl 19259 | Functionality of the multi... |
| srgdi 19260 | Distributive law for the m... |
| srgdir 19261 | Distributive law for the m... |
| srgidcl 19262 | The unit element of a semi... |
| srg0cl 19263 | The zero element of a semi... |
| srgidmlem 19264 | Lemma for ~ srglidm and ~ ... |
| srglidm 19265 | The unit element of a semi... |
| srgridm 19266 | The unit element of a semi... |
| issrgid 19267 | Properties showing that an... |
| srgacl 19268 | Closure of the addition op... |
| srgcom 19269 | Commutativity of the addit... |
| srgrz 19270 | The zero of a semiring is ... |
| srglz 19271 | The zero of a semiring is ... |
| srgisid 19272 | In a semiring, the only le... |
| srg1zr 19273 | The only semiring with a b... |
| srgen1zr 19274 | The only semiring with one... |
| srgmulgass 19275 | An associative property be... |
| srgpcomp 19276 | If two elements of a semir... |
| srgpcompp 19277 | If two elements of a semir... |
| srgpcomppsc 19278 | If two elements of a semir... |
| srglmhm 19279 | Left-multiplication in a s... |
| srgrmhm 19280 | Right-multiplication in a ... |
| srgsummulcr 19281 | A finite semiring sum mult... |
| sgsummulcl 19282 | A finite semiring sum mult... |
| srg1expzeq1 19283 | The exponentiation (by a n... |
| srgbinomlem1 19284 | Lemma 1 for ~ srgbinomlem ... |
| srgbinomlem2 19285 | Lemma 2 for ~ srgbinomlem ... |
| srgbinomlem3 19286 | Lemma 3 for ~ srgbinomlem ... |
| srgbinomlem4 19287 | Lemma 4 for ~ srgbinomlem ... |
| srgbinomlem 19288 | Lemma for ~ srgbinom . In... |
| srgbinom 19289 | The binomial theorem for c... |
| csrgbinom 19290 | The binomial theorem for c... |
| isring 19295 | The predicate "is a (unita... |
| ringgrp 19296 | A ring is a group. (Contr... |
| ringmgp 19297 | A ring is a monoid under m... |
| iscrng 19298 | A commutative ring is a ri... |
| crngmgp 19299 | A commutative ring's multi... |
| ringmnd 19300 | A ring is a monoid under a... |
| ringmgm 19301 | A ring is a magma. (Contr... |
| crngring 19302 | A commutative ring is a ri... |
| mgpf 19303 | Restricted functionality o... |
| ringi 19304 | Properties of a unital rin... |
| ringcl 19305 | Closure of the multiplicat... |
| crngcom 19306 | A commutative ring's multi... |
| iscrng2 19307 | A commutative ring is a ri... |
| ringass 19308 | Associative law for the mu... |
| ringideu 19309 | The unit element of a ring... |
| ringdi 19310 | Distributive law for the m... |
| ringdir 19311 | Distributive law for the m... |
| ringidcl 19312 | The unit element of a ring... |
| ring0cl 19313 | The zero element of a ring... |
| ringidmlem 19314 | Lemma for ~ ringlidm and ~... |
| ringlidm 19315 | The unit element of a ring... |
| ringridm 19316 | The unit element of a ring... |
| isringid 19317 | Properties showing that an... |
| ringid 19318 | The multiplication operati... |
| ringadd2 19319 | A ring element plus itself... |
| rngo2times 19320 | A ring element plus itself... |
| ringidss 19321 | A subset of the multiplica... |
| ringacl 19322 | Closure of the addition op... |
| ringcom 19323 | Commutativity of the addit... |
| ringabl 19324 | A ring is an Abelian group... |
| ringcmn 19325 | A ring is a commutative mo... |
| ringpropd 19326 | If two structures have the... |
| crngpropd 19327 | If two structures have the... |
| ringprop 19328 | If two structures have the... |
| isringd 19329 | Properties that determine ... |
| iscrngd 19330 | Properties that determine ... |
| ringlz 19331 | The zero of a unital ring ... |
| ringrz 19332 | The zero of a unital ring ... |
| ringsrg 19333 | Any ring is also a semirin... |
| ring1eq0 19334 | If one and zero are equal,... |
| ring1ne0 19335 | If a ring has at least two... |
| ringinvnz1ne0 19336 | In a unitary ring, a left ... |
| ringinvnzdiv 19337 | In a unitary ring, a left ... |
| ringnegl 19338 | Negation in a ring is the ... |
| rngnegr 19339 | Negation in a ring is the ... |
| ringmneg1 19340 | Negation of a product in a... |
| ringmneg2 19341 | Negation of a product in a... |
| ringm2neg 19342 | Double negation of a produ... |
| ringsubdi 19343 | Ring multiplication distri... |
| rngsubdir 19344 | Ring multiplication distri... |
| mulgass2 19345 | An associative property be... |
| ring1 19346 | The (smallest) structure r... |
| ringn0 19347 | Rings exist. (Contributed... |
| ringlghm 19348 | Left-multiplication in a r... |
| ringrghm 19349 | Right-multiplication in a ... |
| gsummulc1 19350 | A finite ring sum multipli... |
| gsummulc2 19351 | A finite ring sum multipli... |
| gsummgp0 19352 | If one factor in a finite ... |
| gsumdixp 19353 | Distribute a binary produc... |
| prdsmgp 19354 | The multiplicative monoid ... |
| prdsmulrcl 19355 | A structure product of rin... |
| prdsringd 19356 | A product of rings is a ri... |
| prdscrngd 19357 | A product of commutative r... |
| prds1 19358 | Value of the ring unit in ... |
| pwsring 19359 | A structure power of a rin... |
| pws1 19360 | Value of the ring unit in ... |
| pwscrng 19361 | A structure power of a com... |
| pwsmgp 19362 | The multiplicative group o... |
| imasring 19363 | The image structure of a r... |
| qusring2 19364 | The quotient structure of ... |
| crngbinom 19365 | The binomial theorem for c... |
| opprval 19368 | Value of the opposite ring... |
| opprmulfval 19369 | Value of the multiplicatio... |
| opprmul 19370 | Value of the multiplicatio... |
| crngoppr 19371 | In a commutative ring, the... |
| opprlem 19372 | Lemma for ~ opprbas and ~ ... |
| opprbas 19373 | Base set of an opposite ri... |
| oppradd 19374 | Addition operation of an o... |
| opprring 19375 | An opposite ring is a ring... |
| opprringb 19376 | Bidirectional form of ~ op... |
| oppr0 19377 | Additive identity of an op... |
| oppr1 19378 | Multiplicative identity of... |
| opprneg 19379 | The negative function in a... |
| opprsubg 19380 | Being a subgroup is a symm... |
| mulgass3 19381 | An associative property be... |
| reldvdsr 19388 | The divides relation is a ... |
| dvdsrval 19389 | Value of the divides relat... |
| dvdsr 19390 | Value of the divides relat... |
| dvdsr2 19391 | Value of the divides relat... |
| dvdsrmul 19392 | A left-multiple of ` X ` i... |
| dvdsrcl 19393 | Closure of a dividing elem... |
| dvdsrcl2 19394 | Closure of a dividing elem... |
| dvdsrid 19395 | An element in a (unital) r... |
| dvdsrtr 19396 | Divisibility is transitive... |
| dvdsrmul1 19397 | The divisibility relation ... |
| dvdsrneg 19398 | An element divides its neg... |
| dvdsr01 19399 | In a ring, zero is divisib... |
| dvdsr02 19400 | Only zero is divisible by ... |
| isunit 19401 | Property of being a unit o... |
| 1unit 19402 | The multiplicative identit... |
| unitcl 19403 | A unit is an element of th... |
| unitss 19404 | The set of units is contai... |
| opprunit 19405 | Being a unit is a symmetri... |
| crngunit 19406 | Property of being a unit i... |
| dvdsunit 19407 | A divisor of a unit is a u... |
| unitmulcl 19408 | The product of units is a ... |
| unitmulclb 19409 | Reversal of ~ unitmulcl in... |
| unitgrpbas 19410 | The base set of the group ... |
| unitgrp 19411 | The group of units is a gr... |
| unitabl 19412 | The group of units of a co... |
| unitgrpid 19413 | The identity of the multip... |
| unitsubm 19414 | The group of units is a su... |
| invrfval 19417 | Multiplicative inverse fun... |
| unitinvcl 19418 | The inverse of a unit exis... |
| unitinvinv 19419 | The inverse of the inverse... |
| ringinvcl 19420 | The inverse of a unit is a... |
| unitlinv 19421 | A unit times its inverse i... |
| unitrinv 19422 | A unit times its inverse i... |
| 1rinv 19423 | The inverse of the identit... |
| 0unit 19424 | The additive identity is a... |
| unitnegcl 19425 | The negative of a unit is ... |
| dvrfval 19428 | Division operation in a ri... |
| dvrval 19429 | Division operation in a ri... |
| dvrcl 19430 | Closure of division operat... |
| unitdvcl 19431 | The units are closed under... |
| dvrid 19432 | A cancellation law for div... |
| dvr1 19433 | A cancellation law for div... |
| dvrass 19434 | An associative law for div... |
| dvrcan1 19435 | A cancellation law for div... |
| dvrcan3 19436 | A cancellation law for div... |
| dvreq1 19437 | A cancellation law for div... |
| ringinvdv 19438 | Write the inverse function... |
| rngidpropd 19439 | The ring identity depends ... |
| dvdsrpropd 19440 | The divisibility relation ... |
| unitpropd 19441 | The set of units depends o... |
| invrpropd 19442 | The ring inverse function ... |
| isirred 19443 | An irreducible element of ... |
| isnirred 19444 | The property of being a no... |
| isirred2 19445 | Expand out the class diffe... |
| opprirred 19446 | Irreducibility is symmetri... |
| irredn0 19447 | The additive identity is n... |
| irredcl 19448 | An irreducible element is ... |
| irrednu 19449 | An irreducible element is ... |
| irredn1 19450 | The multiplicative identit... |
| irredrmul 19451 | The product of an irreduci... |
| irredlmul 19452 | The product of a unit and ... |
| irredmul 19453 | If product of two elements... |
| irredneg 19454 | The negative of an irreduc... |
| irrednegb 19455 | An element is irreducible ... |
| dfrhm2 19463 | The property of a ring hom... |
| rhmrcl1 19465 | Reverse closure of a ring ... |
| rhmrcl2 19466 | Reverse closure of a ring ... |
| isrhm 19467 | A function is a ring homom... |
| rhmmhm 19468 | A ring homomorphism is a h... |
| isrim0 19469 | An isomorphism of rings is... |
| rimrcl 19470 | Reverse closure for an iso... |
| rhmghm 19471 | A ring homomorphism is an ... |
| rhmf 19472 | A ring homomorphism is a f... |
| rhmmul 19473 | A homomorphism of rings pr... |
| isrhm2d 19474 | Demonstration of ring homo... |
| isrhmd 19475 | Demonstration of ring homo... |
| rhm1 19476 | Ring homomorphisms are req... |
| idrhm 19477 | The identity homomorphism ... |
| rhmf1o 19478 | A ring homomorphism is bij... |
| isrim 19479 | An isomorphism of rings is... |
| rimf1o 19480 | An isomorphism of rings is... |
| rimrhm 19481 | An isomorphism of rings is... |
| rimgim 19482 | An isomorphism of rings is... |
| rhmco 19483 | The composition of ring ho... |
| pwsco1rhm 19484 | Right composition with a f... |
| pwsco2rhm 19485 | Left composition with a ri... |
| f1ghm0to0 19486 | If a group homomorphism ` ... |
| f1rhm0to0OLD 19487 | Obsolete version of ~ f1gh... |
| f1rhm0to0ALT 19488 | Alternate proof for ~ f1gh... |
| gim0to0 19489 | A group isomorphism maps t... |
| rim0to0OLD 19490 | Obsolete version of ~ gim0... |
| kerf1ghm 19491 | A group homomorphism ` F `... |
| kerf1hrmOLD 19492 | Obsolete version of ~ kerf... |
| brric 19493 | The relation "is isomorphi... |
| brric2 19494 | The relation "is isomorphi... |
| ricgic 19495 | If two rings are (ring) is... |
| isdrng 19500 | The predicate "is a divisi... |
| drngunit 19501 | Elementhood in the set of ... |
| drngui 19502 | The set of units of a divi... |
| drngring 19503 | A division ring is a ring.... |
| drnggrp 19504 | A division ring is a group... |
| isfld 19505 | A field is a commutative d... |
| isdrng2 19506 | A division ring can equiva... |
| drngprop 19507 | If two structures have the... |
| drngmgp 19508 | A division ring contains a... |
| drngmcl 19509 | The product of two nonzero... |
| drngid 19510 | A division ring's unit is ... |
| drngunz 19511 | A division ring's unit is ... |
| drngid2 19512 | Properties showing that an... |
| drnginvrcl 19513 | Closure of the multiplicat... |
| drnginvrn0 19514 | The multiplicative inverse... |
| drnginvrl 19515 | Property of the multiplica... |
| drnginvrr 19516 | Property of the multiplica... |
| drngmul0or 19517 | A product is zero iff one ... |
| drngmulne0 19518 | A product is nonzero iff b... |
| drngmuleq0 19519 | An element is zero iff its... |
| opprdrng 19520 | The opposite of a division... |
| isdrngd 19521 | Properties that characteri... |
| isdrngrd 19522 | Properties that characteri... |
| drngpropd 19523 | If two structures have the... |
| fldpropd 19524 | If two structures have the... |
| issubrg 19529 | The subring predicate. (C... |
| subrgss 19530 | A subring is a subset. (C... |
| subrgid 19531 | Every ring is a subring of... |
| subrgring 19532 | A subring is a ring. (Con... |
| subrgcrng 19533 | A subring of a commutative... |
| subrgrcl 19534 | Reverse closure for a subr... |
| subrgsubg 19535 | A subring is a subgroup. ... |
| subrg0 19536 | A subring always has the s... |
| subrg1cl 19537 | A subring contains the mul... |
| subrgbas 19538 | Base set of a subring stru... |
| subrg1 19539 | A subring always has the s... |
| subrgacl 19540 | A subring is closed under ... |
| subrgmcl 19541 | A subgroup is closed under... |
| subrgsubm 19542 | A subring is a submonoid o... |
| subrgdvds 19543 | If an element divides anot... |
| subrguss 19544 | A unit of a subring is a u... |
| subrginv 19545 | A subring always has the s... |
| subrgdv 19546 | A subring always has the s... |
| subrgunit 19547 | An element of a ring is a ... |
| subrgugrp 19548 | The units of a subring for... |
| issubrg2 19549 | Characterize the subrings ... |
| opprsubrg 19550 | Being a subring is a symme... |
| subrgint 19551 | The intersection of a none... |
| subrgin 19552 | The intersection of two su... |
| subrgmre 19553 | The subrings of a ring are... |
| issubdrg 19554 | Characterize the subfields... |
| subsubrg 19555 | A subring of a subring is ... |
| subsubrg2 19556 | The set of subrings of a s... |
| issubrg3 19557 | A subring is an additive s... |
| resrhm 19558 | Restriction of a ring homo... |
| rhmeql 19559 | The equalizer of two ring ... |
| rhmima 19560 | The homomorphic image of a... |
| rnrhmsubrg 19561 | The range of a ring homomo... |
| cntzsubr 19562 | Centralizers in a ring are... |
| pwsdiagrhm 19563 | Diagonal homomorphism into... |
| subrgpropd 19564 | If two structures have the... |
| rhmpropd 19565 | Ring homomorphism depends ... |
| issdrg 19568 | Property of a division sub... |
| sdrgid 19569 | Every division ring is a d... |
| sdrgss 19570 | A division subring is a su... |
| issdrg2 19571 | Property of a division sub... |
| acsfn1p 19572 | Construction of a closure ... |
| subrgacs 19573 | Closure property of subrin... |
| sdrgacs 19574 | Closure property of divisi... |
| cntzsdrg 19575 | Centralizers in division r... |
| subdrgint 19576 | The intersection of a none... |
| sdrgint 19577 | The intersection of a none... |
| primefld 19578 | The smallest sub division ... |
| primefld0cl 19579 | The prime field contains t... |
| primefld1cl 19580 | The prime field contains t... |
| abvfval 19583 | Value of the set of absolu... |
| isabv 19584 | Elementhood in the set of ... |
| isabvd 19585 | Properties that determine ... |
| abvrcl 19586 | Reverse closure for the ab... |
| abvfge0 19587 | An absolute value is a fun... |
| abvf 19588 | An absolute value is a fun... |
| abvcl 19589 | An absolute value is a fun... |
| abvge0 19590 | The absolute value of a nu... |
| abveq0 19591 | The value of an absolute v... |
| abvne0 19592 | The absolute value of a no... |
| abvgt0 19593 | The absolute value of a no... |
| abvmul 19594 | An absolute value distribu... |
| abvtri 19595 | An absolute value satisfie... |
| abv0 19596 | The absolute value of zero... |
| abv1z 19597 | The absolute value of one ... |
| abv1 19598 | The absolute value of one ... |
| abvneg 19599 | The absolute value of a ne... |
| abvsubtri 19600 | An absolute value satisfie... |
| abvrec 19601 | The absolute value distrib... |
| abvdiv 19602 | The absolute value distrib... |
| abvdom 19603 | Any ring with an absolute ... |
| abvres 19604 | The restriction of an abso... |
| abvtrivd 19605 | The trivial absolute value... |
| abvtriv 19606 | The trivial absolute value... |
| abvpropd 19607 | If two structures have the... |
| staffval 19612 | The functionalization of t... |
| stafval 19613 | The functionalization of t... |
| staffn 19614 | The functionalization is e... |
| issrng 19615 | The predicate "is a star r... |
| srngrhm 19616 | The involution function in... |
| srngring 19617 | A star ring is a ring. (C... |
| srngcnv 19618 | The involution function in... |
| srngf1o 19619 | The involution function in... |
| srngcl 19620 | The involution function in... |
| srngnvl 19621 | The involution function in... |
| srngadd 19622 | The involution function in... |
| srngmul 19623 | The involution function in... |
| srng1 19624 | The conjugate of the ring ... |
| srng0 19625 | The conjugate of the ring ... |
| issrngd 19626 | Properties that determine ... |
| idsrngd 19627 | A commutative ring is a st... |
| islmod 19632 | The predicate "is a left m... |
| lmodlema 19633 | Lemma for properties of a ... |
| islmodd 19634 | Properties that determine ... |
| lmodgrp 19635 | A left module is a group. ... |
| lmodring 19636 | The scalar component of a ... |
| lmodfgrp 19637 | The scalar component of a ... |
| lmodbn0 19638 | The base set of a left mod... |
| lmodacl 19639 | Closure of ring addition f... |
| lmodmcl 19640 | Closure of ring multiplica... |
| lmodsn0 19641 | The set of scalars in a le... |
| lmodvacl 19642 | Closure of vector addition... |
| lmodass 19643 | Left module vector sum is ... |
| lmodlcan 19644 | Left cancellation law for ... |
| lmodvscl 19645 | Closure of scalar product ... |
| scaffval 19646 | The scalar multiplication ... |
| scafval 19647 | The scalar multiplication ... |
| scafeq 19648 | If the scalar multiplicati... |
| scaffn 19649 | The scalar multiplication ... |
| lmodscaf 19650 | The scalar multiplication ... |
| lmodvsdi 19651 | Distributive law for scala... |
| lmodvsdir 19652 | Distributive law for scala... |
| lmodvsass 19653 | Associative law for scalar... |
| lmod0cl 19654 | The ring zero in a left mo... |
| lmod1cl 19655 | The ring unit in a left mo... |
| lmodvs1 19656 | Scalar product with ring u... |
| lmod0vcl 19657 | The zero vector is a vecto... |
| lmod0vlid 19658 | Left identity law for the ... |
| lmod0vrid 19659 | Right identity law for the... |
| lmod0vid 19660 | Identity equivalent to the... |
| lmod0vs 19661 | Zero times a vector is the... |
| lmodvs0 19662 | Anything times the zero ve... |
| lmodvsmmulgdi 19663 | Distributive law for a gro... |
| lmodfopnelem1 19664 | Lemma 1 for ~ lmodfopne . ... |
| lmodfopnelem2 19665 | Lemma 2 for ~ lmodfopne . ... |
| lmodfopne 19666 | The (functionalized) opera... |
| lcomf 19667 | A linear-combination sum i... |
| lcomfsupp 19668 | A linear-combination sum i... |
| lmodvnegcl 19669 | Closure of vector negative... |
| lmodvnegid 19670 | Addition of a vector with ... |
| lmodvneg1 19671 | Minus 1 times a vector is ... |
| lmodvsneg 19672 | Multiplication of a vector... |
| lmodvsubcl 19673 | Closure of vector subtract... |
| lmodcom 19674 | Left module vector sum is ... |
| lmodabl 19675 | A left module is an abelia... |
| lmodcmn 19676 | A left module is a commuta... |
| lmodnegadd 19677 | Distribute negation throug... |
| lmod4 19678 | Commutative/associative la... |
| lmodvsubadd 19679 | Relationship between vecto... |
| lmodvaddsub4 19680 | Vector addition/subtractio... |
| lmodvpncan 19681 | Addition/subtraction cance... |
| lmodvnpcan 19682 | Cancellation law for vecto... |
| lmodvsubval2 19683 | Value of vector subtractio... |
| lmodsubvs 19684 | Subtraction of a scalar pr... |
| lmodsubdi 19685 | Scalar multiplication dist... |
| lmodsubdir 19686 | Scalar multiplication dist... |
| lmodsubeq0 19687 | If the difference between ... |
| lmodsubid 19688 | Subtraction of a vector fr... |
| lmodvsghm 19689 | Scalar multiplication of t... |
| lmodprop2d 19690 | If two structures have the... |
| lmodpropd 19691 | If two structures have the... |
| gsumvsmul 19692 | Pull a scalar multiplicati... |
| mptscmfsupp0 19693 | A mapping to a scalar prod... |
| mptscmfsuppd 19694 | A function mapping to a sc... |
| rmodislmodlem 19695 | Lemma for ~ rmodislmod . ... |
| rmodislmod 19696 | The right module ` R ` ind... |
| lssset 19699 | The set of all (not necess... |
| islss 19700 | The predicate "is a subspa... |
| islssd 19701 | Properties that determine ... |
| lssss 19702 | A subspace is a set of vec... |
| lssel 19703 | A subspace member is a vec... |
| lss1 19704 | The set of vectors in a le... |
| lssuni 19705 | The union of all subspaces... |
| lssn0 19706 | A subspace is not empty. ... |
| 00lss 19707 | The empty structure has no... |
| lsscl 19708 | Closure property of a subs... |
| lssvsubcl 19709 | Closure of vector subtract... |
| lssvancl1 19710 | Non-closure: if one vector... |
| lssvancl2 19711 | Non-closure: if one vector... |
| lss0cl 19712 | The zero vector belongs to... |
| lsssn0 19713 | The singleton of the zero ... |
| lss0ss 19714 | The zero subspace is inclu... |
| lssle0 19715 | No subspace is smaller tha... |
| lssne0 19716 | A nonzero subspace has a n... |
| lssvneln0 19717 | A vector ` X ` which doesn... |
| lssneln0 19718 | A vector ` X ` which doesn... |
| lssssr 19719 | Conclude subspace ordering... |
| lssvacl 19720 | Closure of vector addition... |
| lssvscl 19721 | Closure of scalar product ... |
| lssvnegcl 19722 | Closure of negative vector... |
| lsssubg 19723 | All subspaces are subgroup... |
| lsssssubg 19724 | All subspaces are subgroup... |
| islss3 19725 | A linear subspace of a mod... |
| lsslmod 19726 | A submodule is a module. ... |
| lsslss 19727 | The subspaces of a subspac... |
| islss4 19728 | A linear subspace is a sub... |
| lss1d 19729 | One-dimensional subspace (... |
| lssintcl 19730 | The intersection of a none... |
| lssincl 19731 | The intersection of two su... |
| lssmre 19732 | The subspaces of a module ... |
| lssacs 19733 | Submodules are an algebrai... |
| prdsvscacl 19734 | Pointwise scalar multiplic... |
| prdslmodd 19735 | The product of a family of... |
| pwslmod 19736 | A structure power of a lef... |
| lspfval 19739 | The span function for a le... |
| lspf 19740 | The span operator on a lef... |
| lspval 19741 | The span of a set of vecto... |
| lspcl 19742 | The span of a set of vecto... |
| lspsncl 19743 | The span of a singleton is... |
| lspprcl 19744 | The span of a pair is a su... |
| lsptpcl 19745 | The span of an unordered t... |
| lspsnsubg 19746 | The span of a singleton is... |
| 00lsp 19747 | ~ fvco4i lemma for linear ... |
| lspid 19748 | The span of a subspace is ... |
| lspssv 19749 | A span is a set of vectors... |
| lspss 19750 | Span preserves subset orde... |
| lspssid 19751 | A set of vectors is a subs... |
| lspidm 19752 | The span of a set of vecto... |
| lspun 19753 | The span of union is the s... |
| lspssp 19754 | If a set of vectors is a s... |
| mrclsp 19755 | Moore closure generalizes ... |
| lspsnss 19756 | The span of the singleton ... |
| lspsnel3 19757 | A member of the span of th... |
| lspprss 19758 | The span of a pair of vect... |
| lspsnid 19759 | A vector belongs to the sp... |
| lspsnel6 19760 | Relationship between a vec... |
| lspsnel5 19761 | Relationship between a vec... |
| lspsnel5a 19762 | Relationship between a vec... |
| lspprid1 19763 | A member of a pair of vect... |
| lspprid2 19764 | A member of a pair of vect... |
| lspprvacl 19765 | The sum of two vectors bel... |
| lssats2 19766 | A way to express atomistic... |
| lspsneli 19767 | A scalar product with a ve... |
| lspsn 19768 | Span of the singleton of a... |
| lspsnel 19769 | Member of span of the sing... |
| lspsnvsi 19770 | Span of a scalar product o... |
| lspsnss2 19771 | Comparable spans of single... |
| lspsnneg 19772 | Negation does not change t... |
| lspsnsub 19773 | Swapping subtraction order... |
| lspsn0 19774 | Span of the singleton of t... |
| lsp0 19775 | Span of the empty set. (C... |
| lspuni0 19776 | Union of the span of the e... |
| lspun0 19777 | The span of a union with t... |
| lspsneq0 19778 | Span of the singleton is t... |
| lspsneq0b 19779 | Equal singleton spans impl... |
| lmodindp1 19780 | Two independent (non-colin... |
| lsslsp 19781 | Spans in submodules corres... |
| lss0v 19782 | The zero vector in a submo... |
| lsspropd 19783 | If two structures have the... |
| lsppropd 19784 | If two structures have the... |
| reldmlmhm 19791 | Lemma for module homomorph... |
| lmimfn 19792 | Lemma for module isomorphi... |
| islmhm 19793 | Property of being a homomo... |
| islmhm3 19794 | Property of a module homom... |
| lmhmlem 19795 | Non-quantified consequence... |
| lmhmsca 19796 | A homomorphism of left mod... |
| lmghm 19797 | A homomorphism of left mod... |
| lmhmlmod2 19798 | A homomorphism of left mod... |
| lmhmlmod1 19799 | A homomorphism of left mod... |
| lmhmf 19800 | A homomorphism of left mod... |
| lmhmlin 19801 | A homomorphism of left mod... |
| lmodvsinv 19802 | Multiplication of a vector... |
| lmodvsinv2 19803 | Multiplying a negated vect... |
| islmhm2 19804 | A one-equation proof of li... |
| islmhmd 19805 | Deduction for a module hom... |
| 0lmhm 19806 | The constant zero linear f... |
| idlmhm 19807 | The identity function on a... |
| invlmhm 19808 | The negative function on a... |
| lmhmco 19809 | The composition of two mod... |
| lmhmplusg 19810 | The pointwise sum of two l... |
| lmhmvsca 19811 | The pointwise scalar produ... |
| lmhmf1o 19812 | A bijective module homomor... |
| lmhmima 19813 | The image of a subspace un... |
| lmhmpreima 19814 | The inverse image of a sub... |
| lmhmlsp 19815 | Homomorphisms preserve spa... |
| lmhmrnlss 19816 | The range of a homomorphis... |
| lmhmkerlss 19817 | The kernel of a homomorphi... |
| reslmhm 19818 | Restriction of a homomorph... |
| reslmhm2 19819 | Expansion of the codomain ... |
| reslmhm2b 19820 | Expansion of the codomain ... |
| lmhmeql 19821 | The equalizer of two modul... |
| lspextmo 19822 | A linear function is compl... |
| pwsdiaglmhm 19823 | Diagonal homomorphism into... |
| pwssplit0 19824 | Splitting for structure po... |
| pwssplit1 19825 | Splitting for structure po... |
| pwssplit2 19826 | Splitting for structure po... |
| pwssplit3 19827 | Splitting for structure po... |
| islmim 19828 | An isomorphism of left mod... |
| lmimf1o 19829 | An isomorphism of left mod... |
| lmimlmhm 19830 | An isomorphism of modules ... |
| lmimgim 19831 | An isomorphism of modules ... |
| islmim2 19832 | An isomorphism of left mod... |
| lmimcnv 19833 | The converse of a bijectiv... |
| brlmic 19834 | The relation "is isomorphi... |
| brlmici 19835 | Prove isomorphic by an exp... |
| lmiclcl 19836 | Isomorphism implies the le... |
| lmicrcl 19837 | Isomorphism implies the ri... |
| lmicsym 19838 | Module isomorphism is symm... |
| lmhmpropd 19839 | Module homomorphism depend... |
| islbs 19842 | The predicate " ` B ` is a... |
| lbsss 19843 | A basis is a set of vector... |
| lbsel 19844 | An element of a basis is a... |
| lbssp 19845 | The span of a basis is the... |
| lbsind 19846 | A basis is linearly indepe... |
| lbsind2 19847 | A basis is linearly indepe... |
| lbspss 19848 | No proper subset of a basi... |
| lsmcl 19849 | The sum of two subspaces i... |
| lsmspsn 19850 | Member of subspace sum of ... |
| lsmelval2 19851 | Subspace sum membership in... |
| lsmsp 19852 | Subspace sum in terms of s... |
| lsmsp2 19853 | Subspace sum of spans of s... |
| lsmssspx 19854 | Subspace sum (in its exten... |
| lsmpr 19855 | The span of a pair of vect... |
| lsppreli 19856 | A vector expressed as a su... |
| lsmelpr 19857 | Two ways to say that a vec... |
| lsppr0 19858 | The span of a vector paire... |
| lsppr 19859 | Span of a pair of vectors.... |
| lspprel 19860 | Member of the span of a pa... |
| lspprabs 19861 | Absorption of vector sum i... |
| lspvadd 19862 | The span of a vector sum i... |
| lspsntri 19863 | Triangle-type inequality f... |
| lspsntrim 19864 | Triangle-type inequality f... |
| lbspropd 19865 | If two structures have the... |
| pj1lmhm 19866 | The left projection functi... |
| pj1lmhm2 19867 | The left projection functi... |
| islvec 19870 | The predicate "is a left v... |
| lvecdrng 19871 | The set of scalars of a le... |
| lveclmod 19872 | A left vector space is a l... |
| lsslvec 19873 | A vector subspace is a vec... |
| lvecvs0or 19874 | If a scalar product is zer... |
| lvecvsn0 19875 | A scalar product is nonzer... |
| lssvs0or 19876 | If a scalar product belong... |
| lvecvscan 19877 | Cancellation law for scala... |
| lvecvscan2 19878 | Cancellation law for scala... |
| lvecinv 19879 | Invert coefficient of scal... |
| lspsnvs 19880 | A nonzero scalar product d... |
| lspsneleq 19881 | Membership relation that i... |
| lspsncmp 19882 | Comparable spans of nonzer... |
| lspsnne1 19883 | Two ways to express that v... |
| lspsnne2 19884 | Two ways to express that v... |
| lspsnnecom 19885 | Swap two vectors with diff... |
| lspabs2 19886 | Absorption law for span of... |
| lspabs3 19887 | Absorption law for span of... |
| lspsneq 19888 | Equal spans of singletons ... |
| lspsneu 19889 | Nonzero vectors with equal... |
| lspsnel4 19890 | A member of the span of th... |
| lspdisj 19891 | The span of a vector not i... |
| lspdisjb 19892 | A nonzero vector is not in... |
| lspdisj2 19893 | Unequal spans are disjoint... |
| lspfixed 19894 | Show membership in the spa... |
| lspexch 19895 | Exchange property for span... |
| lspexchn1 19896 | Exchange property for span... |
| lspexchn2 19897 | Exchange property for span... |
| lspindpi 19898 | Partial independence prope... |
| lspindp1 19899 | Alternate way to say 3 vec... |
| lspindp2l 19900 | Alternate way to say 3 vec... |
| lspindp2 19901 | Alternate way to say 3 vec... |
| lspindp3 19902 | Independence of 2 vectors ... |
| lspindp4 19903 | (Partial) independence of ... |
| lvecindp 19904 | Compute the ` X ` coeffici... |
| lvecindp2 19905 | Sums of independent vector... |
| lspsnsubn0 19906 | Unequal singleton spans im... |
| lsmcv 19907 | Subspace sum has the cover... |
| lspsolvlem 19908 | Lemma for ~ lspsolv . (Co... |
| lspsolv 19909 | If ` X ` is in the span of... |
| lssacsex 19910 | In a vector space, subspac... |
| lspsnat 19911 | There is no subspace stric... |
| lspsncv0 19912 | The span of a singleton co... |
| lsppratlem1 19913 | Lemma for ~ lspprat . Let... |
| lsppratlem2 19914 | Lemma for ~ lspprat . Sho... |
| lsppratlem3 19915 | Lemma for ~ lspprat . In ... |
| lsppratlem4 19916 | Lemma for ~ lspprat . In ... |
| lsppratlem5 19917 | Lemma for ~ lspprat . Com... |
| lsppratlem6 19918 | Lemma for ~ lspprat . Neg... |
| lspprat 19919 | A proper subspace of the s... |
| islbs2 19920 | An equivalent formulation ... |
| islbs3 19921 | An equivalent formulation ... |
| lbsacsbs 19922 | Being a basis in a vector ... |
| lvecdim 19923 | The dimension theorem for ... |
| lbsextlem1 19924 | Lemma for ~ lbsext . The ... |
| lbsextlem2 19925 | Lemma for ~ lbsext . Sinc... |
| lbsextlem3 19926 | Lemma for ~ lbsext . A ch... |
| lbsextlem4 19927 | Lemma for ~ lbsext . ~ lbs... |
| lbsextg 19928 | For any linearly independe... |
| lbsext 19929 | For any linearly independe... |
| lbsexg 19930 | Every vector space has a b... |
| lbsex 19931 | Every vector space has a b... |
| lvecprop2d 19932 | If two structures have the... |
| lvecpropd 19933 | If two structures have the... |
| sraval 19942 | Lemma for ~ srabase throug... |
| sralem 19943 | Lemma for ~ srabase and si... |
| srabase 19944 | Base set of a subring alge... |
| sraaddg 19945 | Additive operation of a su... |
| sramulr 19946 | Multiplicative operation o... |
| srasca 19947 | The set of scalars of a su... |
| sravsca 19948 | The scalar product operati... |
| sraip 19949 | The inner product operatio... |
| sratset 19950 | Topology component of a su... |
| sratopn 19951 | Topology component of a su... |
| srads 19952 | Distance function of a sub... |
| sralmod 19953 | The subring algebra is a l... |
| sralmod0 19954 | The subring module inherit... |
| issubrngd2 19955 | Prove a subring by closure... |
| rlmfn 19956 | ` ringLMod ` is a function... |
| rlmval 19957 | Value of the ring module. ... |
| lidlval 19958 | Value of the set of ring i... |
| rspval 19959 | Value of the ring span fun... |
| rlmval2 19960 | Value of the ring module e... |
| rlmbas 19961 | Base set of the ring modul... |
| rlmplusg 19962 | Vector addition in the rin... |
| rlm0 19963 | Zero vector in the ring mo... |
| rlmsub 19964 | Subtraction in the ring mo... |
| rlmmulr 19965 | Ring multiplication in the... |
| rlmsca 19966 | Scalars in the ring module... |
| rlmsca2 19967 | Scalars in the ring module... |
| rlmvsca 19968 | Scalar multiplication in t... |
| rlmtopn 19969 | Topology component of the ... |
| rlmds 19970 | Metric component of the ri... |
| rlmlmod 19971 | The ring module is a modul... |
| rlmlvec 19972 | The ring module over a div... |
| rlmlsm 19973 | Subgroup sum of the ring m... |
| rlmvneg 19974 | Vector negation in the rin... |
| rlmscaf 19975 | Functionalized scalar mult... |
| ixpsnbasval 19976 | The value of an infinite C... |
| lidlss 19977 | An ideal is a subset of th... |
| islidl 19978 | Predicate of being a (left... |
| lidl0cl 19979 | An ideal contains 0. (Con... |
| lidlacl 19980 | An ideal is closed under a... |
| lidlnegcl 19981 | An ideal contains negative... |
| lidlsubg 19982 | An ideal is a subgroup of ... |
| lidlsubcl 19983 | An ideal is closed under s... |
| lidlmcl 19984 | An ideal is closed under l... |
| lidl1el 19985 | An ideal contains 1 iff it... |
| lidl0 19986 | Every ring contains a zero... |
| lidl1 19987 | Every ring contains a unit... |
| lidlacs 19988 | The ideal system is an alg... |
| rspcl 19989 | The span of a set of ring ... |
| rspssid 19990 | The span of a set of ring ... |
| rsp1 19991 | The span of the identity e... |
| rsp0 19992 | The span of the zero eleme... |
| rspssp 19993 | The ideal span of a set of... |
| mrcrsp 19994 | Moore closure generalizes ... |
| lidlnz 19995 | A nonzero ideal contains a... |
| drngnidl 19996 | A division ring has only t... |
| lidlrsppropd 19997 | The left ideals and ring s... |
| 2idlval 20000 | Definition of a two-sided ... |
| 2idlcpbl 20001 | The coset equivalence rela... |
| qus1 20002 | The multiplicative identit... |
| qusring 20003 | If ` S ` is a two-sided id... |
| qusrhm 20004 | If ` S ` is a two-sided id... |
| crngridl 20005 | In a commutative ring, the... |
| crng2idl 20006 | In a commutative ring, a t... |
| quscrng 20007 | The quotient of a commutat... |
| lpival 20012 | Value of the set of princi... |
| islpidl 20013 | Property of being a princi... |
| lpi0 20014 | The zero ideal is always p... |
| lpi1 20015 | The unit ideal is always p... |
| islpir 20016 | Principal ideal rings are ... |
| lpiss 20017 | Principal ideals are a sub... |
| islpir2 20018 | Principal ideal rings are ... |
| lpirring 20019 | Principal ideal rings are ... |
| drnglpir 20020 | Division rings are princip... |
| rspsn 20021 | Membership in principal id... |
| lidldvgen 20022 | An element generates an id... |
| lpigen 20023 | An ideal is principal iff ... |
| isnzr 20026 | Property of a nonzero ring... |
| nzrnz 20027 | One and zero are different... |
| nzrring 20028 | A nonzero ring is a ring. ... |
| drngnzr 20029 | All division rings are non... |
| isnzr2 20030 | Equivalent characterizatio... |
| isnzr2hash 20031 | Equivalent characterizatio... |
| opprnzr 20032 | The opposite of a nonzero ... |
| ringelnzr 20033 | A ring is nonzero if it ha... |
| nzrunit 20034 | A unit is nonzero in any n... |
| subrgnzr 20035 | A subring of a nonzero rin... |
| 0ringnnzr 20036 | A ring is a zero ring iff ... |
| 0ring 20037 | If a ring has only one ele... |
| 0ring01eq 20038 | In a ring with only one el... |
| 01eq0ring 20039 | If the zero and the identi... |
| 0ring01eqbi 20040 | In a unital ring the zero ... |
| rng1nnzr 20041 | The (smallest) structure r... |
| ring1zr 20042 | The only (unital) ring wit... |
| rngen1zr 20043 | The only (unital) ring wit... |
| ringen1zr 20044 | The only unital ring with ... |
| rng1nfld 20045 | The zero ring is not a fie... |
| rrgval 20054 | Value of the set or left-r... |
| isrrg 20055 | Membership in the set of l... |
| rrgeq0i 20056 | Property of a left-regular... |
| rrgeq0 20057 | Left-multiplication by a l... |
| rrgsupp 20058 | Left multiplication by a l... |
| rrgss 20059 | Left-regular elements are ... |
| unitrrg 20060 | Units are regular elements... |
| isdomn 20061 | Expand definition of a dom... |
| domnnzr 20062 | A domain is a nonzero ring... |
| domnring 20063 | A domain is a ring. (Cont... |
| domneq0 20064 | In a domain, a product is ... |
| domnmuln0 20065 | In a domain, a product of ... |
| isdomn2 20066 | A ring is a domain iff all... |
| domnrrg 20067 | In a domain, any nonzero e... |
| opprdomn 20068 | The opposite of a domain i... |
| abvn0b 20069 | Another characterization o... |
| drngdomn 20070 | A division ring is a domai... |
| isidom 20071 | An integral domain is a co... |
| fldidom 20072 | A field is an integral dom... |
| fidomndrnglem 20073 | Lemma for ~ fidomndrng . ... |
| fidomndrng 20074 | A finite domain is a divis... |
| fiidomfld 20075 | A finite integral domain i... |
| isassa 20082 | The properties of an assoc... |
| assalem 20083 | The properties of an assoc... |
| assaass 20084 | Left-associative property ... |
| assaassr 20085 | Right-associative property... |
| assalmod 20086 | An associative algebra is ... |
| assaring 20087 | An associative algebra is ... |
| assasca 20088 | An associative algebra's s... |
| assa2ass 20089 | Left- and right-associativ... |
| isassad 20090 | Sufficient condition for b... |
| issubassa3 20091 | A subring that is also a s... |
| issubassa 20092 | The subalgebras of an asso... |
| sraassa 20093 | The subring algebra over a... |
| rlmassa 20094 | The ring module over a com... |
| assapropd 20095 | If two structures have the... |
| aspval 20096 | Value of the algebraic clo... |
| asplss 20097 | The algebraic span of a se... |
| aspid 20098 | The algebraic span of a su... |
| aspsubrg 20099 | The algebraic span of a se... |
| aspss 20100 | Span preserves subset orde... |
| aspssid 20101 | A set of vectors is a subs... |
| asclfval 20102 | Function value of the alge... |
| asclval 20103 | Value of a mapped algebra ... |
| asclfn 20104 | Unconditional functionalit... |
| asclf 20105 | The algebra scalars functi... |
| asclghm 20106 | The algebra scalars functi... |
| ascl0 20107 | The scalar 0 embedded into... |
| asclmul1 20108 | Left multiplication by a l... |
| asclmul2 20109 | Right multiplication by a ... |
| ascldimul 20110 | The algebra scalars functi... |
| ascldimulOLD 20111 | The algebra scalars functi... |
| asclinvg 20112 | The group inverse (negatio... |
| asclrhm 20113 | The scalar injection is a ... |
| rnascl 20114 | The set of injected scalar... |
| issubassa2 20115 | A subring of a unital alge... |
| rnasclsubrg 20116 | The scalar multiples of th... |
| rnasclmulcl 20117 | (Vector) multiplication is... |
| rnasclassa 20118 | The scalar multiples of th... |
| ressascl 20119 | The injection of scalars i... |
| asclpropd 20120 | If two structures have the... |
| aspval2 20121 | The algebraic closure is t... |
| assamulgscmlem1 20122 | Lemma 1 for ~ assamulgscm ... |
| assamulgscmlem2 20123 | Lemma for ~ assamulgscm (i... |
| assamulgscm 20124 | Exponentiation of a scalar... |
| reldmpsr 20135 | The multivariate power ser... |
| psrval 20136 | Value of the multivariate ... |
| psrvalstr 20137 | The multivariate power ser... |
| psrbag 20138 | Elementhood in the set of ... |
| psrbagf 20139 | A finite bag is a function... |
| snifpsrbag 20140 | A bag containing one eleme... |
| fczpsrbag 20141 | The constant function equa... |
| psrbaglesupp 20142 | The support of a dominated... |
| psrbaglecl 20143 | The set of finite bags is ... |
| psrbagaddcl 20144 | The sum of two finite bags... |
| psrbagcon 20145 | The analogue of the statem... |
| psrbaglefi 20146 | There are finitely many ba... |
| psrbagconcl 20147 | The complement of a bag is... |
| psrbagconf1o 20148 | Bag complementation is a b... |
| gsumbagdiaglem 20149 | Lemma for ~ gsumbagdiag . ... |
| gsumbagdiag 20150 | Two-dimensional commutatio... |
| psrass1lem 20151 | A group sum commutation us... |
| psrbas 20152 | The base set of the multiv... |
| psrelbas 20153 | An element of the set of p... |
| psrelbasfun 20154 | An element of the set of p... |
| psrplusg 20155 | The addition operation of ... |
| psradd 20156 | The addition operation of ... |
| psraddcl 20157 | Closure of the power serie... |
| psrmulr 20158 | The multiplication operati... |
| psrmulfval 20159 | The multiplication operati... |
| psrmulval 20160 | The multiplication operati... |
| psrmulcllem 20161 | Closure of the power serie... |
| psrmulcl 20162 | Closure of the power serie... |
| psrsca 20163 | The scalar field of the mu... |
| psrvscafval 20164 | The scalar multiplication ... |
| psrvsca 20165 | The scalar multiplication ... |
| psrvscaval 20166 | The scalar multiplication ... |
| psrvscacl 20167 | Closure of the power serie... |
| psr0cl 20168 | The zero element of the ri... |
| psr0lid 20169 | The zero element of the ri... |
| psrnegcl 20170 | The negative function in t... |
| psrlinv 20171 | The negative function in t... |
| psrgrp 20172 | The ring of power series i... |
| psr0 20173 | The zero element of the ri... |
| psrneg 20174 | The negative function of t... |
| psrlmod 20175 | The ring of power series i... |
| psr1cl 20176 | The identity element of th... |
| psrlidm 20177 | The identity element of th... |
| psrridm 20178 | The identity element of th... |
| psrass1 20179 | Associative identity for t... |
| psrdi 20180 | Distributive law for the r... |
| psrdir 20181 | Distributive law for the r... |
| psrass23l 20182 | Associative identity for t... |
| psrcom 20183 | Commutative law for the ri... |
| psrass23 20184 | Associative identities for... |
| psrring 20185 | The ring of power series i... |
| psr1 20186 | The identity element of th... |
| psrcrng 20187 | The ring of power series i... |
| psrassa 20188 | The ring of power series i... |
| resspsrbas 20189 | A restricted power series ... |
| resspsradd 20190 | A restricted power series ... |
| resspsrmul 20191 | A restricted power series ... |
| resspsrvsca 20192 | A restricted power series ... |
| subrgpsr 20193 | A subring of the base ring... |
| mvrfval 20194 | Value of the generating el... |
| mvrval 20195 | Value of the generating el... |
| mvrval2 20196 | Value of the generating el... |
| mvrid 20197 | The ` X i ` -th coefficien... |
| mvrf 20198 | The power series variable ... |
| mvrf1 20199 | The power series variable ... |
| mvrcl2 20200 | A power series variable is... |
| reldmmpl 20201 | The multivariate polynomia... |
| mplval 20202 | Value of the set of multiv... |
| mplbas 20203 | Base set of the set of mul... |
| mplelbas 20204 | Property of being a polyno... |
| mplval2 20205 | Self-referential expressio... |
| mplbasss 20206 | The set of polynomials is ... |
| mplelf 20207 | A polynomial is defined as... |
| mplsubglem 20208 | If ` A ` is an ideal of se... |
| mpllsslem 20209 | If ` A ` is an ideal of su... |
| mplsubglem2 20210 | Lemma for ~ mplsubg and ~ ... |
| mplsubg 20211 | The set of polynomials is ... |
| mpllss 20212 | The set of polynomials is ... |
| mplsubrglem 20213 | Lemma for ~ mplsubrg . (C... |
| mplsubrg 20214 | The set of polynomials is ... |
| mpl0 20215 | The zero polynomial. (Con... |
| mpladd 20216 | The addition operation on ... |
| mplmul 20217 | The multiplication operati... |
| mpl1 20218 | The identity element of th... |
| mplsca 20219 | The scalar field of a mult... |
| mplvsca2 20220 | The scalar multiplication ... |
| mplvsca 20221 | The scalar multiplication ... |
| mplvscaval 20222 | The scalar multiplication ... |
| mvrcl 20223 | A power series variable is... |
| mplgrp 20224 | The polynomial ring is a g... |
| mpllmod 20225 | The polynomial ring is a l... |
| mplring 20226 | The polynomial ring is a r... |
| mpllvec 20227 | The polynomial ring is a v... |
| mplcrng 20228 | The polynomial ring is a c... |
| mplassa 20229 | The polynomial ring is an ... |
| ressmplbas2 20230 | The base set of a restrict... |
| ressmplbas 20231 | A restricted polynomial al... |
| ressmpladd 20232 | A restricted polynomial al... |
| ressmplmul 20233 | A restricted polynomial al... |
| ressmplvsca 20234 | A restricted power series ... |
| subrgmpl 20235 | A subring of the base ring... |
| subrgmvr 20236 | The variables in a subring... |
| subrgmvrf 20237 | The variables in a polynom... |
| mplmon 20238 | A monomial is a polynomial... |
| mplmonmul 20239 | The product of two monomia... |
| mplcoe1 20240 | Decompose a polynomial int... |
| mplcoe3 20241 | Decompose a monomial in on... |
| mplcoe5lem 20242 | Lemma for ~ mplcoe4 . (Co... |
| mplcoe5 20243 | Decompose a monomial into ... |
| mplcoe2 20244 | Decompose a monomial into ... |
| mplbas2 20245 | An alternative expression ... |
| ltbval 20246 | Value of the well-order on... |
| ltbwe 20247 | The finite bag order is a ... |
| reldmopsr 20248 | Lemma for ordered power se... |
| opsrval 20249 | The value of the "ordered ... |
| opsrle 20250 | An alternative expression ... |
| opsrval2 20251 | Self-referential expressio... |
| opsrbaslem 20252 | Get a component of the ord... |
| opsrbas 20253 | The base set of the ordere... |
| opsrplusg 20254 | The addition operation of ... |
| opsrmulr 20255 | The multiplication operati... |
| opsrvsca 20256 | The scalar product operati... |
| opsrsca 20257 | The scalar ring of the ord... |
| opsrtoslem1 20258 | Lemma for ~ opsrtos . (Co... |
| opsrtoslem2 20259 | Lemma for ~ opsrtos . (Co... |
| opsrtos 20260 | The ordered power series s... |
| opsrso 20261 | The ordered power series s... |
| opsrcrng 20262 | The ring of ordered power ... |
| opsrassa 20263 | The ring of ordered power ... |
| mplrcl 20264 | Reverse closure for the po... |
| mplelsfi 20265 | A polynomial treated as a ... |
| mvrf2 20266 | The power series/polynomia... |
| mplmon2 20267 | Express a scaled monomial.... |
| psrbag0 20268 | The empty bag is a bag. (... |
| psrbagsn 20269 | A singleton bag is a bag. ... |
| mplascl 20270 | Value of the scalar inject... |
| mplasclf 20271 | The scalar injection is a ... |
| subrgascl 20272 | The scalar injection funct... |
| subrgasclcl 20273 | The scalars in a polynomia... |
| mplmon2cl 20274 | A scaled monomial is a pol... |
| mplmon2mul 20275 | Product of scaled monomial... |
| mplind 20276 | Prove a property of polyno... |
| mplcoe4 20277 | Decompose a polynomial int... |
| evlslem4 20282 | The support of a tensor pr... |
| psrbagfsupp 20283 | Finite bags have finite no... |
| psrbagev1 20284 | A bag of multipliers provi... |
| psrbagev2 20285 | Closure of a sum using a b... |
| evlslem2 20286 | A linear function on the p... |
| evlslem3 20287 | Lemma for ~ evlseu . Poly... |
| evlslem6 20288 | Lemma for ~ evlseu . Fini... |
| evlslem1 20289 | Lemma for ~ evlseu , give ... |
| evlseu 20290 | For a given interpretation... |
| reldmevls 20291 | Well-behaved binary operat... |
| mpfrcl 20292 | Reverse closure for the se... |
| evlsval 20293 | Value of the polynomial ev... |
| evlsval2 20294 | Characterizing properties ... |
| evlsrhm 20295 | Polynomial evaluation is a... |
| evlssca 20296 | Polynomial evaluation maps... |
| evlsvar 20297 | Polynomial evaluation maps... |
| evlsgsumadd 20298 | Polynomial evaluation maps... |
| evlsgsummul 20299 | Polynomial evaluation maps... |
| evlspw 20300 | Polynomial evaluation for ... |
| evlsvarpw 20301 | Polynomial evaluation for ... |
| evlval 20302 | Value of the simple/same r... |
| evlrhm 20303 | The simple evaluation map ... |
| evlsscasrng 20304 | The evaluation of a scalar... |
| evlsca 20305 | Simple polynomial evaluati... |
| evlsvarsrng 20306 | The evaluation of the vari... |
| evlvar 20307 | Simple polynomial evaluati... |
| mpfconst 20308 | Constants are multivariate... |
| mpfproj 20309 | Projections are multivaria... |
| mpfsubrg 20310 | Polynomial functions are a... |
| mpff 20311 | Polynomial functions are f... |
| mpfaddcl 20312 | The sum of multivariate po... |
| mpfmulcl 20313 | The product of multivariat... |
| mpfind 20314 | Prove a property of polyno... |
| selvffval 20323 | Value of the "variable sel... |
| selvfval 20324 | Value of the "variable sel... |
| selvval 20325 | Value of the "variable sel... |
| mhpfval 20326 | Value of the "homogeneous ... |
| mhpval 20327 | Value of the "homogeneous ... |
| ismhp 20328 | Property of being a homoge... |
| mhpmpl 20329 | A homogeneous polynomial i... |
| mhpdeg 20330 | All nonzero terms of a hom... |
| mhp0cl 20331 | The zero polynomial is hom... |
| mhpaddcl 20332 | Homogeneous polynomials ar... |
| mhpinvcl 20333 | Homogeneous polynomials ar... |
| mhpsubg 20334 | Homogeneous polynomials fo... |
| mhpvscacl 20335 | Homogeneous polynomials ar... |
| mhplss 20336 | Homogeneous polynomials fo... |
| psr1baslem 20347 | The set of finite bags on ... |
| psr1val 20348 | Value of the ring of univa... |
| psr1crng 20349 | The ring of univariate pow... |
| psr1assa 20350 | The ring of univariate pow... |
| psr1tos 20351 | The ordered power series s... |
| psr1bas2 20352 | The base set of the ring o... |
| psr1bas 20353 | The base set of the ring o... |
| vr1val 20354 | The value of the generator... |
| vr1cl2 20355 | The variable ` X ` is a me... |
| ply1val 20356 | The value of the set of un... |
| ply1bas 20357 | The value of the base set ... |
| ply1lss 20358 | Univariate polynomials for... |
| ply1subrg 20359 | Univariate polynomials for... |
| ply1crng 20360 | The ring of univariate pol... |
| ply1assa 20361 | The ring of univariate pol... |
| psr1bascl 20362 | A univariate power series ... |
| psr1basf 20363 | Univariate power series ba... |
| ply1basf 20364 | Univariate polynomial base... |
| ply1bascl 20365 | A univariate polynomial is... |
| ply1bascl2 20366 | A univariate polynomial is... |
| coe1fval 20367 | Value of the univariate po... |
| coe1fv 20368 | Value of an evaluated coef... |
| fvcoe1 20369 | Value of a multivariate co... |
| coe1fval3 20370 | Univariate power series co... |
| coe1f2 20371 | Functionality of univariat... |
| coe1fval2 20372 | Univariate polynomial coef... |
| coe1f 20373 | Functionality of univariat... |
| coe1fvalcl 20374 | A coefficient of a univari... |
| coe1sfi 20375 | Finite support of univaria... |
| coe1fsupp 20376 | The coefficient vector of ... |
| mptcoe1fsupp 20377 | A mapping involving coeffi... |
| coe1ae0 20378 | The coefficient vector of ... |
| vr1cl 20379 | The generator of a univari... |
| opsr0 20380 | Zero in the ordered power ... |
| opsr1 20381 | One in the ordered power s... |
| mplplusg 20382 | Value of addition in a pol... |
| mplmulr 20383 | Value of multiplication in... |
| psr1plusg 20384 | Value of addition in a uni... |
| psr1vsca 20385 | Value of scalar multiplica... |
| psr1mulr 20386 | Value of multiplication in... |
| ply1plusg 20387 | Value of addition in a uni... |
| ply1vsca 20388 | Value of scalar multiplica... |
| ply1mulr 20389 | Value of multiplication in... |
| ressply1bas2 20390 | The base set of a restrict... |
| ressply1bas 20391 | A restricted polynomial al... |
| ressply1add 20392 | A restricted polynomial al... |
| ressply1mul 20393 | A restricted polynomial al... |
| ressply1vsca 20394 | A restricted power series ... |
| subrgply1 20395 | A subring of the base ring... |
| gsumply1subr 20396 | Evaluate a group sum in a ... |
| psrbaspropd 20397 | Property deduction for pow... |
| psrplusgpropd 20398 | Property deduction for pow... |
| mplbaspropd 20399 | Property deduction for pol... |
| psropprmul 20400 | Reversing multiplication i... |
| ply1opprmul 20401 | Reversing multiplication i... |
| 00ply1bas 20402 | Lemma for ~ ply1basfvi and... |
| ply1basfvi 20403 | Protection compatibility o... |
| ply1plusgfvi 20404 | Protection compatibility o... |
| ply1baspropd 20405 | Property deduction for uni... |
| ply1plusgpropd 20406 | Property deduction for uni... |
| opsrring 20407 | Ordered power series form ... |
| opsrlmod 20408 | Ordered power series form ... |
| psr1ring 20409 | Univariate power series fo... |
| ply1ring 20410 | Univariate polynomials for... |
| psr1lmod 20411 | Univariate power series fo... |
| psr1sca 20412 | Scalars of a univariate po... |
| psr1sca2 20413 | Scalars of a univariate po... |
| ply1lmod 20414 | Univariate polynomials for... |
| ply1sca 20415 | Scalars of a univariate po... |
| ply1sca2 20416 | Scalars of a univariate po... |
| ply1mpl0 20417 | The univariate polynomial ... |
| ply10s0 20418 | Zero times a univariate po... |
| ply1mpl1 20419 | The univariate polynomial ... |
| ply1ascl 20420 | The univariate polynomial ... |
| subrg1ascl 20421 | The scalar injection funct... |
| subrg1asclcl 20422 | The scalars in a polynomia... |
| subrgvr1 20423 | The variables in a subring... |
| subrgvr1cl 20424 | The variables in a polynom... |
| coe1z 20425 | The coefficient vector of ... |
| coe1add 20426 | The coefficient vector of ... |
| coe1addfv 20427 | A particular coefficient o... |
| coe1subfv 20428 | A particular coefficient o... |
| coe1mul2lem1 20429 | An equivalence for ~ coe1m... |
| coe1mul2lem2 20430 | An equivalence for ~ coe1m... |
| coe1mul2 20431 | The coefficient vector of ... |
| coe1mul 20432 | The coefficient vector of ... |
| ply1moncl 20433 | Closure of the expression ... |
| ply1tmcl 20434 | Closure of the expression ... |
| coe1tm 20435 | Coefficient vector of a po... |
| coe1tmfv1 20436 | Nonzero coefficient of a p... |
| coe1tmfv2 20437 | Zero coefficient of a poly... |
| coe1tmmul2 20438 | Coefficient vector of a po... |
| coe1tmmul 20439 | Coefficient vector of a po... |
| coe1tmmul2fv 20440 | Function value of a right-... |
| coe1pwmul 20441 | Coefficient vector of a po... |
| coe1pwmulfv 20442 | Function value of a right-... |
| ply1scltm 20443 | A scalar is a term with ze... |
| coe1sclmul 20444 | Coefficient vector of a po... |
| coe1sclmulfv 20445 | A single coefficient of a ... |
| coe1sclmul2 20446 | Coefficient vector of a po... |
| ply1sclf 20447 | A scalar polynomial is a p... |
| ply1sclcl 20448 | The value of the algebra s... |
| coe1scl 20449 | Coefficient vector of a sc... |
| ply1sclid 20450 | Recover the base scalar fr... |
| ply1sclf1 20451 | The polynomial scalar func... |
| ply1scl0 20452 | The zero scalar is zero. ... |
| ply1scln0 20453 | Nonzero scalars create non... |
| ply1scl1 20454 | The one scalar is the unit... |
| ply1idvr1 20455 | The identity of a polynomi... |
| cply1mul 20456 | The product of two constan... |
| ply1coefsupp 20457 | The decomposition of a uni... |
| ply1coe 20458 | Decompose a univariate pol... |
| eqcoe1ply1eq 20459 | Two polynomials over the s... |
| ply1coe1eq 20460 | Two polynomials over the s... |
| cply1coe0 20461 | All but the first coeffici... |
| cply1coe0bi 20462 | A polynomial is constant (... |
| coe1fzgsumdlem 20463 | Lemma for ~ coe1fzgsumd (i... |
| coe1fzgsumd 20464 | Value of an evaluated coef... |
| gsumsmonply1 20465 | A finite group sum of scal... |
| gsummoncoe1 20466 | A coefficient of the polyn... |
| gsumply1eq 20467 | Two univariate polynomials... |
| lply1binom 20468 | The binomial theorem for l... |
| lply1binomsc 20469 | The binomial theorem for l... |
| reldmevls1 20474 | Well-behaved binary operat... |
| ply1frcl 20475 | Reverse closure for the se... |
| evls1fval 20476 | Value of the univariate po... |
| evls1val 20477 | Value of the univariate po... |
| evls1rhmlem 20478 | Lemma for ~ evl1rhm and ~ ... |
| evls1rhm 20479 | Polynomial evaluation is a... |
| evls1sca 20480 | Univariate polynomial eval... |
| evls1gsumadd 20481 | Univariate polynomial eval... |
| evls1gsummul 20482 | Univariate polynomial eval... |
| evls1pw 20483 | Univariate polynomial eval... |
| evls1varpw 20484 | Univariate polynomial eval... |
| evl1fval 20485 | Value of the simple/same r... |
| evl1val 20486 | Value of the simple/same r... |
| evl1fval1lem 20487 | Lemma for ~ evl1fval1 . (... |
| evl1fval1 20488 | Value of the simple/same r... |
| evl1rhm 20489 | Polynomial evaluation is a... |
| fveval1fvcl 20490 | The function value of the ... |
| evl1sca 20491 | Polynomial evaluation maps... |
| evl1scad 20492 | Polynomial evaluation buil... |
| evl1var 20493 | Polynomial evaluation maps... |
| evl1vard 20494 | Polynomial evaluation buil... |
| evls1var 20495 | Univariate polynomial eval... |
| evls1scasrng 20496 | The evaluation of a scalar... |
| evls1varsrng 20497 | The evaluation of the vari... |
| evl1addd 20498 | Polynomial evaluation buil... |
| evl1subd 20499 | Polynomial evaluation buil... |
| evl1muld 20500 | Polynomial evaluation buil... |
| evl1vsd 20501 | Polynomial evaluation buil... |
| evl1expd 20502 | Polynomial evaluation buil... |
| pf1const 20503 | Constants are polynomial f... |
| pf1id 20504 | The identity is a polynomi... |
| pf1subrg 20505 | Polynomial functions are a... |
| pf1rcl 20506 | Reverse closure for the se... |
| pf1f 20507 | Polynomial functions are f... |
| mpfpf1 20508 | Convert a multivariate pol... |
| pf1mpf 20509 | Convert a univariate polyn... |
| pf1addcl 20510 | The sum of multivariate po... |
| pf1mulcl 20511 | The product of multivariat... |
| pf1ind 20512 | Prove a property of polyno... |
| evl1gsumdlem 20513 | Lemma for ~ evl1gsumd (ind... |
| evl1gsumd 20514 | Polynomial evaluation buil... |
| evl1gsumadd 20515 | Univariate polynomial eval... |
| evl1gsumaddval 20516 | Value of a univariate poly... |
| evl1gsummul 20517 | Univariate polynomial eval... |
| evl1varpw 20518 | Univariate polynomial eval... |
| evl1varpwval 20519 | Value of a univariate poly... |
| evl1scvarpw 20520 | Univariate polynomial eval... |
| evl1scvarpwval 20521 | Value of a univariate poly... |
| evl1gsummon 20522 | Value of a univariate poly... |
| cnfldstr 20541 | The field of complex numbe... |
| cnfldex 20542 | The field of complex numbe... |
| cnfldbas 20543 | The base set of the field ... |
| cnfldadd 20544 | The addition operation of ... |
| cnfldmul 20545 | The multiplication operati... |
| cnfldcj 20546 | The conjugation operation ... |
| cnfldtset 20547 | The topology component of ... |
| cnfldle 20548 | The ordering of the field ... |
| cnfldds 20549 | The metric of the field of... |
| cnfldunif 20550 | The uniform structure comp... |
| cnfldfun 20551 | The field of complex numbe... |
| cnfldfunALT 20552 | Alternate proof of ~ cnfld... |
| xrsstr 20553 | The extended real structur... |
| xrsex 20554 | The extended real structur... |
| xrsbas 20555 | The base set of the extend... |
| xrsadd 20556 | The addition operation of ... |
| xrsmul 20557 | The multiplication operati... |
| xrstset 20558 | The topology component of ... |
| xrsle 20559 | The ordering of the extend... |
| cncrng 20560 | The complex numbers form a... |
| cnring 20561 | The complex numbers form a... |
| xrsmcmn 20562 | The "multiplicative group"... |
| cnfld0 20563 | Zero is the zero element o... |
| cnfld1 20564 | One is the unit element of... |
| cnfldneg 20565 | The additive inverse in th... |
| cnfldplusf 20566 | The functionalized additio... |
| cnfldsub 20567 | The subtraction operator i... |
| cndrng 20568 | The complex numbers form a... |
| cnflddiv 20569 | The division operation in ... |
| cnfldinv 20570 | The multiplicative inverse... |
| cnfldmulg 20571 | The group multiple functio... |
| cnfldexp 20572 | The exponentiation operato... |
| cnsrng 20573 | The complex numbers form a... |
| xrsmgm 20574 | The "additive group" of th... |
| xrsnsgrp 20575 | The "additive group" of th... |
| xrsmgmdifsgrp 20576 | The "additive group" of th... |
| xrs1mnd 20577 | The extended real numbers,... |
| xrs10 20578 | The zero of the extended r... |
| xrs1cmn 20579 | The extended real numbers ... |
| xrge0subm 20580 | The nonnegative extended r... |
| xrge0cmn 20581 | The nonnegative extended r... |
| xrsds 20582 | The metric of the extended... |
| xrsdsval 20583 | The metric of the extended... |
| xrsdsreval 20584 | The metric of the extended... |
| xrsdsreclblem 20585 | Lemma for ~ xrsdsreclb . ... |
| xrsdsreclb 20586 | The metric of the extended... |
| cnsubmlem 20587 | Lemma for ~ nn0subm and fr... |
| cnsubglem 20588 | Lemma for ~ resubdrg and f... |
| cnsubrglem 20589 | Lemma for ~ resubdrg and f... |
| cnsubdrglem 20590 | Lemma for ~ resubdrg and f... |
| qsubdrg 20591 | The rational numbers form ... |
| zsubrg 20592 | The integers form a subrin... |
| gzsubrg 20593 | The gaussian integers form... |
| nn0subm 20594 | The nonnegative integers f... |
| rege0subm 20595 | The nonnegative reals form... |
| absabv 20596 | The regular absolute value... |
| zsssubrg 20597 | The integers are a subset ... |
| qsssubdrg 20598 | The rational numbers are a... |
| cnsubrg 20599 | There are no subrings of t... |
| cnmgpabl 20600 | The unit group of the comp... |
| cnmgpid 20601 | The group identity element... |
| cnmsubglem 20602 | Lemma for ~ rpmsubg and fr... |
| rpmsubg 20603 | The positive reals form a ... |
| gzrngunitlem 20604 | Lemma for ~ gzrngunit . (... |
| gzrngunit 20605 | The units on ` ZZ [ _i ] `... |
| gsumfsum 20606 | Relate a group sum on ` CC... |
| regsumfsum 20607 | Relate a group sum on ` ( ... |
| expmhm 20608 | Exponentiation is a monoid... |
| nn0srg 20609 | The nonnegative integers f... |
| rge0srg 20610 | The nonnegative real numbe... |
| zringcrng 20613 | The ring of integers is a ... |
| zringring 20614 | The ring of integers is a ... |
| zringabl 20615 | The ring of integers is an... |
| zringgrp 20616 | The ring of integers is an... |
| zringbas 20617 | The integers are the base ... |
| zringplusg 20618 | The addition operation of ... |
| zringmulg 20619 | The multiplication (group ... |
| zringmulr 20620 | The multiplication operati... |
| zring0 20621 | The neutral element of the... |
| zring1 20622 | The multiplicative neutral... |
| zringnzr 20623 | The ring of integers is a ... |
| dvdsrzring 20624 | Ring divisibility in the r... |
| zringlpirlem1 20625 | Lemma for ~ zringlpir . A... |
| zringlpirlem2 20626 | Lemma for ~ zringlpir . A... |
| zringlpirlem3 20627 | Lemma for ~ zringlpir . A... |
| zringinvg 20628 | The additive inverse of an... |
| zringunit 20629 | The units of ` ZZ ` are th... |
| zringlpir 20630 | The integers are a princip... |
| zringndrg 20631 | The integers are not a div... |
| zringcyg 20632 | The integers are a cyclic ... |
| zringmpg 20633 | The multiplication group o... |
| prmirredlem 20634 | A positive integer is irre... |
| dfprm2 20635 | The positive irreducible e... |
| prmirred 20636 | The irreducible elements o... |
| expghm 20637 | Exponentiation is a group ... |
| mulgghm2 20638 | The powers of a group elem... |
| mulgrhm 20639 | The powers of the element ... |
| mulgrhm2 20640 | The powers of the element ... |
| zrhval 20649 | Define the unique homomorp... |
| zrhval2 20650 | Alternate value of the ` Z... |
| zrhmulg 20651 | Value of the ` ZRHom ` hom... |
| zrhrhmb 20652 | The ` ZRHom ` homomorphism... |
| zrhrhm 20653 | The ` ZRHom ` homomorphism... |
| zrh1 20654 | Interpretation of 1 in a r... |
| zrh0 20655 | Interpretation of 0 in a r... |
| zrhpropd 20656 | The ` ZZ ` ring homomorphi... |
| zlmval 20657 | Augment an abelian group w... |
| zlmlem 20658 | Lemma for ~ zlmbas and ~ z... |
| zlmbas 20659 | Base set of a ` ZZ ` -modu... |
| zlmplusg 20660 | Group operation of a ` ZZ ... |
| zlmmulr 20661 | Ring operation of a ` ZZ `... |
| zlmsca 20662 | Scalar ring of a ` ZZ ` -m... |
| zlmvsca 20663 | Scalar multiplication oper... |
| zlmlmod 20664 | The ` ZZ ` -module operati... |
| zlmassa 20665 | The ` ZZ ` -module operati... |
| chrval 20666 | Definition substitution of... |
| chrcl 20667 | Closure of the characteris... |
| chrid 20668 | The canonical ` ZZ ` ring ... |
| chrdvds 20669 | The ` ZZ ` ring homomorphi... |
| chrcong 20670 | If two integers are congru... |
| chrnzr 20671 | Nonzero rings are precisel... |
| chrrhm 20672 | The characteristic restric... |
| domnchr 20673 | The characteristic of a do... |
| znlidl 20674 | The set ` n ZZ ` is an ide... |
| zncrng2 20675 | The value of the ` Z/nZ ` ... |
| znval 20676 | The value of the ` Z/nZ ` ... |
| znle 20677 | The value of the ` Z/nZ ` ... |
| znval2 20678 | Self-referential expressio... |
| znbaslem 20679 | Lemma for ~ znbas . (Cont... |
| znbas2 20680 | The base set of ` Z/nZ ` i... |
| znadd 20681 | The additive structure of ... |
| znmul 20682 | The multiplicative structu... |
| znzrh 20683 | The ` ZZ ` ring homomorphi... |
| znbas 20684 | The base set of ` Z/nZ ` s... |
| zncrng 20685 | ` Z/nZ ` is a commutative ... |
| znzrh2 20686 | The ` ZZ ` ring homomorphi... |
| znzrhval 20687 | The ` ZZ ` ring homomorphi... |
| znzrhfo 20688 | The ` ZZ ` ring homomorphi... |
| zncyg 20689 | The group ` ZZ / n ZZ ` is... |
| zndvds 20690 | Express equality of equiva... |
| zndvds0 20691 | Special case of ~ zndvds w... |
| znf1o 20692 | The function ` F ` enumera... |
| zzngim 20693 | The ` ZZ ` ring homomorphi... |
| znle2 20694 | The ordering of the ` Z/nZ... |
| znleval 20695 | The ordering of the ` Z/nZ... |
| znleval2 20696 | The ordering of the ` Z/nZ... |
| zntoslem 20697 | Lemma for ~ zntos . (Cont... |
| zntos 20698 | The ` Z/nZ ` structure is ... |
| znhash 20699 | The ` Z/nZ ` structure has... |
| znfi 20700 | The ` Z/nZ ` structure is ... |
| znfld 20701 | The ` Z/nZ ` structure is ... |
| znidomb 20702 | The ` Z/nZ ` structure is ... |
| znchr 20703 | Cyclic rings are defined b... |
| znunit 20704 | The units of ` Z/nZ ` are ... |
| znunithash 20705 | The size of the unit group... |
| znrrg 20706 | The regular elements of ` ... |
| cygznlem1 20707 | Lemma for ~ cygzn . (Cont... |
| cygznlem2a 20708 | Lemma for ~ cygzn . (Cont... |
| cygznlem2 20709 | Lemma for ~ cygzn . (Cont... |
| cygznlem3 20710 | A cyclic group with ` n ` ... |
| cygzn 20711 | A cyclic group with ` n ` ... |
| cygth 20712 | The "fundamental theorem o... |
| cyggic 20713 | Cyclic groups are isomorph... |
| frgpcyg 20714 | A free group is cyclic iff... |
| cnmsgnsubg 20715 | The signs form a multiplic... |
| cnmsgnbas 20716 | The base set of the sign s... |
| cnmsgngrp 20717 | The group of signs under m... |
| psgnghm 20718 | The sign is a homomorphism... |
| psgnghm2 20719 | The sign is a homomorphism... |
| psgninv 20720 | The sign of a permutation ... |
| psgnco 20721 | Multiplicativity of the pe... |
| zrhpsgnmhm 20722 | Embedding of permutation s... |
| zrhpsgninv 20723 | The embedded sign of a per... |
| evpmss 20724 | Even permutations are perm... |
| psgnevpmb 20725 | A class is an even permuta... |
| psgnodpm 20726 | A permutation which is odd... |
| psgnevpm 20727 | A permutation which is eve... |
| psgnodpmr 20728 | If a permutation has sign ... |
| zrhpsgnevpm 20729 | The sign of an even permut... |
| zrhpsgnodpm 20730 | The sign of an odd permuta... |
| cofipsgn 20731 | Composition of any class `... |
| zrhpsgnelbas 20732 | Embedding of permutation s... |
| zrhcopsgnelbas 20733 | Embedding of permutation s... |
| evpmodpmf1o 20734 | The function for performin... |
| pmtrodpm 20735 | A transposition is an odd ... |
| psgnfix1 20736 | A permutation of a finite ... |
| psgnfix2 20737 | A permutation of a finite ... |
| psgndiflemB 20738 | Lemma 1 for ~ psgndif . (... |
| psgndiflemA 20739 | Lemma 2 for ~ psgndif . (... |
| psgndif 20740 | Embedding of permutation s... |
| copsgndif 20741 | Embedding of permutation s... |
| rebase 20744 | The base of the field of r... |
| remulg 20745 | The multiplication (group ... |
| resubdrg 20746 | The real numbers form a di... |
| resubgval 20747 | Subtraction in the field o... |
| replusg 20748 | The addition operation of ... |
| remulr 20749 | The multiplication operati... |
| re0g 20750 | The neutral element of the... |
| re1r 20751 | The multiplicative neutral... |
| rele2 20752 | The ordering relation of t... |
| relt 20753 | The ordering relation of t... |
| reds 20754 | The distance of the field ... |
| redvr 20755 | The division operation of ... |
| retos 20756 | The real numbers are a tot... |
| refld 20757 | The real numbers form a fi... |
| refldcj 20758 | The conjugation operation ... |
| recrng 20759 | The real numbers form a st... |
| regsumsupp 20760 | The group sum over the rea... |
| rzgrp 20761 | The quotient group ` RR / ... |
| isphl 20766 | The predicate "is a genera... |
| phllvec 20767 | A pre-Hilbert space is a l... |
| phllmod 20768 | A pre-Hilbert space is a l... |
| phlsrng 20769 | The scalar ring of a pre-H... |
| phllmhm 20770 | The inner product of a pre... |
| ipcl 20771 | Closure of the inner produ... |
| ipcj 20772 | Conjugate of an inner prod... |
| iporthcom 20773 | Orthogonality (meaning inn... |
| ip0l 20774 | Inner product with a zero ... |
| ip0r 20775 | Inner product with a zero ... |
| ipeq0 20776 | The inner product of a vec... |
| ipdir 20777 | Distributive law for inner... |
| ipdi 20778 | Distributive law for inner... |
| ip2di 20779 | Distributive law for inner... |
| ipsubdir 20780 | Distributive law for inner... |
| ipsubdi 20781 | Distributive law for inner... |
| ip2subdi 20782 | Distributive law for inner... |
| ipass 20783 | Associative law for inner ... |
| ipassr 20784 | "Associative" law for seco... |
| ipassr2 20785 | "Associative" law for inne... |
| ipffval 20786 | The inner product operatio... |
| ipfval 20787 | The inner product operatio... |
| ipfeq 20788 | If the inner product opera... |
| ipffn 20789 | The inner product operatio... |
| phlipf 20790 | The inner product operatio... |
| ip2eq 20791 | Two vectors are equal iff ... |
| isphld 20792 | Properties that determine ... |
| phlpropd 20793 | If two structures have the... |
| ssipeq 20794 | The inner product on a sub... |
| phssipval 20795 | The inner product on a sub... |
| phssip 20796 | The inner product (as a fu... |
| phlssphl 20797 | A subspace of an inner pro... |
| ocvfval 20804 | The orthocomplement operat... |
| ocvval 20805 | Value of the orthocompleme... |
| elocv 20806 | Elementhood in the orthoco... |
| ocvi 20807 | Property of a member of th... |
| ocvss 20808 | The orthocomplement of a s... |
| ocvocv 20809 | A set is contained in its ... |
| ocvlss 20810 | The orthocomplement of a s... |
| ocv2ss 20811 | Orthocomplements reverse s... |
| ocvin 20812 | An orthocomplement has tri... |
| ocvsscon 20813 | Two ways to say that ` S `... |
| ocvlsp 20814 | The orthocomplement of a l... |
| ocv0 20815 | The orthocomplement of the... |
| ocvz 20816 | The orthocomplement of the... |
| ocv1 20817 | The orthocomplement of the... |
| unocv 20818 | The orthocomplement of a u... |
| iunocv 20819 | The orthocomplement of an ... |
| cssval 20820 | The set of closed subspace... |
| iscss 20821 | The predicate "is a closed... |
| cssi 20822 | Property of a closed subsp... |
| cssss 20823 | A closed subspace is a sub... |
| iscss2 20824 | It is sufficient to prove ... |
| ocvcss 20825 | The orthocomplement of any... |
| cssincl 20826 | The zero subspace is a clo... |
| css0 20827 | The zero subspace is a clo... |
| css1 20828 | The whole space is a close... |
| csslss 20829 | A closed subspace of a pre... |
| lsmcss 20830 | A subset of a pre-Hilbert ... |
| cssmre 20831 | The closed subspaces of a ... |
| mrccss 20832 | The Moore closure correspo... |
| thlval 20833 | Value of the Hilbert latti... |
| thlbas 20834 | Base set of the Hilbert la... |
| thlle 20835 | Ordering on the Hilbert la... |
| thlleval 20836 | Ordering on the Hilbert la... |
| thloc 20837 | Orthocomplement on the Hil... |
| pjfval 20844 | The value of the projectio... |
| pjdm 20845 | A subspace is in the domai... |
| pjpm 20846 | The projection map is a pa... |
| pjfval2 20847 | Value of the projection ma... |
| pjval 20848 | Value of the projection ma... |
| pjdm2 20849 | A subspace is in the domai... |
| pjff 20850 | A projection is a linear o... |
| pjf 20851 | A projection is a function... |
| pjf2 20852 | A projection is a function... |
| pjfo 20853 | A projection is a surjecti... |
| pjcss 20854 | A projection subspace is a... |
| ocvpj 20855 | The orthocomplement of a p... |
| ishil 20856 | The predicate "is a Hilber... |
| ishil2 20857 | The predicate "is a Hilber... |
| isobs 20858 | The predicate "is an ortho... |
| obsip 20859 | The inner product of two e... |
| obsipid 20860 | A basis element has unit l... |
| obsrcl 20861 | Reverse closure for an ort... |
| obsss 20862 | An orthonormal basis is a ... |
| obsne0 20863 | A basis element is nonzero... |
| obsocv 20864 | An orthonormal basis has t... |
| obs2ocv 20865 | The double orthocomplement... |
| obselocv 20866 | A basis element is in the ... |
| obs2ss 20867 | A basis has no proper subs... |
| obslbs 20868 | An orthogonal basis is a l... |
| reldmdsmm 20871 | The direct sum is a well-b... |
| dsmmval 20872 | Value of the module direct... |
| dsmmbase 20873 | Base set of the module dir... |
| dsmmval2 20874 | Self-referential definitio... |
| dsmmbas2 20875 | Base set of the direct sum... |
| dsmmfi 20876 | For finite products, the d... |
| dsmmelbas 20877 | Membership in the finitely... |
| dsmm0cl 20878 | The all-zero vector is con... |
| dsmmacl 20879 | The finite hull is closed ... |
| prdsinvgd2 20880 | Negation of a single coord... |
| dsmmsubg 20881 | The finite hull of a produ... |
| dsmmlss 20882 | The finite hull of a produ... |
| dsmmlmod 20883 | The direct sum of a family... |
| frlmval 20886 | Value of the "free module"... |
| frlmlmod 20887 | The free module is a modul... |
| frlmpws 20888 | The free module as a restr... |
| frlmlss 20889 | The base set of the free m... |
| frlmpwsfi 20890 | The finite free module is ... |
| frlmsca 20891 | The ring of scalars of a f... |
| frlm0 20892 | Zero in a free module (rin... |
| frlmbas 20893 | Base set of the free modul... |
| frlmelbas 20894 | Membership in the base set... |
| frlmrcl 20895 | If a free module is inhabi... |
| frlmbasfsupp 20896 | Elements of the free modul... |
| frlmbasmap 20897 | Elements of the free modul... |
| frlmbasf 20898 | Elements of the free modul... |
| frlmlvec 20899 | The free module over a div... |
| frlmfibas 20900 | The base set of the finite... |
| elfrlmbasn0 20901 | If the dimension of a free... |
| frlmplusgval 20902 | Addition in a free module.... |
| frlmsubgval 20903 | Subtraction in a free modu... |
| frlmvscafval 20904 | Scalar multiplication in a... |
| frlmvplusgvalc 20905 | Coordinates of a sum with ... |
| frlmvscaval 20906 | Coordinates of a scalar mu... |
| frlmplusgvalb 20907 | Addition in a free module ... |
| frlmvscavalb 20908 | Scalar multiplication in a... |
| frlmvplusgscavalb 20909 | Addition combined with sca... |
| frlmgsum 20910 | Finite commutative sums in... |
| frlmsplit2 20911 | Restriction is homomorphic... |
| frlmsslss 20912 | A subset of a free module ... |
| frlmsslss2 20913 | A subset of a free module ... |
| frlmbas3 20914 | An element of the base set... |
| mpofrlmd 20915 | Elements of the free modul... |
| frlmip 20916 | The inner product of a fre... |
| frlmipval 20917 | The inner product of a fre... |
| frlmphllem 20918 | Lemma for ~ frlmphl . (Co... |
| frlmphl 20919 | Conditions for a free modu... |
| uvcfval 20922 | Value of the unit-vector g... |
| uvcval 20923 | Value of a single unit vec... |
| uvcvval 20924 | Value of a unit vector coo... |
| uvcvvcl 20925 | A coordinate of a unit vec... |
| uvcvvcl2 20926 | A unit vector coordinate i... |
| uvcvv1 20927 | The unit vector is one at ... |
| uvcvv0 20928 | The unit vector is zero at... |
| uvcff 20929 | Domain and range of the un... |
| uvcf1 20930 | In a nonzero ring, each un... |
| uvcresum 20931 | Any element of a free modu... |
| frlmssuvc1 20932 | A scalar multiple of a uni... |
| frlmssuvc2 20933 | A nonzero scalar multiple ... |
| frlmsslsp 20934 | A subset of a free module ... |
| frlmlbs 20935 | The unit vectors comprise ... |
| frlmup1 20936 | Any assignment of unit vec... |
| frlmup2 20937 | The evaluation map has the... |
| frlmup3 20938 | The range of such an evalu... |
| frlmup4 20939 | Universal property of the ... |
| ellspd 20940 | The elements of the span o... |
| elfilspd 20941 | Simplified version of ~ el... |
| rellindf 20946 | The independent-family pre... |
| islinds 20947 | Property of an independent... |
| linds1 20948 | An independent set of vect... |
| linds2 20949 | An independent set of vect... |
| islindf 20950 | Property of an independent... |
| islinds2 20951 | Expanded property of an in... |
| islindf2 20952 | Property of an independent... |
| lindff 20953 | Functional property of a l... |
| lindfind 20954 | A linearly independent fam... |
| lindsind 20955 | A linearly independent set... |
| lindfind2 20956 | In a linearly independent ... |
| lindsind2 20957 | In a linearly independent ... |
| lindff1 20958 | A linearly independent fam... |
| lindfrn 20959 | The range of an independen... |
| f1lindf 20960 | Rearranging and deleting e... |
| lindfres 20961 | Any restriction of an inde... |
| lindsss 20962 | Any subset of an independe... |
| f1linds 20963 | A family constructed from ... |
| islindf3 20964 | In a nonzero ring, indepen... |
| lindfmm 20965 | Linear independence of a f... |
| lindsmm 20966 | Linear independence of a s... |
| lindsmm2 20967 | The monomorphic image of a... |
| lsslindf 20968 | Linear independence is unc... |
| lsslinds 20969 | Linear independence is unc... |
| islbs4 20970 | A basis is an independent ... |
| lbslinds 20971 | A basis is independent. (... |
| islinds3 20972 | A subset is linearly indep... |
| islinds4 20973 | A set is independent in a ... |
| lmimlbs 20974 | The isomorphic image of a ... |
| lmiclbs 20975 | Having a basis is an isomo... |
| islindf4 20976 | A family is independent if... |
| islindf5 20977 | A family is independent if... |
| indlcim 20978 | An independent, spanning f... |
| lbslcic 20979 | A module with a basis is i... |
| lmisfree 20980 | A module has a basis iff i... |
| lvecisfrlm 20981 | Every vector space is isom... |
| lmimco 20982 | The composition of two iso... |
| lmictra 20983 | Module isomorphism is tran... |
| uvcf1o 20984 | In a nonzero ring, the map... |
| uvcendim 20985 | In a nonzero ring, the num... |
| frlmisfrlm 20986 | A free module is isomorphi... |
| frlmiscvec 20987 | Every free module is isomo... |
| mamufval 20990 | Functional value of the ma... |
| mamuval 20991 | Multiplication of two matr... |
| mamufv 20992 | A cell in the multiplicati... |
| mamudm 20993 | The domain of the matrix m... |
| mamufacex 20994 | Every solution of the equa... |
| mamures 20995 | Rows in a matrix product a... |
| mndvcl 20996 | Tuple-wise additive closur... |
| mndvass 20997 | Tuple-wise associativity i... |
| mndvlid 20998 | Tuple-wise left identity i... |
| mndvrid 20999 | Tuple-wise right identity ... |
| grpvlinv 21000 | Tuple-wise left inverse in... |
| grpvrinv 21001 | Tuple-wise right inverse i... |
| mhmvlin 21002 | Tuple extension of monoid ... |
| ringvcl 21003 | Tuple-wise multiplication ... |
| mamucl 21004 | Operation closure of matri... |
| mamuass 21005 | Matrix multiplication is a... |
| mamudi 21006 | Matrix multiplication dist... |
| mamudir 21007 | Matrix multiplication dist... |
| mamuvs1 21008 | Matrix multiplication dist... |
| mamuvs2 21009 | Matrix multiplication dist... |
| matbas0pc 21012 | There is no matrix with a ... |
| matbas0 21013 | There is no matrix for a n... |
| matval 21014 | Value of the matrix algebr... |
| matrcl 21015 | Reverse closure for the ma... |
| matbas 21016 | The matrix ring has the sa... |
| matplusg 21017 | The matrix ring has the sa... |
| matsca 21018 | The matrix ring has the sa... |
| matvsca 21019 | The matrix ring has the sa... |
| mat0 21020 | The matrix ring has the sa... |
| matinvg 21021 | The matrix ring has the sa... |
| mat0op 21022 | Value of a zero matrix as ... |
| matsca2 21023 | The scalars of the matrix ... |
| matbas2 21024 | The base set of the matrix... |
| matbas2i 21025 | A matrix is a function. (... |
| matbas2d 21026 | The base set of the matrix... |
| eqmat 21027 | Two square matrices of the... |
| matecl 21028 | Each entry (according to W... |
| matecld 21029 | Each entry (according to W... |
| matplusg2 21030 | Addition in the matrix rin... |
| matvsca2 21031 | Scalar multiplication in t... |
| matlmod 21032 | The matrix ring is a linea... |
| matgrp 21033 | The matrix ring is a group... |
| matvscl 21034 | Closure of the scalar mult... |
| matsubg 21035 | The matrix ring has the sa... |
| matplusgcell 21036 | Addition in the matrix rin... |
| matsubgcell 21037 | Subtraction in the matrix ... |
| matinvgcell 21038 | Additive inversion in the ... |
| matvscacell 21039 | Scalar multiplication in t... |
| matgsum 21040 | Finite commutative sums in... |
| matmulr 21041 | Multiplication in the matr... |
| mamumat1cl 21042 | The identity matrix (as op... |
| mat1comp 21043 | The components of the iden... |
| mamulid 21044 | The identity matrix (as op... |
| mamurid 21045 | The identity matrix (as op... |
| matring 21046 | Existence of the matrix ri... |
| matassa 21047 | Existence of the matrix al... |
| matmulcell 21048 | Multiplication in the matr... |
| mpomatmul 21049 | Multiplication of two N x ... |
| mat1 21050 | Value of an identity matri... |
| mat1ov 21051 | Entries of an identity mat... |
| mat1bas 21052 | The identity matrix is a m... |
| matsc 21053 | The identity matrix multip... |
| ofco2 21054 | Distribution law for the f... |
| oftpos 21055 | The transposition of the v... |
| mattposcl 21056 | The transpose of a square ... |
| mattpostpos 21057 | The transpose of the trans... |
| mattposvs 21058 | The transposition of a mat... |
| mattpos1 21059 | The transposition of the i... |
| tposmap 21060 | The transposition of an I ... |
| mamutpos 21061 | Behavior of transposes in ... |
| mattposm 21062 | Multiplying two transposed... |
| matgsumcl 21063 | Closure of a group sum ove... |
| madetsumid 21064 | The identity summand in th... |
| matepmcl 21065 | Each entry of a matrix wit... |
| matepm2cl 21066 | Each entry of a matrix wit... |
| madetsmelbas 21067 | A summand of the determina... |
| madetsmelbas2 21068 | A summand of the determina... |
| mat0dimbas0 21069 | The empty set is the one a... |
| mat0dim0 21070 | The zero of the algebra of... |
| mat0dimid 21071 | The identity of the algebr... |
| mat0dimscm 21072 | The scalar multiplication ... |
| mat0dimcrng 21073 | The algebra of matrices wi... |
| mat1dimelbas 21074 | A matrix with dimension 1 ... |
| mat1dimbas 21075 | A matrix with dimension 1 ... |
| mat1dim0 21076 | The zero of the algebra of... |
| mat1dimid 21077 | The identity of the algebr... |
| mat1dimscm 21078 | The scalar multiplication ... |
| mat1dimmul 21079 | The ring multiplication in... |
| mat1dimcrng 21080 | The algebra of matrices wi... |
| mat1f1o 21081 | There is a 1-1 function fr... |
| mat1rhmval 21082 | The value of the ring homo... |
| mat1rhmelval 21083 | The value of the ring homo... |
| mat1rhmcl 21084 | The value of the ring homo... |
| mat1f 21085 | There is a function from a... |
| mat1ghm 21086 | There is a group homomorph... |
| mat1mhm 21087 | There is a monoid homomorp... |
| mat1rhm 21088 | There is a ring homomorphi... |
| mat1rngiso 21089 | There is a ring isomorphis... |
| mat1ric 21090 | A ring is isomorphic to th... |
| dmatval 21095 | The set of ` N ` x ` N ` d... |
| dmatel 21096 | A ` N ` x ` N ` diagonal m... |
| dmatmat 21097 | An ` N ` x ` N ` diagonal ... |
| dmatid 21098 | The identity matrix is a d... |
| dmatelnd 21099 | An extradiagonal entry of ... |
| dmatmul 21100 | The product of two diagona... |
| dmatsubcl 21101 | The difference of two diag... |
| dmatsgrp 21102 | The set of diagonal matric... |
| dmatmulcl 21103 | The product of two diagona... |
| dmatsrng 21104 | The set of diagonal matric... |
| dmatcrng 21105 | The subring of diagonal ma... |
| dmatscmcl 21106 | The multiplication of a di... |
| scmatval 21107 | The set of ` N ` x ` N ` s... |
| scmatel 21108 | An ` N ` x ` N ` scalar ma... |
| scmatscmid 21109 | A scalar matrix can be exp... |
| scmatscmide 21110 | An entry of a scalar matri... |
| scmatscmiddistr 21111 | Distributive law for scala... |
| scmatmat 21112 | An ` N ` x ` N ` scalar ma... |
| scmate 21113 | An entry of an ` N ` x ` N... |
| scmatmats 21114 | The set of an ` N ` x ` N ... |
| scmateALT 21115 | Alternate proof of ~ scmat... |
| scmatscm 21116 | The multiplication of a ma... |
| scmatid 21117 | The identity matrix is a s... |
| scmatdmat 21118 | A scalar matrix is a diago... |
| scmataddcl 21119 | The sum of two scalar matr... |
| scmatsubcl 21120 | The difference of two scal... |
| scmatmulcl 21121 | The product of two scalar ... |
| scmatsgrp 21122 | The set of scalar matrices... |
| scmatsrng 21123 | The set of scalar matrices... |
| scmatcrng 21124 | The subring of scalar matr... |
| scmatsgrp1 21125 | The set of scalar matrices... |
| scmatsrng1 21126 | The set of scalar matrices... |
| smatvscl 21127 | Closure of the scalar mult... |
| scmatlss 21128 | The set of scalar matrices... |
| scmatstrbas 21129 | The set of scalar matrices... |
| scmatrhmval 21130 | The value of the ring homo... |
| scmatrhmcl 21131 | The value of the ring homo... |
| scmatf 21132 | There is a function from a... |
| scmatfo 21133 | There is a function from a... |
| scmatf1 21134 | There is a 1-1 function fr... |
| scmatf1o 21135 | There is a bijection betwe... |
| scmatghm 21136 | There is a group homomorph... |
| scmatmhm 21137 | There is a monoid homomorp... |
| scmatrhm 21138 | There is a ring homomorphi... |
| scmatrngiso 21139 | There is a ring isomorphis... |
| scmatric 21140 | A ring is isomorphic to ev... |
| mat0scmat 21141 | The empty matrix over a ri... |
| mat1scmat 21142 | A 1-dimensional matrix ove... |
| mvmulfval 21145 | Functional value of the ma... |
| mvmulval 21146 | Multiplication of a vector... |
| mvmulfv 21147 | A cell/element in the vect... |
| mavmulval 21148 | Multiplication of a vector... |
| mavmulfv 21149 | A cell/element in the vect... |
| mavmulcl 21150 | Multiplication of an NxN m... |
| 1mavmul 21151 | Multiplication of the iden... |
| mavmulass 21152 | Associativity of the multi... |
| mavmuldm 21153 | The domain of the matrix v... |
| mavmulsolcl 21154 | Every solution of the equa... |
| mavmul0 21155 | Multiplication of a 0-dime... |
| mavmul0g 21156 | The result of the 0-dimens... |
| mvmumamul1 21157 | The multiplication of an M... |
| mavmumamul1 21158 | The multiplication of an N... |
| marrepfval 21163 | First substitution for the... |
| marrepval0 21164 | Second substitution for th... |
| marrepval 21165 | Third substitution for the... |
| marrepeval 21166 | An entry of a matrix with ... |
| marrepcl 21167 | Closure of the row replace... |
| marepvfval 21168 | First substitution for the... |
| marepvval0 21169 | Second substitution for th... |
| marepvval 21170 | Third substitution for the... |
| marepveval 21171 | An entry of a matrix with ... |
| marepvcl 21172 | Closure of the column repl... |
| ma1repvcl 21173 | Closure of the column repl... |
| ma1repveval 21174 | An entry of an identity ma... |
| mulmarep1el 21175 | Element by element multipl... |
| mulmarep1gsum1 21176 | The sum of element by elem... |
| mulmarep1gsum2 21177 | The sum of element by elem... |
| 1marepvmarrepid 21178 | Replacing the ith row by 0... |
| submabas 21181 | Any subset of the index se... |
| submafval 21182 | First substitution for a s... |
| submaval0 21183 | Second substitution for a ... |
| submaval 21184 | Third substitution for a s... |
| submaeval 21185 | An entry of a submatrix of... |
| 1marepvsma1 21186 | The submatrix of the ident... |
| mdetfval 21189 | First substitution for the... |
| mdetleib 21190 | Full substitution of our d... |
| mdetleib2 21191 | Leibniz' formula can also ... |
| nfimdetndef 21192 | The determinant is not def... |
| mdetfval1 21193 | First substitution of an a... |
| mdetleib1 21194 | Full substitution of an al... |
| mdet0pr 21195 | The determinant for 0-dime... |
| mdet0f1o 21196 | The determinant for 0-dime... |
| mdet0fv0 21197 | The determinant of a 0-dim... |
| mdetf 21198 | Functionality of the deter... |
| mdetcl 21199 | The determinant evaluates ... |
| m1detdiag 21200 | The determinant of a 1-dim... |
| mdetdiaglem 21201 | Lemma for ~ mdetdiag . Pr... |
| mdetdiag 21202 | The determinant of a diago... |
| mdetdiagid 21203 | The determinant of a diago... |
| mdet1 21204 | The determinant of the ide... |
| mdetrlin 21205 | The determinant function i... |
| mdetrsca 21206 | The determinant function i... |
| mdetrsca2 21207 | The determinant function i... |
| mdetr0 21208 | The determinant of a matri... |
| mdet0 21209 | The determinant of the zer... |
| mdetrlin2 21210 | The determinant function i... |
| mdetralt 21211 | The determinant function i... |
| mdetralt2 21212 | The determinant function i... |
| mdetero 21213 | The determinant function i... |
| mdettpos 21214 | Determinant is invariant u... |
| mdetunilem1 21215 | Lemma for ~ mdetuni . (Co... |
| mdetunilem2 21216 | Lemma for ~ mdetuni . (Co... |
| mdetunilem3 21217 | Lemma for ~ mdetuni . (Co... |
| mdetunilem4 21218 | Lemma for ~ mdetuni . (Co... |
| mdetunilem5 21219 | Lemma for ~ mdetuni . (Co... |
| mdetunilem6 21220 | Lemma for ~ mdetuni . (Co... |
| mdetunilem7 21221 | Lemma for ~ mdetuni . (Co... |
| mdetunilem8 21222 | Lemma for ~ mdetuni . (Co... |
| mdetunilem9 21223 | Lemma for ~ mdetuni . (Co... |
| mdetuni0 21224 | Lemma for ~ mdetuni . (Co... |
| mdetuni 21225 | According to the definitio... |
| mdetmul 21226 | Multiplicativity of the de... |
| m2detleiblem1 21227 | Lemma 1 for ~ m2detleib . ... |
| m2detleiblem5 21228 | Lemma 5 for ~ m2detleib . ... |
| m2detleiblem6 21229 | Lemma 6 for ~ m2detleib . ... |
| m2detleiblem7 21230 | Lemma 7 for ~ m2detleib . ... |
| m2detleiblem2 21231 | Lemma 2 for ~ m2detleib . ... |
| m2detleiblem3 21232 | Lemma 3 for ~ m2detleib . ... |
| m2detleiblem4 21233 | Lemma 4 for ~ m2detleib . ... |
| m2detleib 21234 | Leibniz' Formula for 2x2-m... |
| mndifsplit 21239 | Lemma for ~ maducoeval2 . ... |
| madufval 21240 | First substitution for the... |
| maduval 21241 | Second substitution for th... |
| maducoeval 21242 | An entry of the adjunct (c... |
| maducoeval2 21243 | An entry of the adjunct (c... |
| maduf 21244 | Creating the adjunct of ma... |
| madutpos 21245 | The adjuct of a transposed... |
| madugsum 21246 | The determinant of a matri... |
| madurid 21247 | Multiplying a matrix with ... |
| madulid 21248 | Multiplying the adjunct of... |
| minmar1fval 21249 | First substitution for the... |
| minmar1val0 21250 | Second substitution for th... |
| minmar1val 21251 | Third substitution for the... |
| minmar1eval 21252 | An entry of a matrix for a... |
| minmar1marrep 21253 | The minor matrix is a spec... |
| minmar1cl 21254 | Closure of the row replace... |
| maducoevalmin1 21255 | The coefficients of an adj... |
| symgmatr01lem 21256 | Lemma for ~ symgmatr01 . ... |
| symgmatr01 21257 | Applying a permutation tha... |
| gsummatr01lem1 21258 | Lemma A for ~ gsummatr01 .... |
| gsummatr01lem2 21259 | Lemma B for ~ gsummatr01 .... |
| gsummatr01lem3 21260 | Lemma 1 for ~ gsummatr01 .... |
| gsummatr01lem4 21261 | Lemma 2 for ~ gsummatr01 .... |
| gsummatr01 21262 | Lemma 1 for ~ smadiadetlem... |
| marep01ma 21263 | Replacing a row of a squar... |
| smadiadetlem0 21264 | Lemma 0 for ~ smadiadet : ... |
| smadiadetlem1 21265 | Lemma 1 for ~ smadiadet : ... |
| smadiadetlem1a 21266 | Lemma 1a for ~ smadiadet :... |
| smadiadetlem2 21267 | Lemma 2 for ~ smadiadet : ... |
| smadiadetlem3lem0 21268 | Lemma 0 for ~ smadiadetlem... |
| smadiadetlem3lem1 21269 | Lemma 1 for ~ smadiadetlem... |
| smadiadetlem3lem2 21270 | Lemma 2 for ~ smadiadetlem... |
| smadiadetlem3 21271 | Lemma 3 for ~ smadiadet . ... |
| smadiadetlem4 21272 | Lemma 4 for ~ smadiadet . ... |
| smadiadet 21273 | The determinant of a subma... |
| smadiadetglem1 21274 | Lemma 1 for ~ smadiadetg .... |
| smadiadetglem2 21275 | Lemma 2 for ~ smadiadetg .... |
| smadiadetg 21276 | The determinant of a squar... |
| smadiadetg0 21277 | Lemma for ~ smadiadetr : v... |
| smadiadetr 21278 | The determinant of a squar... |
| invrvald 21279 | If a matrix multiplied wit... |
| matinv 21280 | The inverse of a matrix is... |
| matunit 21281 | A matrix is a unit in the ... |
| slesolvec 21282 | Every solution of a system... |
| slesolinv 21283 | The solution of a system o... |
| slesolinvbi 21284 | The solution of a system o... |
| slesolex 21285 | Every system of linear equ... |
| cramerimplem1 21286 | Lemma 1 for ~ cramerimp : ... |
| cramerimplem2 21287 | Lemma 2 for ~ cramerimp : ... |
| cramerimplem3 21288 | Lemma 3 for ~ cramerimp : ... |
| cramerimp 21289 | One direction of Cramer's ... |
| cramerlem1 21290 | Lemma 1 for ~ cramer . (C... |
| cramerlem2 21291 | Lemma 2 for ~ cramer . (C... |
| cramerlem3 21292 | Lemma 3 for ~ cramer . (C... |
| cramer0 21293 | Special case of Cramer's r... |
| cramer 21294 | Cramer's rule. According ... |
| pmatring 21295 | The set of polynomial matr... |
| pmatlmod 21296 | The set of polynomial matr... |
| pmat0op 21297 | The zero polynomial matrix... |
| pmat1op 21298 | The identity polynomial ma... |
| pmat1ovd 21299 | Entries of the identity po... |
| pmat0opsc 21300 | The zero polynomial matrix... |
| pmat1opsc 21301 | The identity polynomial ma... |
| pmat1ovscd 21302 | Entries of the identity po... |
| pmatcoe1fsupp 21303 | For a polynomial matrix th... |
| 1pmatscmul 21304 | The scalar product of the ... |
| cpmat 21311 | Value of the constructor o... |
| cpmatpmat 21312 | A constant polynomial matr... |
| cpmatel 21313 | Property of a constant pol... |
| cpmatelimp 21314 | Implication of a set being... |
| cpmatel2 21315 | Another property of a cons... |
| cpmatelimp2 21316 | Another implication of a s... |
| 1elcpmat 21317 | The identity of the ring o... |
| cpmatacl 21318 | The set of all constant po... |
| cpmatinvcl 21319 | The set of all constant po... |
| cpmatmcllem 21320 | Lemma for ~ cpmatmcl . (C... |
| cpmatmcl 21321 | The set of all constant po... |
| cpmatsubgpmat 21322 | The set of all constant po... |
| cpmatsrgpmat 21323 | The set of all constant po... |
| 0elcpmat 21324 | The zero of the ring of al... |
| mat2pmatfval 21325 | Value of the matrix transf... |
| mat2pmatval 21326 | The result of a matrix tra... |
| mat2pmatvalel 21327 | A (matrix) element of the ... |
| mat2pmatbas 21328 | The result of a matrix tra... |
| mat2pmatbas0 21329 | The result of a matrix tra... |
| mat2pmatf 21330 | The matrix transformation ... |
| mat2pmatf1 21331 | The matrix transformation ... |
| mat2pmatghm 21332 | The transformation of matr... |
| mat2pmatmul 21333 | The transformation of matr... |
| mat2pmat1 21334 | The transformation of the ... |
| mat2pmatmhm 21335 | The transformation of matr... |
| mat2pmatrhm 21336 | The transformation of matr... |
| mat2pmatlin 21337 | The transformation of matr... |
| 0mat2pmat 21338 | The transformed zero matri... |
| idmatidpmat 21339 | The transformed identity m... |
| d0mat2pmat 21340 | The transformed empty set ... |
| d1mat2pmat 21341 | The transformation of a ma... |
| mat2pmatscmxcl 21342 | A transformed matrix multi... |
| m2cpm 21343 | The result of a matrix tra... |
| m2cpmf 21344 | The matrix transformation ... |
| m2cpmf1 21345 | The matrix transformation ... |
| m2cpmghm 21346 | The transformation of matr... |
| m2cpmmhm 21347 | The transformation of matr... |
| m2cpmrhm 21348 | The transformation of matr... |
| m2pmfzmap 21349 | The transformed values of ... |
| m2pmfzgsumcl 21350 | Closure of the sum of scal... |
| cpm2mfval 21351 | Value of the inverse matri... |
| cpm2mval 21352 | The result of an inverse m... |
| cpm2mvalel 21353 | A (matrix) element of the ... |
| cpm2mf 21354 | The inverse matrix transfo... |
| m2cpminvid 21355 | The inverse transformation... |
| m2cpminvid2lem 21356 | Lemma for ~ m2cpminvid2 . ... |
| m2cpminvid2 21357 | The transformation applied... |
| m2cpmfo 21358 | The matrix transformation ... |
| m2cpmf1o 21359 | The matrix transformation ... |
| m2cpmrngiso 21360 | The transformation of matr... |
| matcpmric 21361 | The ring of matrices over ... |
| m2cpminv 21362 | The inverse matrix transfo... |
| m2cpminv0 21363 | The inverse matrix transfo... |
| decpmatval0 21366 | The matrix consisting of t... |
| decpmatval 21367 | The matrix consisting of t... |
| decpmate 21368 | An entry of the matrix con... |
| decpmatcl 21369 | Closure of the decompositi... |
| decpmataa0 21370 | The matrix consisting of t... |
| decpmatfsupp 21371 | The mapping to the matrice... |
| decpmatid 21372 | The matrix consisting of t... |
| decpmatmullem 21373 | Lemma for ~ decpmatmul . ... |
| decpmatmul 21374 | The matrix consisting of t... |
| decpmatmulsumfsupp 21375 | Lemma 0 for ~ pm2mpmhm . ... |
| pmatcollpw1lem1 21376 | Lemma 1 for ~ pmatcollpw1 ... |
| pmatcollpw1lem2 21377 | Lemma 2 for ~ pmatcollpw1 ... |
| pmatcollpw1 21378 | Write a polynomial matrix ... |
| pmatcollpw2lem 21379 | Lemma for ~ pmatcollpw2 . ... |
| pmatcollpw2 21380 | Write a polynomial matrix ... |
| monmatcollpw 21381 | The matrix consisting of t... |
| pmatcollpwlem 21382 | Lemma for ~ pmatcollpw . ... |
| pmatcollpw 21383 | Write a polynomial matrix ... |
| pmatcollpwfi 21384 | Write a polynomial matrix ... |
| pmatcollpw3lem 21385 | Lemma for ~ pmatcollpw3 an... |
| pmatcollpw3 21386 | Write a polynomial matrix ... |
| pmatcollpw3fi 21387 | Write a polynomial matrix ... |
| pmatcollpw3fi1lem1 21388 | Lemma 1 for ~ pmatcollpw3f... |
| pmatcollpw3fi1lem2 21389 | Lemma 2 for ~ pmatcollpw3f... |
| pmatcollpw3fi1 21390 | Write a polynomial matrix ... |
| pmatcollpwscmatlem1 21391 | Lemma 1 for ~ pmatcollpwsc... |
| pmatcollpwscmatlem2 21392 | Lemma 2 for ~ pmatcollpwsc... |
| pmatcollpwscmat 21393 | Write a scalar matrix over... |
| pm2mpf1lem 21396 | Lemma for ~ pm2mpf1 . (Co... |
| pm2mpval 21397 | Value of the transformatio... |
| pm2mpfval 21398 | A polynomial matrix transf... |
| pm2mpcl 21399 | The transformation of poly... |
| pm2mpf 21400 | The transformation of poly... |
| pm2mpf1 21401 | The transformation of poly... |
| pm2mpcoe1 21402 | A coefficient of the polyn... |
| idpm2idmp 21403 | The transformation of the ... |
| mptcoe1matfsupp 21404 | The mapping extracting the... |
| mply1topmatcllem 21405 | Lemma for ~ mply1topmatcl ... |
| mply1topmatval 21406 | A polynomial over matrices... |
| mply1topmatcl 21407 | A polynomial over matrices... |
| mp2pm2mplem1 21408 | Lemma 1 for ~ mp2pm2mp . ... |
| mp2pm2mplem2 21409 | Lemma 2 for ~ mp2pm2mp . ... |
| mp2pm2mplem3 21410 | Lemma 3 for ~ mp2pm2mp . ... |
| mp2pm2mplem4 21411 | Lemma 4 for ~ mp2pm2mp . ... |
| mp2pm2mplem5 21412 | Lemma 5 for ~ mp2pm2mp . ... |
| mp2pm2mp 21413 | A polynomial over matrices... |
| pm2mpghmlem2 21414 | Lemma 2 for ~ pm2mpghm . ... |
| pm2mpghmlem1 21415 | Lemma 1 for pm2mpghm . (C... |
| pm2mpfo 21416 | The transformation of poly... |
| pm2mpf1o 21417 | The transformation of poly... |
| pm2mpghm 21418 | The transformation of poly... |
| pm2mpgrpiso 21419 | The transformation of poly... |
| pm2mpmhmlem1 21420 | Lemma 1 for ~ pm2mpmhm . ... |
| pm2mpmhmlem2 21421 | Lemma 2 for ~ pm2mpmhm . ... |
| pm2mpmhm 21422 | The transformation of poly... |
| pm2mprhm 21423 | The transformation of poly... |
| pm2mprngiso 21424 | The transformation of poly... |
| pmmpric 21425 | The ring of polynomial mat... |
| monmat2matmon 21426 | The transformation of a po... |
| pm2mp 21427 | The transformation of a su... |
| chmatcl 21430 | Closure of the characteris... |
| chmatval 21431 | The entries of the charact... |
| chpmatfval 21432 | Value of the characteristi... |
| chpmatval 21433 | The characteristic polynom... |
| chpmatply1 21434 | The characteristic polynom... |
| chpmatval2 21435 | The characteristic polynom... |
| chpmat0d 21436 | The characteristic polynom... |
| chpmat1dlem 21437 | Lemma for ~ chpmat1d . (C... |
| chpmat1d 21438 | The characteristic polynom... |
| chpdmatlem0 21439 | Lemma 0 for ~ chpdmat . (... |
| chpdmatlem1 21440 | Lemma 1 for ~ chpdmat . (... |
| chpdmatlem2 21441 | Lemma 2 for ~ chpdmat . (... |
| chpdmatlem3 21442 | Lemma 3 for ~ chpdmat . (... |
| chpdmat 21443 | The characteristic polynom... |
| chpscmat 21444 | The characteristic polynom... |
| chpscmat0 21445 | The characteristic polynom... |
| chpscmatgsumbin 21446 | The characteristic polynom... |
| chpscmatgsummon 21447 | The characteristic polynom... |
| chp0mat 21448 | The characteristic polynom... |
| chpidmat 21449 | The characteristic polynom... |
| chmaidscmat 21450 | The characteristic polynom... |
| fvmptnn04if 21451 | The function values of a m... |
| fvmptnn04ifa 21452 | The function value of a ma... |
| fvmptnn04ifb 21453 | The function value of a ma... |
| fvmptnn04ifc 21454 | The function value of a ma... |
| fvmptnn04ifd 21455 | The function value of a ma... |
| chfacfisf 21456 | The "characteristic factor... |
| chfacfisfcpmat 21457 | The "characteristic factor... |
| chfacffsupp 21458 | The "characteristic factor... |
| chfacfscmulcl 21459 | Closure of a scaled value ... |
| chfacfscmul0 21460 | A scaled value of the "cha... |
| chfacfscmulfsupp 21461 | A mapping of scaled values... |
| chfacfscmulgsum 21462 | Breaking up a sum of value... |
| chfacfpmmulcl 21463 | Closure of the value of th... |
| chfacfpmmul0 21464 | The value of the "characte... |
| chfacfpmmulfsupp 21465 | A mapping of values of the... |
| chfacfpmmulgsum 21466 | Breaking up a sum of value... |
| chfacfpmmulgsum2 21467 | Breaking up a sum of value... |
| cayhamlem1 21468 | Lemma 1 for ~ cayleyhamilt... |
| cpmadurid 21469 | The right-hand fundamental... |
| cpmidgsum 21470 | Representation of the iden... |
| cpmidgsumm2pm 21471 | Representation of the iden... |
| cpmidpmatlem1 21472 | Lemma 1 for ~ cpmidpmat . ... |
| cpmidpmatlem2 21473 | Lemma 2 for ~ cpmidpmat . ... |
| cpmidpmatlem3 21474 | Lemma 3 for ~ cpmidpmat . ... |
| cpmidpmat 21475 | Representation of the iden... |
| cpmadugsumlemB 21476 | Lemma B for ~ cpmadugsum .... |
| cpmadugsumlemC 21477 | Lemma C for ~ cpmadugsum .... |
| cpmadugsumlemF 21478 | Lemma F for ~ cpmadugsum .... |
| cpmadugsumfi 21479 | The product of the charact... |
| cpmadugsum 21480 | The product of the charact... |
| cpmidgsum2 21481 | Representation of the iden... |
| cpmidg2sum 21482 | Equality of two sums repre... |
| cpmadumatpolylem1 21483 | Lemma 1 for ~ cpmadumatpol... |
| cpmadumatpolylem2 21484 | Lemma 2 for ~ cpmadumatpol... |
| cpmadumatpoly 21485 | The product of the charact... |
| cayhamlem2 21486 | Lemma for ~ cayhamlem3 . ... |
| chcoeffeqlem 21487 | Lemma for ~ chcoeffeq . (... |
| chcoeffeq 21488 | The coefficients of the ch... |
| cayhamlem3 21489 | Lemma for ~ cayhamlem4 . ... |
| cayhamlem4 21490 | Lemma for ~ cayleyhamilton... |
| cayleyhamilton0 21491 | The Cayley-Hamilton theore... |
| cayleyhamilton 21492 | The Cayley-Hamilton theore... |
| cayleyhamiltonALT 21493 | Alternate proof of ~ cayle... |
| cayleyhamilton1 21494 | The Cayley-Hamilton theore... |
| istopg 21497 | Express the predicate " ` ... |
| istop2g 21498 | Express the predicate " ` ... |
| uniopn 21499 | The union of a subset of a... |
| iunopn 21500 | The indexed union of a sub... |
| inopn 21501 | The intersection of two op... |
| fitop 21502 | A topology is closed under... |
| fiinopn 21503 | The intersection of a none... |
| iinopn 21504 | The intersection of a none... |
| unopn 21505 | The union of two open sets... |
| 0opn 21506 | The empty set is an open s... |
| 0ntop 21507 | The empty set is not a top... |
| topopn 21508 | The underlying set of a to... |
| eltopss 21509 | A member of a topology is ... |
| riinopn 21510 | A finite indexed relative ... |
| rintopn 21511 | A finite relative intersec... |
| istopon 21514 | Property of being a topolo... |
| topontop 21515 | A topology on a given base... |
| toponuni 21516 | The base set of a topology... |
| topontopi 21517 | A topology on a given base... |
| toponunii 21518 | The base set of a topology... |
| toptopon 21519 | Alternative definition of ... |
| toptopon2 21520 | A topology is the same thi... |
| topontopon 21521 | A topology on a set is a t... |
| funtopon 21522 | The class ` TopOn ` is a f... |
| toponrestid 21523 | Given a topology on a set,... |
| toponsspwpw 21524 | The set of topologies on a... |
| dmtopon 21525 | The domain of ` TopOn ` is... |
| fntopon 21526 | The class ` TopOn ` is a f... |
| toprntopon 21527 | A topology is the same thi... |
| toponmax 21528 | The base set of a topology... |
| toponss 21529 | A member of a topology is ... |
| toponcom 21530 | If ` K ` is a topology on ... |
| toponcomb 21531 | Biconditional form of ~ to... |
| topgele 21532 | The topologies over the sa... |
| topsn 21533 | The only topology on a sin... |
| istps 21536 | Express the predicate "is ... |
| istps2 21537 | Express the predicate "is ... |
| tpsuni 21538 | The base set of a topologi... |
| tpstop 21539 | The topology extractor on ... |
| tpspropd 21540 | A topological space depend... |
| tpsprop2d 21541 | A topological space depend... |
| topontopn 21542 | Express the predicate "is ... |
| tsettps 21543 | If the topology component ... |
| istpsi 21544 | Properties that determine ... |
| eltpsg 21545 | Properties that determine ... |
| eltpsi 21546 | Properties that determine ... |
| isbasisg 21549 | Express the predicate "the... |
| isbasis2g 21550 | Express the predicate "the... |
| isbasis3g 21551 | Express the predicate "the... |
| basis1 21552 | Property of a basis. (Con... |
| basis2 21553 | Property of a basis. (Con... |
| fiinbas 21554 | If a set is closed under f... |
| basdif0 21555 | A basis is not affected by... |
| baspartn 21556 | A disjoint system of sets ... |
| tgval 21557 | The topology generated by ... |
| tgval2 21558 | Definition of a topology g... |
| eltg 21559 | Membership in a topology g... |
| eltg2 21560 | Membership in a topology g... |
| eltg2b 21561 | Membership in a topology g... |
| eltg4i 21562 | An open set in a topology ... |
| eltg3i 21563 | The union of a set of basi... |
| eltg3 21564 | Membership in a topology g... |
| tgval3 21565 | Alternate expression for t... |
| tg1 21566 | Property of a member of a ... |
| tg2 21567 | Property of a member of a ... |
| bastg 21568 | A member of a basis is a s... |
| unitg 21569 | The topology generated by ... |
| tgss 21570 | Subset relation for genera... |
| tgcl 21571 | Show that a basis generate... |
| tgclb 21572 | The property ~ tgcl can be... |
| tgtopon 21573 | A basis generates a topolo... |
| topbas 21574 | A topology is its own basi... |
| tgtop 21575 | A topology is its own basi... |
| eltop 21576 | Membership in a topology, ... |
| eltop2 21577 | Membership in a topology. ... |
| eltop3 21578 | Membership in a topology. ... |
| fibas 21579 | A collection of finite int... |
| tgdom 21580 | A space has no more open s... |
| tgiun 21581 | The indexed union of a set... |
| tgidm 21582 | The topology generator fun... |
| bastop 21583 | Two ways to express that a... |
| tgtop11 21584 | The topology generation fu... |
| 0top 21585 | The singleton of the empty... |
| en1top 21586 | ` { (/) } ` is the only to... |
| en2top 21587 | If a topology has two elem... |
| tgss3 21588 | A criterion for determinin... |
| tgss2 21589 | A criterion for determinin... |
| basgen 21590 | Given a topology ` J ` , s... |
| basgen2 21591 | Given a topology ` J ` , s... |
| 2basgen 21592 | Conditions that determine ... |
| tgfiss 21593 | If a subbase is included i... |
| tgdif0 21594 | A generated topology is no... |
| bastop1 21595 | A subset of a topology is ... |
| bastop2 21596 | A version of ~ bastop1 tha... |
| distop 21597 | The discrete topology on a... |
| topnex 21598 | The class of all topologie... |
| distopon 21599 | The discrete topology on a... |
| sn0topon 21600 | The singleton of the empty... |
| sn0top 21601 | The singleton of the empty... |
| indislem 21602 | A lemma to eliminate some ... |
| indistopon 21603 | The indiscrete topology on... |
| indistop 21604 | The indiscrete topology on... |
| indisuni 21605 | The base set of the indisc... |
| fctop 21606 | The finite complement topo... |
| fctop2 21607 | The finite complement topo... |
| cctop 21608 | The countable complement t... |
| ppttop 21609 | The particular point topol... |
| pptbas 21610 | The particular point topol... |
| epttop 21611 | The excluded point topolog... |
| indistpsx 21612 | The indiscrete topology on... |
| indistps 21613 | The indiscrete topology on... |
| indistps2 21614 | The indiscrete topology on... |
| indistpsALT 21615 | The indiscrete topology on... |
| indistps2ALT 21616 | The indiscrete topology on... |
| distps 21617 | The discrete topology on a... |
| fncld 21624 | The closed-set generator i... |
| cldval 21625 | The set of closed sets of ... |
| ntrfval 21626 | The interior function on t... |
| clsfval 21627 | The closure function on th... |
| cldrcl 21628 | Reverse closure of the clo... |
| iscld 21629 | The predicate "the class `... |
| iscld2 21630 | A subset of the underlying... |
| cldss 21631 | A closed set is a subset o... |
| cldss2 21632 | The set of closed sets is ... |
| cldopn 21633 | The complement of a closed... |
| isopn2 21634 | A subset of the underlying... |
| opncld 21635 | The complement of an open ... |
| difopn 21636 | The difference of a closed... |
| topcld 21637 | The underlying set of a to... |
| ntrval 21638 | The interior of a subset o... |
| clsval 21639 | The closure of a subset of... |
| 0cld 21640 | The empty set is closed. ... |
| iincld 21641 | The indexed intersection o... |
| intcld 21642 | The intersection of a set ... |
| uncld 21643 | The union of two closed se... |
| cldcls 21644 | A closed subset equals its... |
| incld 21645 | The intersection of two cl... |
| riincld 21646 | An indexed relative inters... |
| iuncld 21647 | A finite indexed union of ... |
| unicld 21648 | A finite union of closed s... |
| clscld 21649 | The closure of a subset of... |
| clsf 21650 | The closure function is a ... |
| ntropn 21651 | The interior of a subset o... |
| clsval2 21652 | Express closure in terms o... |
| ntrval2 21653 | Interior expressed in term... |
| ntrdif 21654 | An interior of a complemen... |
| clsdif 21655 | A closure of a complement ... |
| clsss 21656 | Subset relationship for cl... |
| ntrss 21657 | Subset relationship for in... |
| sscls 21658 | A subset of a topology's u... |
| ntrss2 21659 | A subset includes its inte... |
| ssntr 21660 | An open subset of a set is... |
| clsss3 21661 | The closure of a subset of... |
| ntrss3 21662 | The interior of a subset o... |
| ntrin 21663 | A pairwise intersection of... |
| cmclsopn 21664 | The complement of a closur... |
| cmntrcld 21665 | The complement of an inter... |
| iscld3 21666 | A subset is closed iff it ... |
| iscld4 21667 | A subset is closed iff it ... |
| isopn3 21668 | A subset is open iff it eq... |
| clsidm 21669 | The closure operation is i... |
| ntridm 21670 | The interior operation is ... |
| clstop 21671 | The closure of a topology'... |
| ntrtop 21672 | The interior of a topology... |
| 0ntr 21673 | A subset with an empty int... |
| clsss2 21674 | If a subset is included in... |
| elcls 21675 | Membership in a closure. ... |
| elcls2 21676 | Membership in a closure. ... |
| clsndisj 21677 | Any open set containing a ... |
| ntrcls0 21678 | A subset whose closure has... |
| ntreq0 21679 | Two ways to say that a sub... |
| cldmre 21680 | The closed sets of a topol... |
| mrccls 21681 | Moore closure generalizes ... |
| cls0 21682 | The closure of the empty s... |
| ntr0 21683 | The interior of the empty ... |
| isopn3i 21684 | An open subset equals its ... |
| elcls3 21685 | Membership in a closure in... |
| opncldf1 21686 | A bijection useful for con... |
| opncldf2 21687 | The values of the open-clo... |
| opncldf3 21688 | The values of the converse... |
| isclo 21689 | A set ` A ` is clopen iff ... |
| isclo2 21690 | A set ` A ` is clopen iff ... |
| discld 21691 | The open sets of a discret... |
| sn0cld 21692 | The closed sets of the top... |
| indiscld 21693 | The closed sets of an indi... |
| mretopd 21694 | A Moore collection which i... |
| toponmre 21695 | The topologies over a give... |
| cldmreon 21696 | The closed sets of a topol... |
| iscldtop 21697 | A family is the closed set... |
| mreclatdemoBAD 21698 | The closed subspaces of a ... |
| neifval 21701 | Value of the neighborhood ... |
| neif 21702 | The neighborhood function ... |
| neiss2 21703 | A set with a neighborhood ... |
| neival 21704 | Value of the set of neighb... |
| isnei 21705 | The predicate "the class `... |
| neiint 21706 | An intuitive definition of... |
| isneip 21707 | The predicate "the class `... |
| neii1 21708 | A neighborhood is included... |
| neisspw 21709 | The neighborhoods of any s... |
| neii2 21710 | Property of a neighborhood... |
| neiss 21711 | Any neighborhood of a set ... |
| ssnei 21712 | A set is included in any o... |
| elnei 21713 | A point belongs to any of ... |
| 0nnei 21714 | The empty set is not a nei... |
| neips 21715 | A neighborhood of a set is... |
| opnneissb 21716 | An open set is a neighborh... |
| opnssneib 21717 | Any superset of an open se... |
| ssnei2 21718 | Any subset ` M ` of ` X ` ... |
| neindisj 21719 | Any neighborhood of an ele... |
| opnneiss 21720 | An open set is a neighborh... |
| opnneip 21721 | An open set is a neighborh... |
| opnnei 21722 | A set is open iff it is a ... |
| tpnei 21723 | The underlying set of a to... |
| neiuni 21724 | The union of the neighborh... |
| neindisj2 21725 | A point ` P ` belongs to t... |
| topssnei 21726 | A finer topology has more ... |
| innei 21727 | The intersection of two ne... |
| opnneiid 21728 | Only an open set is a neig... |
| neissex 21729 | For any neighborhood ` N `... |
| 0nei 21730 | The empty set is a neighbo... |
| neipeltop 21731 | Lemma for ~ neiptopreu . ... |
| neiptopuni 21732 | Lemma for ~ neiptopreu . ... |
| neiptoptop 21733 | Lemma for ~ neiptopreu . ... |
| neiptopnei 21734 | Lemma for ~ neiptopreu . ... |
| neiptopreu 21735 | If, to each element ` P ` ... |
| lpfval 21740 | The limit point function o... |
| lpval 21741 | The set of limit points of... |
| islp 21742 | The predicate "the class `... |
| lpsscls 21743 | The limit points of a subs... |
| lpss 21744 | The limit points of a subs... |
| lpdifsn 21745 | ` P ` is a limit point of ... |
| lpss3 21746 | Subset relationship for li... |
| islp2 21747 | The predicate " ` P ` is a... |
| islp3 21748 | The predicate " ` P ` is a... |
| maxlp 21749 | A point is a limit point o... |
| clslp 21750 | The closure of a subset of... |
| islpi 21751 | A point belonging to a set... |
| cldlp 21752 | A subset of a topological ... |
| isperf 21753 | Definition of a perfect sp... |
| isperf2 21754 | Definition of a perfect sp... |
| isperf3 21755 | A perfect space is a topol... |
| perflp 21756 | The limit points of a perf... |
| perfi 21757 | Property of a perfect spac... |
| perftop 21758 | A perfect space is a topol... |
| restrcl 21759 | Reverse closure for the su... |
| restbas 21760 | A subspace topology basis ... |
| tgrest 21761 | A subspace can be generate... |
| resttop 21762 | A subspace topology is a t... |
| resttopon 21763 | A subspace topology is a t... |
| restuni 21764 | The underlying set of a su... |
| stoig 21765 | The topological space buil... |
| restco 21766 | Composition of subspaces. ... |
| restabs 21767 | Equivalence of being a sub... |
| restin 21768 | When the subspace region i... |
| restuni2 21769 | The underlying set of a su... |
| resttopon2 21770 | The underlying set of a su... |
| rest0 21771 | The subspace topology indu... |
| restsn 21772 | The only subspace topology... |
| restsn2 21773 | The subspace topology indu... |
| restcld 21774 | A closed set of a subspace... |
| restcldi 21775 | A closed set is closed in ... |
| restcldr 21776 | A set which is closed in t... |
| restopnb 21777 | If ` B ` is an open subset... |
| ssrest 21778 | If ` K ` is a finer topolo... |
| restopn2 21779 | If ` A ` is open, then ` B... |
| restdis 21780 | A subspace of a discrete t... |
| restfpw 21781 | The restriction of the set... |
| neitr 21782 | The neighborhood of a trac... |
| restcls 21783 | A closure in a subspace to... |
| restntr 21784 | An interior in a subspace ... |
| restlp 21785 | The limit points of a subs... |
| restperf 21786 | Perfection of a subspace. ... |
| perfopn 21787 | An open subset of a perfec... |
| resstopn 21788 | The topology of a restrict... |
| resstps 21789 | A restricted topological s... |
| ordtbaslem 21790 | Lemma for ~ ordtbas . In ... |
| ordtval 21791 | Value of the order topolog... |
| ordtuni 21792 | Value of the order topolog... |
| ordtbas2 21793 | Lemma for ~ ordtbas . (Co... |
| ordtbas 21794 | In a total order, the fini... |
| ordttopon 21795 | Value of the order topolog... |
| ordtopn1 21796 | An upward ray ` ( P , +oo ... |
| ordtopn2 21797 | A downward ray ` ( -oo , P... |
| ordtopn3 21798 | An open interval ` ( A , B... |
| ordtcld1 21799 | A downward ray ` ( -oo , P... |
| ordtcld2 21800 | An upward ray ` [ P , +oo ... |
| ordtcld3 21801 | A closed interval ` [ A , ... |
| ordttop 21802 | The order topology is a to... |
| ordtcnv 21803 | The order dual generates t... |
| ordtrest 21804 | The subspace topology of a... |
| ordtrest2lem 21805 | Lemma for ~ ordtrest2 . (... |
| ordtrest2 21806 | An interval-closed set ` A... |
| letopon 21807 | The topology of the extend... |
| letop 21808 | The topology of the extend... |
| letopuni 21809 | The topology of the extend... |
| xrstopn 21810 | The topology component of ... |
| xrstps 21811 | The extended real number s... |
| leordtvallem1 21812 | Lemma for ~ leordtval . (... |
| leordtvallem2 21813 | Lemma for ~ leordtval . (... |
| leordtval2 21814 | The topology of the extend... |
| leordtval 21815 | The topology of the extend... |
| iccordt 21816 | A closed interval is close... |
| iocpnfordt 21817 | An unbounded above open in... |
| icomnfordt 21818 | An unbounded above open in... |
| iooordt 21819 | An open interval is open i... |
| reordt 21820 | The real numbers are an op... |
| lecldbas 21821 | The set of closed interval... |
| pnfnei 21822 | A neighborhood of ` +oo ` ... |
| mnfnei 21823 | A neighborhood of ` -oo ` ... |
| ordtrestixx 21824 | The restriction of the les... |
| ordtresticc 21825 | The restriction of the les... |
| lmrel 21832 | The topological space conv... |
| lmrcl 21833 | Reverse closure for the co... |
| lmfval 21834 | The relation "sequence ` f... |
| cnfval 21835 | The set of all continuous ... |
| cnpfval 21836 | The function mapping the p... |
| iscn 21837 | The predicate "the class `... |
| cnpval 21838 | The set of all functions f... |
| iscnp 21839 | The predicate "the class `... |
| iscn2 21840 | The predicate "the class `... |
| iscnp2 21841 | The predicate "the class `... |
| cntop1 21842 | Reverse closure for a cont... |
| cntop2 21843 | Reverse closure for a cont... |
| cnptop1 21844 | Reverse closure for a func... |
| cnptop2 21845 | Reverse closure for a func... |
| iscnp3 21846 | The predicate "the class `... |
| cnprcl 21847 | Reverse closure for a func... |
| cnf 21848 | A continuous function is a... |
| cnpf 21849 | A continuous function at p... |
| cnpcl 21850 | The value of a continuous ... |
| cnf2 21851 | A continuous function is a... |
| cnpf2 21852 | A continuous function at p... |
| cnprcl2 21853 | Reverse closure for a func... |
| tgcn 21854 | The continuity predicate w... |
| tgcnp 21855 | The "continuous at a point... |
| subbascn 21856 | The continuity predicate w... |
| ssidcn 21857 | The identity function is a... |
| cnpimaex 21858 | Property of a function con... |
| idcn 21859 | A restricted identity func... |
| lmbr 21860 | Express the binary relatio... |
| lmbr2 21861 | Express the binary relatio... |
| lmbrf 21862 | Express the binary relatio... |
| lmconst 21863 | A constant sequence conver... |
| lmcvg 21864 | Convergence property of a ... |
| iscnp4 21865 | The predicate "the class `... |
| cnpnei 21866 | A condition for continuity... |
| cnima 21867 | An open subset of the codo... |
| cnco 21868 | The composition of two con... |
| cnpco 21869 | The composition of a funct... |
| cnclima 21870 | A closed subset of the cod... |
| iscncl 21871 | A characterization of a co... |
| cncls2i 21872 | Property of the preimage o... |
| cnntri 21873 | Property of the preimage o... |
| cnclsi 21874 | Property of the image of a... |
| cncls2 21875 | Continuity in terms of clo... |
| cncls 21876 | Continuity in terms of clo... |
| cnntr 21877 | Continuity in terms of int... |
| cnss1 21878 | If the topology ` K ` is f... |
| cnss2 21879 | If the topology ` K ` is f... |
| cncnpi 21880 | A continuous function is c... |
| cnsscnp 21881 | The set of continuous func... |
| cncnp 21882 | A continuous function is c... |
| cncnp2 21883 | A continuous function is c... |
| cnnei 21884 | Continuity in terms of nei... |
| cnconst2 21885 | A constant function is con... |
| cnconst 21886 | A constant function is con... |
| cnrest 21887 | Continuity of a restrictio... |
| cnrest2 21888 | Equivalence of continuity ... |
| cnrest2r 21889 | Equivalence of continuity ... |
| cnpresti 21890 | One direction of ~ cnprest... |
| cnprest 21891 | Equivalence of continuity ... |
| cnprest2 21892 | Equivalence of point-conti... |
| cndis 21893 | Every function is continuo... |
| cnindis 21894 | Every function is continuo... |
| cnpdis 21895 | If ` A ` is an isolated po... |
| paste 21896 | Pasting lemma. If ` A ` a... |
| lmfpm 21897 | If ` F ` converges, then `... |
| lmfss 21898 | Inclusion of a function ha... |
| lmcl 21899 | Closure of a limit. (Cont... |
| lmss 21900 | Limit on a subspace. (Con... |
| sslm 21901 | A finer topology has fewer... |
| lmres 21902 | A function converges iff i... |
| lmff 21903 | If ` F ` converges, there ... |
| lmcls 21904 | Any convergent sequence of... |
| lmcld 21905 | Any convergent sequence of... |
| lmcnp 21906 | The image of a convergent ... |
| lmcn 21907 | The image of a convergent ... |
| ist0 21922 | The predicate "is a T_0 sp... |
| ist1 21923 | The predicate "is a T_1 sp... |
| ishaus 21924 | The predicate "is a Hausdo... |
| iscnrm 21925 | The property of being comp... |
| t0sep 21926 | Any two topologically indi... |
| t0dist 21927 | Any two distinct points in... |
| t1sncld 21928 | In a T_1 space, singletons... |
| t1ficld 21929 | In a T_1 space, finite set... |
| hausnei 21930 | Neighborhood property of a... |
| t0top 21931 | A T_0 space is a topologic... |
| t1top 21932 | A T_1 space is a topologic... |
| haustop 21933 | A Hausdorff space is a top... |
| isreg 21934 | The predicate "is a regula... |
| regtop 21935 | A regular space is a topol... |
| regsep 21936 | In a regular space, every ... |
| isnrm 21937 | The predicate "is a normal... |
| nrmtop 21938 | A normal space is a topolo... |
| cnrmtop 21939 | A completely normal space ... |
| iscnrm2 21940 | The property of being comp... |
| ispnrm 21941 | The property of being perf... |
| pnrmnrm 21942 | A perfectly normal space i... |
| pnrmtop 21943 | A perfectly normal space i... |
| pnrmcld 21944 | A closed set in a perfectl... |
| pnrmopn 21945 | An open set in a perfectly... |
| ist0-2 21946 | The predicate "is a T_0 sp... |
| ist0-3 21947 | The predicate "is a T_0 sp... |
| cnt0 21948 | The preimage of a T_0 topo... |
| ist1-2 21949 | An alternate characterizat... |
| t1t0 21950 | A T_1 space is a T_0 space... |
| ist1-3 21951 | A space is T_1 iff every p... |
| cnt1 21952 | The preimage of a T_1 topo... |
| ishaus2 21953 | Express the predicate " ` ... |
| haust1 21954 | A Hausdorff space is a T_1... |
| hausnei2 21955 | The Hausdorff condition st... |
| cnhaus 21956 | The preimage of a Hausdorf... |
| nrmsep3 21957 | In a normal space, given a... |
| nrmsep2 21958 | In a normal space, any two... |
| nrmsep 21959 | In a normal space, disjoin... |
| isnrm2 21960 | An alternate characterizat... |
| isnrm3 21961 | A topological space is nor... |
| cnrmi 21962 | A subspace of a completely... |
| cnrmnrm 21963 | A completely normal space ... |
| restcnrm 21964 | A subspace of a completely... |
| resthauslem 21965 | Lemma for ~ resthaus and s... |
| lpcls 21966 | The limit points of the cl... |
| perfcls 21967 | A subset of a perfect spac... |
| restt0 21968 | A subspace of a T_0 topolo... |
| restt1 21969 | A subspace of a T_1 topolo... |
| resthaus 21970 | A subspace of a Hausdorff ... |
| t1sep2 21971 | Any two points in a T_1 sp... |
| t1sep 21972 | Any two distinct points in... |
| sncld 21973 | A singleton is closed in a... |
| sshauslem 21974 | Lemma for ~ sshaus and sim... |
| sst0 21975 | A topology finer than a T_... |
| sst1 21976 | A topology finer than a T_... |
| sshaus 21977 | A topology finer than a Ha... |
| regsep2 21978 | In a regular space, a clos... |
| isreg2 21979 | A topological space is reg... |
| dnsconst 21980 | If a continuous mapping to... |
| ordtt1 21981 | The order topology is T_1 ... |
| lmmo 21982 | A sequence in a Hausdorff ... |
| lmfun 21983 | The convergence relation i... |
| dishaus 21984 | A discrete topology is Hau... |
| ordthauslem 21985 | Lemma for ~ ordthaus . (C... |
| ordthaus 21986 | The order topology of a to... |
| xrhaus 21987 | The topology of the extend... |
| iscmp 21990 | The predicate "is a compac... |
| cmpcov 21991 | An open cover of a compact... |
| cmpcov2 21992 | Rewrite ~ cmpcov for the c... |
| cmpcovf 21993 | Combine ~ cmpcov with ~ ac... |
| cncmp 21994 | Compactness is respected b... |
| fincmp 21995 | A finite topology is compa... |
| 0cmp 21996 | The singleton of the empty... |
| cmptop 21997 | A compact topology is a to... |
| rncmp 21998 | The image of a compact set... |
| imacmp 21999 | The image of a compact set... |
| discmp 22000 | A discrete topology is com... |
| cmpsublem 22001 | Lemma for ~ cmpsub . (Con... |
| cmpsub 22002 | Two equivalent ways of des... |
| tgcmp 22003 | A topology generated by a ... |
| cmpcld 22004 | A closed subset of a compa... |
| uncmp 22005 | The union of two compact s... |
| fiuncmp 22006 | A finite union of compact ... |
| sscmp 22007 | A subset of a compact topo... |
| hauscmplem 22008 | Lemma for ~ hauscmp . (Co... |
| hauscmp 22009 | A compact subspace of a T2... |
| cmpfi 22010 | If a topology is compact a... |
| cmpfii 22011 | In a compact topology, a s... |
| bwth 22012 | The glorious Bolzano-Weier... |
| isconn 22015 | The predicate ` J ` is a c... |
| isconn2 22016 | The predicate ` J ` is a c... |
| connclo 22017 | The only nonempty clopen s... |
| conndisj 22018 | If a topology is connected... |
| conntop 22019 | A connected topology is a ... |
| indisconn 22020 | The indiscrete topology (o... |
| dfconn2 22021 | An alternate definition of... |
| connsuba 22022 | Connectedness for a subspa... |
| connsub 22023 | Two equivalent ways of say... |
| cnconn 22024 | Connectedness is respected... |
| nconnsubb 22025 | Disconnectedness for a sub... |
| connsubclo 22026 | If a clopen set meets a co... |
| connima 22027 | The image of a connected s... |
| conncn 22028 | A continuous function from... |
| iunconnlem 22029 | Lemma for ~ iunconn . (Co... |
| iunconn 22030 | The indexed union of conne... |
| unconn 22031 | The union of two connected... |
| clsconn 22032 | The closure of a connected... |
| conncompid 22033 | The connected component co... |
| conncompconn 22034 | The connected component co... |
| conncompss 22035 | The connected component co... |
| conncompcld 22036 | The connected component co... |
| conncompclo 22037 | The connected component co... |
| t1connperf 22038 | A connected T_1 space is p... |
| is1stc 22043 | The predicate "is a first-... |
| is1stc2 22044 | An equivalent way of sayin... |
| 1stctop 22045 | A first-countable topology... |
| 1stcclb 22046 | A property of points in a ... |
| 1stcfb 22047 | For any point ` A ` in a f... |
| is2ndc 22048 | The property of being seco... |
| 2ndctop 22049 | A second-countable topolog... |
| 2ndci 22050 | A countable basis generate... |
| 2ndcsb 22051 | Having a countable subbase... |
| 2ndcredom 22052 | A second-countable space h... |
| 2ndc1stc 22053 | A second-countable space i... |
| 1stcrestlem 22054 | Lemma for ~ 1stcrest . (C... |
| 1stcrest 22055 | A subspace of a first-coun... |
| 2ndcrest 22056 | A subspace of a second-cou... |
| 2ndcctbss 22057 | If a topology is second-co... |
| 2ndcdisj 22058 | Any disjoint family of ope... |
| 2ndcdisj2 22059 | Any disjoint collection of... |
| 2ndcomap 22060 | A surjective continuous op... |
| 2ndcsep 22061 | A second-countable topolog... |
| dis2ndc 22062 | A discrete space is second... |
| 1stcelcls 22063 | A point belongs to the clo... |
| 1stccnp 22064 | A mapping is continuous at... |
| 1stccn 22065 | A mapping ` X --> Y ` , wh... |
| islly 22070 | The property of being a lo... |
| isnlly 22071 | The property of being an n... |
| llyeq 22072 | Equality theorem for the `... |
| nllyeq 22073 | Equality theorem for the `... |
| llytop 22074 | A locally ` A ` space is a... |
| nllytop 22075 | A locally ` A ` space is a... |
| llyi 22076 | The property of a locally ... |
| nllyi 22077 | The property of an n-local... |
| nlly2i 22078 | Eliminate the neighborhood... |
| llynlly 22079 | A locally ` A ` space is n... |
| llyssnlly 22080 | A locally ` A ` space is n... |
| llyss 22081 | The "locally" predicate re... |
| nllyss 22082 | The "n-locally" predicate ... |
| subislly 22083 | The property of a subspace... |
| restnlly 22084 | If the property ` A ` pass... |
| restlly 22085 | If the property ` A ` pass... |
| islly2 22086 | An alternative expression ... |
| llyrest 22087 | An open subspace of a loca... |
| nllyrest 22088 | An open subspace of an n-l... |
| loclly 22089 | If ` A ` is a local proper... |
| llyidm 22090 | Idempotence of the "locall... |
| nllyidm 22091 | Idempotence of the "n-loca... |
| toplly 22092 | A topology is locally a to... |
| topnlly 22093 | A topology is n-locally a ... |
| hauslly 22094 | A Hausdorff space is local... |
| hausnlly 22095 | A Hausdorff space is n-loc... |
| hausllycmp 22096 | A compact Hausdorff space ... |
| cldllycmp 22097 | A closed subspace of a loc... |
| lly1stc 22098 | First-countability is a lo... |
| dislly 22099 | The discrete space ` ~P X ... |
| disllycmp 22100 | A discrete space is locall... |
| dis1stc 22101 | A discrete space is first-... |
| hausmapdom 22102 | If ` X ` is a first-counta... |
| hauspwdom 22103 | Simplify the cardinal ` A ... |
| refrel 22110 | Refinement is a relation. ... |
| isref 22111 | The property of being a re... |
| refbas 22112 | A refinement covers the sa... |
| refssex 22113 | Every set in a refinement ... |
| ssref 22114 | A subcover is a refinement... |
| refref 22115 | Reflexivity of refinement.... |
| reftr 22116 | Refinement is transitive. ... |
| refun0 22117 | Adding the empty set prese... |
| isptfin 22118 | The statement "is a point-... |
| islocfin 22119 | The statement "is a locall... |
| finptfin 22120 | A finite cover is a point-... |
| ptfinfin 22121 | A point covered by a point... |
| finlocfin 22122 | A finite cover of a topolo... |
| locfintop 22123 | A locally finite cover cov... |
| locfinbas 22124 | A locally finite cover mus... |
| locfinnei 22125 | A point covered by a local... |
| lfinpfin 22126 | A locally finite cover is ... |
| lfinun 22127 | Adding a finite set preser... |
| locfincmp 22128 | For a compact space, the l... |
| unisngl 22129 | Taking the union of the se... |
| dissnref 22130 | The set of singletons is a... |
| dissnlocfin 22131 | The set of singletons is l... |
| locfindis 22132 | The locally finite covers ... |
| locfincf 22133 | A locally finite cover in ... |
| comppfsc 22134 | A space where every open c... |
| kgenval 22137 | Value of the compact gener... |
| elkgen 22138 | Value of the compact gener... |
| kgeni 22139 | Property of the open sets ... |
| kgentopon 22140 | The compact generator gene... |
| kgenuni 22141 | The base set of the compac... |
| kgenftop 22142 | The compact generator gene... |
| kgenf 22143 | The compact generator is a... |
| kgentop 22144 | A compactly generated spac... |
| kgenss 22145 | The compact generator gene... |
| kgenhaus 22146 | The compact generator gene... |
| kgencmp 22147 | The compact generator topo... |
| kgencmp2 22148 | The compact generator topo... |
| kgenidm 22149 | The compact generator is i... |
| iskgen2 22150 | A space is compactly gener... |
| iskgen3 22151 | Derive the usual definitio... |
| llycmpkgen2 22152 | A locally compact space is... |
| cmpkgen 22153 | A compact space is compact... |
| llycmpkgen 22154 | A locally compact space is... |
| 1stckgenlem 22155 | The one-point compactifica... |
| 1stckgen 22156 | A first-countable space is... |
| kgen2ss 22157 | The compact generator pres... |
| kgencn 22158 | A function from a compactl... |
| kgencn2 22159 | A function ` F : J --> K `... |
| kgencn3 22160 | The set of continuous func... |
| kgen2cn 22161 | A continuous function is a... |
| txval 22166 | Value of the binary topolo... |
| txuni2 22167 | The underlying set of the ... |
| txbasex 22168 | The basis for the product ... |
| txbas 22169 | The set of Cartesian produ... |
| eltx 22170 | A set in a product is open... |
| txtop 22171 | The product of two topolog... |
| ptval 22172 | The value of the product t... |
| ptpjpre1 22173 | The preimage of a projecti... |
| elpt 22174 | Elementhood in the bases o... |
| elptr 22175 | A basic open set in the pr... |
| elptr2 22176 | A basic open set in the pr... |
| ptbasid 22177 | The base set of the produc... |
| ptuni2 22178 | The base set for the produ... |
| ptbasin 22179 | The basis for a product to... |
| ptbasin2 22180 | The basis for a product to... |
| ptbas 22181 | The basis for a product to... |
| ptpjpre2 22182 | The basis for a product to... |
| ptbasfi 22183 | The basis for the product ... |
| pttop 22184 | The product topology is a ... |
| ptopn 22185 | A basic open set in the pr... |
| ptopn2 22186 | A sub-basic open set in th... |
| xkotf 22187 | Functionality of function ... |
| xkobval 22188 | Alternative expression for... |
| xkoval 22189 | Value of the compact-open ... |
| xkotop 22190 | The compact-open topology ... |
| xkoopn 22191 | A basic open set of the co... |
| txtopi 22192 | The product of two topolog... |
| txtopon 22193 | The underlying set of the ... |
| txuni 22194 | The underlying set of the ... |
| txunii 22195 | The underlying set of the ... |
| ptuni 22196 | The base set for the produ... |
| ptunimpt 22197 | Base set of a product topo... |
| pttopon 22198 | The base set for the produ... |
| pttoponconst 22199 | The base set for a product... |
| ptuniconst 22200 | The base set for a product... |
| xkouni 22201 | The base set of the compac... |
| xkotopon 22202 | The base set of the compac... |
| ptval2 22203 | The value of the product t... |
| txopn 22204 | The product of two open se... |
| txcld 22205 | The product of two closed ... |
| txcls 22206 | Closure of a rectangle in ... |
| txss12 22207 | Subset property of the top... |
| txbasval 22208 | It is sufficient to consid... |
| neitx 22209 | The Cartesian product of t... |
| txcnpi 22210 | Continuity of a two-argume... |
| tx1cn 22211 | Continuity of the first pr... |
| tx2cn 22212 | Continuity of the second p... |
| ptpjcn 22213 | Continuity of a projection... |
| ptpjopn 22214 | The projection map is an o... |
| ptcld 22215 | A closed box in the produc... |
| ptcldmpt 22216 | A closed box in the produc... |
| ptclsg 22217 | The closure of a box in th... |
| ptcls 22218 | The closure of a box in th... |
| dfac14lem 22219 | Lemma for ~ dfac14 . By e... |
| dfac14 22220 | Theorem ~ ptcls is an equi... |
| xkoccn 22221 | The "constant function" fu... |
| txcnp 22222 | If two functions are conti... |
| ptcnplem 22223 | Lemma for ~ ptcnp . (Cont... |
| ptcnp 22224 | If every projection of a f... |
| upxp 22225 | Universal property of the ... |
| txcnmpt 22226 | A map into the product of ... |
| uptx 22227 | Universal property of the ... |
| txcn 22228 | A map into the product of ... |
| ptcn 22229 | If every projection of a f... |
| prdstopn 22230 | Topology of a structure pr... |
| prdstps 22231 | A structure product of top... |
| pwstps 22232 | A structure power of a top... |
| txrest 22233 | The subspace of a topologi... |
| txdis 22234 | The topological product of... |
| txindislem 22235 | Lemma for ~ txindis . (Co... |
| txindis 22236 | The topological product of... |
| txdis1cn 22237 | A function is jointly cont... |
| txlly 22238 | If the property ` A ` is p... |
| txnlly 22239 | If the property ` A ` is p... |
| pthaus 22240 | The product of a collectio... |
| ptrescn 22241 | Restriction is a continuou... |
| txtube 22242 | The "tube lemma". If ` X ... |
| txcmplem1 22243 | Lemma for ~ txcmp . (Cont... |
| txcmplem2 22244 | Lemma for ~ txcmp . (Cont... |
| txcmp 22245 | The topological product of... |
| txcmpb 22246 | The topological product of... |
| hausdiag 22247 | A topology is Hausdorff if... |
| hauseqlcld 22248 | In a Hausdorff topology, t... |
| txhaus 22249 | The topological product of... |
| txlm 22250 | Two sequences converge iff... |
| lmcn2 22251 | The image of a convergent ... |
| tx1stc 22252 | The topological product of... |
| tx2ndc 22253 | The topological product of... |
| txkgen 22254 | The topological product of... |
| xkohaus 22255 | If the codomain space is H... |
| xkoptsub 22256 | The compact-open topology ... |
| xkopt 22257 | The compact-open topology ... |
| xkopjcn 22258 | Continuity of a projection... |
| xkoco1cn 22259 | If ` F ` is a continuous f... |
| xkoco2cn 22260 | If ` F ` is a continuous f... |
| xkococnlem 22261 | Continuity of the composit... |
| xkococn 22262 | Continuity of the composit... |
| cnmptid 22263 | The identity function is c... |
| cnmptc 22264 | A constant function is con... |
| cnmpt11 22265 | The composition of continu... |
| cnmpt11f 22266 | The composition of continu... |
| cnmpt1t 22267 | The composition of continu... |
| cnmpt12f 22268 | The composition of continu... |
| cnmpt12 22269 | The composition of continu... |
| cnmpt1st 22270 | The projection onto the fi... |
| cnmpt2nd 22271 | The projection onto the se... |
| cnmpt2c 22272 | A constant function is con... |
| cnmpt21 22273 | The composition of continu... |
| cnmpt21f 22274 | The composition of continu... |
| cnmpt2t 22275 | The composition of continu... |
| cnmpt22 22276 | The composition of continu... |
| cnmpt22f 22277 | The composition of continu... |
| cnmpt1res 22278 | The restriction of a conti... |
| cnmpt2res 22279 | The restriction of a conti... |
| cnmptcom 22280 | The argument converse of a... |
| cnmptkc 22281 | The curried first projecti... |
| cnmptkp 22282 | The evaluation of the inne... |
| cnmptk1 22283 | The composition of a curri... |
| cnmpt1k 22284 | The composition of a one-a... |
| cnmptkk 22285 | The composition of two cur... |
| xkofvcn 22286 | Joint continuity of the fu... |
| cnmptk1p 22287 | The evaluation of a currie... |
| cnmptk2 22288 | The uncurrying of a currie... |
| xkoinjcn 22289 | Continuity of "injection",... |
| cnmpt2k 22290 | The currying of a two-argu... |
| txconn 22291 | The topological product of... |
| imasnopn 22292 | If a relation graph is ope... |
| imasncld 22293 | If a relation graph is clo... |
| imasncls 22294 | If a relation graph is clo... |
| qtopval 22297 | Value of the quotient topo... |
| qtopval2 22298 | Value of the quotient topo... |
| elqtop 22299 | Value of the quotient topo... |
| qtopres 22300 | The quotient topology is u... |
| qtoptop2 22301 | The quotient topology is a... |
| qtoptop 22302 | The quotient topology is a... |
| elqtop2 22303 | Value of the quotient topo... |
| qtopuni 22304 | The base set of the quotie... |
| elqtop3 22305 | Value of the quotient topo... |
| qtoptopon 22306 | The base set of the quotie... |
| qtopid 22307 | A quotient map is a contin... |
| idqtop 22308 | The quotient topology indu... |
| qtopcmplem 22309 | Lemma for ~ qtopcmp and ~ ... |
| qtopcmp 22310 | A quotient of a compact sp... |
| qtopconn 22311 | A quotient of a connected ... |
| qtopkgen 22312 | A quotient of a compactly ... |
| basqtop 22313 | An injection maps bases to... |
| tgqtop 22314 | An injection maps generate... |
| qtopcld 22315 | The property of being a cl... |
| qtopcn 22316 | Universal property of a qu... |
| qtopss 22317 | A surjective continuous fu... |
| qtopeu 22318 | Universal property of the ... |
| qtoprest 22319 | If ` A ` is a saturated op... |
| qtopomap 22320 | If ` F ` is a surjective c... |
| qtopcmap 22321 | If ` F ` is a surjective c... |
| imastopn 22322 | The topology of an image s... |
| imastps 22323 | The image of a topological... |
| qustps 22324 | A quotient structure is a ... |
| kqfval 22325 | Value of the function appe... |
| kqfeq 22326 | Two points in the Kolmogor... |
| kqffn 22327 | The topological indistingu... |
| kqval 22328 | Value of the quotient topo... |
| kqtopon 22329 | The Kolmogorov quotient is... |
| kqid 22330 | The topological indistingu... |
| ist0-4 22331 | The topological indistingu... |
| kqfvima 22332 | When the image set is open... |
| kqsat 22333 | Any open set is saturated ... |
| kqdisj 22334 | A version of ~ imain for t... |
| kqcldsat 22335 | Any closed set is saturate... |
| kqopn 22336 | The topological indistingu... |
| kqcld 22337 | The topological indistingu... |
| kqt0lem 22338 | Lemma for ~ kqt0 . (Contr... |
| isr0 22339 | The property " ` J ` is an... |
| r0cld 22340 | The analogue of the T_1 ax... |
| regr1lem 22341 | Lemma for ~ regr1 . (Cont... |
| regr1lem2 22342 | A Kolmogorov quotient of a... |
| kqreglem1 22343 | A Kolmogorov quotient of a... |
| kqreglem2 22344 | If the Kolmogorov quotient... |
| kqnrmlem1 22345 | A Kolmogorov quotient of a... |
| kqnrmlem2 22346 | If the Kolmogorov quotient... |
| kqtop 22347 | The Kolmogorov quotient is... |
| kqt0 22348 | The Kolmogorov quotient is... |
| kqf 22349 | The Kolmogorov quotient is... |
| r0sep 22350 | The separation property of... |
| nrmr0reg 22351 | A normal R_0 space is also... |
| regr1 22352 | A regular space is R_1, wh... |
| kqreg 22353 | The Kolmogorov quotient of... |
| kqnrm 22354 | The Kolmogorov quotient of... |
| hmeofn 22359 | The set of homeomorphisms ... |
| hmeofval 22360 | The set of all the homeomo... |
| ishmeo 22361 | The predicate F is a homeo... |
| hmeocn 22362 | A homeomorphism is continu... |
| hmeocnvcn 22363 | The converse of a homeomor... |
| hmeocnv 22364 | The converse of a homeomor... |
| hmeof1o2 22365 | A homeomorphism is a 1-1-o... |
| hmeof1o 22366 | A homeomorphism is a 1-1-o... |
| hmeoima 22367 | The image of an open set b... |
| hmeoopn 22368 | Homeomorphisms preserve op... |
| hmeocld 22369 | Homeomorphisms preserve cl... |
| hmeocls 22370 | Homeomorphisms preserve cl... |
| hmeontr 22371 | Homeomorphisms preserve in... |
| hmeoimaf1o 22372 | The function mapping open ... |
| hmeores 22373 | The restriction of a homeo... |
| hmeoco 22374 | The composite of two homeo... |
| idhmeo 22375 | The identity function is a... |
| hmeocnvb 22376 | The converse of a homeomor... |
| hmeoqtop 22377 | A homeomorphism is a quoti... |
| hmph 22378 | Express the predicate ` J ... |
| hmphi 22379 | If there is a homeomorphis... |
| hmphtop 22380 | Reverse closure for the ho... |
| hmphtop1 22381 | The relation "being homeom... |
| hmphtop2 22382 | The relation "being homeom... |
| hmphref 22383 | "Is homeomorphic to" is re... |
| hmphsym 22384 | "Is homeomorphic to" is sy... |
| hmphtr 22385 | "Is homeomorphic to" is tr... |
| hmpher 22386 | "Is homeomorphic to" is an... |
| hmphen 22387 | Homeomorphisms preserve th... |
| hmphsymb 22388 | "Is homeomorphic to" is sy... |
| haushmphlem 22389 | Lemma for ~ haushmph and s... |
| cmphmph 22390 | Compactness is a topologic... |
| connhmph 22391 | Connectedness is a topolog... |
| t0hmph 22392 | T_0 is a topological prope... |
| t1hmph 22393 | T_1 is a topological prope... |
| haushmph 22394 | Hausdorff-ness is a topolo... |
| reghmph 22395 | Regularity is a topologica... |
| nrmhmph 22396 | Normality is a topological... |
| hmph0 22397 | A topology homeomorphic to... |
| hmphdis 22398 | Homeomorphisms preserve to... |
| hmphindis 22399 | Homeomorphisms preserve to... |
| indishmph 22400 | Equinumerous sets equipped... |
| hmphen2 22401 | Homeomorphisms preserve th... |
| cmphaushmeo 22402 | A continuous bijection fro... |
| ordthmeolem 22403 | Lemma for ~ ordthmeo . (C... |
| ordthmeo 22404 | An order isomorphism is a ... |
| txhmeo 22405 | Lift a pair of homeomorphi... |
| txswaphmeolem 22406 | Show inverse for the "swap... |
| txswaphmeo 22407 | There is a homeomorphism f... |
| pt1hmeo 22408 | The canonical homeomorphis... |
| ptuncnv 22409 | Exhibit the converse funct... |
| ptunhmeo 22410 | Define a homeomorphism fro... |
| xpstopnlem1 22411 | The function ` F ` used in... |
| xpstps 22412 | A binary product of topolo... |
| xpstopnlem2 22413 | Lemma for ~ xpstopn . (Co... |
| xpstopn 22414 | The topology on a binary p... |
| ptcmpfi 22415 | A topological product of f... |
| xkocnv 22416 | The inverse of the "curryi... |
| xkohmeo 22417 | The Exponential Law for to... |
| qtopf1 22418 | If a quotient map is injec... |
| qtophmeo 22419 | If two functions on a base... |
| t0kq 22420 | A topological space is T_0... |
| kqhmph 22421 | A topological space is T_0... |
| ist1-5lem 22422 | Lemma for ~ ist1-5 and sim... |
| t1r0 22423 | A T_1 space is R_0. That ... |
| ist1-5 22424 | A topological space is T_1... |
| ishaus3 22425 | A topological space is Hau... |
| nrmreg 22426 | A normal T_1 space is regu... |
| reghaus 22427 | A regular T_0 space is Hau... |
| nrmhaus 22428 | A T_1 normal space is Haus... |
| elmptrab 22429 | Membership in a one-parame... |
| elmptrab2 22430 | Membership in a one-parame... |
| isfbas 22431 | The predicate " ` F ` is a... |
| fbasne0 22432 | There are no empty filter ... |
| 0nelfb 22433 | No filter base contains th... |
| fbsspw 22434 | A filter base on a set is ... |
| fbelss 22435 | An element of the filter b... |
| fbdmn0 22436 | The domain of a filter bas... |
| isfbas2 22437 | The predicate " ` F ` is a... |
| fbasssin 22438 | A filter base contains sub... |
| fbssfi 22439 | A filter base contains sub... |
| fbssint 22440 | A filter base contains sub... |
| fbncp 22441 | A filter base does not con... |
| fbun 22442 | A necessary and sufficient... |
| fbfinnfr 22443 | No filter base containing ... |
| opnfbas 22444 | The collection of open sup... |
| trfbas2 22445 | Conditions for the trace o... |
| trfbas 22446 | Conditions for the trace o... |
| isfil 22449 | The predicate "is a filter... |
| filfbas 22450 | A filter is a filter base.... |
| 0nelfil 22451 | The empty set doesn't belo... |
| fileln0 22452 | An element of a filter is ... |
| filsspw 22453 | A filter is a subset of th... |
| filelss 22454 | An element of a filter is ... |
| filss 22455 | A filter is closed under t... |
| filin 22456 | A filter is closed under t... |
| filtop 22457 | The underlying set belongs... |
| isfil2 22458 | Derive the standard axioms... |
| isfildlem 22459 | Lemma for ~ isfild . (Con... |
| isfild 22460 | Sufficient condition for a... |
| filfi 22461 | A filter is closed under t... |
| filinn0 22462 | The intersection of two el... |
| filintn0 22463 | A filter has the finite in... |
| filn0 22464 | The empty set is not a fil... |
| infil 22465 | The intersection of two fi... |
| snfil 22466 | A singleton is a filter. ... |
| fbasweak 22467 | A filter base on any set i... |
| snfbas 22468 | Condition for a singleton ... |
| fsubbas 22469 | A condition for a set to g... |
| fbasfip 22470 | A filter base has the fini... |
| fbunfip 22471 | A helpful lemma for showin... |
| fgval 22472 | The filter generating clas... |
| elfg 22473 | A condition for elements o... |
| ssfg 22474 | A filter base is a subset ... |
| fgss 22475 | A bigger base generates a ... |
| fgss2 22476 | A condition for a filter t... |
| fgfil 22477 | A filter generates itself.... |
| elfilss 22478 | An element belongs to a fi... |
| filfinnfr 22479 | No filter containing a fin... |
| fgcl 22480 | A generated filter is a fi... |
| fgabs 22481 | Absorption law for filter ... |
| neifil 22482 | The neighborhoods of a non... |
| filunibas 22483 | Recover the base set from ... |
| filunirn 22484 | Two ways to express a filt... |
| filconn 22485 | A filter gives rise to a c... |
| fbasrn 22486 | Given a filter on a domain... |
| filuni 22487 | The union of a nonempty se... |
| trfil1 22488 | Conditions for the trace o... |
| trfil2 22489 | Conditions for the trace o... |
| trfil3 22490 | Conditions for the trace o... |
| trfilss 22491 | If ` A ` is a member of th... |
| fgtr 22492 | If ` A ` is a member of th... |
| trfg 22493 | The trace operation and th... |
| trnei 22494 | The trace, over a set ` A ... |
| cfinfil 22495 | Relative complements of th... |
| csdfil 22496 | The set of all elements wh... |
| supfil 22497 | The supersets of a nonempt... |
| zfbas 22498 | The set of upper sets of i... |
| uzrest 22499 | The restriction of the set... |
| uzfbas 22500 | The set of upper sets of i... |
| isufil 22505 | The property of being an u... |
| ufilfil 22506 | An ultrafilter is a filter... |
| ufilss 22507 | For any subset of the base... |
| ufilb 22508 | The complement is in an ul... |
| ufilmax 22509 | Any filter finer than an u... |
| isufil2 22510 | The maximal property of an... |
| ufprim 22511 | An ultrafilter is a prime ... |
| trufil 22512 | Conditions for the trace o... |
| filssufilg 22513 | A filter is contained in s... |
| filssufil 22514 | A filter is contained in s... |
| isufl 22515 | Define the (strong) ultraf... |
| ufli 22516 | Property of a set that sat... |
| numufl 22517 | Consequence of ~ filssufil... |
| fiufl 22518 | A finite set satisfies the... |
| acufl 22519 | The axiom of choice implie... |
| ssufl 22520 | If ` Y ` is a subset of ` ... |
| ufileu 22521 | If the ultrafilter contain... |
| filufint 22522 | A filter is equal to the i... |
| uffix 22523 | Lemma for ~ fixufil and ~ ... |
| fixufil 22524 | The condition describing a... |
| uffixfr 22525 | An ultrafilter is either f... |
| uffix2 22526 | A classification of fixed ... |
| uffixsn 22527 | The singleton of the gener... |
| ufildom1 22528 | An ultrafilter is generate... |
| uffinfix 22529 | An ultrafilter containing ... |
| cfinufil 22530 | An ultrafilter is free iff... |
| ufinffr 22531 | An infinite subset is cont... |
| ufilen 22532 | Any infinite set has an ul... |
| ufildr 22533 | An ultrafilter gives rise ... |
| fin1aufil 22534 | There are no definable fre... |
| fmval 22545 | Introduce a function that ... |
| fmfil 22546 | A mapping filter is a filt... |
| fmf 22547 | Pushing-forward via a func... |
| fmss 22548 | A finer filter produces a ... |
| elfm 22549 | An element of a mapping fi... |
| elfm2 22550 | An element of a mapping fi... |
| fmfg 22551 | The image filter of a filt... |
| elfm3 22552 | An alternate formulation o... |
| imaelfm 22553 | An image of a filter eleme... |
| rnelfmlem 22554 | Lemma for ~ rnelfm . (Con... |
| rnelfm 22555 | A condition for a filter t... |
| fmfnfmlem1 22556 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem2 22557 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem3 22558 | Lemma for ~ fmfnfm . (Con... |
| fmfnfmlem4 22559 | Lemma for ~ fmfnfm . (Con... |
| fmfnfm 22560 | A filter finer than an ima... |
| fmufil 22561 | An image filter of an ultr... |
| fmid 22562 | The filter map applied to ... |
| fmco 22563 | Composition of image filte... |
| ufldom 22564 | The ultrafilter lemma prop... |
| flimval 22565 | The set of limit points of... |
| elflim2 22566 | The predicate "is a limit ... |
| flimtop 22567 | Reverse closure for the li... |
| flimneiss 22568 | A filter contains the neig... |
| flimnei 22569 | A filter contains all of t... |
| flimelbas 22570 | A limit point of a filter ... |
| flimfil 22571 | Reverse closure for the li... |
| flimtopon 22572 | Reverse closure for the li... |
| elflim 22573 | The predicate "is a limit ... |
| flimss2 22574 | A limit point of a filter ... |
| flimss1 22575 | A limit point of a filter ... |
| neiflim 22576 | A point is a limit point o... |
| flimopn 22577 | The condition for being a ... |
| fbflim 22578 | A condition for a filter t... |
| fbflim2 22579 | A condition for a filter b... |
| flimclsi 22580 | The convergent points of a... |
| hausflimlem 22581 | If ` A ` and ` B ` are bot... |
| hausflimi 22582 | One direction of ~ hausfli... |
| hausflim 22583 | A condition for a topology... |
| flimcf 22584 | Fineness is properly chara... |
| flimrest 22585 | The set of limit points in... |
| flimclslem 22586 | Lemma for ~ flimcls . (Co... |
| flimcls 22587 | Closure in terms of filter... |
| flimsncls 22588 | If ` A ` is a limit point ... |
| hauspwpwf1 22589 | Lemma for ~ hauspwpwdom . ... |
| hauspwpwdom 22590 | If ` X ` is a Hausdorff sp... |
| flffval 22591 | Given a topology and a fil... |
| flfval 22592 | Given a function from a fi... |
| flfnei 22593 | The property of being a li... |
| flfneii 22594 | A neighborhood of a limit ... |
| isflf 22595 | The property of being a li... |
| flfelbas 22596 | A limit point of a functio... |
| flffbas 22597 | Limit points of a function... |
| flftg 22598 | Limit points of a function... |
| hausflf 22599 | If a function has its valu... |
| hausflf2 22600 | If a convergent function h... |
| cnpflfi 22601 | Forward direction of ~ cnp... |
| cnpflf2 22602 | ` F ` is continuous at poi... |
| cnpflf 22603 | Continuity of a function a... |
| cnflf 22604 | A function is continuous i... |
| cnflf2 22605 | A function is continuous i... |
| flfcnp 22606 | A continuous function pres... |
| lmflf 22607 | The topological limit rela... |
| txflf 22608 | Two sequences converge in ... |
| flfcnp2 22609 | The image of a convergent ... |
| fclsval 22610 | The set of all cluster poi... |
| isfcls 22611 | A cluster point of a filte... |
| fclsfil 22612 | Reverse closure for the cl... |
| fclstop 22613 | Reverse closure for the cl... |
| fclstopon 22614 | Reverse closure for the cl... |
| isfcls2 22615 | A cluster point of a filte... |
| fclsopn 22616 | Write the cluster point co... |
| fclsopni 22617 | An open neighborhood of a ... |
| fclselbas 22618 | A cluster point is in the ... |
| fclsneii 22619 | A neighborhood of a cluste... |
| fclssscls 22620 | The set of cluster points ... |
| fclsnei 22621 | Cluster points in terms of... |
| supnfcls 22622 | The filter of supersets of... |
| fclsbas 22623 | Cluster points in terms of... |
| fclsss1 22624 | A finer topology has fewer... |
| fclsss2 22625 | A finer filter has fewer c... |
| fclsrest 22626 | The set of cluster points ... |
| fclscf 22627 | Characterization of finene... |
| flimfcls 22628 | A limit point is a cluster... |
| fclsfnflim 22629 | A filter clusters at a poi... |
| flimfnfcls 22630 | A filter converges to a po... |
| fclscmpi 22631 | Forward direction of ~ fcl... |
| fclscmp 22632 | A space is compact iff eve... |
| uffclsflim 22633 | The cluster points of an u... |
| ufilcmp 22634 | A space is compact iff eve... |
| fcfval 22635 | The set of cluster points ... |
| isfcf 22636 | The property of being a cl... |
| fcfnei 22637 | The property of being a cl... |
| fcfelbas 22638 | A cluster point of a funct... |
| fcfneii 22639 | A neighborhood of a cluste... |
| flfssfcf 22640 | A limit point of a functio... |
| uffcfflf 22641 | If the domain filter is an... |
| cnpfcfi 22642 | Lemma for ~ cnpfcf . If a... |
| cnpfcf 22643 | A function ` F ` is contin... |
| cnfcf 22644 | Continuity of a function i... |
| flfcntr 22645 | A continuous function's va... |
| alexsublem 22646 | Lemma for ~ alexsub . (Co... |
| alexsub 22647 | The Alexander Subbase Theo... |
| alexsubb 22648 | Biconditional form of the ... |
| alexsubALTlem1 22649 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem2 22650 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem3 22651 | Lemma for ~ alexsubALT . ... |
| alexsubALTlem4 22652 | Lemma for ~ alexsubALT . ... |
| alexsubALT 22653 | The Alexander Subbase Theo... |
| ptcmplem1 22654 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem2 22655 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem3 22656 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem4 22657 | Lemma for ~ ptcmp . (Cont... |
| ptcmplem5 22658 | Lemma for ~ ptcmp . (Cont... |
| ptcmpg 22659 | Tychonoff's theorem: The ... |
| ptcmp 22660 | Tychonoff's theorem: The ... |
| cnextval 22663 | The function applying cont... |
| cnextfval 22664 | The continuous extension o... |
| cnextrel 22665 | In the general case, a con... |
| cnextfun 22666 | If the target space is Hau... |
| cnextfvval 22667 | The value of the continuou... |
| cnextf 22668 | Extension by continuity. ... |
| cnextcn 22669 | Extension by continuity. ... |
| cnextfres1 22670 | ` F ` and its extension by... |
| cnextfres 22671 | ` F ` and its extension by... |
| istmd 22676 | The predicate "is a topolo... |
| tmdmnd 22677 | A topological monoid is a ... |
| tmdtps 22678 | A topological monoid is a ... |
| istgp 22679 | The predicate "is a topolo... |
| tgpgrp 22680 | A topological group is a g... |
| tgptmd 22681 | A topological group is a t... |
| tgptps 22682 | A topological group is a t... |
| tmdtopon 22683 | The topology of a topologi... |
| tgptopon 22684 | The topology of a topologi... |
| tmdcn 22685 | In a topological monoid, t... |
| tgpcn 22686 | In a topological group, th... |
| tgpinv 22687 | In a topological group, th... |
| grpinvhmeo 22688 | The inverse function in a ... |
| cnmpt1plusg 22689 | Continuity of the group su... |
| cnmpt2plusg 22690 | Continuity of the group su... |
| tmdcn2 22691 | Write out the definition o... |
| tgpsubcn 22692 | In a topological group, th... |
| istgp2 22693 | A group with a topology is... |
| tmdmulg 22694 | In a topological monoid, t... |
| tgpmulg 22695 | In a topological group, th... |
| tgpmulg2 22696 | In a topological monoid, t... |
| tmdgsum 22697 | In a topological monoid, t... |
| tmdgsum2 22698 | For any neighborhood ` U `... |
| oppgtmd 22699 | The opposite of a topologi... |
| oppgtgp 22700 | The opposite of a topologi... |
| distgp 22701 | Any group equipped with th... |
| indistgp 22702 | Any group equipped with th... |
| efmndtmd 22703 | The monoid of endofunction... |
| tmdlactcn 22704 | The left group action of e... |
| tgplacthmeo 22705 | The left group action of e... |
| submtmd 22706 | A submonoid of a topologic... |
| subgtgp 22707 | A subgroup of a topologica... |
| symgtgp 22708 | The symmetric group is a t... |
| subgntr 22709 | A subgroup of a topologica... |
| opnsubg 22710 | An open subgroup of a topo... |
| clssubg 22711 | The closure of a subgroup ... |
| clsnsg 22712 | The closure of a normal su... |
| cldsubg 22713 | A subgroup of finite index... |
| tgpconncompeqg 22714 | The connected component co... |
| tgpconncomp 22715 | The identity component, th... |
| tgpconncompss 22716 | The identity component is ... |
| ghmcnp 22717 | A group homomorphism on to... |
| snclseqg 22718 | The coset of the closure o... |
| tgphaus 22719 | A topological group is Hau... |
| tgpt1 22720 | Hausdorff and T1 are equiv... |
| tgpt0 22721 | Hausdorff and T0 are equiv... |
| qustgpopn 22722 | A quotient map in a topolo... |
| qustgplem 22723 | Lemma for ~ qustgp . (Con... |
| qustgp 22724 | The quotient of a topologi... |
| qustgphaus 22725 | The quotient of a topologi... |
| prdstmdd 22726 | The product of a family of... |
| prdstgpd 22727 | The product of a family of... |
| tsmsfbas 22730 | The collection of all sets... |
| tsmslem1 22731 | The finite partial sums of... |
| tsmsval2 22732 | Definition of the topologi... |
| tsmsval 22733 | Definition of the topologi... |
| tsmspropd 22734 | The group sum depends only... |
| eltsms 22735 | The property of being a su... |
| tsmsi 22736 | The property of being a su... |
| tsmscl 22737 | A sum in a topological gro... |
| haustsms 22738 | In a Hausdorff topological... |
| haustsms2 22739 | In a Hausdorff topological... |
| tsmscls 22740 | One half of ~ tgptsmscls ,... |
| tsmsgsum 22741 | The convergent points of a... |
| tsmsid 22742 | If a sum is finite, the us... |
| haustsmsid 22743 | In a Hausdorff topological... |
| tsms0 22744 | The sum of zero is zero. ... |
| tsmssubm 22745 | Evaluate an infinite group... |
| tsmsres 22746 | Extend an infinite group s... |
| tsmsf1o 22747 | Re-index an infinite group... |
| tsmsmhm 22748 | Apply a continuous group h... |
| tsmsadd 22749 | The sum of two infinite gr... |
| tsmsinv 22750 | Inverse of an infinite gro... |
| tsmssub 22751 | The difference of two infi... |
| tgptsmscls 22752 | A sum in a topological gro... |
| tgptsmscld 22753 | The set of limit points to... |
| tsmssplit 22754 | Split a topological group ... |
| tsmsxplem1 22755 | Lemma for ~ tsmsxp . (Con... |
| tsmsxplem2 22756 | Lemma for ~ tsmsxp . (Con... |
| tsmsxp 22757 | Write a sum over a two-dim... |
| istrg 22766 | Express the predicate " ` ... |
| trgtmd 22767 | The multiplicative monoid ... |
| istdrg 22768 | Express the predicate " ` ... |
| tdrgunit 22769 | The unit group of a topolo... |
| trgtgp 22770 | A topological ring is a to... |
| trgtmd2 22771 | A topological ring is a to... |
| trgtps 22772 | A topological ring is a to... |
| trgring 22773 | A topological ring is a ri... |
| trggrp 22774 | A topological ring is a gr... |
| tdrgtrg 22775 | A topological division rin... |
| tdrgdrng 22776 | A topological division rin... |
| tdrgring 22777 | A topological division rin... |
| tdrgtmd 22778 | A topological division rin... |
| tdrgtps 22779 | A topological division rin... |
| istdrg2 22780 | A topological-ring divisio... |
| mulrcn 22781 | The functionalization of t... |
| invrcn2 22782 | The multiplicative inverse... |
| invrcn 22783 | The multiplicative inverse... |
| cnmpt1mulr 22784 | Continuity of ring multipl... |
| cnmpt2mulr 22785 | Continuity of ring multipl... |
| dvrcn 22786 | The division function is c... |
| istlm 22787 | The predicate " ` W ` is a... |
| vscacn 22788 | The scalar multiplication ... |
| tlmtmd 22789 | A topological module is a ... |
| tlmtps 22790 | A topological module is a ... |
| tlmlmod 22791 | A topological module is a ... |
| tlmtrg 22792 | The scalar ring of a topol... |
| tlmscatps 22793 | The scalar ring of a topol... |
| istvc 22794 | A topological vector space... |
| tvctdrg 22795 | The scalar field of a topo... |
| cnmpt1vsca 22796 | Continuity of scalar multi... |
| cnmpt2vsca 22797 | Continuity of scalar multi... |
| tlmtgp 22798 | A topological vector space... |
| tvctlm 22799 | A topological vector space... |
| tvclmod 22800 | A topological vector space... |
| tvclvec 22801 | A topological vector space... |
| ustfn 22804 | The defined uniform struct... |
| ustval 22805 | The class of all uniform s... |
| isust 22806 | The predicate " ` U ` is a... |
| ustssxp 22807 | Entourages are subsets of ... |
| ustssel 22808 | A uniform structure is upw... |
| ustbasel 22809 | The full set is always an ... |
| ustincl 22810 | A uniform structure is clo... |
| ustdiag 22811 | The diagonal set is includ... |
| ustinvel 22812 | If ` V ` is an entourage, ... |
| ustexhalf 22813 | For each entourage ` V ` t... |
| ustrel 22814 | The elements of uniform st... |
| ustfilxp 22815 | A uniform structure on a n... |
| ustne0 22816 | A uniform structure cannot... |
| ustssco 22817 | In an uniform structure, a... |
| ustexsym 22818 | In an uniform structure, f... |
| ustex2sym 22819 | In an uniform structure, f... |
| ustex3sym 22820 | In an uniform structure, f... |
| ustref 22821 | Any element of the base se... |
| ust0 22822 | The unique uniform structu... |
| ustn0 22823 | The empty set is not an un... |
| ustund 22824 | If two intersecting sets `... |
| ustelimasn 22825 | Any point ` A ` is near en... |
| ustneism 22826 | For a point ` A ` in ` X `... |
| elrnust 22827 | First direction for ~ ustb... |
| ustbas2 22828 | Second direction for ~ ust... |
| ustuni 22829 | The set union of a uniform... |
| ustbas 22830 | Recover the base of an uni... |
| ustimasn 22831 | Lemma for ~ ustuqtop . (C... |
| trust 22832 | The trace of a uniform str... |
| utopval 22835 | The topology induced by a ... |
| elutop 22836 | Open sets in the topology ... |
| utoptop 22837 | The topology induced by a ... |
| utopbas 22838 | The base of the topology i... |
| utoptopon 22839 | Topology induced by a unif... |
| restutop 22840 | Restriction of a topology ... |
| restutopopn 22841 | The restriction of the top... |
| ustuqtoplem 22842 | Lemma for ~ ustuqtop . (C... |
| ustuqtop0 22843 | Lemma for ~ ustuqtop . (C... |
| ustuqtop1 22844 | Lemma for ~ ustuqtop , sim... |
| ustuqtop2 22845 | Lemma for ~ ustuqtop . (C... |
| ustuqtop3 22846 | Lemma for ~ ustuqtop , sim... |
| ustuqtop4 22847 | Lemma for ~ ustuqtop . (C... |
| ustuqtop5 22848 | Lemma for ~ ustuqtop . (C... |
| ustuqtop 22849 | For a given uniform struct... |
| utopsnneiplem 22850 | The neighborhoods of a poi... |
| utopsnneip 22851 | The neighborhoods of a poi... |
| utopsnnei 22852 | Images of singletons by en... |
| utop2nei 22853 | For any symmetrical entour... |
| utop3cls 22854 | Relation between a topolog... |
| utopreg 22855 | All Hausdorff uniform spac... |
| ussval 22862 | The uniform structure on u... |
| ussid 22863 | In case the base of the ` ... |
| isusp 22864 | The predicate ` W ` is a u... |
| ressunif 22865 | ` UnifSet ` is unaffected ... |
| ressuss 22866 | Value of the uniform struc... |
| ressust 22867 | The uniform structure of a... |
| ressusp 22868 | The restriction of a unifo... |
| tusval 22869 | The value of the uniform s... |
| tuslem 22870 | Lemma for ~ tusbas , ~ tus... |
| tusbas 22871 | The base set of a construc... |
| tusunif 22872 | The uniform structure of a... |
| tususs 22873 | The uniform structure of a... |
| tustopn 22874 | The topology induced by a ... |
| tususp 22875 | A constructed uniform spac... |
| tustps 22876 | A constructed uniform spac... |
| uspreg 22877 | If a uniform space is Haus... |
| ucnval 22880 | The set of all uniformly c... |
| isucn 22881 | The predicate " ` F ` is a... |
| isucn2 22882 | The predicate " ` F ` is a... |
| ucnimalem 22883 | Reformulate the ` G ` func... |
| ucnima 22884 | An equivalent statement of... |
| ucnprima 22885 | The preimage by a uniforml... |
| iducn 22886 | The identity is uniformly ... |
| cstucnd 22887 | A constant function is uni... |
| ucncn 22888 | Uniform continuity implies... |
| iscfilu 22891 | The predicate " ` F ` is a... |
| cfilufbas 22892 | A Cauchy filter base is a ... |
| cfiluexsm 22893 | For a Cauchy filter base a... |
| fmucndlem 22894 | Lemma for ~ fmucnd . (Con... |
| fmucnd 22895 | The image of a Cauchy filt... |
| cfilufg 22896 | The filter generated by a ... |
| trcfilu 22897 | Condition for the trace of... |
| cfiluweak 22898 | A Cauchy filter base is al... |
| neipcfilu 22899 | In an uniform space, a nei... |
| iscusp 22902 | The predicate " ` W ` is a... |
| cuspusp 22903 | A complete uniform space i... |
| cuspcvg 22904 | In a complete uniform spac... |
| iscusp2 22905 | The predicate " ` W ` is a... |
| cnextucn 22906 | Extension by continuity. ... |
| ucnextcn 22907 | Extension by continuity. ... |
| ispsmet 22908 | Express the predicate " ` ... |
| psmetdmdm 22909 | Recover the base set from ... |
| psmetf 22910 | The distance function of a... |
| psmetcl 22911 | Closure of the distance fu... |
| psmet0 22912 | The distance function of a... |
| psmettri2 22913 | Triangle inequality for th... |
| psmetsym 22914 | The distance function of a... |
| psmettri 22915 | Triangle inequality for th... |
| psmetge0 22916 | The distance function of a... |
| psmetxrge0 22917 | The distance function of a... |
| psmetres2 22918 | Restriction of a pseudomet... |
| psmetlecl 22919 | Real closure of an extende... |
| distspace 22920 | A set ` X ` together with ... |
| ismet 22927 | Express the predicate " ` ... |
| isxmet 22928 | Express the predicate " ` ... |
| ismeti 22929 | Properties that determine ... |
| isxmetd 22930 | Properties that determine ... |
| isxmet2d 22931 | It is safe to only require... |
| metflem 22932 | Lemma for ~ metf and other... |
| xmetf 22933 | Mapping of the distance fu... |
| metf 22934 | Mapping of the distance fu... |
| xmetcl 22935 | Closure of the distance fu... |
| metcl 22936 | Closure of the distance fu... |
| ismet2 22937 | An extended metric is a me... |
| metxmet 22938 | A metric is an extended me... |
| xmetdmdm 22939 | Recover the base set from ... |
| metdmdm 22940 | Recover the base set from ... |
| xmetunirn 22941 | Two ways to express an ext... |
| xmeteq0 22942 | The value of an extended m... |
| meteq0 22943 | The value of a metric is z... |
| xmettri2 22944 | Triangle inequality for th... |
| mettri2 22945 | Triangle inequality for th... |
| xmet0 22946 | The distance function of a... |
| met0 22947 | The distance function of a... |
| xmetge0 22948 | The distance function of a... |
| metge0 22949 | The distance function of a... |
| xmetlecl 22950 | Real closure of an extende... |
| xmetsym 22951 | The distance function of a... |
| xmetpsmet 22952 | An extended metric is a ps... |
| xmettpos 22953 | The distance function of a... |
| metsym 22954 | The distance function of a... |
| xmettri 22955 | Triangle inequality for th... |
| mettri 22956 | Triangle inequality for th... |
| xmettri3 22957 | Triangle inequality for th... |
| mettri3 22958 | Triangle inequality for th... |
| xmetrtri 22959 | One half of the reverse tr... |
| xmetrtri2 22960 | The reverse triangle inequ... |
| metrtri 22961 | Reverse triangle inequalit... |
| xmetgt0 22962 | The distance function of a... |
| metgt0 22963 | The distance function of a... |
| metn0 22964 | A metric space is nonempty... |
| xmetres2 22965 | Restriction of an extended... |
| metreslem 22966 | Lemma for ~ metres . (Con... |
| metres2 22967 | Lemma for ~ metres . (Con... |
| xmetres 22968 | A restriction of an extend... |
| metres 22969 | A restriction of a metric ... |
| 0met 22970 | The empty metric. (Contri... |
| prdsdsf 22971 | The product metric is a fu... |
| prdsxmetlem 22972 | The product metric is an e... |
| prdsxmet 22973 | The product metric is an e... |
| prdsmet 22974 | The product metric is a me... |
| ressprdsds 22975 | Restriction of a product m... |
| resspwsds 22976 | Restriction of a power met... |
| imasdsf1olem 22977 | Lemma for ~ imasdsf1o . (... |
| imasdsf1o 22978 | The distance function is t... |
| imasf1oxmet 22979 | The image of an extended m... |
| imasf1omet 22980 | The image of a metric is a... |
| xpsdsfn 22981 | Closure of the metric in a... |
| xpsdsfn2 22982 | Closure of the metric in a... |
| xpsxmetlem 22983 | Lemma for ~ xpsxmet . (Co... |
| xpsxmet 22984 | A product metric of extend... |
| xpsdsval 22985 | Value of the metric in a b... |
| xpsmet 22986 | The direct product of two ... |
| blfvalps 22987 | The value of the ball func... |
| blfval 22988 | The value of the ball func... |
| blvalps 22989 | The ball around a point ` ... |
| blval 22990 | The ball around a point ` ... |
| elblps 22991 | Membership in a ball. (Co... |
| elbl 22992 | Membership in a ball. (Co... |
| elbl2ps 22993 | Membership in a ball. (Co... |
| elbl2 22994 | Membership in a ball. (Co... |
| elbl3ps 22995 | Membership in a ball, with... |
| elbl3 22996 | Membership in a ball, with... |
| blcomps 22997 | Commute the arguments to t... |
| blcom 22998 | Commute the arguments to t... |
| xblpnfps 22999 | The infinity ball in an ex... |
| xblpnf 23000 | The infinity ball in an ex... |
| blpnf 23001 | The infinity ball in a sta... |
| bldisj 23002 | Two balls are disjoint if ... |
| blgt0 23003 | A nonempty ball implies th... |
| bl2in 23004 | Two balls are disjoint if ... |
| xblss2ps 23005 | One ball is contained in a... |
| xblss2 23006 | One ball is contained in a... |
| blss2ps 23007 | One ball is contained in a... |
| blss2 23008 | One ball is contained in a... |
| blhalf 23009 | A ball of radius ` R / 2 `... |
| blfps 23010 | Mapping of a ball. (Contr... |
| blf 23011 | Mapping of a ball. (Contr... |
| blrnps 23012 | Membership in the range of... |
| blrn 23013 | Membership in the range of... |
| xblcntrps 23014 | A ball contains its center... |
| xblcntr 23015 | A ball contains its center... |
| blcntrps 23016 | A ball contains its center... |
| blcntr 23017 | A ball contains its center... |
| xbln0 23018 | A ball is nonempty iff the... |
| bln0 23019 | A ball is not empty. (Con... |
| blelrnps 23020 | A ball belongs to the set ... |
| blelrn 23021 | A ball belongs to the set ... |
| blssm 23022 | A ball is a subset of the ... |
| unirnblps 23023 | The union of the set of ba... |
| unirnbl 23024 | The union of the set of ba... |
| blin 23025 | The intersection of two ba... |
| ssblps 23026 | The size of a ball increas... |
| ssbl 23027 | The size of a ball increas... |
| blssps 23028 | Any point ` P ` in a ball ... |
| blss 23029 | Any point ` P ` in a ball ... |
| blssexps 23030 | Two ways to express the ex... |
| blssex 23031 | Two ways to express the ex... |
| ssblex 23032 | A nested ball exists whose... |
| blin2 23033 | Given any two balls and a ... |
| blbas 23034 | The balls of a metric spac... |
| blres 23035 | A ball in a restricted met... |
| xmeterval 23036 | Value of the "finitely sep... |
| xmeter 23037 | The "finitely separated" r... |
| xmetec 23038 | The equivalence classes un... |
| blssec 23039 | A ball centered at ` P ` i... |
| blpnfctr 23040 | The infinity ball in an ex... |
| xmetresbl 23041 | An extended metric restric... |
| mopnval 23042 | An open set is a subset of... |
| mopntopon 23043 | The set of open sets of a ... |
| mopntop 23044 | The set of open sets of a ... |
| mopnuni 23045 | The union of all open sets... |
| elmopn 23046 | The defining property of a... |
| mopnfss 23047 | The family of open sets of... |
| mopnm 23048 | The base set of a metric s... |
| elmopn2 23049 | A defining property of an ... |
| mopnss 23050 | An open set of a metric sp... |
| isxms 23051 | Express the predicate " ` ... |
| isxms2 23052 | Express the predicate " ` ... |
| isms 23053 | Express the predicate " ` ... |
| isms2 23054 | Express the predicate " ` ... |
| xmstopn 23055 | The topology component of ... |
| mstopn 23056 | The topology component of ... |
| xmstps 23057 | An extended metric space i... |
| msxms 23058 | A metric space is an exten... |
| mstps 23059 | A metric space is a topolo... |
| xmsxmet 23060 | The distance function, sui... |
| msmet 23061 | The distance function, sui... |
| msf 23062 | The distance function of a... |
| xmsxmet2 23063 | The distance function, sui... |
| msmet2 23064 | The distance function, sui... |
| mscl 23065 | Closure of the distance fu... |
| xmscl 23066 | Closure of the distance fu... |
| xmsge0 23067 | The distance function in a... |
| xmseq0 23068 | The distance between two p... |
| xmssym 23069 | The distance function in a... |
| xmstri2 23070 | Triangle inequality for th... |
| mstri2 23071 | Triangle inequality for th... |
| xmstri 23072 | Triangle inequality for th... |
| mstri 23073 | Triangle inequality for th... |
| xmstri3 23074 | Triangle inequality for th... |
| mstri3 23075 | Triangle inequality for th... |
| msrtri 23076 | Reverse triangle inequalit... |
| xmspropd 23077 | Property deduction for an ... |
| mspropd 23078 | Property deduction for a m... |
| setsmsbas 23079 | The base set of a construc... |
| setsmsds 23080 | The distance function of a... |
| setsmstset 23081 | The topology of a construc... |
| setsmstopn 23082 | The topology of a construc... |
| setsxms 23083 | The constructed metric spa... |
| setsms 23084 | The constructed metric spa... |
| tmsval 23085 | For any metric there is an... |
| tmslem 23086 | Lemma for ~ tmsbas , ~ tms... |
| tmsbas 23087 | The base set of a construc... |
| tmsds 23088 | The metric of a constructe... |
| tmstopn 23089 | The topology of a construc... |
| tmsxms 23090 | The constructed metric spa... |
| tmsms 23091 | The constructed metric spa... |
| imasf1obl 23092 | The image of a metric spac... |
| imasf1oxms 23093 | The image of a metric spac... |
| imasf1oms 23094 | The image of a metric spac... |
| prdsbl 23095 | A ball in the product metr... |
| mopni 23096 | An open set of a metric sp... |
| mopni2 23097 | An open set of a metric sp... |
| mopni3 23098 | An open set of a metric sp... |
| blssopn 23099 | The balls of a metric spac... |
| unimopn 23100 | The union of a collection ... |
| mopnin 23101 | The intersection of two op... |
| mopn0 23102 | The empty set is an open s... |
| rnblopn 23103 | A ball of a metric space i... |
| blopn 23104 | A ball of a metric space i... |
| neibl 23105 | The neighborhoods around a... |
| blnei 23106 | A ball around a point is a... |
| lpbl 23107 | Every ball around a limit ... |
| blsscls2 23108 | A smaller closed ball is c... |
| blcld 23109 | A "closed ball" in a metri... |
| blcls 23110 | The closure of an open bal... |
| blsscls 23111 | If two concentric balls ha... |
| metss 23112 | Two ways of saying that me... |
| metequiv 23113 | Two ways of saying that tw... |
| metequiv2 23114 | If there is a sequence of ... |
| metss2lem 23115 | Lemma for ~ metss2 . (Con... |
| metss2 23116 | If the metric ` D ` is "st... |
| comet 23117 | The composition of an exte... |
| stdbdmetval 23118 | Value of the standard boun... |
| stdbdxmet 23119 | The standard bounded metri... |
| stdbdmet 23120 | The standard bounded metri... |
| stdbdbl 23121 | The standard bounded metri... |
| stdbdmopn 23122 | The standard bounded metri... |
| mopnex 23123 | The topology generated by ... |
| methaus 23124 | The topology generated by ... |
| met1stc 23125 | The topology generated by ... |
| met2ndci 23126 | A separable metric space (... |
| met2ndc 23127 | A metric space is second-c... |
| metrest 23128 | Two alternate formulations... |
| ressxms 23129 | The restriction of a metri... |
| ressms 23130 | The restriction of a metri... |
| prdsmslem1 23131 | Lemma for ~ prdsms . The ... |
| prdsxmslem1 23132 | Lemma for ~ prdsms . The ... |
| prdsxmslem2 23133 | Lemma for ~ prdsxms . The... |
| prdsxms 23134 | The indexed product struct... |
| prdsms 23135 | The indexed product struct... |
| pwsxms 23136 | A power of an extended met... |
| pwsms 23137 | A power of a metric space ... |
| xpsxms 23138 | A binary product of metric... |
| xpsms 23139 | A binary product of metric... |
| tmsxps 23140 | Express the product of two... |
| tmsxpsmopn 23141 | Express the product of two... |
| tmsxpsval 23142 | Value of the product of tw... |
| tmsxpsval2 23143 | Value of the product of tw... |
| metcnp3 23144 | Two ways to express that `... |
| metcnp 23145 | Two ways to say a mapping ... |
| metcnp2 23146 | Two ways to say a mapping ... |
| metcn 23147 | Two ways to say a mapping ... |
| metcnpi 23148 | Epsilon-delta property of ... |
| metcnpi2 23149 | Epsilon-delta property of ... |
| metcnpi3 23150 | Epsilon-delta property of ... |
| txmetcnp 23151 | Continuity of a binary ope... |
| txmetcn 23152 | Continuity of a binary ope... |
| metuval 23153 | Value of the uniform struc... |
| metustel 23154 | Define a filter base ` F `... |
| metustss 23155 | Range of the elements of t... |
| metustrel 23156 | Elements of the filter bas... |
| metustto 23157 | Any two elements of the fi... |
| metustid 23158 | The identity diagonal is i... |
| metustsym 23159 | Elements of the filter bas... |
| metustexhalf 23160 | For any element ` A ` of t... |
| metustfbas 23161 | The filter base generated ... |
| metust 23162 | The uniform structure gene... |
| cfilucfil 23163 | Given a metric ` D ` and a... |
| metuust 23164 | The uniform structure gene... |
| cfilucfil2 23165 | Given a metric ` D ` and a... |
| blval2 23166 | The ball around a point ` ... |
| elbl4 23167 | Membership in a ball, alte... |
| metuel 23168 | Elementhood in the uniform... |
| metuel2 23169 | Elementhood in the uniform... |
| metustbl 23170 | The "section" image of an ... |
| psmetutop 23171 | The topology induced by a ... |
| xmetutop 23172 | The topology induced by a ... |
| xmsusp 23173 | If the uniform set of a me... |
| restmetu 23174 | The uniform structure gene... |
| metucn 23175 | Uniform continuity in metr... |
| dscmet 23176 | The discrete metric on any... |
| dscopn 23177 | The discrete metric genera... |
| nrmmetd 23178 | Show that a group norm gen... |
| abvmet 23179 | An absolute value ` F ` ge... |
| nmfval 23192 | The value of the norm func... |
| nmval 23193 | The value of the norm func... |
| nmfval2 23194 | The value of the norm func... |
| nmval2 23195 | The value of the norm func... |
| nmf2 23196 | The norm is a function fro... |
| nmpropd 23197 | Weak property deduction fo... |
| nmpropd2 23198 | Strong property deduction ... |
| isngp 23199 | The property of being a no... |
| isngp2 23200 | The property of being a no... |
| isngp3 23201 | The property of being a no... |
| ngpgrp 23202 | A normed group is a group.... |
| ngpms 23203 | A normed group is a metric... |
| ngpxms 23204 | A normed group is a metric... |
| ngptps 23205 | A normed group is a topolo... |
| ngpmet 23206 | The (induced) metric of a ... |
| ngpds 23207 | Value of the distance func... |
| ngpdsr 23208 | Value of the distance func... |
| ngpds2 23209 | Write the distance between... |
| ngpds2r 23210 | Write the distance between... |
| ngpds3 23211 | Write the distance between... |
| ngpds3r 23212 | Write the distance between... |
| ngprcan 23213 | Cancel right addition insi... |
| ngplcan 23214 | Cancel left addition insid... |
| isngp4 23215 | Express the property of be... |
| ngpinvds 23216 | Two elements are the same ... |
| ngpsubcan 23217 | Cancel right subtraction i... |
| nmf 23218 | The norm on a normed group... |
| nmcl 23219 | The norm of a normed group... |
| nmge0 23220 | The norm of a normed group... |
| nmeq0 23221 | The identity is the only e... |
| nmne0 23222 | The norm of a nonzero elem... |
| nmrpcl 23223 | The norm of a nonzero elem... |
| nminv 23224 | The norm of a negated elem... |
| nmmtri 23225 | The triangle inequality fo... |
| nmsub 23226 | The norm of the difference... |
| nmrtri 23227 | Reverse triangle inequalit... |
| nm2dif 23228 | Inequality for the differe... |
| nmtri 23229 | The triangle inequality fo... |
| nmtri2 23230 | Triangle inequality for th... |
| ngpi 23231 | The properties of a normed... |
| nm0 23232 | Norm of the identity eleme... |
| nmgt0 23233 | The norm of a nonzero elem... |
| sgrim 23234 | The induced metric on a su... |
| sgrimval 23235 | The induced metric on a su... |
| subgnm 23236 | The norm in a subgroup. (... |
| subgnm2 23237 | A substructure assigns the... |
| subgngp 23238 | A normed group restricted ... |
| ngptgp 23239 | A normed abelian group is ... |
| ngppropd 23240 | Property deduction for a n... |
| reldmtng 23241 | The function ` toNrmGrp ` ... |
| tngval 23242 | Value of the function whic... |
| tnglem 23243 | Lemma for ~ tngbas and sim... |
| tngbas 23244 | The base set of a structur... |
| tngplusg 23245 | The group addition of a st... |
| tng0 23246 | The group identity of a st... |
| tngmulr 23247 | The ring multiplication of... |
| tngsca 23248 | The scalar ring of a struc... |
| tngvsca 23249 | The scalar multiplication ... |
| tngip 23250 | The inner product operatio... |
| tngds 23251 | The metric function of a s... |
| tngtset 23252 | The topology generated by ... |
| tngtopn 23253 | The topology generated by ... |
| tngnm 23254 | The topology generated by ... |
| tngngp2 23255 | A norm turns a group into ... |
| tngngpd 23256 | Derive the axioms for a no... |
| tngngp 23257 | Derive the axioms for a no... |
| tnggrpr 23258 | If a structure equipped wi... |
| tngngp3 23259 | Alternate definition of a ... |
| nrmtngdist 23260 | The augmentation of a norm... |
| nrmtngnrm 23261 | The augmentation of a norm... |
| tngngpim 23262 | The induced metric of a no... |
| isnrg 23263 | A normed ring is a ring wi... |
| nrgabv 23264 | The norm of a normed ring ... |
| nrgngp 23265 | A normed ring is a normed ... |
| nrgring 23266 | A normed ring is a ring. ... |
| nmmul 23267 | The norm of a product in a... |
| nrgdsdi 23268 | Distribute a distance calc... |
| nrgdsdir 23269 | Distribute a distance calc... |
| nm1 23270 | The norm of one in a nonze... |
| unitnmn0 23271 | The norm of a unit is nonz... |
| nminvr 23272 | The norm of an inverse in ... |
| nmdvr 23273 | The norm of a division in ... |
| nrgdomn 23274 | A nonzero normed ring is a... |
| nrgtgp 23275 | A normed ring is a topolog... |
| subrgnrg 23276 | A normed ring restricted t... |
| tngnrg 23277 | Given any absolute value o... |
| isnlm 23278 | A normed (left) module is ... |
| nmvs 23279 | Defining property of a nor... |
| nlmngp 23280 | A normed module is a norme... |
| nlmlmod 23281 | A normed module is a left ... |
| nlmnrg 23282 | The scalar component of a ... |
| nlmngp2 23283 | The scalar component of a ... |
| nlmdsdi 23284 | Distribute a distance calc... |
| nlmdsdir 23285 | Distribute a distance calc... |
| nlmmul0or 23286 | If a scalar product is zer... |
| sranlm 23287 | The subring algebra over a... |
| nlmvscnlem2 23288 | Lemma for ~ nlmvscn . Com... |
| nlmvscnlem1 23289 | Lemma for ~ nlmvscn . (Co... |
| nlmvscn 23290 | The scalar multiplication ... |
| rlmnlm 23291 | The ring module over a nor... |
| rlmnm 23292 | The norm function in the r... |
| nrgtrg 23293 | A normed ring is a topolog... |
| nrginvrcnlem 23294 | Lemma for ~ nrginvrcn . C... |
| nrginvrcn 23295 | The ring inverse function ... |
| nrgtdrg 23296 | A normed division ring is ... |
| nlmtlm 23297 | A normed module is a topol... |
| isnvc 23298 | A normed vector space is j... |
| nvcnlm 23299 | A normed vector space is a... |
| nvclvec 23300 | A normed vector space is a... |
| nvclmod 23301 | A normed vector space is a... |
| isnvc2 23302 | A normed vector space is j... |
| nvctvc 23303 | A normed vector space is a... |
| lssnlm 23304 | A subspace of a normed mod... |
| lssnvc 23305 | A subspace of a normed vec... |
| rlmnvc 23306 | The ring module over a nor... |
| ngpocelbl 23307 | Membership of an off-cente... |
| nmoffn 23314 | The function producing ope... |
| reldmnghm 23315 | Lemma for normed group hom... |
| reldmnmhm 23316 | Lemma for module homomorph... |
| nmofval 23317 | Value of the operator norm... |
| nmoval 23318 | Value of the operator norm... |
| nmogelb 23319 | Property of the operator n... |
| nmolb 23320 | Any upper bound on the val... |
| nmolb2d 23321 | Any upper bound on the val... |
| nmof 23322 | The operator norm is a fun... |
| nmocl 23323 | The operator norm of an op... |
| nmoge0 23324 | The operator norm of an op... |
| nghmfval 23325 | A normed group homomorphis... |
| isnghm 23326 | A normed group homomorphis... |
| isnghm2 23327 | A normed group homomorphis... |
| isnghm3 23328 | A normed group homomorphis... |
| bddnghm 23329 | A bounded group homomorphi... |
| nghmcl 23330 | A normed group homomorphis... |
| nmoi 23331 | The operator norm achieves... |
| nmoix 23332 | The operator norm is a bou... |
| nmoi2 23333 | The operator norm is a bou... |
| nmoleub 23334 | The operator norm, defined... |
| nghmrcl1 23335 | Reverse closure for a norm... |
| nghmrcl2 23336 | Reverse closure for a norm... |
| nghmghm 23337 | A normed group homomorphis... |
| nmo0 23338 | The operator norm of the z... |
| nmoeq0 23339 | The operator norm is zero ... |
| nmoco 23340 | An upper bound on the oper... |
| nghmco 23341 | The composition of normed ... |
| nmotri 23342 | Triangle inequality for th... |
| nghmplusg 23343 | The sum of two bounded lin... |
| 0nghm 23344 | The zero operator is a nor... |
| nmoid 23345 | The operator norm of the i... |
| idnghm 23346 | The identity operator is a... |
| nmods 23347 | Upper bound for the distan... |
| nghmcn 23348 | A normed group homomorphis... |
| isnmhm 23349 | A normed module homomorphi... |
| nmhmrcl1 23350 | Reverse closure for a norm... |
| nmhmrcl2 23351 | Reverse closure for a norm... |
| nmhmlmhm 23352 | A normed module homomorphi... |
| nmhmnghm 23353 | A normed module homomorphi... |
| nmhmghm 23354 | A normed module homomorphi... |
| isnmhm2 23355 | A normed module homomorphi... |
| nmhmcl 23356 | A normed module homomorphi... |
| idnmhm 23357 | The identity operator is a... |
| 0nmhm 23358 | The zero operator is a bou... |
| nmhmco 23359 | The composition of bounded... |
| nmhmplusg 23360 | The sum of two bounded lin... |
| qtopbaslem 23361 | The set of open intervals ... |
| qtopbas 23362 | The set of open intervals ... |
| retopbas 23363 | A basis for the standard t... |
| retop 23364 | The standard topology on t... |
| uniretop 23365 | The underlying set of the ... |
| retopon 23366 | The standard topology on t... |
| retps 23367 | The standard topological s... |
| iooretop 23368 | Open intervals are open se... |
| icccld 23369 | Closed intervals are close... |
| icopnfcld 23370 | Right-unbounded closed int... |
| iocmnfcld 23371 | Left-unbounded closed inte... |
| qdensere 23372 | ` QQ ` is dense in the sta... |
| cnmetdval 23373 | Value of the distance func... |
| cnmet 23374 | The absolute value metric ... |
| cnxmet 23375 | The absolute value metric ... |
| cnbl0 23376 | Two ways to write the open... |
| cnblcld 23377 | Two ways to write the clos... |
| cnfldms 23378 | The complex number field i... |
| cnfldxms 23379 | The complex number field i... |
| cnfldtps 23380 | The complex number field i... |
| cnfldnm 23381 | The norm of the field of c... |
| cnngp 23382 | The complex numbers form a... |
| cnnrg 23383 | The complex numbers form a... |
| cnfldtopn 23384 | The topology of the comple... |
| cnfldtopon 23385 | The topology of the comple... |
| cnfldtop 23386 | The topology of the comple... |
| cnfldhaus 23387 | The topology of the comple... |
| unicntop 23388 | The underlying set of the ... |
| cnopn 23389 | The set of complex numbers... |
| zringnrg 23390 | The ring of integers is a ... |
| remetdval 23391 | Value of the distance func... |
| remet 23392 | The absolute value metric ... |
| rexmet 23393 | The absolute value metric ... |
| bl2ioo 23394 | A ball in terms of an open... |
| ioo2bl 23395 | An open interval of reals ... |
| ioo2blex 23396 | An open interval of reals ... |
| blssioo 23397 | The balls of the standard ... |
| tgioo 23398 | The topology generated by ... |
| qdensere2 23399 | ` QQ ` is dense in ` RR ` ... |
| blcvx 23400 | An open ball in the comple... |
| rehaus 23401 | The standard topology on t... |
| tgqioo 23402 | The topology generated by ... |
| re2ndc 23403 | The standard topology on t... |
| resubmet 23404 | The subspace topology indu... |
| tgioo2 23405 | The standard topology on t... |
| rerest 23406 | The subspace topology indu... |
| tgioo3 23407 | The standard topology on t... |
| xrtgioo 23408 | The topology on the extend... |
| xrrest 23409 | The subspace topology indu... |
| xrrest2 23410 | The subspace topology indu... |
| xrsxmet 23411 | The metric on the extended... |
| xrsdsre 23412 | The metric on the extended... |
| xrsblre 23413 | Any ball of the metric of ... |
| xrsmopn 23414 | The metric on the extended... |
| zcld 23415 | The integers are a closed ... |
| recld2 23416 | The real numbers are a clo... |
| zcld2 23417 | The integers are a closed ... |
| zdis 23418 | The integers are a discret... |
| sszcld 23419 | Every subset of the intege... |
| reperflem 23420 | A subset of the real numbe... |
| reperf 23421 | The real numbers are a per... |
| cnperf 23422 | The complex numbers are a ... |
| iccntr 23423 | The interior of a closed i... |
| icccmplem1 23424 | Lemma for ~ icccmp . (Con... |
| icccmplem2 23425 | Lemma for ~ icccmp . (Con... |
| icccmplem3 23426 | Lemma for ~ icccmp . (Con... |
| icccmp 23427 | A closed interval in ` RR ... |
| reconnlem1 23428 | Lemma for ~ reconn . Conn... |
| reconnlem2 23429 | Lemma for ~ reconn . (Con... |
| reconn 23430 | A subset of the reals is c... |
| retopconn 23431 | Corollary of ~ reconn . T... |
| iccconn 23432 | A closed interval is conne... |
| opnreen 23433 | Every nonempty open set is... |
| rectbntr0 23434 | A countable subset of the ... |
| xrge0gsumle 23435 | A finite sum in the nonneg... |
| xrge0tsms 23436 | Any finite or infinite sum... |
| xrge0tsms2 23437 | Any finite or infinite sum... |
| metdcnlem 23438 | The metric function of a m... |
| xmetdcn2 23439 | The metric function of an ... |
| xmetdcn 23440 | The metric function of an ... |
| metdcn2 23441 | The metric function of a m... |
| metdcn 23442 | The metric function of a m... |
| msdcn 23443 | The metric function of a m... |
| cnmpt1ds 23444 | Continuity of the metric f... |
| cnmpt2ds 23445 | Continuity of the metric f... |
| nmcn 23446 | The norm of a normed group... |
| ngnmcncn 23447 | The norm of a normed group... |
| abscn 23448 | The absolute value functio... |
| metdsval 23449 | Value of the "distance to ... |
| metdsf 23450 | The distance from a point ... |
| metdsge 23451 | The distance from the poin... |
| metds0 23452 | If a point is in a set, it... |
| metdstri 23453 | A generalization of the tr... |
| metdsle 23454 | The distance from a point ... |
| metdsre 23455 | The distance from a point ... |
| metdseq0 23456 | The distance from a point ... |
| metdscnlem 23457 | Lemma for ~ metdscn . (Co... |
| metdscn 23458 | The function ` F ` which g... |
| metdscn2 23459 | The function ` F ` which g... |
| metnrmlem1a 23460 | Lemma for ~ metnrm . (Con... |
| metnrmlem1 23461 | Lemma for ~ metnrm . (Con... |
| metnrmlem2 23462 | Lemma for ~ metnrm . (Con... |
| metnrmlem3 23463 | Lemma for ~ metnrm . (Con... |
| metnrm 23464 | A metric space is normal. ... |
| metreg 23465 | A metric space is regular.... |
| addcnlem 23466 | Lemma for ~ addcn , ~ subc... |
| addcn 23467 | Complex number addition is... |
| subcn 23468 | Complex number subtraction... |
| mulcn 23469 | Complex number multiplicat... |
| divcn 23470 | Complex number division is... |
| cnfldtgp 23471 | The complex numbers form a... |
| fsumcn 23472 | A finite sum of functions ... |
| fsum2cn 23473 | Version of ~ fsumcn for tw... |
| expcn 23474 | The power function on comp... |
| divccn 23475 | Division by a nonzero cons... |
| sqcn 23476 | The square function on com... |
| iitopon 23481 | The unit interval is a top... |
| iitop 23482 | The unit interval is a top... |
| iiuni 23483 | The base set of the unit i... |
| dfii2 23484 | Alternate definition of th... |
| dfii3 23485 | Alternate definition of th... |
| dfii4 23486 | Alternate definition of th... |
| dfii5 23487 | The unit interval expresse... |
| iicmp 23488 | The unit interval is compa... |
| iiconn 23489 | The unit interval is conne... |
| cncfval 23490 | The value of the continuou... |
| elcncf 23491 | Membership in the set of c... |
| elcncf2 23492 | Version of ~ elcncf with a... |
| cncfrss 23493 | Reverse closure of the con... |
| cncfrss2 23494 | Reverse closure of the con... |
| cncff 23495 | A continuous complex funct... |
| cncfi 23496 | Defining property of a con... |
| elcncf1di 23497 | Membership in the set of c... |
| elcncf1ii 23498 | Membership in the set of c... |
| rescncf 23499 | A continuous complex funct... |
| cncffvrn 23500 | Change the codomain of a c... |
| cncfss 23501 | The set of continuous func... |
| climcncf 23502 | Image of a limit under a c... |
| abscncf 23503 | Absolute value is continuo... |
| recncf 23504 | Real part is continuous. ... |
| imcncf 23505 | Imaginary part is continuo... |
| cjcncf 23506 | Complex conjugate is conti... |
| mulc1cncf 23507 | Multiplication by a consta... |
| divccncf 23508 | Division by a constant is ... |
| cncfco 23509 | The composition of two con... |
| cncfmet 23510 | Relate complex function co... |
| cncfcn 23511 | Relate complex function co... |
| cncfcn1 23512 | Relate complex function co... |
| cncfmptc 23513 | A constant function is a c... |
| cncfmptid 23514 | The identity function is a... |
| cncfmpt1f 23515 | Composition of continuous ... |
| cncfmpt2f 23516 | Composition of continuous ... |
| cncfmpt2ss 23517 | Composition of continuous ... |
| addccncf 23518 | Adding a constant is a con... |
| cdivcncf 23519 | Division with a constant n... |
| negcncf 23520 | The negative function is c... |
| negfcncf 23521 | The negative of a continuo... |
| abscncfALT 23522 | Absolute value is continuo... |
| cncfcnvcn 23523 | Rewrite ~ cmphaushmeo for ... |
| expcncf 23524 | The power function on comp... |
| cnmptre 23525 | Lemma for ~ iirevcn and re... |
| cnmpopc 23526 | Piecewise definition of a ... |
| iirev 23527 | Reverse the unit interval.... |
| iirevcn 23528 | The reversion function is ... |
| iihalf1 23529 | Map the first half of ` II... |
| iihalf1cn 23530 | The first half function is... |
| iihalf2 23531 | Map the second half of ` I... |
| iihalf2cn 23532 | The second half function i... |
| elii1 23533 | Divide the unit interval i... |
| elii2 23534 | Divide the unit interval i... |
| iimulcl 23535 | The unit interval is close... |
| iimulcn 23536 | Multiplication is a contin... |
| icoopnst 23537 | A half-open interval start... |
| iocopnst 23538 | A half-open interval endin... |
| icchmeo 23539 | The natural bijection from... |
| icopnfcnv 23540 | Define a bijection from ` ... |
| icopnfhmeo 23541 | The defined bijection from... |
| iccpnfcnv 23542 | Define a bijection from ` ... |
| iccpnfhmeo 23543 | The defined bijection from... |
| xrhmeo 23544 | The bijection from ` [ -u ... |
| xrhmph 23545 | The extended reals are hom... |
| xrcmp 23546 | The topology of the extend... |
| xrconn 23547 | The topology of the extend... |
| icccvx 23548 | A linear combination of tw... |
| oprpiece1res1 23549 | Restriction to the first p... |
| oprpiece1res2 23550 | Restriction to the second ... |
| cnrehmeo 23551 | The canonical bijection fr... |
| cnheiborlem 23552 | Lemma for ~ cnheibor . (C... |
| cnheibor 23553 | Heine-Borel theorem for co... |
| cnllycmp 23554 | The topology on the comple... |
| rellycmp 23555 | The topology on the reals ... |
| bndth 23556 | The Boundedness Theorem. ... |
| evth 23557 | The Extreme Value Theorem.... |
| evth2 23558 | The Extreme Value Theorem,... |
| lebnumlem1 23559 | Lemma for ~ lebnum . The ... |
| lebnumlem2 23560 | Lemma for ~ lebnum . As a... |
| lebnumlem3 23561 | Lemma for ~ lebnum . By t... |
| lebnum 23562 | The Lebesgue number lemma,... |
| xlebnum 23563 | Generalize ~ lebnum to ext... |
| lebnumii 23564 | Specialize the Lebesgue nu... |
| ishtpy 23570 | Membership in the class of... |
| htpycn 23571 | A homotopy is a continuous... |
| htpyi 23572 | A homotopy evaluated at it... |
| ishtpyd 23573 | Deduction for membership i... |
| htpycom 23574 | Given a homotopy from ` F ... |
| htpyid 23575 | A homotopy from a function... |
| htpyco1 23576 | Compose a homotopy with a ... |
| htpyco2 23577 | Compose a homotopy with a ... |
| htpycc 23578 | Concatenate two homotopies... |
| isphtpy 23579 | Membership in the class of... |
| phtpyhtpy 23580 | A path homotopy is a homot... |
| phtpycn 23581 | A path homotopy is a conti... |
| phtpyi 23582 | Membership in the class of... |
| phtpy01 23583 | Two path-homotopic paths h... |
| isphtpyd 23584 | Deduction for membership i... |
| isphtpy2d 23585 | Deduction for membership i... |
| phtpycom 23586 | Given a homotopy from ` F ... |
| phtpyid 23587 | A homotopy from a path to ... |
| phtpyco2 23588 | Compose a path homotopy wi... |
| phtpycc 23589 | Concatenate two path homot... |
| phtpcrel 23591 | The path homotopy relation... |
| isphtpc 23592 | The relation "is path homo... |
| phtpcer 23593 | Path homotopy is an equiva... |
| phtpc01 23594 | Path homotopic paths have ... |
| reparphti 23595 | Lemma for ~ reparpht . (C... |
| reparpht 23596 | Reparametrization lemma. ... |
| phtpcco2 23597 | Compose a path homotopy wi... |
| pcofval 23608 | The value of the path conc... |
| pcoval 23609 | The concatenation of two p... |
| pcovalg 23610 | Evaluate the concatenation... |
| pcoval1 23611 | Evaluate the concatenation... |
| pco0 23612 | The starting point of a pa... |
| pco1 23613 | The ending point of a path... |
| pcoval2 23614 | Evaluate the concatenation... |
| pcocn 23615 | The concatenation of two p... |
| copco 23616 | The composition of a conca... |
| pcohtpylem 23617 | Lemma for ~ pcohtpy . (Co... |
| pcohtpy 23618 | Homotopy invariance of pat... |
| pcoptcl 23619 | A constant function is a p... |
| pcopt 23620 | Concatenation with a point... |
| pcopt2 23621 | Concatenation with a point... |
| pcoass 23622 | Order of concatenation doe... |
| pcorevcl 23623 | Closure for a reversed pat... |
| pcorevlem 23624 | Lemma for ~ pcorev . Prov... |
| pcorev 23625 | Concatenation with the rev... |
| pcorev2 23626 | Concatenation with the rev... |
| pcophtb 23627 | The path homotopy equivale... |
| om1val 23628 | The definition of the loop... |
| om1bas 23629 | The base set of the loop s... |
| om1elbas 23630 | Elementhood in the base se... |
| om1addcl 23631 | Closure of the group opera... |
| om1plusg 23632 | The group operation (which... |
| om1tset 23633 | The topology of the loop s... |
| om1opn 23634 | The topology of the loop s... |
| pi1val 23635 | The definition of the fund... |
| pi1bas 23636 | The base set of the fundam... |
| pi1blem 23637 | Lemma for ~ pi1buni . (Co... |
| pi1buni 23638 | Another way to write the l... |
| pi1bas2 23639 | The base set of the fundam... |
| pi1eluni 23640 | Elementhood in the base se... |
| pi1bas3 23641 | The base set of the fundam... |
| pi1cpbl 23642 | The group operation, loop ... |
| elpi1 23643 | The elements of the fundam... |
| elpi1i 23644 | The elements of the fundam... |
| pi1addf 23645 | The group operation of ` p... |
| pi1addval 23646 | The concatenation of two p... |
| pi1grplem 23647 | Lemma for ~ pi1grp . (Con... |
| pi1grp 23648 | The fundamental group is a... |
| pi1id 23649 | The identity element of th... |
| pi1inv 23650 | An inverse in the fundamen... |
| pi1xfrf 23651 | Functionality of the loop ... |
| pi1xfrval 23652 | The value of the loop tran... |
| pi1xfr 23653 | Given a path ` F ` and its... |
| pi1xfrcnvlem 23654 | Given a path ` F ` between... |
| pi1xfrcnv 23655 | Given a path ` F ` between... |
| pi1xfrgim 23656 | The mapping ` G ` between ... |
| pi1cof 23657 | Functionality of the loop ... |
| pi1coval 23658 | The value of the loop tran... |
| pi1coghm 23659 | The mapping ` G ` between ... |
| isclm 23662 | A subcomplex module is a l... |
| clmsca 23663 | The ring of scalars ` F ` ... |
| clmsubrg 23664 | The base set of the ring o... |
| clmlmod 23665 | A subcomplex module is a l... |
| clmgrp 23666 | A subcomplex module is an ... |
| clmabl 23667 | A subcomplex module is an ... |
| clmring 23668 | The scalar ring of a subco... |
| clmfgrp 23669 | The scalar ring of a subco... |
| clm0 23670 | The zero of the scalar rin... |
| clm1 23671 | The identity of the scalar... |
| clmadd 23672 | The addition of the scalar... |
| clmmul 23673 | The multiplication of the ... |
| clmcj 23674 | The conjugation of the sca... |
| isclmi 23675 | Reverse direction of ~ isc... |
| clmzss 23676 | The scalar ring of a subco... |
| clmsscn 23677 | The scalar ring of a subco... |
| clmsub 23678 | Subtraction in the scalar ... |
| clmneg 23679 | Negation in the scalar rin... |
| clmneg1 23680 | Minus one is in the scalar... |
| clmabs 23681 | Norm in the scalar ring of... |
| clmacl 23682 | Closure of ring addition f... |
| clmmcl 23683 | Closure of ring multiplica... |
| clmsubcl 23684 | Closure of ring subtractio... |
| lmhmclm 23685 | The domain of a linear ope... |
| clmvscl 23686 | Closure of scalar product ... |
| clmvsass 23687 | Associative law for scalar... |
| clmvscom 23688 | Commutative law for the sc... |
| clmvsdir 23689 | Distributive law for scala... |
| clmvsdi 23690 | Distributive law for scala... |
| clmvs1 23691 | Scalar product with ring u... |
| clmvs2 23692 | A vector plus itself is tw... |
| clm0vs 23693 | Zero times a vector is the... |
| clmopfne 23694 | The (functionalized) opera... |
| isclmp 23695 | The predicate "is a subcom... |
| isclmi0 23696 | Properties that determine ... |
| clmvneg1 23697 | Minus 1 times a vector is ... |
| clmvsneg 23698 | Multiplication of a vector... |
| clmmulg 23699 | The group multiple functio... |
| clmsubdir 23700 | Scalar multiplication dist... |
| clmpm1dir 23701 | Subtractive distributive l... |
| clmnegneg 23702 | Double negative of a vecto... |
| clmnegsubdi2 23703 | Distribution of negative o... |
| clmsub4 23704 | Rearrangement of 4 terms i... |
| clmvsrinv 23705 | A vector minus itself. (C... |
| clmvslinv 23706 | Minus a vector plus itself... |
| clmvsubval 23707 | Value of vector subtractio... |
| clmvsubval2 23708 | Value of vector subtractio... |
| clmvz 23709 | Two ways to express the ne... |
| zlmclm 23710 | The ` ZZ ` -module operati... |
| clmzlmvsca 23711 | The scalar product of a su... |
| nmoleub2lem 23712 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem3 23713 | Lemma for ~ nmoleub2a and ... |
| nmoleub2lem2 23714 | Lemma for ~ nmoleub2a and ... |
| nmoleub2a 23715 | The operator norm is the s... |
| nmoleub2b 23716 | The operator norm is the s... |
| nmoleub3 23717 | The operator norm is the s... |
| nmhmcn 23718 | A linear operator over a n... |
| cmodscexp 23719 | The powers of ` _i ` belon... |
| cmodscmulexp 23720 | The scalar product of a ve... |
| cvslvec 23723 | A subcomplex vector space ... |
| cvsclm 23724 | A subcomplex vector space ... |
| iscvs 23725 | A subcomplex vector space ... |
| iscvsp 23726 | The predicate "is a subcom... |
| iscvsi 23727 | Properties that determine ... |
| cvsi 23728 | The properties of a subcom... |
| cvsunit 23729 | Unit group of the scalar r... |
| cvsdiv 23730 | Division of the scalar rin... |
| cvsdivcl 23731 | The scalar field of a subc... |
| cvsmuleqdivd 23732 | An equality involving rati... |
| cvsdiveqd 23733 | An equality involving rati... |
| cnlmodlem1 23734 | Lemma 1 for ~ cnlmod . (C... |
| cnlmodlem2 23735 | Lemma 2 for ~ cnlmod . (C... |
| cnlmodlem3 23736 | Lemma 3 for ~ cnlmod . (C... |
| cnlmod4 23737 | Lemma 4 for ~ cnlmod . (C... |
| cnlmod 23738 | The set of complex numbers... |
| cnstrcvs 23739 | The set of complex numbers... |
| cnrbas 23740 | The set of complex numbers... |
| cnrlmod 23741 | The complex left module of... |
| cnrlvec 23742 | The complex left module of... |
| cncvs 23743 | The complex left module of... |
| recvs 23744 | The field of the real numb... |
| qcvs 23745 | The field of rational numb... |
| zclmncvs 23746 | The ring of integers as le... |
| isncvsngp 23747 | A normed subcomplex vector... |
| isncvsngpd 23748 | Properties that determine ... |
| ncvsi 23749 | The properties of a normed... |
| ncvsprp 23750 | Proportionality property o... |
| ncvsge0 23751 | The norm of a scalar produ... |
| ncvsm1 23752 | The norm of the opposite o... |
| ncvsdif 23753 | The norm of the difference... |
| ncvspi 23754 | The norm of a vector plus ... |
| ncvs1 23755 | From any nonzero vector of... |
| cnrnvc 23756 | The module of complex numb... |
| cnncvs 23757 | The module of complex numb... |
| cnnm 23758 | The norm of the normed sub... |
| ncvspds 23759 | Value of the distance func... |
| cnindmet 23760 | The metric induced on the ... |
| cnncvsaddassdemo 23761 | Derive the associative law... |
| cnncvsmulassdemo 23762 | Derive the associative law... |
| cnncvsabsnegdemo 23763 | Derive the absolute value ... |
| iscph 23768 | A subcomplex pre-Hilbert s... |
| cphphl 23769 | A subcomplex pre-Hilbert s... |
| cphnlm 23770 | A subcomplex pre-Hilbert s... |
| cphngp 23771 | A subcomplex pre-Hilbert s... |
| cphlmod 23772 | A subcomplex pre-Hilbert s... |
| cphlvec 23773 | A subcomplex pre-Hilbert s... |
| cphnvc 23774 | A subcomplex pre-Hilbert s... |
| cphsubrglem 23775 | Lemma for ~ cphsubrg . (C... |
| cphreccllem 23776 | Lemma for ~ cphreccl . (C... |
| cphsca 23777 | A subcomplex pre-Hilbert s... |
| cphsubrg 23778 | The scalar field of a subc... |
| cphreccl 23779 | The scalar field of a subc... |
| cphdivcl 23780 | The scalar field of a subc... |
| cphcjcl 23781 | The scalar field of a subc... |
| cphsqrtcl 23782 | The scalar field of a subc... |
| cphabscl 23783 | The scalar field of a subc... |
| cphsqrtcl2 23784 | The scalar field of a subc... |
| cphsqrtcl3 23785 | If the scalar field of a s... |
| cphqss 23786 | The scalar field of a subc... |
| cphclm 23787 | A subcomplex pre-Hilbert s... |
| cphnmvs 23788 | Norm of a scalar product. ... |
| cphipcl 23789 | An inner product is a memb... |
| cphnmfval 23790 | The value of the norm in a... |
| cphnm 23791 | The square of the norm is ... |
| nmsq 23792 | The square of the norm is ... |
| cphnmf 23793 | The norm of a vector is a ... |
| cphnmcl 23794 | The norm of a vector is a ... |
| reipcl 23795 | An inner product of an ele... |
| ipge0 23796 | The inner product in a sub... |
| cphipcj 23797 | Conjugate of an inner prod... |
| cphipipcj 23798 | An inner product times its... |
| cphorthcom 23799 | Orthogonality (meaning inn... |
| cphip0l 23800 | Inner product with a zero ... |
| cphip0r 23801 | Inner product with a zero ... |
| cphipeq0 23802 | The inner product of a vec... |
| cphdir 23803 | Distributive law for inner... |
| cphdi 23804 | Distributive law for inner... |
| cph2di 23805 | Distributive law for inner... |
| cphsubdir 23806 | Distributive law for inner... |
| cphsubdi 23807 | Distributive law for inner... |
| cph2subdi 23808 | Distributive law for inner... |
| cphass 23809 | Associative law for inner ... |
| cphassr 23810 | "Associative" law for seco... |
| cph2ass 23811 | Move scalar multiplication... |
| cphassi 23812 | Associative law for the fi... |
| cphassir 23813 | "Associative" law for the ... |
| tcphex 23814 | Lemma for ~ tcphbas and si... |
| tcphval 23815 | Define a function to augme... |
| tcphbas 23816 | The base set of a subcompl... |
| tchplusg 23817 | The addition operation of ... |
| tcphsub 23818 | The subtraction operation ... |
| tcphmulr 23819 | The ring operation of a su... |
| tcphsca 23820 | The scalar field of a subc... |
| tcphvsca 23821 | The scalar multiplication ... |
| tcphip 23822 | The inner product of a sub... |
| tcphtopn 23823 | The topology of a subcompl... |
| tcphphl 23824 | Augmentation of a subcompl... |
| tchnmfval 23825 | The norm of a subcomplex p... |
| tcphnmval 23826 | The norm of a subcomplex p... |
| cphtcphnm 23827 | The norm of a norm-augment... |
| tcphds 23828 | The distance of a pre-Hilb... |
| phclm 23829 | A pre-Hilbert space whose ... |
| tcphcphlem3 23830 | Lemma for ~ tcphcph : real... |
| ipcau2 23831 | The Cauchy-Schwarz inequal... |
| tcphcphlem1 23832 | Lemma for ~ tcphcph : the ... |
| tcphcphlem2 23833 | Lemma for ~ tcphcph : homo... |
| tcphcph 23834 | The standard definition of... |
| ipcau 23835 | The Cauchy-Schwarz inequal... |
| nmparlem 23836 | Lemma for ~ nmpar . (Cont... |
| nmpar 23837 | A subcomplex pre-Hilbert s... |
| cphipval2 23838 | Value of the inner product... |
| 4cphipval2 23839 | Four times the inner produ... |
| cphipval 23840 | Value of the inner product... |
| ipcnlem2 23841 | The inner product operatio... |
| ipcnlem1 23842 | The inner product operatio... |
| ipcn 23843 | The inner product operatio... |
| cnmpt1ip 23844 | Continuity of inner produc... |
| cnmpt2ip 23845 | Continuity of inner produc... |
| csscld 23846 | A "closed subspace" in a s... |
| clsocv 23847 | The orthogonal complement ... |
| cphsscph 23848 | A subspace of a subcomplex... |
| lmmbr 23855 | Express the binary relatio... |
| lmmbr2 23856 | Express the binary relatio... |
| lmmbr3 23857 | Express the binary relatio... |
| lmmcvg 23858 | Convergence property of a ... |
| lmmbrf 23859 | Express the binary relatio... |
| lmnn 23860 | A condition that implies c... |
| cfilfval 23861 | The set of Cauchy filters ... |
| iscfil 23862 | The property of being a Ca... |
| iscfil2 23863 | The property of being a Ca... |
| cfilfil 23864 | A Cauchy filter is a filte... |
| cfili 23865 | Property of a Cauchy filte... |
| cfil3i 23866 | A Cauchy filter contains b... |
| cfilss 23867 | A filter finer than a Cauc... |
| fgcfil 23868 | The Cauchy filter conditio... |
| fmcfil 23869 | The Cauchy filter conditio... |
| iscfil3 23870 | A filter is Cauchy iff it ... |
| cfilfcls 23871 | Similar to ultrafilters ( ... |
| caufval 23872 | The set of Cauchy sequence... |
| iscau 23873 | Express the property " ` F... |
| iscau2 23874 | Express the property " ` F... |
| iscau3 23875 | Express the Cauchy sequenc... |
| iscau4 23876 | Express the property " ` F... |
| iscauf 23877 | Express the property " ` F... |
| caun0 23878 | A metric with a Cauchy seq... |
| caufpm 23879 | Inclusion of a Cauchy sequ... |
| caucfil 23880 | A Cauchy sequence predicat... |
| iscmet 23881 | The property " ` D ` is a ... |
| cmetcvg 23882 | The convergence of a Cauch... |
| cmetmet 23883 | A complete metric space is... |
| cmetmeti 23884 | A complete metric space is... |
| cmetcaulem 23885 | Lemma for ~ cmetcau . (Co... |
| cmetcau 23886 | The convergence of a Cauch... |
| iscmet3lem3 23887 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem1 23888 | Lemma for ~ iscmet3 . (Co... |
| iscmet3lem2 23889 | Lemma for ~ iscmet3 . (Co... |
| iscmet3 23890 | The property " ` D ` is a ... |
| iscmet2 23891 | A metric ` D ` is complete... |
| cfilresi 23892 | A Cauchy filter on a metri... |
| cfilres 23893 | Cauchy filter on a metric ... |
| caussi 23894 | Cauchy sequence on a metri... |
| causs 23895 | Cauchy sequence on a metri... |
| equivcfil 23896 | If the metric ` D ` is "st... |
| equivcau 23897 | If the metric ` D ` is "st... |
| lmle 23898 | If the distance from each ... |
| nglmle 23899 | If the norm of each member... |
| lmclim 23900 | Relate a limit on the metr... |
| lmclimf 23901 | Relate a limit on the metr... |
| metelcls 23902 | A point belongs to the clo... |
| metcld 23903 | A subset of a metric space... |
| metcld2 23904 | A subset of a metric space... |
| caubl 23905 | Sufficient condition to en... |
| caublcls 23906 | The convergent point of a ... |
| metcnp4 23907 | Two ways to say a mapping ... |
| metcn4 23908 | Two ways to say a mapping ... |
| iscmet3i 23909 | Properties that determine ... |
| lmcau 23910 | Every convergent sequence ... |
| flimcfil 23911 | Every convergent filter in... |
| metsscmetcld 23912 | A complete subspace of a m... |
| cmetss 23913 | A subspace of a complete m... |
| equivcmet 23914 | If two metrics are strongl... |
| relcmpcmet 23915 | If ` D ` is a metric space... |
| cmpcmet 23916 | A compact metric space is ... |
| cfilucfil3 23917 | Given a metric ` D ` and a... |
| cfilucfil4 23918 | Given a metric ` D ` and a... |
| cncmet 23919 | The set of complex numbers... |
| recmet 23920 | The real numbers are a com... |
| bcthlem1 23921 | Lemma for ~ bcth . Substi... |
| bcthlem2 23922 | Lemma for ~ bcth . The ba... |
| bcthlem3 23923 | Lemma for ~ bcth . The li... |
| bcthlem4 23924 | Lemma for ~ bcth . Given ... |
| bcthlem5 23925 | Lemma for ~ bcth . The pr... |
| bcth 23926 | Baire's Category Theorem. ... |
| bcth2 23927 | Baire's Category Theorem, ... |
| bcth3 23928 | Baire's Category Theorem, ... |
| isbn 23935 | A Banach space is a normed... |
| bnsca 23936 | The scalar field of a Bana... |
| bnnvc 23937 | A Banach space is a normed... |
| bnnlm 23938 | A Banach space is a normed... |
| bnngp 23939 | A Banach space is a normed... |
| bnlmod 23940 | A Banach space is a left m... |
| bncms 23941 | A Banach space is a comple... |
| iscms 23942 | A complete metric space is... |
| cmscmet 23943 | The induced metric on a co... |
| bncmet 23944 | The induced metric on Bana... |
| cmsms 23945 | A complete metric space is... |
| cmspropd 23946 | Property deduction for a c... |
| cmssmscld 23947 | The restriction of a metri... |
| cmsss 23948 | The restriction of a compl... |
| lssbn 23949 | A subspace of a Banach spa... |
| cmetcusp1 23950 | If the uniform set of a co... |
| cmetcusp 23951 | The uniform space generate... |
| cncms 23952 | The field of complex numbe... |
| cnflduss 23953 | The uniform structure of t... |
| cnfldcusp 23954 | The field of complex numbe... |
| resscdrg 23955 | The real numbers are a sub... |
| cncdrg 23956 | The only complete subfield... |
| srabn 23957 | The subring algebra over a... |
| rlmbn 23958 | The ring module over a com... |
| ishl 23959 | The predicate "is a subcom... |
| hlbn 23960 | Every subcomplex Hilbert s... |
| hlcph 23961 | Every subcomplex Hilbert s... |
| hlphl 23962 | Every subcomplex Hilbert s... |
| hlcms 23963 | Every subcomplex Hilbert s... |
| hlprlem 23964 | Lemma for ~ hlpr . (Contr... |
| hlress 23965 | The scalar field of a subc... |
| hlpr 23966 | The scalar field of a subc... |
| ishl2 23967 | A Hilbert space is a compl... |
| cphssphl 23968 | A Banach subspace of a sub... |
| cmslssbn 23969 | A complete linear subspace... |
| cmscsscms 23970 | A closed subspace of a com... |
| bncssbn 23971 | A closed subspace of a Ban... |
| cssbn 23972 | A complete subspace of a n... |
| csschl 23973 | A complete subspace of a c... |
| cmslsschl 23974 | A complete linear subspace... |
| chlcsschl 23975 | A closed subspace of a sub... |
| retopn 23976 | The topology of the real n... |
| recms 23977 | The real numbers form a co... |
| reust 23978 | The Uniform structure of t... |
| recusp 23979 | The real numbers form a co... |
| rrxval 23984 | Value of the generalized E... |
| rrxbase 23985 | The base of the generalize... |
| rrxprds 23986 | Expand the definition of t... |
| rrxip 23987 | The inner product of the g... |
| rrxnm 23988 | The norm of the generalize... |
| rrxcph 23989 | Generalized Euclidean real... |
| rrxds 23990 | The distance over generali... |
| rrxvsca 23991 | The scalar product over ge... |
| rrxplusgvscavalb 23992 | The result of the addition... |
| rrxsca 23993 | The field of real numbers ... |
| rrx0 23994 | The zero ("origin") in a g... |
| rrx0el 23995 | The zero ("origin") in a g... |
| csbren 23996 | Cauchy-Schwarz-Bunjakovsky... |
| trirn 23997 | Triangle inequality in R^n... |
| rrxf 23998 | Euclidean vectors as funct... |
| rrxfsupp 23999 | Euclidean vectors are of f... |
| rrxsuppss 24000 | Support of Euclidean vecto... |
| rrxmvallem 24001 | Support of the function us... |
| rrxmval 24002 | The value of the Euclidean... |
| rrxmfval 24003 | The value of the Euclidean... |
| rrxmetlem 24004 | Lemma for ~ rrxmet . (Con... |
| rrxmet 24005 | Euclidean space is a metri... |
| rrxdstprj1 24006 | The distance between two p... |
| rrxbasefi 24007 | The base of the generalize... |
| rrxdsfi 24008 | The distance over generali... |
| rrxmetfi 24009 | Euclidean space is a metri... |
| rrxdsfival 24010 | The value of the Euclidean... |
| ehlval 24011 | Value of the Euclidean spa... |
| ehlbase 24012 | The base of the Euclidean ... |
| ehl0base 24013 | The base of the Euclidean ... |
| ehl0 24014 | The Euclidean space of dim... |
| ehleudis 24015 | The Euclidean distance fun... |
| ehleudisval 24016 | The value of the Euclidean... |
| ehl1eudis 24017 | The Euclidean distance fun... |
| ehl1eudisval 24018 | The value of the Euclidean... |
| ehl2eudis 24019 | The Euclidean distance fun... |
| ehl2eudisval 24020 | The value of the Euclidean... |
| minveclem1 24021 | Lemma for ~ minvec . The ... |
| minveclem4c 24022 | Lemma for ~ minvec . The ... |
| minveclem2 24023 | Lemma for ~ minvec . Any ... |
| minveclem3a 24024 | Lemma for ~ minvec . ` D `... |
| minveclem3b 24025 | Lemma for ~ minvec . The ... |
| minveclem3 24026 | Lemma for ~ minvec . The ... |
| minveclem4a 24027 | Lemma for ~ minvec . ` F `... |
| minveclem4b 24028 | Lemma for ~ minvec . The ... |
| minveclem4 24029 | Lemma for ~ minvec . The ... |
| minveclem5 24030 | Lemma for ~ minvec . Disc... |
| minveclem6 24031 | Lemma for ~ minvec . Any ... |
| minveclem7 24032 | Lemma for ~ minvec . Sinc... |
| minvec 24033 | Minimizing vector theorem,... |
| pjthlem1 24034 | Lemma for ~ pjth . (Contr... |
| pjthlem2 24035 | Lemma for ~ pjth . (Contr... |
| pjth 24036 | Projection Theorem: Any H... |
| pjth2 24037 | Projection Theorem with ab... |
| cldcss 24038 | Corollary of the Projectio... |
| cldcss2 24039 | Corollary of the Projectio... |
| hlhil 24040 | Corollary of the Projectio... |
| mulcncf 24041 | The multiplication of two ... |
| divcncf 24042 | The quotient of two contin... |
| pmltpclem1 24043 | Lemma for ~ pmltpc . (Con... |
| pmltpclem2 24044 | Lemma for ~ pmltpc . (Con... |
| pmltpc 24045 | Any function on the reals ... |
| ivthlem1 24046 | Lemma for ~ ivth . The se... |
| ivthlem2 24047 | Lemma for ~ ivth . Show t... |
| ivthlem3 24048 | Lemma for ~ ivth , the int... |
| ivth 24049 | The intermediate value the... |
| ivth2 24050 | The intermediate value the... |
| ivthle 24051 | The intermediate value the... |
| ivthle2 24052 | The intermediate value the... |
| ivthicc 24053 | The interval between any t... |
| evthicc 24054 | Specialization of the Extr... |
| evthicc2 24055 | Combine ~ ivthicc with ~ e... |
| cniccbdd 24056 | A continuous function on a... |
| ovolfcl 24061 | Closure for the interval e... |
| ovolfioo 24062 | Unpack the interval coveri... |
| ovolficc 24063 | Unpack the interval coveri... |
| ovolficcss 24064 | Any (closed) interval cove... |
| ovolfsval 24065 | The value of the interval ... |
| ovolfsf 24066 | Closure for the interval l... |
| ovolsf 24067 | Closure for the partial su... |
| ovolval 24068 | The value of the outer mea... |
| elovolmlem 24069 | Lemma for ~ elovolm and re... |
| elovolm 24070 | Elementhood in the set ` M... |
| elovolmr 24071 | Sufficient condition for e... |
| ovolmge0 24072 | The set ` M ` is composed ... |
| ovolcl 24073 | The volume of a set is an ... |
| ovollb 24074 | The outer volume is a lowe... |
| ovolgelb 24075 | The outer volume is the gr... |
| ovolge0 24076 | The volume of a set is alw... |
| ovolf 24077 | The domain and range of th... |
| ovollecl 24078 | If an outer volume is boun... |
| ovolsslem 24079 | Lemma for ~ ovolss . (Con... |
| ovolss 24080 | The volume of a set is mon... |
| ovolsscl 24081 | If a set is contained in a... |
| ovolssnul 24082 | A subset of a nullset is n... |
| ovollb2lem 24083 | Lemma for ~ ovollb2 . (Co... |
| ovollb2 24084 | It is often more convenien... |
| ovolctb 24085 | The volume of a denumerabl... |
| ovolq 24086 | The rational numbers have ... |
| ovolctb2 24087 | The volume of a countable ... |
| ovol0 24088 | The empty set has 0 outer ... |
| ovolfi 24089 | A finite set has 0 outer L... |
| ovolsn 24090 | A singleton has 0 outer Le... |
| ovolunlem1a 24091 | Lemma for ~ ovolun . (Con... |
| ovolunlem1 24092 | Lemma for ~ ovolun . (Con... |
| ovolunlem2 24093 | Lemma for ~ ovolun . (Con... |
| ovolun 24094 | The Lebesgue outer measure... |
| ovolunnul 24095 | Adding a nullset does not ... |
| ovolfiniun 24096 | The Lebesgue outer measure... |
| ovoliunlem1 24097 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem2 24098 | Lemma for ~ ovoliun . (Co... |
| ovoliunlem3 24099 | Lemma for ~ ovoliun . (Co... |
| ovoliun 24100 | The Lebesgue outer measure... |
| ovoliun2 24101 | The Lebesgue outer measure... |
| ovoliunnul 24102 | A countable union of nulls... |
| shft2rab 24103 | If ` B ` is a shift of ` A... |
| ovolshftlem1 24104 | Lemma for ~ ovolshft . (C... |
| ovolshftlem2 24105 | Lemma for ~ ovolshft . (C... |
| ovolshft 24106 | The Lebesgue outer measure... |
| sca2rab 24107 | If ` B ` is a scale of ` A... |
| ovolscalem1 24108 | Lemma for ~ ovolsca . (Co... |
| ovolscalem2 24109 | Lemma for ~ ovolshft . (C... |
| ovolsca 24110 | The Lebesgue outer measure... |
| ovolicc1 24111 | The measure of a closed in... |
| ovolicc2lem1 24112 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem2 24113 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem3 24114 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem4 24115 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2lem5 24116 | Lemma for ~ ovolicc2 . (C... |
| ovolicc2 24117 | The measure of a closed in... |
| ovolicc 24118 | The measure of a closed in... |
| ovolicopnf 24119 | The measure of a right-unb... |
| ovolre 24120 | The measure of the real nu... |
| ismbl 24121 | The predicate " ` A ` is L... |
| ismbl2 24122 | From ~ ovolun , it suffice... |
| volres 24123 | A self-referencing abbrevi... |
| volf 24124 | The domain and range of th... |
| mblvol 24125 | The volume of a measurable... |
| mblss 24126 | A measurable set is a subs... |
| mblsplit 24127 | The defining property of m... |
| volss 24128 | The Lebesgue measure is mo... |
| cmmbl 24129 | The complement of a measur... |
| nulmbl 24130 | A nullset is measurable. ... |
| nulmbl2 24131 | A set of outer measure zer... |
| unmbl 24132 | A union of measurable sets... |
| shftmbl 24133 | A shift of a measurable se... |
| 0mbl 24134 | The empty set is measurabl... |
| rembl 24135 | The set of all real number... |
| unidmvol 24136 | The union of the Lebesgue ... |
| inmbl 24137 | An intersection of measura... |
| difmbl 24138 | A difference of measurable... |
| finiunmbl 24139 | A finite union of measurab... |
| volun 24140 | The Lebesgue measure funct... |
| volinun 24141 | Addition of non-disjoint s... |
| volfiniun 24142 | The volume of a disjoint f... |
| iundisj 24143 | Rewrite a countable union ... |
| iundisj2 24144 | A disjoint union is disjoi... |
| voliunlem1 24145 | Lemma for ~ voliun . (Con... |
| voliunlem2 24146 | Lemma for ~ voliun . (Con... |
| voliunlem3 24147 | Lemma for ~ voliun . (Con... |
| iunmbl 24148 | The measurable sets are cl... |
| voliun 24149 | The Lebesgue measure funct... |
| volsuplem 24150 | Lemma for ~ volsup . (Con... |
| volsup 24151 | The volume of the limit of... |
| iunmbl2 24152 | The measurable sets are cl... |
| ioombl1lem1 24153 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem2 24154 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem3 24155 | Lemma for ~ ioombl1 . (Co... |
| ioombl1lem4 24156 | Lemma for ~ ioombl1 . (Co... |
| ioombl1 24157 | An open right-unbounded in... |
| icombl1 24158 | A closed unbounded-above i... |
| icombl 24159 | A closed-below, open-above... |
| ioombl 24160 | An open real interval is m... |
| iccmbl 24161 | A closed real interval is ... |
| iccvolcl 24162 | A closed real interval has... |
| ovolioo 24163 | The measure of an open int... |
| volioo 24164 | The measure of an open int... |
| ioovolcl 24165 | An open real interval has ... |
| ovolfs2 24166 | Alternative expression for... |
| ioorcl2 24167 | An open interval with fini... |
| ioorf 24168 | Define a function from ope... |
| ioorval 24169 | Define a function from ope... |
| ioorinv2 24170 | The function ` F ` is an "... |
| ioorinv 24171 | The function ` F ` is an "... |
| ioorcl 24172 | The function ` F ` does no... |
| uniiccdif 24173 | A union of closed interval... |
| uniioovol 24174 | A disjoint union of open i... |
| uniiccvol 24175 | An almost-disjoint union o... |
| uniioombllem1 24176 | Lemma for ~ uniioombl . (... |
| uniioombllem2a 24177 | Lemma for ~ uniioombl . (... |
| uniioombllem2 24178 | Lemma for ~ uniioombl . (... |
| uniioombllem3a 24179 | Lemma for ~ uniioombl . (... |
| uniioombllem3 24180 | Lemma for ~ uniioombl . (... |
| uniioombllem4 24181 | Lemma for ~ uniioombl . (... |
| uniioombllem5 24182 | Lemma for ~ uniioombl . (... |
| uniioombllem6 24183 | Lemma for ~ uniioombl . (... |
| uniioombl 24184 | A disjoint union of open i... |
| uniiccmbl 24185 | An almost-disjoint union o... |
| dyadf 24186 | The function ` F ` returns... |
| dyadval 24187 | Value of the dyadic ration... |
| dyadovol 24188 | Volume of a dyadic rationa... |
| dyadss 24189 | Two closed dyadic rational... |
| dyaddisjlem 24190 | Lemma for ~ dyaddisj . (C... |
| dyaddisj 24191 | Two closed dyadic rational... |
| dyadmaxlem 24192 | Lemma for ~ dyadmax . (Co... |
| dyadmax 24193 | Any nonempty set of dyadic... |
| dyadmbllem 24194 | Lemma for ~ dyadmbl . (Co... |
| dyadmbl 24195 | Any union of dyadic ration... |
| opnmbllem 24196 | Lemma for ~ opnmbl . (Con... |
| opnmbl 24197 | All open sets are measurab... |
| opnmblALT 24198 | All open sets are measurab... |
| subopnmbl 24199 | Sets which are open in a m... |
| volsup2 24200 | The volume of ` A ` is the... |
| volcn 24201 | The function formed by res... |
| volivth 24202 | The Intermediate Value The... |
| vitalilem1 24203 | Lemma for ~ vitali . (Con... |
| vitalilem2 24204 | Lemma for ~ vitali . (Con... |
| vitalilem3 24205 | Lemma for ~ vitali . (Con... |
| vitalilem4 24206 | Lemma for ~ vitali . (Con... |
| vitalilem5 24207 | Lemma for ~ vitali . (Con... |
| vitali 24208 | If the reals can be well-o... |
| ismbf1 24219 | The predicate " ` F ` is a... |
| mbff 24220 | A measurable function is a... |
| mbfdm 24221 | The domain of a measurable... |
| mbfconstlem 24222 | Lemma for ~ mbfconst and r... |
| ismbf 24223 | The predicate " ` F ` is a... |
| ismbfcn 24224 | A complex function is meas... |
| mbfima 24225 | Definitional property of a... |
| mbfimaicc 24226 | The preimage of any closed... |
| mbfimasn 24227 | The preimage of a point un... |
| mbfconst 24228 | A constant function is mea... |
| mbf0 24229 | The empty function is meas... |
| mbfid 24230 | The identity function is m... |
| mbfmptcl 24231 | Lemma for the ` MblFn ` pr... |
| mbfdm2 24232 | The domain of a measurable... |
| ismbfcn2 24233 | A complex function is meas... |
| ismbfd 24234 | Deduction to prove measura... |
| ismbf2d 24235 | Deduction to prove measura... |
| mbfeqalem1 24236 | Lemma for ~ mbfeqalem2 . ... |
| mbfeqalem2 24237 | Lemma for ~ mbfeqa . (Con... |
| mbfeqa 24238 | If two functions are equal... |
| mbfres 24239 | The restriction of a measu... |
| mbfres2 24240 | Measurability of a piecewi... |
| mbfss 24241 | Change the domain of a mea... |
| mbfmulc2lem 24242 | Multiplication by a consta... |
| mbfmulc2re 24243 | Multiplication by a consta... |
| mbfmax 24244 | The maximum of two functio... |
| mbfneg 24245 | The negative of a measurab... |
| mbfpos 24246 | The positive part of a mea... |
| mbfposr 24247 | Converse to ~ mbfpos . (C... |
| mbfposb 24248 | A function is measurable i... |
| ismbf3d 24249 | Simplified form of ~ ismbf... |
| mbfimaopnlem 24250 | Lemma for ~ mbfimaopn . (... |
| mbfimaopn 24251 | The preimage of any open s... |
| mbfimaopn2 24252 | The preimage of any set op... |
| cncombf 24253 | The composition of a conti... |
| cnmbf 24254 | A continuous function is m... |
| mbfaddlem 24255 | The sum of two measurable ... |
| mbfadd 24256 | The sum of two measurable ... |
| mbfsub 24257 | The difference of two meas... |
| mbfmulc2 24258 | A complex constant times a... |
| mbfsup 24259 | The supremum of a sequence... |
| mbfinf 24260 | The infimum of a sequence ... |
| mbflimsup 24261 | The limit supremum of a se... |
| mbflimlem 24262 | The pointwise limit of a s... |
| mbflim 24263 | The pointwise limit of a s... |
| 0pval 24266 | The zero function evaluate... |
| 0plef 24267 | Two ways to say that the f... |
| 0pledm 24268 | Adjust the domain of the l... |
| isi1f 24269 | The predicate " ` F ` is a... |
| i1fmbf 24270 | Simple functions are measu... |
| i1ff 24271 | A simple function is a fun... |
| i1frn 24272 | A simple function has fini... |
| i1fima 24273 | Any preimage of a simple f... |
| i1fima2 24274 | Any preimage of a simple f... |
| i1fima2sn 24275 | Preimage of a singleton. ... |
| i1fd 24276 | A simplified set of assump... |
| i1f0rn 24277 | Any simple function takes ... |
| itg1val 24278 | The value of the integral ... |
| itg1val2 24279 | The value of the integral ... |
| itg1cl 24280 | Closure of the integral on... |
| itg1ge0 24281 | Closure of the integral on... |
| i1f0 24282 | The zero function is simpl... |
| itg10 24283 | The zero function has zero... |
| i1f1lem 24284 | Lemma for ~ i1f1 and ~ itg... |
| i1f1 24285 | Base case simple functions... |
| itg11 24286 | The integral of an indicat... |
| itg1addlem1 24287 | Decompose a preimage, whic... |
| i1faddlem 24288 | Decompose the preimage of ... |
| i1fmullem 24289 | Decompose the preimage of ... |
| i1fadd 24290 | The sum of two simple func... |
| i1fmul 24291 | The pointwise product of t... |
| itg1addlem2 24292 | Lemma for ~ itg1add . The... |
| itg1addlem3 24293 | Lemma for ~ itg1add . (Co... |
| itg1addlem4 24294 | Lemma for itg1add . (Cont... |
| itg1addlem5 24295 | Lemma for itg1add . (Cont... |
| itg1add 24296 | The integral of a sum of s... |
| i1fmulclem 24297 | Decompose the preimage of ... |
| i1fmulc 24298 | A nonnegative constant tim... |
| itg1mulc 24299 | The integral of a constant... |
| i1fres 24300 | The "restriction" of a sim... |
| i1fpos 24301 | The positive part of a sim... |
| i1fposd 24302 | Deduction form of ~ i1fpos... |
| i1fsub 24303 | The difference of two simp... |
| itg1sub 24304 | The integral of a differen... |
| itg10a 24305 | The integral of a simple f... |
| itg1ge0a 24306 | The integral of an almost ... |
| itg1lea 24307 | Approximate version of ~ i... |
| itg1le 24308 | If one simple function dom... |
| itg1climres 24309 | Restricting the simple fun... |
| mbfi1fseqlem1 24310 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem2 24311 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem3 24312 | Lemma for ~ mbfi1fseq . (... |
| mbfi1fseqlem4 24313 | Lemma for ~ mbfi1fseq . T... |
| mbfi1fseqlem5 24314 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseqlem6 24315 | Lemma for ~ mbfi1fseq . V... |
| mbfi1fseq 24316 | A characterization of meas... |
| mbfi1flimlem 24317 | Lemma for ~ mbfi1flim . (... |
| mbfi1flim 24318 | Any real measurable functi... |
| mbfmullem2 24319 | Lemma for ~ mbfmul . (Con... |
| mbfmullem 24320 | Lemma for ~ mbfmul . (Con... |
| mbfmul 24321 | The product of two measura... |
| itg2lcl 24322 | The set of lower sums is a... |
| itg2val 24323 | Value of the integral on n... |
| itg2l 24324 | Elementhood in the set ` L... |
| itg2lr 24325 | Sufficient condition for e... |
| xrge0f 24326 | A real function is a nonne... |
| itg2cl 24327 | The integral of a nonnegat... |
| itg2ub 24328 | The integral of a nonnegat... |
| itg2leub 24329 | Any upper bound on the int... |
| itg2ge0 24330 | The integral of a nonnegat... |
| itg2itg1 24331 | The integral of a nonnegat... |
| itg20 24332 | The integral of the zero f... |
| itg2lecl 24333 | If an ` S.2 ` integral is ... |
| itg2le 24334 | If one function dominates ... |
| itg2const 24335 | Integral of a constant fun... |
| itg2const2 24336 | When the base set of a con... |
| itg2seq 24337 | Definitional property of t... |
| itg2uba 24338 | Approximate version of ~ i... |
| itg2lea 24339 | Approximate version of ~ i... |
| itg2eqa 24340 | Approximate equality of in... |
| itg2mulclem 24341 | Lemma for ~ itg2mulc . (C... |
| itg2mulc 24342 | The integral of a nonnegat... |
| itg2splitlem 24343 | Lemma for ~ itg2split . (... |
| itg2split 24344 | The ` S.2 ` integral split... |
| itg2monolem1 24345 | Lemma for ~ itg2mono . We... |
| itg2monolem2 24346 | Lemma for ~ itg2mono . (C... |
| itg2monolem3 24347 | Lemma for ~ itg2mono . (C... |
| itg2mono 24348 | The Monotone Convergence T... |
| itg2i1fseqle 24349 | Subject to the conditions ... |
| itg2i1fseq 24350 | Subject to the conditions ... |
| itg2i1fseq2 24351 | In an extension to the res... |
| itg2i1fseq3 24352 | Special case of ~ itg2i1fs... |
| itg2addlem 24353 | Lemma for ~ itg2add . (Co... |
| itg2add 24354 | The ` S.2 ` integral is li... |
| itg2gt0 24355 | If the function ` F ` is s... |
| itg2cnlem1 24356 | Lemma for ~ itgcn . (Cont... |
| itg2cnlem2 24357 | Lemma for ~ itgcn . (Cont... |
| itg2cn 24358 | A sort of absolute continu... |
| ibllem 24359 | Conditioned equality theor... |
| isibl 24360 | The predicate " ` F ` is i... |
| isibl2 24361 | The predicate " ` F ` is i... |
| iblmbf 24362 | An integrable function is ... |
| iblitg 24363 | If a function is integrabl... |
| dfitg 24364 | Evaluate the class substit... |
| itgex 24365 | An integral is a set. (Co... |
| itgeq1f 24366 | Equality theorem for an in... |
| itgeq1 24367 | Equality theorem for an in... |
| nfitg1 24368 | Bound-variable hypothesis ... |
| nfitg 24369 | Bound-variable hypothesis ... |
| cbvitg 24370 | Change bound variable in a... |
| cbvitgv 24371 | Change bound variable in a... |
| itgeq2 24372 | Equality theorem for an in... |
| itgresr 24373 | The domain of an integral ... |
| itg0 24374 | The integral of anything o... |
| itgz 24375 | The integral of zero on an... |
| itgeq2dv 24376 | Equality theorem for an in... |
| itgmpt 24377 | Change bound variable in a... |
| itgcl 24378 | The integral of an integra... |
| itgvallem 24379 | Substitution lemma. (Cont... |
| itgvallem3 24380 | Lemma for ~ itgposval and ... |
| ibl0 24381 | The zero function is integ... |
| iblcnlem1 24382 | Lemma for ~ iblcnlem . (C... |
| iblcnlem 24383 | Expand out the forall in ~... |
| itgcnlem 24384 | Expand out the sum in ~ df... |
| iblrelem 24385 | Integrability of a real fu... |
| iblposlem 24386 | Lemma for ~ iblpos . (Con... |
| iblpos 24387 | Integrability of a nonnega... |
| iblre 24388 | Integrability of a real fu... |
| itgrevallem1 24389 | Lemma for ~ itgposval and ... |
| itgposval 24390 | The integral of a nonnegat... |
| itgreval 24391 | Decompose the integral of ... |
| itgrecl 24392 | Real closure of an integra... |
| iblcn 24393 | Integrability of a complex... |
| itgcnval 24394 | Decompose the integral of ... |
| itgre 24395 | Real part of an integral. ... |
| itgim 24396 | Imaginary part of an integ... |
| iblneg 24397 | The negative of an integra... |
| itgneg 24398 | Negation of an integral. ... |
| iblss 24399 | A subset of an integrable ... |
| iblss2 24400 | Change the domain of an in... |
| itgitg2 24401 | Transfer an integral using... |
| i1fibl 24402 | A simple function is integ... |
| itgitg1 24403 | Transfer an integral using... |
| itgle 24404 | Monotonicity of an integra... |
| itgge0 24405 | The integral of a positive... |
| itgss 24406 | Expand the set of an integ... |
| itgss2 24407 | Expand the set of an integ... |
| itgeqa 24408 | Approximate equality of in... |
| itgss3 24409 | Expand the set of an integ... |
| itgioo 24410 | Equality of integrals on o... |
| itgless 24411 | Expand the integral of a n... |
| iblconst 24412 | A constant function is int... |
| itgconst 24413 | Integral of a constant fun... |
| ibladdlem 24414 | Lemma for ~ ibladd . (Con... |
| ibladd 24415 | Add two integrals over the... |
| iblsub 24416 | Subtract two integrals ove... |
| itgaddlem1 24417 | Lemma for ~ itgadd . (Con... |
| itgaddlem2 24418 | Lemma for ~ itgadd . (Con... |
| itgadd 24419 | Add two integrals over the... |
| itgsub 24420 | Subtract two integrals ove... |
| itgfsum 24421 | Take a finite sum of integ... |
| iblabslem 24422 | Lemma for ~ iblabs . (Con... |
| iblabs 24423 | The absolute value of an i... |
| iblabsr 24424 | A measurable function is i... |
| iblmulc2 24425 | Multiply an integral by a ... |
| itgmulc2lem1 24426 | Lemma for ~ itgmulc2 : pos... |
| itgmulc2lem2 24427 | Lemma for ~ itgmulc2 : rea... |
| itgmulc2 24428 | Multiply an integral by a ... |
| itgabs 24429 | The triangle inequality fo... |
| itgsplit 24430 | The ` S. ` integral splits... |
| itgspliticc 24431 | The ` S. ` integral splits... |
| itgsplitioo 24432 | The ` S. ` integral splits... |
| bddmulibl 24433 | A bounded function times a... |
| bddibl 24434 | A bounded function is inte... |
| cniccibl 24435 | A continuous function on a... |
| itggt0 24436 | The integral of a strictly... |
| itgcn 24437 | Transfer ~ itg2cn to the f... |
| ditgeq1 24440 | Equality theorem for the d... |
| ditgeq2 24441 | Equality theorem for the d... |
| ditgeq3 24442 | Equality theorem for the d... |
| ditgeq3dv 24443 | Equality theorem for the d... |
| ditgex 24444 | A directed integral is a s... |
| ditg0 24445 | Value of the directed inte... |
| cbvditg 24446 | Change bound variable in a... |
| cbvditgv 24447 | Change bound variable in a... |
| ditgpos 24448 | Value of the directed inte... |
| ditgneg 24449 | Value of the directed inte... |
| ditgcl 24450 | Closure of a directed inte... |
| ditgswap 24451 | Reverse a directed integra... |
| ditgsplitlem 24452 | Lemma for ~ ditgsplit . (... |
| ditgsplit 24453 | This theorem is the raison... |
| reldv 24462 | The derivative function is... |
| limcvallem 24463 | Lemma for ~ ellimc . (Con... |
| limcfval 24464 | Value and set bounds on th... |
| ellimc 24465 | Value of the limit predica... |
| limcrcl 24466 | Reverse closure for the li... |
| limccl 24467 | Closure of the limit opera... |
| limcdif 24468 | It suffices to consider fu... |
| ellimc2 24469 | Write the definition of a ... |
| limcnlp 24470 | If ` B ` is not a limit po... |
| ellimc3 24471 | Write the epsilon-delta de... |
| limcflflem 24472 | Lemma for ~ limcflf . (Co... |
| limcflf 24473 | The limit operator can be ... |
| limcmo 24474 | If ` B ` is a limit point ... |
| limcmpt 24475 | Express the limit operator... |
| limcmpt2 24476 | Express the limit operator... |
| limcresi 24477 | Any limit of ` F ` is also... |
| limcres 24478 | If ` B ` is an interior po... |
| cnplimc 24479 | A function is continuous a... |
| cnlimc 24480 | ` F ` is a continuous func... |
| cnlimci 24481 | If ` F ` is a continuous f... |
| cnmptlimc 24482 | If ` F ` is a continuous f... |
| limccnp 24483 | If the limit of ` F ` at `... |
| limccnp2 24484 | The image of a convergent ... |
| limcco 24485 | Composition of two limits.... |
| limciun 24486 | A point is a limit of ` F ... |
| limcun 24487 | A point is a limit of ` F ... |
| dvlem 24488 | Closure for a difference q... |
| dvfval 24489 | Value and set bounds on th... |
| eldv 24490 | The differentiable predica... |
| dvcl 24491 | The derivative function ta... |
| dvbssntr 24492 | The set of differentiable ... |
| dvbss 24493 | The set of differentiable ... |
| dvbsss 24494 | The set of differentiable ... |
| perfdvf 24495 | The derivative is a functi... |
| recnprss 24496 | Both ` RR ` and ` CC ` are... |
| recnperf 24497 | Both ` RR ` and ` CC ` are... |
| dvfg 24498 | Explicitly write out the f... |
| dvf 24499 | The derivative is a functi... |
| dvfcn 24500 | The derivative is a functi... |
| dvreslem 24501 | Lemma for ~ dvres . (Cont... |
| dvres2lem 24502 | Lemma for ~ dvres2 . (Con... |
| dvres 24503 | Restriction of a derivativ... |
| dvres2 24504 | Restriction of the base se... |
| dvres3 24505 | Restriction of a complex d... |
| dvres3a 24506 | Restriction of a complex d... |
| dvidlem 24507 | Lemma for ~ dvid and ~ dvc... |
| dvconst 24508 | Derivative of a constant f... |
| dvid 24509 | Derivative of the identity... |
| dvcnp 24510 | The difference quotient is... |
| dvcnp2 24511 | A function is continuous a... |
| dvcn 24512 | A differentiable function ... |
| dvnfval 24513 | Value of the iterated deri... |
| dvnff 24514 | The iterated derivative is... |
| dvn0 24515 | Zero times iterated deriva... |
| dvnp1 24516 | Successor iterated derivat... |
| dvn1 24517 | One times iterated derivat... |
| dvnf 24518 | The N-times derivative is ... |
| dvnbss 24519 | The set of N-times differe... |
| dvnadd 24520 | The ` N ` -th derivative o... |
| dvn2bss 24521 | An N-times differentiable ... |
| dvnres 24522 | Multiple derivative versio... |
| cpnfval 24523 | Condition for n-times cont... |
| fncpn 24524 | The ` C^n ` object is a fu... |
| elcpn 24525 | Condition for n-times cont... |
| cpnord 24526 | ` C^n ` conditions are ord... |
| cpncn 24527 | A ` C^n ` function is cont... |
| cpnres 24528 | The restriction of a ` C^n... |
| dvaddbr 24529 | The sum rule for derivativ... |
| dvmulbr 24530 | The product rule for deriv... |
| dvadd 24531 | The sum rule for derivativ... |
| dvmul 24532 | The product rule for deriv... |
| dvaddf 24533 | The sum rule for everywher... |
| dvmulf 24534 | The product rule for every... |
| dvcmul 24535 | The product rule when one ... |
| dvcmulf 24536 | The product rule when one ... |
| dvcobr 24537 | The chain rule for derivat... |
| dvco 24538 | The chain rule for derivat... |
| dvcof 24539 | The chain rule for everywh... |
| dvcjbr 24540 | The derivative of the conj... |
| dvcj 24541 | The derivative of the conj... |
| dvfre 24542 | The derivative of a real f... |
| dvnfre 24543 | The ` N ` -th derivative o... |
| dvexp 24544 | Derivative of a power func... |
| dvexp2 24545 | Derivative of an exponenti... |
| dvrec 24546 | Derivative of the reciproc... |
| dvmptres3 24547 | Function-builder for deriv... |
| dvmptid 24548 | Function-builder for deriv... |
| dvmptc 24549 | Function-builder for deriv... |
| dvmptcl 24550 | Closure lemma for ~ dvmptc... |
| dvmptadd 24551 | Function-builder for deriv... |
| dvmptmul 24552 | Function-builder for deriv... |
| dvmptres2 24553 | Function-builder for deriv... |
| dvmptres 24554 | Function-builder for deriv... |
| dvmptcmul 24555 | Function-builder for deriv... |
| dvmptdivc 24556 | Function-builder for deriv... |
| dvmptneg 24557 | Function-builder for deriv... |
| dvmptsub 24558 | Function-builder for deriv... |
| dvmptcj 24559 | Function-builder for deriv... |
| dvmptre 24560 | Function-builder for deriv... |
| dvmptim 24561 | Function-builder for deriv... |
| dvmptntr 24562 | Function-builder for deriv... |
| dvmptco 24563 | Function-builder for deriv... |
| dvrecg 24564 | Derivative of the reciproc... |
| dvmptdiv 24565 | Function-builder for deriv... |
| dvmptfsum 24566 | Function-builder for deriv... |
| dvcnvlem 24567 | Lemma for ~ dvcnvre . (Co... |
| dvcnv 24568 | A weak version of ~ dvcnvr... |
| dvexp3 24569 | Derivative of an exponenti... |
| dveflem 24570 | Derivative of the exponent... |
| dvef 24571 | Derivative of the exponent... |
| dvsincos 24572 | Derivative of the sine and... |
| dvsin 24573 | Derivative of the sine fun... |
| dvcos 24574 | Derivative of the cosine f... |
| dvferm1lem 24575 | Lemma for ~ dvferm . (Con... |
| dvferm1 24576 | One-sided version of ~ dvf... |
| dvferm2lem 24577 | Lemma for ~ dvferm . (Con... |
| dvferm2 24578 | One-sided version of ~ dvf... |
| dvferm 24579 | Fermat's theorem on statio... |
| rollelem 24580 | Lemma for ~ rolle . (Cont... |
| rolle 24581 | Rolle's theorem. If ` F `... |
| cmvth 24582 | Cauchy's Mean Value Theore... |
| mvth 24583 | The Mean Value Theorem. I... |
| dvlip 24584 | A function with derivative... |
| dvlipcn 24585 | A complex function with de... |
| dvlip2 24586 | Combine the results of ~ d... |
| c1liplem1 24587 | Lemma for ~ c1lip1 . (Con... |
| c1lip1 24588 | C^1 functions are Lipschit... |
| c1lip2 24589 | C^1 functions are Lipschit... |
| c1lip3 24590 | C^1 functions are Lipschit... |
| dveq0 24591 | If a continuous function h... |
| dv11cn 24592 | Two functions defined on a... |
| dvgt0lem1 24593 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0lem2 24594 | Lemma for ~ dvgt0 and ~ dv... |
| dvgt0 24595 | A function on a closed int... |
| dvlt0 24596 | A function on a closed int... |
| dvge0 24597 | A function on a closed int... |
| dvle 24598 | If ` A ( x ) , C ( x ) ` a... |
| dvivthlem1 24599 | Lemma for ~ dvivth . (Con... |
| dvivthlem2 24600 | Lemma for ~ dvivth . (Con... |
| dvivth 24601 | Darboux' theorem, or the i... |
| dvne0 24602 | A function on a closed int... |
| dvne0f1 24603 | A function on a closed int... |
| lhop1lem 24604 | Lemma for ~ lhop1 . (Cont... |
| lhop1 24605 | L'Hôpital's Rule for... |
| lhop2 24606 | L'Hôpital's Rule for... |
| lhop 24607 | L'Hôpital's Rule. I... |
| dvcnvrelem1 24608 | Lemma for ~ dvcnvre . (Co... |
| dvcnvrelem2 24609 | Lemma for ~ dvcnvre . (Co... |
| dvcnvre 24610 | The derivative rule for in... |
| dvcvx 24611 | A real function with stric... |
| dvfsumle 24612 | Compare a finite sum to an... |
| dvfsumge 24613 | Compare a finite sum to an... |
| dvfsumabs 24614 | Compare a finite sum to an... |
| dvmptrecl 24615 | Real closure of a derivati... |
| dvfsumrlimf 24616 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem1 24617 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem2 24618 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem3 24619 | Lemma for ~ dvfsumrlim . ... |
| dvfsumlem4 24620 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlimge0 24621 | Lemma for ~ dvfsumrlim . ... |
| dvfsumrlim 24622 | Compare a finite sum to an... |
| dvfsumrlim2 24623 | Compare a finite sum to an... |
| dvfsumrlim3 24624 | Conjoin the statements of ... |
| dvfsum2 24625 | The reverse of ~ dvfsumrli... |
| ftc1lem1 24626 | Lemma for ~ ftc1a and ~ ft... |
| ftc1lem2 24627 | Lemma for ~ ftc1 . (Contr... |
| ftc1a 24628 | The Fundamental Theorem of... |
| ftc1lem3 24629 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem4 24630 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem5 24631 | Lemma for ~ ftc1 . (Contr... |
| ftc1lem6 24632 | Lemma for ~ ftc1 . (Contr... |
| ftc1 24633 | The Fundamental Theorem of... |
| ftc1cn 24634 | Strengthen the assumptions... |
| ftc2 24635 | The Fundamental Theorem of... |
| ftc2ditglem 24636 | Lemma for ~ ftc2ditg . (C... |
| ftc2ditg 24637 | Directed integral analogue... |
| itgparts 24638 | Integration by parts. If ... |
| itgsubstlem 24639 | Lemma for ~ itgsubst . (C... |
| itgsubst 24640 | Integration by ` u ` -subs... |
| reldmmdeg 24645 | Multivariate degree is a b... |
| tdeglem1 24646 | Functionality of the total... |
| tdeglem3 24647 | Additivity of the total de... |
| tdeglem4 24648 | There is only one multi-in... |
| tdeglem2 24649 | Simplification of total de... |
| mdegfval 24650 | Value of the multivariate ... |
| mdegval 24651 | Value of the multivariate ... |
| mdegleb 24652 | Property of being of limit... |
| mdeglt 24653 | If there is an upper limit... |
| mdegldg 24654 | A nonzero polynomial has s... |
| mdegxrcl 24655 | Closure of polynomial degr... |
| mdegxrf 24656 | Functionality of polynomia... |
| mdegcl 24657 | Sharp closure for multivar... |
| mdeg0 24658 | Degree of the zero polynom... |
| mdegnn0cl 24659 | Degree of a nonzero polyno... |
| degltlem1 24660 | Theorem on arithmetic of e... |
| degltp1le 24661 | Theorem on arithmetic of e... |
| mdegaddle 24662 | The degree of a sum is at ... |
| mdegvscale 24663 | The degree of a scalar mul... |
| mdegvsca 24664 | The degree of a scalar mul... |
| mdegle0 24665 | A polynomial has nonpositi... |
| mdegmullem 24666 | Lemma for ~ mdegmulle2 . ... |
| mdegmulle2 24667 | The multivariate degree of... |
| deg1fval 24668 | Relate univariate polynomi... |
| deg1xrf 24669 | Functionality of univariat... |
| deg1xrcl 24670 | Closure of univariate poly... |
| deg1cl 24671 | Sharp closure of univariat... |
| mdegpropd 24672 | Property deduction for pol... |
| deg1fvi 24673 | Univariate polynomial degr... |
| deg1propd 24674 | Property deduction for pol... |
| deg1z 24675 | Degree of the zero univari... |
| deg1nn0cl 24676 | Degree of a nonzero univar... |
| deg1n0ima 24677 | Degree image of a set of p... |
| deg1nn0clb 24678 | A polynomial is nonzero if... |
| deg1lt0 24679 | A polynomial is zero iff i... |
| deg1ldg 24680 | A nonzero univariate polyn... |
| deg1ldgn 24681 | An index at which a polyno... |
| deg1ldgdomn 24682 | A nonzero univariate polyn... |
| deg1leb 24683 | Property of being of limit... |
| deg1val 24684 | Value of the univariate de... |
| deg1lt 24685 | If the degree of a univari... |
| deg1ge 24686 | Conversely, a nonzero coef... |
| coe1mul3 24687 | The coefficient vector of ... |
| coe1mul4 24688 | Value of the "leading" coe... |
| deg1addle 24689 | The degree of a sum is at ... |
| deg1addle2 24690 | If both factors have degre... |
| deg1add 24691 | Exact degree of a sum of t... |
| deg1vscale 24692 | The degree of a scalar tim... |
| deg1vsca 24693 | The degree of a scalar tim... |
| deg1invg 24694 | The degree of the negated ... |
| deg1suble 24695 | The degree of a difference... |
| deg1sub 24696 | Exact degree of a differen... |
| deg1mulle2 24697 | Produce a bound on the pro... |
| deg1sublt 24698 | Subtraction of two polynom... |
| deg1le0 24699 | A polynomial has nonpositi... |
| deg1sclle 24700 | A scalar polynomial has no... |
| deg1scl 24701 | A nonzero scalar polynomia... |
| deg1mul2 24702 | Degree of multiplication o... |
| deg1mul3 24703 | Degree of multiplication o... |
| deg1mul3le 24704 | Degree of multiplication o... |
| deg1tmle 24705 | Limiting degree of a polyn... |
| deg1tm 24706 | Exact degree of a polynomi... |
| deg1pwle 24707 | Limiting degree of a varia... |
| deg1pw 24708 | Exact degree of a variable... |
| ply1nz 24709 | Univariate polynomials ove... |
| ply1nzb 24710 | Univariate polynomials are... |
| ply1domn 24711 | Corollary of ~ deg1mul2 : ... |
| ply1idom 24712 | The ring of univariate pol... |
| ply1divmo 24723 | Uniqueness of a quotient i... |
| ply1divex 24724 | Lemma for ~ ply1divalg : e... |
| ply1divalg 24725 | The division algorithm for... |
| ply1divalg2 24726 | Reverse the order of multi... |
| uc1pval 24727 | Value of the set of unitic... |
| isuc1p 24728 | Being a unitic polynomial.... |
| mon1pval 24729 | Value of the set of monic ... |
| ismon1p 24730 | Being a monic polynomial. ... |
| uc1pcl 24731 | Unitic polynomials are pol... |
| mon1pcl 24732 | Monic polynomials are poly... |
| uc1pn0 24733 | Unitic polynomials are not... |
| mon1pn0 24734 | Monic polynomials are not ... |
| uc1pdeg 24735 | Unitic polynomials have no... |
| uc1pldg 24736 | Unitic polynomials have un... |
| mon1pldg 24737 | Unitic polynomials have on... |
| mon1puc1p 24738 | Monic polynomials are unit... |
| uc1pmon1p 24739 | Make a unitic polynomial m... |
| deg1submon1p 24740 | The difference of two moni... |
| q1pval 24741 | Value of the univariate po... |
| q1peqb 24742 | Characterizing property of... |
| q1pcl 24743 | Closure of the quotient by... |
| r1pval 24744 | Value of the polynomial re... |
| r1pcl 24745 | Closure of remainder follo... |
| r1pdeglt 24746 | The remainder has a degree... |
| r1pid 24747 | Express the original polyn... |
| dvdsq1p 24748 | Divisibility in a polynomi... |
| dvdsr1p 24749 | Divisibility in a polynomi... |
| ply1remlem 24750 | A term of the form ` x - N... |
| ply1rem 24751 | The polynomial remainder t... |
| facth1 24752 | The factor theorem and its... |
| fta1glem1 24753 | Lemma for ~ fta1g . (Cont... |
| fta1glem2 24754 | Lemma for ~ fta1g . (Cont... |
| fta1g 24755 | The one-sided fundamental ... |
| fta1blem 24756 | Lemma for ~ fta1b . (Cont... |
| fta1b 24757 | The assumption that ` R ` ... |
| drnguc1p 24758 | Over a division ring, all ... |
| ig1peu 24759 | There is a unique monic po... |
| ig1pval 24760 | Substitutions for the poly... |
| ig1pval2 24761 | Generator of the zero idea... |
| ig1pval3 24762 | Characterizing properties ... |
| ig1pcl 24763 | The monic generator of an ... |
| ig1pdvds 24764 | The monic generator of an ... |
| ig1prsp 24765 | Any ideal of polynomials o... |
| ply1lpir 24766 | The ring of polynomials ov... |
| ply1pid 24767 | The polynomials over a fie... |
| plyco0 24776 | Two ways to say that a fun... |
| plyval 24777 | Value of the polynomial se... |
| plybss 24778 | Reverse closure of the par... |
| elply 24779 | Definition of a polynomial... |
| elply2 24780 | The coefficient function c... |
| plyun0 24781 | The set of polynomials is ... |
| plyf 24782 | The polynomial is a functi... |
| plyss 24783 | The polynomial set functio... |
| plyssc 24784 | Every polynomial ring is c... |
| elplyr 24785 | Sufficient condition for e... |
| elplyd 24786 | Sufficient condition for e... |
| ply1termlem 24787 | Lemma for ~ ply1term . (C... |
| ply1term 24788 | A one-term polynomial. (C... |
| plypow 24789 | A power is a polynomial. ... |
| plyconst 24790 | A constant function is a p... |
| ne0p 24791 | A test to show that a poly... |
| ply0 24792 | The zero function is a pol... |
| plyid 24793 | The identity function is a... |
| plyeq0lem 24794 | Lemma for ~ plyeq0 . If `... |
| plyeq0 24795 | If a polynomial is zero at... |
| plypf1 24796 | Write the set of complex p... |
| plyaddlem1 24797 | Derive the coefficient fun... |
| plymullem1 24798 | Derive the coefficient fun... |
| plyaddlem 24799 | Lemma for ~ plyadd . (Con... |
| plymullem 24800 | Lemma for ~ plymul . (Con... |
| plyadd 24801 | The sum of two polynomials... |
| plymul 24802 | The product of two polynom... |
| plysub 24803 | The difference of two poly... |
| plyaddcl 24804 | The sum of two polynomials... |
| plymulcl 24805 | The product of two polynom... |
| plysubcl 24806 | The difference of two poly... |
| coeval 24807 | Value of the coefficient f... |
| coeeulem 24808 | Lemma for ~ coeeu . (Cont... |
| coeeu 24809 | Uniqueness of the coeffici... |
| coelem 24810 | Lemma for properties of th... |
| coeeq 24811 | If ` A ` satisfies the pro... |
| dgrval 24812 | Value of the degree functi... |
| dgrlem 24813 | Lemma for ~ dgrcl and simi... |
| coef 24814 | The domain and range of th... |
| coef2 24815 | The domain and range of th... |
| coef3 24816 | The domain and range of th... |
| dgrcl 24817 | The degree of any polynomi... |
| dgrub 24818 | If the ` M ` -th coefficie... |
| dgrub2 24819 | All the coefficients above... |
| dgrlb 24820 | If all the coefficients ab... |
| coeidlem 24821 | Lemma for ~ coeid . (Cont... |
| coeid 24822 | Reconstruct a polynomial a... |
| coeid2 24823 | Reconstruct a polynomial a... |
| coeid3 24824 | Reconstruct a polynomial a... |
| plyco 24825 | The composition of two pol... |
| coeeq2 24826 | Compute the coefficient fu... |
| dgrle 24827 | Given an explicit expressi... |
| dgreq 24828 | If the highest term in a p... |
| 0dgr 24829 | A constant function has de... |
| 0dgrb 24830 | A function has degree zero... |
| dgrnznn 24831 | A nonzero polynomial with ... |
| coefv0 24832 | The result of evaluating a... |
| coeaddlem 24833 | Lemma for ~ coeadd and ~ d... |
| coemullem 24834 | Lemma for ~ coemul and ~ d... |
| coeadd 24835 | The coefficient function o... |
| coemul 24836 | A coefficient of a product... |
| coe11 24837 | The coefficient function i... |
| coemulhi 24838 | The leading coefficient of... |
| coemulc 24839 | The coefficient function i... |
| coe0 24840 | The coefficients of the ze... |
| coesub 24841 | The coefficient function o... |
| coe1termlem 24842 | The coefficient function o... |
| coe1term 24843 | The coefficient function o... |
| dgr1term 24844 | The degree of a monomial. ... |
| plycn 24845 | A polynomial is a continuo... |
| dgr0 24846 | The degree of the zero pol... |
| coeidp 24847 | The coefficients of the id... |
| dgrid 24848 | The degree of the identity... |
| dgreq0 24849 | The leading coefficient of... |
| dgrlt 24850 | Two ways to say that the d... |
| dgradd 24851 | The degree of a sum of pol... |
| dgradd2 24852 | The degree of a sum of pol... |
| dgrmul2 24853 | The degree of a product of... |
| dgrmul 24854 | The degree of a product of... |
| dgrmulc 24855 | Scalar multiplication by a... |
| dgrsub 24856 | The degree of a difference... |
| dgrcolem1 24857 | The degree of a compositio... |
| dgrcolem2 24858 | Lemma for ~ dgrco . (Cont... |
| dgrco 24859 | The degree of a compositio... |
| plycjlem 24860 | Lemma for ~ plycj and ~ co... |
| plycj 24861 | The double conjugation of ... |
| coecj 24862 | Double conjugation of a po... |
| plyrecj 24863 | A polynomial with real coe... |
| plymul0or 24864 | Polynomial multiplication ... |
| ofmulrt 24865 | The set of roots of a prod... |
| plyreres 24866 | Real-coefficient polynomia... |
| dvply1 24867 | Derivative of a polynomial... |
| dvply2g 24868 | The derivative of a polyno... |
| dvply2 24869 | The derivative of a polyno... |
| dvnply2 24870 | Polynomials have polynomia... |
| dvnply 24871 | Polynomials have polynomia... |
| plycpn 24872 | Polynomials are smooth. (... |
| quotval 24875 | Value of the quotient func... |
| plydivlem1 24876 | Lemma for ~ plydivalg . (... |
| plydivlem2 24877 | Lemma for ~ plydivalg . (... |
| plydivlem3 24878 | Lemma for ~ plydivex . Ba... |
| plydivlem4 24879 | Lemma for ~ plydivex . In... |
| plydivex 24880 | Lemma for ~ plydivalg . (... |
| plydiveu 24881 | Lemma for ~ plydivalg . (... |
| plydivalg 24882 | The division algorithm on ... |
| quotlem 24883 | Lemma for properties of th... |
| quotcl 24884 | The quotient of two polyno... |
| quotcl2 24885 | Closure of the quotient fu... |
| quotdgr 24886 | Remainder property of the ... |
| plyremlem 24887 | Closure of a linear factor... |
| plyrem 24888 | The polynomial remainder t... |
| facth 24889 | The factor theorem. If a ... |
| fta1lem 24890 | Lemma for ~ fta1 . (Contr... |
| fta1 24891 | The easy direction of the ... |
| quotcan 24892 | Exact division with a mult... |
| vieta1lem1 24893 | Lemma for ~ vieta1 . (Con... |
| vieta1lem2 24894 | Lemma for ~ vieta1 : induc... |
| vieta1 24895 | The first-order Vieta's fo... |
| plyexmo 24896 | An infinite set of values ... |
| elaa 24899 | Elementhood in the set of ... |
| aacn 24900 | An algebraic number is a c... |
| aasscn 24901 | The algebraic numbers are ... |
| elqaalem1 24902 | Lemma for ~ elqaa . The f... |
| elqaalem2 24903 | Lemma for ~ elqaa . (Cont... |
| elqaalem3 24904 | Lemma for ~ elqaa . (Cont... |
| elqaa 24905 | The set of numbers generat... |
| qaa 24906 | Every rational number is a... |
| qssaa 24907 | The rational numbers are c... |
| iaa 24908 | The imaginary unit is alge... |
| aareccl 24909 | The reciprocal of an algeb... |
| aacjcl 24910 | The conjugate of an algebr... |
| aannenlem1 24911 | Lemma for ~ aannen . (Con... |
| aannenlem2 24912 | Lemma for ~ aannen . (Con... |
| aannenlem3 24913 | The algebraic numbers are ... |
| aannen 24914 | The algebraic numbers are ... |
| aalioulem1 24915 | Lemma for ~ aaliou . An i... |
| aalioulem2 24916 | Lemma for ~ aaliou . (Con... |
| aalioulem3 24917 | Lemma for ~ aaliou . (Con... |
| aalioulem4 24918 | Lemma for ~ aaliou . (Con... |
| aalioulem5 24919 | Lemma for ~ aaliou . (Con... |
| aalioulem6 24920 | Lemma for ~ aaliou . (Con... |
| aaliou 24921 | Liouville's theorem on dio... |
| geolim3 24922 | Geometric series convergen... |
| aaliou2 24923 | Liouville's approximation ... |
| aaliou2b 24924 | Liouville's approximation ... |
| aaliou3lem1 24925 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem2 24926 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem3 24927 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem8 24928 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem4 24929 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem5 24930 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem6 24931 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem7 24932 | Lemma for ~ aaliou3 . (Co... |
| aaliou3lem9 24933 | Example of a "Liouville nu... |
| aaliou3 24934 | Example of a "Liouville nu... |
| taylfvallem1 24939 | Lemma for ~ taylfval . (C... |
| taylfvallem 24940 | Lemma for ~ taylfval . (C... |
| taylfval 24941 | Define the Taylor polynomi... |
| eltayl 24942 | Value of the Taylor series... |
| taylf 24943 | The Taylor series defines ... |
| tayl0 24944 | The Taylor series is alway... |
| taylplem1 24945 | Lemma for ~ taylpfval and ... |
| taylplem2 24946 | Lemma for ~ taylpfval and ... |
| taylpfval 24947 | Define the Taylor polynomi... |
| taylpf 24948 | The Taylor polynomial is a... |
| taylpval 24949 | Value of the Taylor polyno... |
| taylply2 24950 | The Taylor polynomial is a... |
| taylply 24951 | The Taylor polynomial is a... |
| dvtaylp 24952 | The derivative of the Tayl... |
| dvntaylp 24953 | The ` M ` -th derivative o... |
| dvntaylp0 24954 | The first ` N ` derivative... |
| taylthlem1 24955 | Lemma for ~ taylth . This... |
| taylthlem2 24956 | Lemma for ~ taylth . (Con... |
| taylth 24957 | Taylor's theorem. The Tay... |
| ulmrel 24960 | The uniform limit relation... |
| ulmscl 24961 | Closure of the base set in... |
| ulmval 24962 | Express the predicate: Th... |
| ulmcl 24963 | Closure of a uniform limit... |
| ulmf 24964 | Closure of a uniform limit... |
| ulmpm 24965 | Closure of a uniform limit... |
| ulmf2 24966 | Closure of a uniform limit... |
| ulm2 24967 | Simplify ~ ulmval when ` F... |
| ulmi 24968 | The uniform limit property... |
| ulmclm 24969 | A uniform limit of functio... |
| ulmres 24970 | A sequence of functions co... |
| ulmshftlem 24971 | Lemma for ~ ulmshft . (Co... |
| ulmshft 24972 | A sequence of functions co... |
| ulm0 24973 | Every function converges u... |
| ulmuni 24974 | A sequence of functions un... |
| ulmdm 24975 | Two ways to express that a... |
| ulmcaulem 24976 | Lemma for ~ ulmcau and ~ u... |
| ulmcau 24977 | A sequence of functions co... |
| ulmcau2 24978 | A sequence of functions co... |
| ulmss 24979 | A uniform limit of functio... |
| ulmbdd 24980 | A uniform limit of bounded... |
| ulmcn 24981 | A uniform limit of continu... |
| ulmdvlem1 24982 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem2 24983 | Lemma for ~ ulmdv . (Cont... |
| ulmdvlem3 24984 | Lemma for ~ ulmdv . (Cont... |
| ulmdv 24985 | If ` F ` is a sequence of ... |
| mtest 24986 | The Weierstrass M-test. I... |
| mtestbdd 24987 | Given the hypotheses of th... |
| mbfulm 24988 | A uniform limit of measura... |
| iblulm 24989 | A uniform limit of integra... |
| itgulm 24990 | A uniform limit of integra... |
| itgulm2 24991 | A uniform limit of integra... |
| pserval 24992 | Value of the function ` G ... |
| pserval2 24993 | Value of the function ` G ... |
| psergf 24994 | The sequence of terms in t... |
| radcnvlem1 24995 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem2 24996 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnvlem3 24997 | Lemma for ~ radcnvlt1 , ~ ... |
| radcnv0 24998 | Zero is always a convergen... |
| radcnvcl 24999 | The radius of convergence ... |
| radcnvlt1 25000 | If ` X ` is within the ope... |
| radcnvlt2 25001 | If ` X ` is within the ope... |
| radcnvle 25002 | If ` X ` is a convergent p... |
| dvradcnv 25003 | The radius of convergence ... |
| pserulm 25004 | If ` S ` is a region conta... |
| psercn2 25005 | Since by ~ pserulm the ser... |
| psercnlem2 25006 | Lemma for ~ psercn . (Con... |
| psercnlem1 25007 | Lemma for ~ psercn . (Con... |
| psercn 25008 | An infinite series converg... |
| pserdvlem1 25009 | Lemma for ~ pserdv . (Con... |
| pserdvlem2 25010 | Lemma for ~ pserdv . (Con... |
| pserdv 25011 | The derivative of a power ... |
| pserdv2 25012 | The derivative of a power ... |
| abelthlem1 25013 | Lemma for ~ abelth . (Con... |
| abelthlem2 25014 | Lemma for ~ abelth . The ... |
| abelthlem3 25015 | Lemma for ~ abelth . (Con... |
| abelthlem4 25016 | Lemma for ~ abelth . (Con... |
| abelthlem5 25017 | Lemma for ~ abelth . (Con... |
| abelthlem6 25018 | Lemma for ~ abelth . (Con... |
| abelthlem7a 25019 | Lemma for ~ abelth . (Con... |
| abelthlem7 25020 | Lemma for ~ abelth . (Con... |
| abelthlem8 25021 | Lemma for ~ abelth . (Con... |
| abelthlem9 25022 | Lemma for ~ abelth . By a... |
| abelth 25023 | Abel's theorem. If the po... |
| abelth2 25024 | Abel's theorem, restricted... |
| efcn 25025 | The exponential function i... |
| sincn 25026 | Sine is continuous. (Cont... |
| coscn 25027 | Cosine is continuous. (Co... |
| reeff1olem 25028 | Lemma for ~ reeff1o . (Co... |
| reeff1o 25029 | The real exponential funct... |
| reefiso 25030 | The exponential function o... |
| efcvx 25031 | The exponential function o... |
| reefgim 25032 | The exponential function i... |
| pilem1 25033 | Lemma for ~ pire , ~ pigt2... |
| pilem2 25034 | Lemma for ~ pire , ~ pigt2... |
| pilem3 25035 | Lemma for ~ pire , ~ pigt2... |
| pigt2lt4 25036 | ` _pi ` is between 2 and 4... |
| sinpi 25037 | The sine of ` _pi ` is 0. ... |
| pire 25038 | ` _pi ` is a real number. ... |
| picn 25039 | ` _pi ` is a complex numbe... |
| pipos 25040 | ` _pi ` is positive. (Con... |
| pirp 25041 | ` _pi ` is a positive real... |
| negpicn 25042 | ` -u _pi ` is a real numbe... |
| sinhalfpilem 25043 | Lemma for ~ sinhalfpi and ... |
| halfpire 25044 | ` _pi / 2 ` is real. (Con... |
| neghalfpire 25045 | ` -u _pi / 2 ` is real. (... |
| neghalfpirx 25046 | ` -u _pi / 2 ` is an exten... |
| pidiv2halves 25047 | Adding ` _pi / 2 ` to itse... |
| sinhalfpi 25048 | The sine of ` _pi / 2 ` is... |
| coshalfpi 25049 | The cosine of ` _pi / 2 ` ... |
| cosneghalfpi 25050 | The cosine of ` -u _pi / 2... |
| efhalfpi 25051 | The exponential of ` _i _p... |
| cospi 25052 | The cosine of ` _pi ` is `... |
| efipi 25053 | The exponential of ` _i x.... |
| eulerid 25054 | Euler's identity. (Contri... |
| sin2pi 25055 | The sine of ` 2 _pi ` is 0... |
| cos2pi 25056 | The cosine of ` 2 _pi ` is... |
| ef2pi 25057 | The exponential of ` 2 _pi... |
| ef2kpi 25058 | If ` K ` is an integer, th... |
| efper 25059 | The exponential function i... |
| sinperlem 25060 | Lemma for ~ sinper and ~ c... |
| sinper 25061 | The sine function is perio... |
| cosper 25062 | The cosine function is per... |
| sin2kpi 25063 | If ` K ` is an integer, th... |
| cos2kpi 25064 | If ` K ` is an integer, th... |
| sin2pim 25065 | Sine of a number subtracte... |
| cos2pim 25066 | Cosine of a number subtrac... |
| sinmpi 25067 | Sine of a number less ` _p... |
| cosmpi 25068 | Cosine of a number less ` ... |
| sinppi 25069 | Sine of a number plus ` _p... |
| cosppi 25070 | Cosine of a number plus ` ... |
| efimpi 25071 | The exponential function a... |
| sinhalfpip 25072 | The sine of ` _pi / 2 ` pl... |
| sinhalfpim 25073 | The sine of ` _pi / 2 ` mi... |
| coshalfpip 25074 | The cosine of ` _pi / 2 ` ... |
| coshalfpim 25075 | The cosine of ` _pi / 2 ` ... |
| ptolemy 25076 | Ptolemy's Theorem. This t... |
| sincosq1lem 25077 | Lemma for ~ sincosq1sgn . ... |
| sincosq1sgn 25078 | The signs of the sine and ... |
| sincosq2sgn 25079 | The signs of the sine and ... |
| sincosq3sgn 25080 | The signs of the sine and ... |
| sincosq4sgn 25081 | The signs of the sine and ... |
| coseq00topi 25082 | Location of the zeroes of ... |
| coseq0negpitopi 25083 | Location of the zeroes of ... |
| tanrpcl 25084 | Positive real closure of t... |
| tangtx 25085 | The tangent function is gr... |
| tanabsge 25086 | The tangent function is gr... |
| sinq12gt0 25087 | The sine of a number stric... |
| sinq12ge0 25088 | The sine of a number betwe... |
| sinq34lt0t 25089 | The sine of a number stric... |
| cosq14gt0 25090 | The cosine of a number str... |
| cosq14ge0 25091 | The cosine of a number bet... |
| sincosq1eq 25092 | Complementarity of the sin... |
| sincos4thpi 25093 | The sine and cosine of ` _... |
| tan4thpi 25094 | The tangent of ` _pi / 4 `... |
| sincos6thpi 25095 | The sine and cosine of ` _... |
| sincos3rdpi 25096 | The sine and cosine of ` _... |
| pigt3 25097 | ` _pi ` is greater than 3.... |
| pige3 25098 | ` _pi ` is greater than or... |
| pige3ALT 25099 | Alternate proof of ~ pige3... |
| abssinper 25100 | The absolute value of sine... |
| sinkpi 25101 | The sine of an integer mul... |
| coskpi 25102 | The absolute value of the ... |
| sineq0 25103 | A complex number whose sin... |
| coseq1 25104 | A complex number whose cos... |
| cos02pilt1 25105 | Cosine is less than one be... |
| cosq34lt1 25106 | Cosine is less than one in... |
| efeq1 25107 | A complex number whose exp... |
| cosne0 25108 | The cosine function has no... |
| cosordlem 25109 | Lemma for ~ cosord . (Con... |
| cosord 25110 | Cosine is decreasing over ... |
| cos11 25111 | Cosine is one-to-one over ... |
| sinord 25112 | Sine is increasing over th... |
| recosf1o 25113 | The cosine function is a b... |
| resinf1o 25114 | The sine function is a bij... |
| tanord1 25115 | The tangent function is st... |
| tanord 25116 | The tangent function is st... |
| tanregt0 25117 | The real part of the tange... |
| negpitopissre 25118 | The interval ` ( -u _pi (,... |
| efgh 25119 | The exponential function o... |
| efif1olem1 25120 | Lemma for ~ efif1o . (Con... |
| efif1olem2 25121 | Lemma for ~ efif1o . (Con... |
| efif1olem3 25122 | Lemma for ~ efif1o . (Con... |
| efif1olem4 25123 | The exponential function o... |
| efif1o 25124 | The exponential function o... |
| efifo 25125 | The exponential function o... |
| eff1olem 25126 | The exponential function m... |
| eff1o 25127 | The exponential function m... |
| efabl 25128 | The image of a subgroup of... |
| efsubm 25129 | The image of a subgroup of... |
| circgrp 25130 | The circle group ` T ` is ... |
| circsubm 25131 | The circle group ` T ` is ... |
| logrn 25136 | The range of the natural l... |
| ellogrn 25137 | Write out the property ` A... |
| dflog2 25138 | The natural logarithm func... |
| relogrn 25139 | The range of the natural l... |
| logrncn 25140 | The range of the natural l... |
| eff1o2 25141 | The exponential function r... |
| logf1o 25142 | The natural logarithm func... |
| dfrelog 25143 | The natural logarithm func... |
| relogf1o 25144 | The natural logarithm func... |
| logrncl 25145 | Closure of the natural log... |
| logcl 25146 | Closure of the natural log... |
| logimcl 25147 | Closure of the imaginary p... |
| logcld 25148 | The logarithm of a nonzero... |
| logimcld 25149 | The imaginary part of the ... |
| logimclad 25150 | The imaginary part of the ... |
| abslogimle 25151 | The imaginary part of the ... |
| logrnaddcl 25152 | The range of the natural l... |
| relogcl 25153 | Closure of the natural log... |
| eflog 25154 | Relationship between the n... |
| logeq0im1 25155 | If the logarithm of a numb... |
| logccne0 25156 | The logarithm isn't 0 if i... |
| logne0 25157 | Logarithm of a non-1 posit... |
| reeflog 25158 | Relationship between the n... |
| logef 25159 | Relationship between the n... |
| relogef 25160 | Relationship between the n... |
| logeftb 25161 | Relationship between the n... |
| relogeftb 25162 | Relationship between the n... |
| log1 25163 | The natural logarithm of `... |
| loge 25164 | The natural logarithm of `... |
| logneg 25165 | The natural logarithm of a... |
| logm1 25166 | The natural logarithm of n... |
| lognegb 25167 | If a number has imaginary ... |
| relogoprlem 25168 | Lemma for ~ relogmul and ~... |
| relogmul 25169 | The natural logarithm of t... |
| relogdiv 25170 | The natural logarithm of t... |
| explog 25171 | Exponentiation of a nonzer... |
| reexplog 25172 | Exponentiation of a positi... |
| relogexp 25173 | The natural logarithm of p... |
| relog 25174 | Real part of a logarithm. ... |
| relogiso 25175 | The natural logarithm func... |
| reloggim 25176 | The natural logarithm is a... |
| logltb 25177 | The natural logarithm func... |
| logfac 25178 | The logarithm of a factori... |
| eflogeq 25179 | Solve an equation involvin... |
| logleb 25180 | Natural logarithm preserve... |
| rplogcl 25181 | Closure of the logarithm f... |
| logge0 25182 | The logarithm of a number ... |
| logcj 25183 | The natural logarithm dist... |
| efiarg 25184 | The exponential of the "ar... |
| cosargd 25185 | The cosine of the argument... |
| cosarg0d 25186 | The cosine of the argument... |
| argregt0 25187 | Closure of the argument of... |
| argrege0 25188 | Closure of the argument of... |
| argimgt0 25189 | Closure of the argument of... |
| argimlt0 25190 | Closure of the argument of... |
| logimul 25191 | Multiplying a number by ` ... |
| logneg2 25192 | The logarithm of the negat... |
| logmul2 25193 | Generalization of ~ relogm... |
| logdiv2 25194 | Generalization of ~ relogd... |
| abslogle 25195 | Bound on the magnitude of ... |
| tanarg 25196 | The basic relation between... |
| logdivlti 25197 | The ` log x / x ` function... |
| logdivlt 25198 | The ` log x / x ` function... |
| logdivle 25199 | The ` log x / x ` function... |
| relogcld 25200 | Closure of the natural log... |
| reeflogd 25201 | Relationship between the n... |
| relogmuld 25202 | The natural logarithm of t... |
| relogdivd 25203 | The natural logarithm of t... |
| logled 25204 | Natural logarithm preserve... |
| relogefd 25205 | Relationship between the n... |
| rplogcld 25206 | Closure of the logarithm f... |
| logge0d 25207 | The logarithm of a number ... |
| logge0b 25208 | The logarithm of a number ... |
| loggt0b 25209 | The logarithm of a number ... |
| logle1b 25210 | The logarithm of a number ... |
| loglt1b 25211 | The logarithm of a number ... |
| divlogrlim 25212 | The inverse logarithm func... |
| logno1 25213 | The logarithm function is ... |
| dvrelog 25214 | The derivative of the real... |
| relogcn 25215 | The real logarithm functio... |
| ellogdm 25216 | Elementhood in the "contin... |
| logdmn0 25217 | A number in the continuous... |
| logdmnrp 25218 | A number in the continuous... |
| logdmss 25219 | The continuity domain of `... |
| logcnlem2 25220 | Lemma for ~ logcn . (Cont... |
| logcnlem3 25221 | Lemma for ~ logcn . (Cont... |
| logcnlem4 25222 | Lemma for ~ logcn . (Cont... |
| logcnlem5 25223 | Lemma for ~ logcn . (Cont... |
| logcn 25224 | The logarithm function is ... |
| dvloglem 25225 | Lemma for ~ dvlog . (Cont... |
| logdmopn 25226 | The "continuous domain" of... |
| logf1o2 25227 | The logarithm maps its con... |
| dvlog 25228 | The derivative of the comp... |
| dvlog2lem 25229 | Lemma for ~ dvlog2 . (Con... |
| dvlog2 25230 | The derivative of the comp... |
| advlog 25231 | The antiderivative of the ... |
| advlogexp 25232 | The antiderivative of a po... |
| efopnlem1 25233 | Lemma for ~ efopn . (Cont... |
| efopnlem2 25234 | Lemma for ~ efopn . (Cont... |
| efopn 25235 | The exponential map is an ... |
| logtayllem 25236 | Lemma for ~ logtayl . (Co... |
| logtayl 25237 | The Taylor series for ` -u... |
| logtaylsum 25238 | The Taylor series for ` -u... |
| logtayl2 25239 | Power series expression fo... |
| logccv 25240 | The natural logarithm func... |
| cxpval 25241 | Value of the complex power... |
| cxpef 25242 | Value of the complex power... |
| 0cxp 25243 | Value of the complex power... |
| cxpexpz 25244 | Relate the complex power f... |
| cxpexp 25245 | Relate the complex power f... |
| logcxp 25246 | Logarithm of a complex pow... |
| cxp0 25247 | Value of the complex power... |
| cxp1 25248 | Value of the complex power... |
| 1cxp 25249 | Value of the complex power... |
| ecxp 25250 | Write the exponential func... |
| cxpcl 25251 | Closure of the complex pow... |
| recxpcl 25252 | Real closure of the comple... |
| rpcxpcl 25253 | Positive real closure of t... |
| cxpne0 25254 | Complex exponentiation is ... |
| cxpeq0 25255 | Complex exponentiation is ... |
| cxpadd 25256 | Sum of exponents law for c... |
| cxpp1 25257 | Value of a nonzero complex... |
| cxpneg 25258 | Value of a complex number ... |
| cxpsub 25259 | Exponent subtraction law f... |
| cxpge0 25260 | Nonnegative exponentiation... |
| mulcxplem 25261 | Lemma for ~ mulcxp . (Con... |
| mulcxp 25262 | Complex exponentiation of ... |
| cxprec 25263 | Complex exponentiation of ... |
| divcxp 25264 | Complex exponentiation of ... |
| cxpmul 25265 | Product of exponents law f... |
| cxpmul2 25266 | Product of exponents law f... |
| cxproot 25267 | The complex power function... |
| cxpmul2z 25268 | Generalize ~ cxpmul2 to ne... |
| abscxp 25269 | Absolute value of a power,... |
| abscxp2 25270 | Absolute value of a power,... |
| cxplt 25271 | Ordering property for comp... |
| cxple 25272 | Ordering property for comp... |
| cxplea 25273 | Ordering property for comp... |
| cxple2 25274 | Ordering property for comp... |
| cxplt2 25275 | Ordering property for comp... |
| cxple2a 25276 | Ordering property for comp... |
| cxplt3 25277 | Ordering property for comp... |
| cxple3 25278 | Ordering property for comp... |
| cxpsqrtlem 25279 | Lemma for ~ cxpsqrt . (Co... |
| cxpsqrt 25280 | The complex exponential fu... |
| logsqrt 25281 | Logarithm of a square root... |
| cxp0d 25282 | Value of the complex power... |
| cxp1d 25283 | Value of the complex power... |
| 1cxpd 25284 | Value of the complex power... |
| cxpcld 25285 | Closure of the complex pow... |
| cxpmul2d 25286 | Product of exponents law f... |
| 0cxpd 25287 | Value of the complex power... |
| cxpexpzd 25288 | Relate the complex power f... |
| cxpefd 25289 | Value of the complex power... |
| cxpne0d 25290 | Complex exponentiation is ... |
| cxpp1d 25291 | Value of a nonzero complex... |
| cxpnegd 25292 | Value of a complex number ... |
| cxpmul2zd 25293 | Generalize ~ cxpmul2 to ne... |
| cxpaddd 25294 | Sum of exponents law for c... |
| cxpsubd 25295 | Exponent subtraction law f... |
| cxpltd 25296 | Ordering property for comp... |
| cxpled 25297 | Ordering property for comp... |
| cxplead 25298 | Ordering property for comp... |
| divcxpd 25299 | Complex exponentiation of ... |
| recxpcld 25300 | Positive real closure of t... |
| cxpge0d 25301 | Nonnegative exponentiation... |
| cxple2ad 25302 | Ordering property for comp... |
| cxplt2d 25303 | Ordering property for comp... |
| cxple2d 25304 | Ordering property for comp... |
| mulcxpd 25305 | Complex exponentiation of ... |
| cxpsqrtth 25306 | Square root theorem over t... |
| 2irrexpq 25307 | There exist irrational num... |
| cxprecd 25308 | Complex exponentiation of ... |
| rpcxpcld 25309 | Positive real closure of t... |
| logcxpd 25310 | Logarithm of a complex pow... |
| cxplt3d 25311 | Ordering property for comp... |
| cxple3d 25312 | Ordering property for comp... |
| cxpmuld 25313 | Product of exponents law f... |
| cxpcom 25314 | Commutative law for real e... |
| dvcxp1 25315 | The derivative of a comple... |
| dvcxp2 25316 | The derivative of a comple... |
| dvsqrt 25317 | The derivative of the real... |
| dvcncxp1 25318 | Derivative of complex powe... |
| dvcnsqrt 25319 | Derivative of square root ... |
| cxpcn 25320 | Domain of continuity of th... |
| cxpcn2 25321 | Continuity of the complex ... |
| cxpcn3lem 25322 | Lemma for ~ cxpcn3 . (Con... |
| cxpcn3 25323 | Extend continuity of the c... |
| resqrtcn 25324 | Continuity of the real squ... |
| sqrtcn 25325 | Continuity of the square r... |
| cxpaddlelem 25326 | Lemma for ~ cxpaddle . (C... |
| cxpaddle 25327 | Ordering property for comp... |
| abscxpbnd 25328 | Bound on the absolute valu... |
| root1id 25329 | Property of an ` N ` -th r... |
| root1eq1 25330 | The only powers of an ` N ... |
| root1cj 25331 | Within the ` N ` -th roots... |
| cxpeq 25332 | Solve an equation involvin... |
| loglesqrt 25333 | An upper bound on the loga... |
| logreclem 25334 | Symmetry of the natural lo... |
| logrec 25335 | Logarithm of a reciprocal ... |
| logbval 25338 | Define the value of the ` ... |
| logbcl 25339 | General logarithm closure.... |
| logbid1 25340 | General logarithm is 1 whe... |
| logb1 25341 | The logarithm of ` 1 ` to ... |
| elogb 25342 | The general logarithm of a... |
| logbchbase 25343 | Change of base for logarit... |
| relogbval 25344 | Value of the general logar... |
| relogbcl 25345 | Closure of the general log... |
| relogbzcl 25346 | Closure of the general log... |
| relogbreexp 25347 | Power law for the general ... |
| relogbzexp 25348 | Power law for the general ... |
| relogbmul 25349 | The logarithm of the produ... |
| relogbmulexp 25350 | The logarithm of the produ... |
| relogbdiv 25351 | The logarithm of the quoti... |
| relogbexp 25352 | Identity law for general l... |
| nnlogbexp 25353 | Identity law for general l... |
| logbrec 25354 | Logarithm of a reciprocal ... |
| logbleb 25355 | The general logarithm func... |
| logblt 25356 | The general logarithm func... |
| relogbcxp 25357 | Identity law for the gener... |
| cxplogb 25358 | Identity law for the gener... |
| relogbcxpb 25359 | The logarithm is the inver... |
| logbmpt 25360 | The general logarithm to a... |
| logbf 25361 | The general logarithm to a... |
| logbfval 25362 | The general logarithm of a... |
| relogbf 25363 | The general logarithm to a... |
| logblog 25364 | The general logarithm to t... |
| logbgt0b 25365 | The logarithm of a positiv... |
| logbgcd1irr 25366 | The logarithm of an intege... |
| 2logb9irr 25367 | Example for ~ logbgcd1irr ... |
| logbprmirr 25368 | The logarithm of a prime t... |
| 2logb3irr 25369 | Example for ~ logbprmirr .... |
| 2logb9irrALT 25370 | Alternate proof of ~ 2logb... |
| sqrt2cxp2logb9e3 25371 | The square root of two to ... |
| 2irrexpqALT 25372 | Alternate proof of ~ 2irre... |
| angval 25373 | Define the angle function,... |
| angcan 25374 | Cancel a constant multipli... |
| angneg 25375 | Cancel a negative sign in ... |
| angvald 25376 | The (signed) angle between... |
| angcld 25377 | The (signed) angle between... |
| angrteqvd 25378 | Two vectors are at a right... |
| cosangneg2d 25379 | The cosine of the angle be... |
| angrtmuld 25380 | Perpendicularity of two ve... |
| ang180lem1 25381 | Lemma for ~ ang180 . Show... |
| ang180lem2 25382 | Lemma for ~ ang180 . Show... |
| ang180lem3 25383 | Lemma for ~ ang180 . Sinc... |
| ang180lem4 25384 | Lemma for ~ ang180 . Redu... |
| ang180lem5 25385 | Lemma for ~ ang180 : Redu... |
| ang180 25386 | The sum of angles ` m A B ... |
| lawcoslem1 25387 | Lemma for ~ lawcos . Here... |
| lawcos 25388 | Law of cosines (also known... |
| pythag 25389 | Pythagorean theorem. Give... |
| isosctrlem1 25390 | Lemma for ~ isosctr . (Co... |
| isosctrlem2 25391 | Lemma for ~ isosctr . Cor... |
| isosctrlem3 25392 | Lemma for ~ isosctr . Cor... |
| isosctr 25393 | Isosceles triangle theorem... |
| ssscongptld 25394 | If two triangles have equa... |
| affineequiv 25395 | Equivalence between two wa... |
| affineequiv2 25396 | Equivalence between two wa... |
| affineequiv3 25397 | Equivalence between two wa... |
| affineequiv4 25398 | Equivalence between two wa... |
| affineequivne 25399 | Equivalence between two wa... |
| angpieqvdlem 25400 | Equivalence used in the pr... |
| angpieqvdlem2 25401 | Equivalence used in ~ angp... |
| angpined 25402 | If the angle at ABC is ` _... |
| angpieqvd 25403 | The angle ABC is ` _pi ` i... |
| chordthmlem 25404 | If M is the midpoint of AB... |
| chordthmlem2 25405 | If M is the midpoint of AB... |
| chordthmlem3 25406 | If M is the midpoint of AB... |
| chordthmlem4 25407 | If P is on the segment AB ... |
| chordthmlem5 25408 | If P is on the segment AB ... |
| chordthm 25409 | The intersecting chords th... |
| heron 25410 | Heron's formula gives the ... |
| quad2 25411 | The quadratic equation, wi... |
| quad 25412 | The quadratic equation. (... |
| 1cubrlem 25413 | The cube roots of unity. ... |
| 1cubr 25414 | The cube roots of unity. ... |
| dcubic1lem 25415 | Lemma for ~ dcubic1 and ~ ... |
| dcubic2 25416 | Reverse direction of ~ dcu... |
| dcubic1 25417 | Forward direction of ~ dcu... |
| dcubic 25418 | Solutions to the depressed... |
| mcubic 25419 | Solutions to a monic cubic... |
| cubic2 25420 | The solution to the genera... |
| cubic 25421 | The cubic equation, which ... |
| binom4 25422 | Work out a quartic binomia... |
| dquartlem1 25423 | Lemma for ~ dquart . (Con... |
| dquartlem2 25424 | Lemma for ~ dquart . (Con... |
| dquart 25425 | Solve a depressed quartic ... |
| quart1cl 25426 | Closure lemmas for ~ quart... |
| quart1lem 25427 | Lemma for ~ quart1 . (Con... |
| quart1 25428 | Depress a quartic equation... |
| quartlem1 25429 | Lemma for ~ quart . (Cont... |
| quartlem2 25430 | Closure lemmas for ~ quart... |
| quartlem3 25431 | Closure lemmas for ~ quart... |
| quartlem4 25432 | Closure lemmas for ~ quart... |
| quart 25433 | The quartic equation, writ... |
| asinlem 25440 | The argument to the logari... |
| asinlem2 25441 | The argument to the logari... |
| asinlem3a 25442 | Lemma for ~ asinlem3 . (C... |
| asinlem3 25443 | The argument to the logari... |
| asinf 25444 | Domain and range of the ar... |
| asincl 25445 | Closure for the arcsin fun... |
| acosf 25446 | Domain and range of the ar... |
| acoscl 25447 | Closure for the arccos fun... |
| atandm 25448 | Since the property is a li... |
| atandm2 25449 | This form of ~ atandm is a... |
| atandm3 25450 | A compact form of ~ atandm... |
| atandm4 25451 | A compact form of ~ atandm... |
| atanf 25452 | Domain and range of the ar... |
| atancl 25453 | Closure for the arctan fun... |
| asinval 25454 | Value of the arcsin functi... |
| acosval 25455 | Value of the arccos functi... |
| atanval 25456 | Value of the arctan functi... |
| atanre 25457 | A real number is in the do... |
| asinneg 25458 | The arcsine function is od... |
| acosneg 25459 | The negative symmetry rela... |
| efiasin 25460 | The exponential of the arc... |
| sinasin 25461 | The arcsine function is an... |
| cosacos 25462 | The arccosine function is ... |
| asinsinlem 25463 | Lemma for ~ asinsin . (Co... |
| asinsin 25464 | The arcsine function compo... |
| acoscos 25465 | The arccosine function is ... |
| asin1 25466 | The arcsine of ` 1 ` is ` ... |
| acos1 25467 | The arcsine of ` 1 ` is ` ... |
| reasinsin 25468 | The arcsine function compo... |
| asinsinb 25469 | Relationship between sine ... |
| acoscosb 25470 | Relationship between sine ... |
| asinbnd 25471 | The arcsine function has r... |
| acosbnd 25472 | The arccosine function has... |
| asinrebnd 25473 | Bounds on the arcsine func... |
| asinrecl 25474 | The arcsine function is re... |
| acosrecl 25475 | The arccosine function is ... |
| cosasin 25476 | The cosine of the arcsine ... |
| sinacos 25477 | The sine of the arccosine ... |
| atandmneg 25478 | The domain of the arctange... |
| atanneg 25479 | The arctangent function is... |
| atan0 25480 | The arctangent of zero is ... |
| atandmcj 25481 | The arctangent function di... |
| atancj 25482 | The arctangent function di... |
| atanrecl 25483 | The arctangent function is... |
| efiatan 25484 | Value of the exponential o... |
| atanlogaddlem 25485 | Lemma for ~ atanlogadd . ... |
| atanlogadd 25486 | The rule ` sqrt ( z w ) = ... |
| atanlogsublem 25487 | Lemma for ~ atanlogsub . ... |
| atanlogsub 25488 | A variation on ~ atanlogad... |
| efiatan2 25489 | Value of the exponential o... |
| 2efiatan 25490 | Value of the exponential o... |
| tanatan 25491 | The arctangent function is... |
| atandmtan 25492 | The tangent function has r... |
| cosatan 25493 | The cosine of an arctangen... |
| cosatanne0 25494 | The arctangent function ha... |
| atantan 25495 | The arctangent function is... |
| atantanb 25496 | Relationship between tange... |
| atanbndlem 25497 | Lemma for ~ atanbnd . (Co... |
| atanbnd 25498 | The arctangent function is... |
| atanord 25499 | The arctangent function is... |
| atan1 25500 | The arctangent of ` 1 ` is... |
| bndatandm 25501 | A point in the open unit d... |
| atans 25502 | The "domain of continuity"... |
| atans2 25503 | It suffices to show that `... |
| atansopn 25504 | The domain of continuity o... |
| atansssdm 25505 | The domain of continuity o... |
| ressatans 25506 | The real number line is a ... |
| dvatan 25507 | The derivative of the arct... |
| atancn 25508 | The arctangent is a contin... |
| atantayl 25509 | The Taylor series for ` ar... |
| atantayl2 25510 | The Taylor series for ` ar... |
| atantayl3 25511 | The Taylor series for ` ar... |
| leibpilem1 25512 | Lemma for ~ leibpi . (Con... |
| leibpilem2 25513 | The Leibniz formula for ` ... |
| leibpi 25514 | The Leibniz formula for ` ... |
| leibpisum 25515 | The Leibniz formula for ` ... |
| log2cnv 25516 | Using the Taylor series fo... |
| log2tlbnd 25517 | Bound the error term in th... |
| log2ublem1 25518 | Lemma for ~ log2ub . The ... |
| log2ublem2 25519 | Lemma for ~ log2ub . (Con... |
| log2ublem3 25520 | Lemma for ~ log2ub . In d... |
| log2ub 25521 | ` log 2 ` is less than ` 2... |
| log2le1 25522 | ` log 2 ` is less than ` 1... |
| birthdaylem1 25523 | Lemma for ~ birthday . (C... |
| birthdaylem2 25524 | For general ` N ` and ` K ... |
| birthdaylem3 25525 | For general ` N ` and ` K ... |
| birthday 25526 | The Birthday Problem. The... |
| dmarea 25529 | The domain of the area fun... |
| areambl 25530 | The fibers of a measurable... |
| areass 25531 | A measurable region is a s... |
| dfarea 25532 | Rewrite ~ df-area self-ref... |
| areaf 25533 | Area measurement is a func... |
| areacl 25534 | The area of a measurable r... |
| areage0 25535 | The area of a measurable r... |
| areaval 25536 | The area of a measurable r... |
| rlimcnp 25537 | Relate a limit of a real-v... |
| rlimcnp2 25538 | Relate a limit of a real-v... |
| rlimcnp3 25539 | Relate a limit of a real-v... |
| xrlimcnp 25540 | Relate a limit of a real-v... |
| efrlim 25541 | The limit of the sequence ... |
| dfef2 25542 | The limit of the sequence ... |
| cxplim 25543 | A power to a negative expo... |
| sqrtlim 25544 | The inverse square root fu... |
| rlimcxp 25545 | Any power to a positive ex... |
| o1cxp 25546 | An eventually bounded func... |
| cxp2limlem 25547 | A linear factor grows slow... |
| cxp2lim 25548 | Any power grows slower tha... |
| cxploglim 25549 | The logarithm grows slower... |
| cxploglim2 25550 | Every power of the logarit... |
| divsqrtsumlem 25551 | Lemma for ~ divsqrsum and ... |
| divsqrsumf 25552 | The function ` F ` used in... |
| divsqrsum 25553 | The sum ` sum_ n <_ x ( 1 ... |
| divsqrtsum2 25554 | A bound on the distance of... |
| divsqrtsumo1 25555 | The sum ` sum_ n <_ x ( 1 ... |
| cvxcl 25556 | Closure of a 0-1 linear co... |
| scvxcvx 25557 | A strictly convex function... |
| jensenlem1 25558 | Lemma for ~ jensen . (Con... |
| jensenlem2 25559 | Lemma for ~ jensen . (Con... |
| jensen 25560 | Jensen's inequality, a fin... |
| amgmlem 25561 | Lemma for ~ amgm . (Contr... |
| amgm 25562 | Inequality of arithmetic a... |
| logdifbnd 25565 | Bound on the difference of... |
| logdiflbnd 25566 | Lower bound on the differe... |
| emcllem1 25567 | Lemma for ~ emcl . The se... |
| emcllem2 25568 | Lemma for ~ emcl . ` F ` i... |
| emcllem3 25569 | Lemma for ~ emcl . The fu... |
| emcllem4 25570 | Lemma for ~ emcl . The di... |
| emcllem5 25571 | Lemma for ~ emcl . The pa... |
| emcllem6 25572 | Lemma for ~ emcl . By the... |
| emcllem7 25573 | Lemma for ~ emcl and ~ har... |
| emcl 25574 | Closure and bounds for the... |
| harmonicbnd 25575 | A bound on the harmonic se... |
| harmonicbnd2 25576 | A bound on the harmonic se... |
| emre 25577 | The Euler-Mascheroni const... |
| emgt0 25578 | The Euler-Mascheroni const... |
| harmonicbnd3 25579 | A bound on the harmonic se... |
| harmoniclbnd 25580 | A bound on the harmonic se... |
| harmonicubnd 25581 | A bound on the harmonic se... |
| harmonicbnd4 25582 | The asymptotic behavior of... |
| fsumharmonic 25583 | Bound a finite sum based o... |
| zetacvg 25586 | The zeta series is converg... |
| eldmgm 25593 | Elementhood in the set of ... |
| dmgmaddn0 25594 | If ` A ` is not a nonposit... |
| dmlogdmgm 25595 | If ` A ` is in the continu... |
| rpdmgm 25596 | A positive real number is ... |
| dmgmn0 25597 | If ` A ` is not a nonposit... |
| dmgmaddnn0 25598 | If ` A ` is not a nonposit... |
| dmgmdivn0 25599 | Lemma for ~ lgamf . (Cont... |
| lgamgulmlem1 25600 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem2 25601 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem3 25602 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem4 25603 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem5 25604 | Lemma for ~ lgamgulm . (C... |
| lgamgulmlem6 25605 | The series ` G ` is unifor... |
| lgamgulm 25606 | The series ` G ` is unifor... |
| lgamgulm2 25607 | Rewrite the limit of the s... |
| lgambdd 25608 | The log-Gamma function is ... |
| lgamucov 25609 | The ` U ` regions used in ... |
| lgamucov2 25610 | The ` U ` regions used in ... |
| lgamcvglem 25611 | Lemma for ~ lgamf and ~ lg... |
| lgamcl 25612 | The log-Gamma function is ... |
| lgamf 25613 | The log-Gamma function is ... |
| gamf 25614 | The Gamma function is a co... |
| gamcl 25615 | The exponential of the log... |
| eflgam 25616 | The exponential of the log... |
| gamne0 25617 | The Gamma function is neve... |
| igamval 25618 | Value of the inverse Gamma... |
| igamz 25619 | Value of the inverse Gamma... |
| igamgam 25620 | Value of the inverse Gamma... |
| igamlgam 25621 | Value of the inverse Gamma... |
| igamf 25622 | Closure of the inverse Gam... |
| igamcl 25623 | Closure of the inverse Gam... |
| gamigam 25624 | The Gamma function is the ... |
| lgamcvg 25625 | The series ` G ` converges... |
| lgamcvg2 25626 | The series ` G ` converges... |
| gamcvg 25627 | The pointwise exponential ... |
| lgamp1 25628 | The functional equation of... |
| gamp1 25629 | The functional equation of... |
| gamcvg2lem 25630 | Lemma for ~ gamcvg2 . (Co... |
| gamcvg2 25631 | An infinite product expres... |
| regamcl 25632 | The Gamma function is real... |
| relgamcl 25633 | The log-Gamma function is ... |
| rpgamcl 25634 | The log-Gamma function is ... |
| lgam1 25635 | The log-Gamma function at ... |
| gam1 25636 | The log-Gamma function at ... |
| facgam 25637 | The Gamma function general... |
| gamfac 25638 | The Gamma function general... |
| wilthlem1 25639 | The only elements that are... |
| wilthlem2 25640 | Lemma for ~ wilth : induct... |
| wilthlem3 25641 | Lemma for ~ wilth . Here ... |
| wilth 25642 | Wilson's theorem. A numbe... |
| wilthimp 25643 | The forward implication of... |
| ftalem1 25644 | Lemma for ~ fta : "growth... |
| ftalem2 25645 | Lemma for ~ fta . There e... |
| ftalem3 25646 | Lemma for ~ fta . There e... |
| ftalem4 25647 | Lemma for ~ fta : Closure... |
| ftalem5 25648 | Lemma for ~ fta : Main pr... |
| ftalem6 25649 | Lemma for ~ fta : Dischar... |
| ftalem7 25650 | Lemma for ~ fta . Shift t... |
| fta 25651 | The Fundamental Theorem of... |
| basellem1 25652 | Lemma for ~ basel . Closu... |
| basellem2 25653 | Lemma for ~ basel . Show ... |
| basellem3 25654 | Lemma for ~ basel . Using... |
| basellem4 25655 | Lemma for ~ basel . By ~ ... |
| basellem5 25656 | Lemma for ~ basel . Using... |
| basellem6 25657 | Lemma for ~ basel . The f... |
| basellem7 25658 | Lemma for ~ basel . The f... |
| basellem8 25659 | Lemma for ~ basel . The f... |
| basellem9 25660 | Lemma for ~ basel . Since... |
| basel 25661 | The sum of the inverse squ... |
| efnnfsumcl 25674 | Finite sum closure in the ... |
| ppisval 25675 | The set of primes less tha... |
| ppisval2 25676 | The set of primes less tha... |
| ppifi 25677 | The set of primes less tha... |
| prmdvdsfi 25678 | The set of prime divisors ... |
| chtf 25679 | Domain and range of the Ch... |
| chtcl 25680 | Real closure of the Chebys... |
| chtval 25681 | Value of the Chebyshev fun... |
| efchtcl 25682 | The Chebyshev function is ... |
| chtge0 25683 | The Chebyshev function is ... |
| vmaval 25684 | Value of the von Mangoldt ... |
| isppw 25685 | Two ways to say that ` A `... |
| isppw2 25686 | Two ways to say that ` A `... |
| vmappw 25687 | Value of the von Mangoldt ... |
| vmaprm 25688 | Value of the von Mangoldt ... |
| vmacl 25689 | Closure for the von Mangol... |
| vmaf 25690 | Functionality of the von M... |
| efvmacl 25691 | The von Mangoldt is closed... |
| vmage0 25692 | The von Mangoldt function ... |
| chpval 25693 | Value of the second Chebys... |
| chpf 25694 | Functionality of the secon... |
| chpcl 25695 | Closure for the second Che... |
| efchpcl 25696 | The second Chebyshev funct... |
| chpge0 25697 | The second Chebyshev funct... |
| ppival 25698 | Value of the prime-countin... |
| ppival2 25699 | Value of the prime-countin... |
| ppival2g 25700 | Value of the prime-countin... |
| ppif 25701 | Domain and range of the pr... |
| ppicl 25702 | Real closure of the prime-... |
| muval 25703 | The value of the Möbi... |
| muval1 25704 | The value of the Möbi... |
| muval2 25705 | The value of the Möbi... |
| isnsqf 25706 | Two ways to say that a num... |
| issqf 25707 | Two ways to say that a num... |
| sqfpc 25708 | The prime count of a squar... |
| dvdssqf 25709 | A divisor of a squarefree ... |
| sqf11 25710 | A squarefree number is com... |
| muf 25711 | The Möbius function i... |
| mucl 25712 | Closure of the Möbius... |
| sgmval 25713 | The value of the divisor f... |
| sgmval2 25714 | The value of the divisor f... |
| 0sgm 25715 | The value of the sum-of-di... |
| sgmf 25716 | The divisor function is a ... |
| sgmcl 25717 | Closure of the divisor fun... |
| sgmnncl 25718 | Closure of the divisor fun... |
| mule1 25719 | The Möbius function t... |
| chtfl 25720 | The Chebyshev function doe... |
| chpfl 25721 | The second Chebyshev funct... |
| ppiprm 25722 | The prime-counting functio... |
| ppinprm 25723 | The prime-counting functio... |
| chtprm 25724 | The Chebyshev function at ... |
| chtnprm 25725 | The Chebyshev function at ... |
| chpp1 25726 | The second Chebyshev funct... |
| chtwordi 25727 | The Chebyshev function is ... |
| chpwordi 25728 | The second Chebyshev funct... |
| chtdif 25729 | The difference of the Cheb... |
| efchtdvds 25730 | The exponentiated Chebyshe... |
| ppifl 25731 | The prime-counting functio... |
| ppip1le 25732 | The prime-counting functio... |
| ppiwordi 25733 | The prime-counting functio... |
| ppidif 25734 | The difference of the prim... |
| ppi1 25735 | The prime-counting functio... |
| cht1 25736 | The Chebyshev function at ... |
| vma1 25737 | The von Mangoldt function ... |
| chp1 25738 | The second Chebyshev funct... |
| ppi1i 25739 | Inference form of ~ ppiprm... |
| ppi2i 25740 | Inference form of ~ ppinpr... |
| ppi2 25741 | The prime-counting functio... |
| ppi3 25742 | The prime-counting functio... |
| cht2 25743 | The Chebyshev function at ... |
| cht3 25744 | The Chebyshev function at ... |
| ppinncl 25745 | Closure of the prime-count... |
| chtrpcl 25746 | Closure of the Chebyshev f... |
| ppieq0 25747 | The prime-counting functio... |
| ppiltx 25748 | The prime-counting functio... |
| prmorcht 25749 | Relate the primorial (prod... |
| mumullem1 25750 | Lemma for ~ mumul . A mul... |
| mumullem2 25751 | Lemma for ~ mumul . The p... |
| mumul 25752 | The Möbius function i... |
| sqff1o 25753 | There is a bijection from ... |
| fsumdvdsdiaglem 25754 | A "diagonal commutation" o... |
| fsumdvdsdiag 25755 | A "diagonal commutation" o... |
| fsumdvdscom 25756 | A double commutation of di... |
| dvdsppwf1o 25757 | A bijection from the divis... |
| dvdsflf1o 25758 | A bijection from the numbe... |
| dvdsflsumcom 25759 | A sum commutation from ` s... |
| fsumfldivdiaglem 25760 | Lemma for ~ fsumfldivdiag ... |
| fsumfldivdiag 25761 | The right-hand side of ~ d... |
| musum 25762 | The sum of the Möbius... |
| musumsum 25763 | Evaluate a collapsing sum ... |
| muinv 25764 | The Möbius inversion ... |
| dvdsmulf1o 25765 | If ` M ` and ` N ` are two... |
| fsumdvdsmul 25766 | Product of two divisor sum... |
| sgmppw 25767 | The value of the divisor f... |
| 0sgmppw 25768 | A prime power ` P ^ K ` ha... |
| 1sgmprm 25769 | The sum of divisors for a ... |
| 1sgm2ppw 25770 | The sum of the divisors of... |
| sgmmul 25771 | The divisor function for f... |
| ppiublem1 25772 | Lemma for ~ ppiub . (Cont... |
| ppiublem2 25773 | A prime greater than ` 3 `... |
| ppiub 25774 | An upper bound on the prim... |
| vmalelog 25775 | The von Mangoldt function ... |
| chtlepsi 25776 | The first Chebyshev functi... |
| chprpcl 25777 | Closure of the second Cheb... |
| chpeq0 25778 | The second Chebyshev funct... |
| chteq0 25779 | The first Chebyshev functi... |
| chtleppi 25780 | Upper bound on the ` theta... |
| chtublem 25781 | Lemma for ~ chtub . (Cont... |
| chtub 25782 | An upper bound on the Cheb... |
| fsumvma 25783 | Rewrite a sum over the von... |
| fsumvma2 25784 | Apply ~ fsumvma for the co... |
| pclogsum 25785 | The logarithmic analogue o... |
| vmasum 25786 | The sum of the von Mangold... |
| logfac2 25787 | Another expression for the... |
| chpval2 25788 | Express the second Chebysh... |
| chpchtsum 25789 | The second Chebyshev funct... |
| chpub 25790 | An upper bound on the seco... |
| logfacubnd 25791 | A simple upper bound on th... |
| logfaclbnd 25792 | A lower bound on the logar... |
| logfacbnd3 25793 | Show the stronger statemen... |
| logfacrlim 25794 | Combine the estimates ~ lo... |
| logexprlim 25795 | The sum ` sum_ n <_ x , lo... |
| logfacrlim2 25796 | Write out ~ logfacrlim as ... |
| mersenne 25797 | A Mersenne prime is a prim... |
| perfect1 25798 | Euclid's contribution to t... |
| perfectlem1 25799 | Lemma for ~ perfect . (Co... |
| perfectlem2 25800 | Lemma for ~ perfect . (Co... |
| perfect 25801 | The Euclid-Euler theorem, ... |
| dchrval 25804 | Value of the group of Diri... |
| dchrbas 25805 | Base set of the group of D... |
| dchrelbas 25806 | A Dirichlet character is a... |
| dchrelbas2 25807 | A Dirichlet character is a... |
| dchrelbas3 25808 | A Dirichlet character is a... |
| dchrelbasd 25809 | A Dirichlet character is a... |
| dchrrcl 25810 | Reverse closure for a Diri... |
| dchrmhm 25811 | A Dirichlet character is a... |
| dchrf 25812 | A Dirichlet character is a... |
| dchrelbas4 25813 | A Dirichlet character is a... |
| dchrzrh1 25814 | Value of a Dirichlet chara... |
| dchrzrhcl 25815 | A Dirichlet character take... |
| dchrzrhmul 25816 | A Dirichlet character is c... |
| dchrplusg 25817 | Group operation on the gro... |
| dchrmul 25818 | Group operation on the gro... |
| dchrmulcl 25819 | Closure of the group opera... |
| dchrn0 25820 | A Dirichlet character is n... |
| dchr1cl 25821 | Closure of the principal D... |
| dchrmulid2 25822 | Left identity for the prin... |
| dchrinvcl 25823 | Closure of the group inver... |
| dchrabl 25824 | The set of Dirichlet chara... |
| dchrfi 25825 | The group of Dirichlet cha... |
| dchrghm 25826 | A Dirichlet character rest... |
| dchr1 25827 | Value of the principal Dir... |
| dchreq 25828 | A Dirichlet character is d... |
| dchrresb 25829 | A Dirichlet character is d... |
| dchrabs 25830 | A Dirichlet character take... |
| dchrinv 25831 | The inverse of a Dirichlet... |
| dchrabs2 25832 | A Dirichlet character take... |
| dchr1re 25833 | The principal Dirichlet ch... |
| dchrptlem1 25834 | Lemma for ~ dchrpt . (Con... |
| dchrptlem2 25835 | Lemma for ~ dchrpt . (Con... |
| dchrptlem3 25836 | Lemma for ~ dchrpt . (Con... |
| dchrpt 25837 | For any element other than... |
| dchrsum2 25838 | An orthogonality relation ... |
| dchrsum 25839 | An orthogonality relation ... |
| sumdchr2 25840 | Lemma for ~ sumdchr . (Co... |
| dchrhash 25841 | There are exactly ` phi ( ... |
| sumdchr 25842 | An orthogonality relation ... |
| dchr2sum 25843 | An orthogonality relation ... |
| sum2dchr 25844 | An orthogonality relation ... |
| bcctr 25845 | Value of the central binom... |
| pcbcctr 25846 | Prime count of a central b... |
| bcmono 25847 | The binomial coefficient i... |
| bcmax 25848 | The binomial coefficient t... |
| bcp1ctr 25849 | Ratio of two central binom... |
| bclbnd 25850 | A bound on the binomial co... |
| efexple 25851 | Convert a bound on a power... |
| bpos1lem 25852 | Lemma for ~ bpos1 . (Cont... |
| bpos1 25853 | Bertrand's postulate, chec... |
| bposlem1 25854 | An upper bound on the prim... |
| bposlem2 25855 | There are no odd primes in... |
| bposlem3 25856 | Lemma for ~ bpos . Since ... |
| bposlem4 25857 | Lemma for ~ bpos . (Contr... |
| bposlem5 25858 | Lemma for ~ bpos . Bound ... |
| bposlem6 25859 | Lemma for ~ bpos . By usi... |
| bposlem7 25860 | Lemma for ~ bpos . The fu... |
| bposlem8 25861 | Lemma for ~ bpos . Evalua... |
| bposlem9 25862 | Lemma for ~ bpos . Derive... |
| bpos 25863 | Bertrand's postulate: ther... |
| zabsle1 25866 | ` { -u 1 , 0 , 1 } ` is th... |
| lgslem1 25867 | When ` a ` is coprime to t... |
| lgslem2 25868 | The set ` Z ` of all integ... |
| lgslem3 25869 | The set ` Z ` of all integ... |
| lgslem4 25870 | Lemma for ~ lgsfcl2 . (Co... |
| lgsval 25871 | Value of the Legendre symb... |
| lgsfval 25872 | Value of the function ` F ... |
| lgsfcl2 25873 | The function ` F ` is clos... |
| lgscllem 25874 | The Legendre symbol is an ... |
| lgsfcl 25875 | Closure of the function ` ... |
| lgsfle1 25876 | The function ` F ` has mag... |
| lgsval2lem 25877 | Lemma for ~ lgsval2 . (Co... |
| lgsval4lem 25878 | Lemma for ~ lgsval4 . (Co... |
| lgscl2 25879 | The Legendre symbol is an ... |
| lgs0 25880 | The Legendre symbol when t... |
| lgscl 25881 | The Legendre symbol is an ... |
| lgsle1 25882 | The Legendre symbol has ab... |
| lgsval2 25883 | The Legendre symbol at a p... |
| lgs2 25884 | The Legendre symbol at ` 2... |
| lgsval3 25885 | The Legendre symbol at an ... |
| lgsvalmod 25886 | The Legendre symbol is equ... |
| lgsval4 25887 | Restate ~ lgsval for nonze... |
| lgsfcl3 25888 | Closure of the function ` ... |
| lgsval4a 25889 | Same as ~ lgsval4 for posi... |
| lgscl1 25890 | The value of the Legendre ... |
| lgsneg 25891 | The Legendre symbol is eit... |
| lgsneg1 25892 | The Legendre symbol for no... |
| lgsmod 25893 | The Legendre (Jacobi) symb... |
| lgsdilem 25894 | Lemma for ~ lgsdi and ~ lg... |
| lgsdir2lem1 25895 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem2 25896 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem3 25897 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem4 25898 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2lem5 25899 | Lemma for ~ lgsdir2 . (Co... |
| lgsdir2 25900 | The Legendre symbol is com... |
| lgsdirprm 25901 | The Legendre symbol is com... |
| lgsdir 25902 | The Legendre symbol is com... |
| lgsdilem2 25903 | Lemma for ~ lgsdi . (Cont... |
| lgsdi 25904 | The Legendre symbol is com... |
| lgsne0 25905 | The Legendre symbol is non... |
| lgsabs1 25906 | The Legendre symbol is non... |
| lgssq 25907 | The Legendre symbol at a s... |
| lgssq2 25908 | The Legendre symbol at a s... |
| lgsprme0 25909 | The Legendre symbol at any... |
| 1lgs 25910 | The Legendre symbol at ` 1... |
| lgs1 25911 | The Legendre symbol at ` 1... |
| lgsmodeq 25912 | The Legendre (Jacobi) symb... |
| lgsmulsqcoprm 25913 | The Legendre (Jacobi) symb... |
| lgsdirnn0 25914 | Variation on ~ lgsdir vali... |
| lgsdinn0 25915 | Variation on ~ lgsdi valid... |
| lgsqrlem1 25916 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem2 25917 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem3 25918 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem4 25919 | Lemma for ~ lgsqr . (Cont... |
| lgsqrlem5 25920 | Lemma for ~ lgsqr . (Cont... |
| lgsqr 25921 | The Legendre symbol for od... |
| lgsqrmod 25922 | If the Legendre symbol of ... |
| lgsqrmodndvds 25923 | If the Legendre symbol of ... |
| lgsdchrval 25924 | The Legendre symbol functi... |
| lgsdchr 25925 | The Legendre symbol functi... |
| gausslemma2dlem0a 25926 | Auxiliary lemma 1 for ~ ga... |
| gausslemma2dlem0b 25927 | Auxiliary lemma 2 for ~ ga... |
| gausslemma2dlem0c 25928 | Auxiliary lemma 3 for ~ ga... |
| gausslemma2dlem0d 25929 | Auxiliary lemma 4 for ~ ga... |
| gausslemma2dlem0e 25930 | Auxiliary lemma 5 for ~ ga... |
| gausslemma2dlem0f 25931 | Auxiliary lemma 6 for ~ ga... |
| gausslemma2dlem0g 25932 | Auxiliary lemma 7 for ~ ga... |
| gausslemma2dlem0h 25933 | Auxiliary lemma 8 for ~ ga... |
| gausslemma2dlem0i 25934 | Auxiliary lemma 9 for ~ ga... |
| gausslemma2dlem1a 25935 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem1 25936 | Lemma 1 for ~ gausslemma2d... |
| gausslemma2dlem2 25937 | Lemma 2 for ~ gausslemma2d... |
| gausslemma2dlem3 25938 | Lemma 3 for ~ gausslemma2d... |
| gausslemma2dlem4 25939 | Lemma 4 for ~ gausslemma2d... |
| gausslemma2dlem5a 25940 | Lemma for ~ gausslemma2dle... |
| gausslemma2dlem5 25941 | Lemma 5 for ~ gausslemma2d... |
| gausslemma2dlem6 25942 | Lemma 6 for ~ gausslemma2d... |
| gausslemma2dlem7 25943 | Lemma 7 for ~ gausslemma2d... |
| gausslemma2d 25944 | Gauss' Lemma (see also the... |
| lgseisenlem1 25945 | Lemma for ~ lgseisen . If... |
| lgseisenlem2 25946 | Lemma for ~ lgseisen . Th... |
| lgseisenlem3 25947 | Lemma for ~ lgseisen . (C... |
| lgseisenlem4 25948 | Lemma for ~ lgseisen . Th... |
| lgseisen 25949 | Eisenstein's lemma, an exp... |
| lgsquadlem1 25950 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem2 25951 | Lemma for ~ lgsquad . Cou... |
| lgsquadlem3 25952 | Lemma for ~ lgsquad . (Co... |
| lgsquad 25953 | The Law of Quadratic Recip... |
| lgsquad2lem1 25954 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2lem2 25955 | Lemma for ~ lgsquad2 . (C... |
| lgsquad2 25956 | Extend ~ lgsquad to coprim... |
| lgsquad3 25957 | Extend ~ lgsquad2 to integ... |
| m1lgs 25958 | The first supplement to th... |
| 2lgslem1a1 25959 | Lemma 1 for ~ 2lgslem1a . ... |
| 2lgslem1a2 25960 | Lemma 2 for ~ 2lgslem1a . ... |
| 2lgslem1a 25961 | Lemma 1 for ~ 2lgslem1 . ... |
| 2lgslem1b 25962 | Lemma 2 for ~ 2lgslem1 . ... |
| 2lgslem1c 25963 | Lemma 3 for ~ 2lgslem1 . ... |
| 2lgslem1 25964 | Lemma 1 for ~ 2lgs . (Con... |
| 2lgslem2 25965 | Lemma 2 for ~ 2lgs . (Con... |
| 2lgslem3a 25966 | Lemma for ~ 2lgslem3a1 . ... |
| 2lgslem3b 25967 | Lemma for ~ 2lgslem3b1 . ... |
| 2lgslem3c 25968 | Lemma for ~ 2lgslem3c1 . ... |
| 2lgslem3d 25969 | Lemma for ~ 2lgslem3d1 . ... |
| 2lgslem3a1 25970 | Lemma 1 for ~ 2lgslem3 . ... |
| 2lgslem3b1 25971 | Lemma 2 for ~ 2lgslem3 . ... |
| 2lgslem3c1 25972 | Lemma 3 for ~ 2lgslem3 . ... |
| 2lgslem3d1 25973 | Lemma 4 for ~ 2lgslem3 . ... |
| 2lgslem3 25974 | Lemma 3 for ~ 2lgs . (Con... |
| 2lgs2 25975 | The Legendre symbol for ` ... |
| 2lgslem4 25976 | Lemma 4 for ~ 2lgs : speci... |
| 2lgs 25977 | The second supplement to t... |
| 2lgsoddprmlem1 25978 | Lemma 1 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem2 25979 | Lemma 2 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem3a 25980 | Lemma 1 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3b 25981 | Lemma 2 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3c 25982 | Lemma 3 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3d 25983 | Lemma 4 for ~ 2lgsoddprmle... |
| 2lgsoddprmlem3 25984 | Lemma 3 for ~ 2lgsoddprm .... |
| 2lgsoddprmlem4 25985 | Lemma 4 for ~ 2lgsoddprm .... |
| 2lgsoddprm 25986 | The second supplement to t... |
| 2sqlem1 25987 | Lemma for ~ 2sq . (Contri... |
| 2sqlem2 25988 | Lemma for ~ 2sq . (Contri... |
| mul2sq 25989 | Fibonacci's identity (actu... |
| 2sqlem3 25990 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem4 25991 | Lemma for ~ 2sqlem5 . (Co... |
| 2sqlem5 25992 | Lemma for ~ 2sq . If a nu... |
| 2sqlem6 25993 | Lemma for ~ 2sq . If a nu... |
| 2sqlem7 25994 | Lemma for ~ 2sq . (Contri... |
| 2sqlem8a 25995 | Lemma for ~ 2sqlem8 . (Co... |
| 2sqlem8 25996 | Lemma for ~ 2sq . (Contri... |
| 2sqlem9 25997 | Lemma for ~ 2sq . (Contri... |
| 2sqlem10 25998 | Lemma for ~ 2sq . Every f... |
| 2sqlem11 25999 | Lemma for ~ 2sq . (Contri... |
| 2sq 26000 | All primes of the form ` 4... |
| 2sqblem 26001 | Lemma for ~ 2sqb . (Contr... |
| 2sqb 26002 | The converse to ~ 2sq . (... |
| 2sq2 26003 | ` 2 ` is the sum of square... |
| 2sqn0 26004 | If the sum of two squares ... |
| 2sqcoprm 26005 | If the sum of two squares ... |
| 2sqmod 26006 | Given two decompositions o... |
| 2sqmo 26007 | There exists at most one d... |
| 2sqnn0 26008 | All primes of the form ` 4... |
| 2sqnn 26009 | All primes of the form ` 4... |
| addsq2reu 26010 | For each complex number ` ... |
| addsqn2reu 26011 | For each complex number ` ... |
| addsqrexnreu 26012 | For each complex number, t... |
| addsqnreup 26013 | There is no unique decompo... |
| addsq2nreurex 26014 | For each complex number ` ... |
| addsqn2reurex2 26015 | For each complex number ` ... |
| 2sqreulem1 26016 | Lemma 1 for ~ 2sqreu . (C... |
| 2sqreultlem 26017 | Lemma for ~ 2sqreult . (C... |
| 2sqreultblem 26018 | Lemma for ~ 2sqreultb . (... |
| 2sqreunnlem1 26019 | Lemma 1 for ~ 2sqreunn . ... |
| 2sqreunnltlem 26020 | Lemma for ~ 2sqreunnlt . ... |
| 2sqreunnltblem 26021 | Lemma for ~ 2sqreunnltb . ... |
| 2sqreulem2 26022 | Lemma 2 for ~ 2sqreu etc. ... |
| 2sqreulem3 26023 | Lemma 3 for ~ 2sqreu etc. ... |
| 2sqreulem4 26024 | Lemma 4 for ~ 2sqreu et. ... |
| 2sqreunnlem2 26025 | Lemma 2 for ~ 2sqreunn . ... |
| 2sqreu 26026 | There exists a unique deco... |
| 2sqreunn 26027 | There exists a unique deco... |
| 2sqreult 26028 | There exists a unique deco... |
| 2sqreultb 26029 | There exists a unique deco... |
| 2sqreunnlt 26030 | There exists a unique deco... |
| 2sqreunnltb 26031 | There exists a unique deco... |
| 2sqreuop 26032 | There exists a unique deco... |
| 2sqreuopnn 26033 | There exists a unique deco... |
| 2sqreuoplt 26034 | There exists a unique deco... |
| 2sqreuopltb 26035 | There exists a unique deco... |
| 2sqreuopnnlt 26036 | There exists a unique deco... |
| 2sqreuopnnltb 26037 | There exists a unique deco... |
| 2sqreuopb 26038 | There exists a unique deco... |
| chebbnd1lem1 26039 | Lemma for ~ chebbnd1 : sho... |
| chebbnd1lem2 26040 | Lemma for ~ chebbnd1 : Sh... |
| chebbnd1lem3 26041 | Lemma for ~ chebbnd1 : get... |
| chebbnd1 26042 | The Chebyshev bound: The ... |
| chtppilimlem1 26043 | Lemma for ~ chtppilim . (... |
| chtppilimlem2 26044 | Lemma for ~ chtppilim . (... |
| chtppilim 26045 | The ` theta ` function is ... |
| chto1ub 26046 | The ` theta ` function is ... |
| chebbnd2 26047 | The Chebyshev bound, part ... |
| chto1lb 26048 | The ` theta ` function is ... |
| chpchtlim 26049 | The ` psi ` and ` theta ` ... |
| chpo1ub 26050 | The ` psi ` function is up... |
| chpo1ubb 26051 | The ` psi ` function is up... |
| vmadivsum 26052 | The sum of the von Mangold... |
| vmadivsumb 26053 | Give a total bound on the ... |
| rplogsumlem1 26054 | Lemma for ~ rplogsum . (C... |
| rplogsumlem2 26055 | Lemma for ~ rplogsum . Eq... |
| dchrisum0lem1a 26056 | Lemma for ~ dchrisum0lem1 ... |
| rpvmasumlem 26057 | Lemma for ~ rpvmasum . Ca... |
| dchrisumlema 26058 | Lemma for ~ dchrisum . Le... |
| dchrisumlem1 26059 | Lemma for ~ dchrisum . Le... |
| dchrisumlem2 26060 | Lemma for ~ dchrisum . Le... |
| dchrisumlem3 26061 | Lemma for ~ dchrisum . Le... |
| dchrisum 26062 | If ` n e. [ M , +oo ) |-> ... |
| dchrmusumlema 26063 | Lemma for ~ dchrmusum and ... |
| dchrmusum2 26064 | The sum of the Möbius... |
| dchrvmasumlem1 26065 | An alternative expression ... |
| dchrvmasum2lem 26066 | Give an expression for ` l... |
| dchrvmasum2if 26067 | Combine the results of ~ d... |
| dchrvmasumlem2 26068 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlem3 26069 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumlema 26070 | Lemma for ~ dchrvmasum and... |
| dchrvmasumiflem1 26071 | Lemma for ~ dchrvmasumif .... |
| dchrvmasumiflem2 26072 | Lemma for ~ dchrvmasum . ... |
| dchrvmasumif 26073 | An asymptotic approximatio... |
| dchrvmaeq0 26074 | The set ` W ` is the colle... |
| dchrisum0fval 26075 | Value of the function ` F ... |
| dchrisum0fmul 26076 | The function ` F ` , the d... |
| dchrisum0ff 26077 | The function ` F ` is a re... |
| dchrisum0flblem1 26078 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flblem2 26079 | Lemma for ~ dchrisum0flb .... |
| dchrisum0flb 26080 | The divisor sum of a real ... |
| dchrisum0fno1 26081 | The sum ` sum_ k <_ x , F ... |
| rpvmasum2 26082 | A partial result along the... |
| dchrisum0re 26083 | Suppose ` X ` is a non-pri... |
| dchrisum0lema 26084 | Lemma for ~ dchrisum0 . A... |
| dchrisum0lem1b 26085 | Lemma for ~ dchrisum0lem1 ... |
| dchrisum0lem1 26086 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2a 26087 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem2 26088 | Lemma for ~ dchrisum0 . (... |
| dchrisum0lem3 26089 | Lemma for ~ dchrisum0 . (... |
| dchrisum0 26090 | The sum ` sum_ n e. NN , X... |
| dchrisumn0 26091 | The sum ` sum_ n e. NN , X... |
| dchrmusumlem 26092 | The sum of the Möbius... |
| dchrvmasumlem 26093 | The sum of the Möbius... |
| dchrmusum 26094 | The sum of the Möbius... |
| dchrvmasum 26095 | The sum of the von Mangold... |
| rpvmasum 26096 | The sum of the von Mangold... |
| rplogsum 26097 | The sum of ` log p / p ` o... |
| dirith2 26098 | Dirichlet's theorem: there... |
| dirith 26099 | Dirichlet's theorem: there... |
| mudivsum 26100 | Asymptotic formula for ` s... |
| mulogsumlem 26101 | Lemma for ~ mulogsum . (C... |
| mulogsum 26102 | Asymptotic formula for ... |
| logdivsum 26103 | Asymptotic analysis of ... |
| mulog2sumlem1 26104 | Asymptotic formula for ... |
| mulog2sumlem2 26105 | Lemma for ~ mulog2sum . (... |
| mulog2sumlem3 26106 | Lemma for ~ mulog2sum . (... |
| mulog2sum 26107 | Asymptotic formula for ... |
| vmalogdivsum2 26108 | The sum ` sum_ n <_ x , La... |
| vmalogdivsum 26109 | The sum ` sum_ n <_ x , La... |
| 2vmadivsumlem 26110 | Lemma for ~ 2vmadivsum . ... |
| 2vmadivsum 26111 | The sum ` sum_ m n <_ x , ... |
| logsqvma 26112 | A formula for ` log ^ 2 ( ... |
| logsqvma2 26113 | The Möbius inverse of... |
| log2sumbnd 26114 | Bound on the difference be... |
| selberglem1 26115 | Lemma for ~ selberg . Est... |
| selberglem2 26116 | Lemma for ~ selberg . (Co... |
| selberglem3 26117 | Lemma for ~ selberg . Est... |
| selberg 26118 | Selberg's symmetry formula... |
| selbergb 26119 | Convert eventual boundedne... |
| selberg2lem 26120 | Lemma for ~ selberg2 . Eq... |
| selberg2 26121 | Selberg's symmetry formula... |
| selberg2b 26122 | Convert eventual boundedne... |
| chpdifbndlem1 26123 | Lemma for ~ chpdifbnd . (... |
| chpdifbndlem2 26124 | Lemma for ~ chpdifbnd . (... |
| chpdifbnd 26125 | A bound on the difference ... |
| logdivbnd 26126 | A bound on a sum of logs, ... |
| selberg3lem1 26127 | Introduce a log weighting ... |
| selberg3lem2 26128 | Lemma for ~ selberg3 . Eq... |
| selberg3 26129 | Introduce a log weighting ... |
| selberg4lem1 26130 | Lemma for ~ selberg4 . Eq... |
| selberg4 26131 | The Selberg symmetry formu... |
| pntrval 26132 | Define the residual of the... |
| pntrf 26133 | Functionality of the resid... |
| pntrmax 26134 | There is a bound on the re... |
| pntrsumo1 26135 | A bound on a sum over ` R ... |
| pntrsumbnd 26136 | A bound on a sum over ` R ... |
| pntrsumbnd2 26137 | A bound on a sum over ` R ... |
| selbergr 26138 | Selberg's symmetry formula... |
| selberg3r 26139 | Selberg's symmetry formula... |
| selberg4r 26140 | Selberg's symmetry formula... |
| selberg34r 26141 | The sum of ~ selberg3r and... |
| pntsval 26142 | Define the "Selberg functi... |
| pntsf 26143 | Functionality of the Selbe... |
| selbergs 26144 | Selberg's symmetry formula... |
| selbergsb 26145 | Selberg's symmetry formula... |
| pntsval2 26146 | The Selberg function can b... |
| pntrlog2bndlem1 26147 | The sum of ~ selberg3r and... |
| pntrlog2bndlem2 26148 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem3 26149 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem4 26150 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem5 26151 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bndlem6a 26152 | Lemma for ~ pntrlog2bndlem... |
| pntrlog2bndlem6 26153 | Lemma for ~ pntrlog2bnd . ... |
| pntrlog2bnd 26154 | A bound on ` R ( x ) log ^... |
| pntpbnd1a 26155 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd1 26156 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd2 26157 | Lemma for ~ pntpbnd . (Co... |
| pntpbnd 26158 | Lemma for ~ pnt . Establi... |
| pntibndlem1 26159 | Lemma for ~ pntibnd . (Co... |
| pntibndlem2a 26160 | Lemma for ~ pntibndlem2 . ... |
| pntibndlem2 26161 | Lemma for ~ pntibnd . The... |
| pntibndlem3 26162 | Lemma for ~ pntibnd . Pac... |
| pntibnd 26163 | Lemma for ~ pnt . Establi... |
| pntlemd 26164 | Lemma for ~ pnt . Closure... |
| pntlemc 26165 | Lemma for ~ pnt . Closure... |
| pntlema 26166 | Lemma for ~ pnt . Closure... |
| pntlemb 26167 | Lemma for ~ pnt . Unpack ... |
| pntlemg 26168 | Lemma for ~ pnt . Closure... |
| pntlemh 26169 | Lemma for ~ pnt . Bounds ... |
| pntlemn 26170 | Lemma for ~ pnt . The "na... |
| pntlemq 26171 | Lemma for ~ pntlemj . (Co... |
| pntlemr 26172 | Lemma for ~ pntlemj . (Co... |
| pntlemj 26173 | Lemma for ~ pnt . The ind... |
| pntlemi 26174 | Lemma for ~ pnt . Elimina... |
| pntlemf 26175 | Lemma for ~ pnt . Add up ... |
| pntlemk 26176 | Lemma for ~ pnt . Evaluat... |
| pntlemo 26177 | Lemma for ~ pnt . Combine... |
| pntleme 26178 | Lemma for ~ pnt . Package... |
| pntlem3 26179 | Lemma for ~ pnt . Equatio... |
| pntlemp 26180 | Lemma for ~ pnt . Wrappin... |
| pntleml 26181 | Lemma for ~ pnt . Equatio... |
| pnt3 26182 | The Prime Number Theorem, ... |
| pnt2 26183 | The Prime Number Theorem, ... |
| pnt 26184 | The Prime Number Theorem: ... |
| abvcxp 26185 | Raising an absolute value ... |
| padicfval 26186 | Value of the p-adic absolu... |
| padicval 26187 | Value of the p-adic absolu... |
| ostth2lem1 26188 | Lemma for ~ ostth2 , altho... |
| qrngbas 26189 | The base set of the field ... |
| qdrng 26190 | The rationals form a divis... |
| qrng0 26191 | The zero element of the fi... |
| qrng1 26192 | The unit element of the fi... |
| qrngneg 26193 | The additive inverse in th... |
| qrngdiv 26194 | The division operation in ... |
| qabvle 26195 | By using induction on ` N ... |
| qabvexp 26196 | Induct the product rule ~ ... |
| ostthlem1 26197 | Lemma for ~ ostth . If tw... |
| ostthlem2 26198 | Lemma for ~ ostth . Refin... |
| qabsabv 26199 | The regular absolute value... |
| padicabv 26200 | The p-adic absolute value ... |
| padicabvf 26201 | The p-adic absolute value ... |
| padicabvcxp 26202 | All positive powers of the... |
| ostth1 26203 | - Lemma for ~ ostth : triv... |
| ostth2lem2 26204 | Lemma for ~ ostth2 . (Con... |
| ostth2lem3 26205 | Lemma for ~ ostth2 . (Con... |
| ostth2lem4 26206 | Lemma for ~ ostth2 . (Con... |
| ostth2 26207 | - Lemma for ~ ostth : regu... |
| ostth3 26208 | - Lemma for ~ ostth : p-ad... |
| ostth 26209 | Ostrowski's theorem, which... |
| itvndx 26220 | Index value of the Interva... |
| lngndx 26221 | Index value of the "line" ... |
| itvid 26222 | Utility theorem: index-ind... |
| lngid 26223 | Utility theorem: index-ind... |
| trkgstr 26224 | Functionality of a Tarski ... |
| trkgbas 26225 | The base set of a Tarski g... |
| trkgdist 26226 | The measure of a distance ... |
| trkgitv 26227 | The congruence relation in... |
| istrkgc 26234 | Property of being a Tarski... |
| istrkgb 26235 | Property of being a Tarski... |
| istrkgcb 26236 | Property of being a Tarski... |
| istrkge 26237 | Property of fulfilling Euc... |
| istrkgl 26238 | Building lines from the se... |
| istrkgld 26239 | Property of fulfilling the... |
| istrkg2ld 26240 | Property of fulfilling the... |
| istrkg3ld 26241 | Property of fulfilling the... |
| axtgcgrrflx 26242 | Axiom of reflexivity of co... |
| axtgcgrid 26243 | Axiom of identity of congr... |
| axtgsegcon 26244 | Axiom of segment construct... |
| axtg5seg 26245 | Five segments axiom, Axiom... |
| axtgbtwnid 26246 | Identity of Betweenness. ... |
| axtgpasch 26247 | Axiom of (Inner) Pasch, Ax... |
| axtgcont1 26248 | Axiom of Continuity. Axio... |
| axtgcont 26249 | Axiom of Continuity. Axio... |
| axtglowdim2 26250 | Lower dimension axiom for ... |
| axtgupdim2 26251 | Upper dimension axiom for ... |
| axtgeucl 26252 | Euclid's Axiom. Axiom A10... |
| tgjustf 26253 | Given any function ` F ` ,... |
| tgjustr 26254 | Given any equivalence rela... |
| tgjustc1 26255 | A justification for using ... |
| tgjustc2 26256 | A justification for using ... |
| tgcgrcomimp 26257 | Congruence commutes on the... |
| tgcgrcomr 26258 | Congruence commutes on the... |
| tgcgrcoml 26259 | Congruence commutes on the... |
| tgcgrcomlr 26260 | Congruence commutes on bot... |
| tgcgreqb 26261 | Congruence and equality. ... |
| tgcgreq 26262 | Congruence and equality. ... |
| tgcgrneq 26263 | Congruence and equality. ... |
| tgcgrtriv 26264 | Degenerate segments are co... |
| tgcgrextend 26265 | Link congruence over a pai... |
| tgsegconeq 26266 | Two points that satisfy th... |
| tgbtwntriv2 26267 | Betweenness always holds f... |
| tgbtwncom 26268 | Betweenness commutes. The... |
| tgbtwncomb 26269 | Betweenness commutes, bico... |
| tgbtwnne 26270 | Betweenness and inequality... |
| tgbtwntriv1 26271 | Betweenness always holds f... |
| tgbtwnswapid 26272 | If you can swap the first ... |
| tgbtwnintr 26273 | Inner transitivity law for... |
| tgbtwnexch3 26274 | Exchange the first endpoin... |
| tgbtwnouttr2 26275 | Outer transitivity law for... |
| tgbtwnexch2 26276 | Exchange the outer point o... |
| tgbtwnouttr 26277 | Outer transitivity law for... |
| tgbtwnexch 26278 | Outer transitivity law for... |
| tgtrisegint 26279 | A line segment between two... |
| tglowdim1 26280 | Lower dimension axiom for ... |
| tglowdim1i 26281 | Lower dimension axiom for ... |
| tgldimor 26282 | Excluded-middle like state... |
| tgldim0eq 26283 | In dimension zero, any two... |
| tgldim0itv 26284 | In dimension zero, any two... |
| tgldim0cgr 26285 | In dimension zero, any two... |
| tgbtwndiff 26286 | There is always a ` c ` di... |
| tgdim01 26287 | In geometries of dimension... |
| tgifscgr 26288 | Inner five segment congrue... |
| tgcgrsub 26289 | Removing identical parts f... |
| iscgrg 26292 | The congruence property fo... |
| iscgrgd 26293 | The property for two seque... |
| iscgrglt 26294 | The property for two seque... |
| trgcgrg 26295 | The property for two trian... |
| trgcgr 26296 | Triangle congruence. (Con... |
| ercgrg 26297 | The shape congruence relat... |
| tgcgrxfr 26298 | A line segment can be divi... |
| cgr3id 26299 | Reflexivity law for three-... |
| cgr3simp1 26300 | Deduce segment congruence ... |
| cgr3simp2 26301 | Deduce segment congruence ... |
| cgr3simp3 26302 | Deduce segment congruence ... |
| cgr3swap12 26303 | Permutation law for three-... |
| cgr3swap23 26304 | Permutation law for three-... |
| cgr3swap13 26305 | Permutation law for three-... |
| cgr3rotr 26306 | Permutation law for three-... |
| cgr3rotl 26307 | Permutation law for three-... |
| trgcgrcom 26308 | Commutative law for three-... |
| cgr3tr 26309 | Transitivity law for three... |
| tgbtwnxfr 26310 | A condition for extending ... |
| tgcgr4 26311 | Two quadrilaterals to be c... |
| isismt 26314 | Property of being an isome... |
| ismot 26315 | Property of being an isome... |
| motcgr 26316 | Property of a motion: dist... |
| idmot 26317 | The identity is a motion. ... |
| motf1o 26318 | Motions are bijections. (... |
| motcl 26319 | Closure of motions. (Cont... |
| motco 26320 | The composition of two mot... |
| cnvmot 26321 | The converse of a motion i... |
| motplusg 26322 | The operation for motions ... |
| motgrp 26323 | The motions of a geometry ... |
| motcgrg 26324 | Property of a motion: dist... |
| motcgr3 26325 | Property of a motion: dist... |
| tglng 26326 | Lines of a Tarski Geometry... |
| tglnfn 26327 | Lines as functions. (Cont... |
| tglnunirn 26328 | Lines are sets of points. ... |
| tglnpt 26329 | Lines are sets of points. ... |
| tglngne 26330 | It takes two different poi... |
| tglngval 26331 | The line going through poi... |
| tglnssp 26332 | Lines are subset of the ge... |
| tgellng 26333 | Property of lying on the l... |
| tgcolg 26334 | We choose the notation ` (... |
| btwncolg1 26335 | Betweenness implies coline... |
| btwncolg2 26336 | Betweenness implies coline... |
| btwncolg3 26337 | Betweenness implies coline... |
| colcom 26338 | Swapping the points defini... |
| colrot1 26339 | Rotating the points defini... |
| colrot2 26340 | Rotating the points defini... |
| ncolcom 26341 | Swapping non-colinear poin... |
| ncolrot1 26342 | Rotating non-colinear poin... |
| ncolrot2 26343 | Rotating non-colinear poin... |
| tgdim01ln 26344 | In geometries of dimension... |
| ncoltgdim2 26345 | If there are three non-col... |
| lnxfr 26346 | Transfer law for colineari... |
| lnext 26347 | Extend a line with a missi... |
| tgfscgr 26348 | Congruence law for the gen... |
| lncgr 26349 | Congruence rule for lines.... |
| lnid 26350 | Identity law for points on... |
| tgidinside 26351 | Law for finding a point in... |
| tgbtwnconn1lem1 26352 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem2 26353 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1lem3 26354 | Lemma for ~ tgbtwnconn1 . ... |
| tgbtwnconn1 26355 | Connectivity law for betwe... |
| tgbtwnconn2 26356 | Another connectivity law f... |
| tgbtwnconn3 26357 | Inner connectivity law for... |
| tgbtwnconnln3 26358 | Derive colinearity from be... |
| tgbtwnconn22 26359 | Double connectivity law fo... |
| tgbtwnconnln1 26360 | Derive colinearity from be... |
| tgbtwnconnln2 26361 | Derive colinearity from be... |
| legval 26364 | Value of the less-than rel... |
| legov 26365 | Value of the less-than rel... |
| legov2 26366 | An equivalent definition o... |
| legid 26367 | Reflexivity of the less-th... |
| btwnleg 26368 | Betweenness implies less-t... |
| legtrd 26369 | Transitivity of the less-t... |
| legtri3 26370 | Equality from the less-tha... |
| legtrid 26371 | Trichotomy law for the les... |
| leg0 26372 | Degenerated (zero-length) ... |
| legeq 26373 | Deduce equality from "less... |
| legbtwn 26374 | Deduce betweenness from "l... |
| tgcgrsub2 26375 | Removing identical parts f... |
| ltgseg 26376 | The set ` E ` denotes the ... |
| ltgov 26377 | Strict "shorter than" geom... |
| legov3 26378 | An equivalent definition o... |
| legso 26379 | The "shorter than" relatio... |
| ishlg 26382 | Rays : Definition 6.1 of ... |
| hlcomb 26383 | The half-line relation com... |
| hlcomd 26384 | The half-line relation com... |
| hlne1 26385 | The half-line relation imp... |
| hlne2 26386 | The half-line relation imp... |
| hlln 26387 | The half-line relation imp... |
| hleqnid 26388 | The endpoint does not belo... |
| hlid 26389 | The half-line relation is ... |
| hltr 26390 | The half-line relation is ... |
| hlbtwn 26391 | Betweenness is a sufficien... |
| btwnhl1 26392 | Deduce half-line from betw... |
| btwnhl2 26393 | Deduce half-line from betw... |
| btwnhl 26394 | Swap betweenness for a hal... |
| lnhl 26395 | Either a point ` C ` on th... |
| hlcgrex 26396 | Construct a point on a hal... |
| hlcgreulem 26397 | Lemma for ~ hlcgreu . (Co... |
| hlcgreu 26398 | The point constructed in ~... |
| btwnlng1 26399 | Betweenness implies coline... |
| btwnlng2 26400 | Betweenness implies coline... |
| btwnlng3 26401 | Betweenness implies coline... |
| lncom 26402 | Swapping the points defini... |
| lnrot1 26403 | Rotating the points defini... |
| lnrot2 26404 | Rotating the points defini... |
| ncolne1 26405 | Non-colinear points are di... |
| ncolne2 26406 | Non-colinear points are di... |
| tgisline 26407 | The property of being a pr... |
| tglnne 26408 | It takes two different poi... |
| tglndim0 26409 | There are no lines in dime... |
| tgelrnln 26410 | The property of being a pr... |
| tglineeltr 26411 | Transitivity law for lines... |
| tglineelsb2 26412 | If ` S ` lies on PQ , then... |
| tglinerflx1 26413 | Reflexivity law for line m... |
| tglinerflx2 26414 | Reflexivity law for line m... |
| tglinecom 26415 | Commutativity law for line... |
| tglinethru 26416 | If ` A ` is a line contain... |
| tghilberti1 26417 | There is a line through an... |
| tghilberti2 26418 | There is at most one line ... |
| tglinethrueu 26419 | There is a unique line goi... |
| tglnne0 26420 | A line ` A ` has at least ... |
| tglnpt2 26421 | Find a second point on a l... |
| tglineintmo 26422 | Two distinct lines interse... |
| tglineineq 26423 | Two distinct lines interse... |
| tglineneq 26424 | Given three non-colinear p... |
| tglineinteq 26425 | Two distinct lines interse... |
| ncolncol 26426 | Deduce non-colinearity fro... |
| coltr 26427 | A transitivity law for col... |
| coltr3 26428 | A transitivity law for col... |
| colline 26429 | Three points are colinear ... |
| tglowdim2l 26430 | Reformulation of the lower... |
| tglowdim2ln 26431 | There is always one point ... |
| mirreu3 26434 | Existential uniqueness of ... |
| mirval 26435 | Value of the point inversi... |
| mirfv 26436 | Value of the point inversi... |
| mircgr 26437 | Property of the image by t... |
| mirbtwn 26438 | Property of the image by t... |
| ismir 26439 | Property of the image by t... |
| mirf 26440 | Point inversion as functio... |
| mircl 26441 | Closure of the point inver... |
| mirmir 26442 | The point inversion functi... |
| mircom 26443 | Variation on ~ mirmir . (... |
| mirreu 26444 | Any point has a unique ant... |
| mireq 26445 | Equality deduction for poi... |
| mirinv 26446 | The only invariant point o... |
| mirne 26447 | Mirror of non-center point... |
| mircinv 26448 | The center point is invari... |
| mirf1o 26449 | The point inversion functi... |
| miriso 26450 | The point inversion functi... |
| mirbtwni 26451 | Point inversion preserves ... |
| mirbtwnb 26452 | Point inversion preserves ... |
| mircgrs 26453 | Point inversion preserves ... |
| mirmir2 26454 | Point inversion of a point... |
| mirmot 26455 | Point investion is a motio... |
| mirln 26456 | If two points are on the s... |
| mirln2 26457 | If a point and its mirror ... |
| mirconn 26458 | Point inversion of connect... |
| mirhl 26459 | If two points ` X ` and ` ... |
| mirbtwnhl 26460 | If the center of the point... |
| mirhl2 26461 | Deduce half-line relation ... |
| mircgrextend 26462 | Link congruence over a pai... |
| mirtrcgr 26463 | Point inversion of one poi... |
| mirauto 26464 | Point inversion preserves ... |
| miduniq 26465 | Uniqueness of the middle p... |
| miduniq1 26466 | Uniqueness of the middle p... |
| miduniq2 26467 | If two point inversions co... |
| colmid 26468 | Colinearity and equidistan... |
| symquadlem 26469 | Lemma of the symetrial qua... |
| krippenlem 26470 | Lemma for ~ krippen . We ... |
| krippen 26471 | Krippenlemma (German for c... |
| midexlem 26472 | Lemma for the existence of... |
| israg 26477 | Property for 3 points A, B... |
| ragcom 26478 | Commutative rule for right... |
| ragcol 26479 | The right angle property i... |
| ragmir 26480 | Right angle property is pr... |
| mirrag 26481 | Right angle is conserved b... |
| ragtrivb 26482 | Trivial right angle. Theo... |
| ragflat2 26483 | Deduce equality from two r... |
| ragflat 26484 | Deduce equality from two r... |
| ragtriva 26485 | Trivial right angle. Theo... |
| ragflat3 26486 | Right angle and colinearit... |
| ragcgr 26487 | Right angle and colinearit... |
| motrag 26488 | Right angles are preserved... |
| ragncol 26489 | Right angle implies non-co... |
| perpln1 26490 | Derive a line from perpend... |
| perpln2 26491 | Derive a line from perpend... |
| isperp 26492 | Property for 2 lines A, B ... |
| perpcom 26493 | The "perpendicular" relati... |
| perpneq 26494 | Two perpendicular lines ar... |
| isperp2 26495 | Property for 2 lines A, B,... |
| isperp2d 26496 | One direction of ~ isperp2... |
| ragperp 26497 | Deduce that two lines are ... |
| footexALT 26498 | Alternative version of ~ f... |
| footexlem1 26499 | Lemma for ~ footex (Contri... |
| footexlem2 26500 | Lemma for ~ footex (Contri... |
| footex 26501 | From a point ` C ` outside... |
| foot 26502 | From a point ` C ` outside... |
| footne 26503 | Uniqueness of the foot poi... |
| footeq 26504 | Uniqueness of the foot poi... |
| hlperpnel 26505 | A point on a half-line whi... |
| perprag 26506 | Deduce a right angle from ... |
| perpdragALT 26507 | Deduce a right angle from ... |
| perpdrag 26508 | Deduce a right angle from ... |
| colperp 26509 | Deduce a perpendicularity ... |
| colperpexlem1 26510 | Lemma for ~ colperp . Fir... |
| colperpexlem2 26511 | Lemma for ~ colperpex . S... |
| colperpexlem3 26512 | Lemma for ~ colperpex . C... |
| colperpex 26513 | In dimension 2 and above, ... |
| mideulem2 26514 | Lemma for ~ opphllem , whi... |
| opphllem 26515 | Lemma 8.24 of [Schwabhause... |
| mideulem 26516 | Lemma for ~ mideu . We ca... |
| midex 26517 | Existence of the midpoint,... |
| mideu 26518 | Existence and uniqueness o... |
| islnopp 26519 | The property for two point... |
| islnoppd 26520 | Deduce that ` A ` and ` B ... |
| oppne1 26521 | Points lying on opposite s... |
| oppne2 26522 | Points lying on opposite s... |
| oppne3 26523 | Points lying on opposite s... |
| oppcom 26524 | Commutativity rule for "op... |
| opptgdim2 26525 | If two points opposite to ... |
| oppnid 26526 | The "opposite to a line" r... |
| opphllem1 26527 | Lemma for ~ opphl . (Cont... |
| opphllem2 26528 | Lemma for ~ opphl . Lemma... |
| opphllem3 26529 | Lemma for ~ opphl : We as... |
| opphllem4 26530 | Lemma for ~ opphl . (Cont... |
| opphllem5 26531 | Second part of Lemma 9.4 o... |
| opphllem6 26532 | First part of Lemma 9.4 of... |
| oppperpex 26533 | Restating ~ colperpex usin... |
| opphl 26534 | If two points ` A ` and ` ... |
| outpasch 26535 | Axiom of Pasch, outer form... |
| hlpasch 26536 | An application of the axio... |
| ishpg 26539 | Value of the half-plane re... |
| hpgbr 26540 | Half-planes : property for... |
| hpgne1 26541 | Points on the open half pl... |
| hpgne2 26542 | Points on the open half pl... |
| lnopp2hpgb 26543 | Theorem 9.8 of [Schwabhaus... |
| lnoppnhpg 26544 | If two points lie on the o... |
| hpgerlem 26545 | Lemma for the proof that t... |
| hpgid 26546 | The half-plane relation is... |
| hpgcom 26547 | The half-plane relation co... |
| hpgtr 26548 | The half-plane relation is... |
| colopp 26549 | Opposite sides of a line f... |
| colhp 26550 | Half-plane relation for co... |
| hphl 26551 | If two points are on the s... |
| midf 26556 | Midpoint as a function. (... |
| midcl 26557 | Closure of the midpoint. ... |
| ismidb 26558 | Property of the midpoint. ... |
| midbtwn 26559 | Betweenness of midpoint. ... |
| midcgr 26560 | Congruence of midpoint. (... |
| midid 26561 | Midpoint of a null segment... |
| midcom 26562 | Commutativity rule for the... |
| mirmid 26563 | Point inversion preserves ... |
| lmieu 26564 | Uniqueness of the line mir... |
| lmif 26565 | Line mirror as a function.... |
| lmicl 26566 | Closure of the line mirror... |
| islmib 26567 | Property of the line mirro... |
| lmicom 26568 | The line mirroring functio... |
| lmilmi 26569 | Line mirroring is an invol... |
| lmireu 26570 | Any point has a unique ant... |
| lmieq 26571 | Equality deduction for lin... |
| lmiinv 26572 | The invariants of the line... |
| lmicinv 26573 | The mirroring line is an i... |
| lmimid 26574 | If we have a right angle, ... |
| lmif1o 26575 | The line mirroring functio... |
| lmiisolem 26576 | Lemma for ~ lmiiso . (Con... |
| lmiiso 26577 | The line mirroring functio... |
| lmimot 26578 | Line mirroring is a motion... |
| hypcgrlem1 26579 | Lemma for ~ hypcgr , case ... |
| hypcgrlem2 26580 | Lemma for ~ hypcgr , case ... |
| hypcgr 26581 | If the catheti of two righ... |
| lmiopp 26582 | Line mirroring produces po... |
| lnperpex 26583 | Existence of a perpendicul... |
| trgcopy 26584 | Triangle construction: a c... |
| trgcopyeulem 26585 | Lemma for ~ trgcopyeu . (... |
| trgcopyeu 26586 | Triangle construction: a c... |
| iscgra 26589 | Property for two angles AB... |
| iscgra1 26590 | A special version of ~ isc... |
| iscgrad 26591 | Sufficient conditions for ... |
| cgrane1 26592 | Angles imply inequality. ... |
| cgrane2 26593 | Angles imply inequality. ... |
| cgrane3 26594 | Angles imply inequality. ... |
| cgrane4 26595 | Angles imply inequality. ... |
| cgrahl1 26596 | Angle congruence is indepe... |
| cgrahl2 26597 | Angle congruence is indepe... |
| cgracgr 26598 | First direction of proposi... |
| cgraid 26599 | Angle congruence is reflex... |
| cgraswap 26600 | Swap rays in a congruence ... |
| cgrcgra 26601 | Triangle congruence implie... |
| cgracom 26602 | Angle congruence commutes.... |
| cgratr 26603 | Angle congruence is transi... |
| flatcgra 26604 | Flat angles are congruent.... |
| cgraswaplr 26605 | Swap both side of angle co... |
| cgrabtwn 26606 | Angle congruence preserves... |
| cgrahl 26607 | Angle congruence preserves... |
| cgracol 26608 | Angle congruence preserves... |
| cgrancol 26609 | Angle congruence preserves... |
| dfcgra2 26610 | This is the full statement... |
| sacgr 26611 | Supplementary angles of co... |
| oacgr 26612 | Vertical angle theorem. V... |
| acopy 26613 | Angle construction. Theor... |
| acopyeu 26614 | Angle construction. Theor... |
| isinag 26618 | Property for point ` X ` t... |
| isinagd 26619 | Sufficient conditions for ... |
| inagflat 26620 | Any point lies in a flat a... |
| inagswap 26621 | Swap the order of the half... |
| inagne1 26622 | Deduce inequality from the... |
| inagne2 26623 | Deduce inequality from the... |
| inagne3 26624 | Deduce inequality from the... |
| inaghl 26625 | The "point lie in angle" r... |
| isleag 26627 | Geometrical "less than" pr... |
| isleagd 26628 | Sufficient condition for "... |
| leagne1 26629 | Deduce inequality from the... |
| leagne2 26630 | Deduce inequality from the... |
| leagne3 26631 | Deduce inequality from the... |
| leagne4 26632 | Deduce inequality from the... |
| cgrg3col4 26633 | Lemma 11.28 of [Schwabhaus... |
| tgsas1 26634 | First congruence theorem: ... |
| tgsas 26635 | First congruence theorem: ... |
| tgsas2 26636 | First congruence theorem: ... |
| tgsas3 26637 | First congruence theorem: ... |
| tgasa1 26638 | Second congruence theorem:... |
| tgasa 26639 | Second congruence theorem:... |
| tgsss1 26640 | Third congruence theorem: ... |
| tgsss2 26641 | Third congruence theorem: ... |
| tgsss3 26642 | Third congruence theorem: ... |
| dfcgrg2 26643 | Congruence for two triangl... |
| isoas 26644 | Congruence theorem for iso... |
| iseqlg 26647 | Property of a triangle bei... |
| iseqlgd 26648 | Condition for a triangle t... |
| f1otrgds 26649 | Convenient lemma for ~ f1o... |
| f1otrgitv 26650 | Convenient lemma for ~ f1o... |
| f1otrg 26651 | A bijection between bases ... |
| f1otrge 26652 | A bijection between bases ... |
| ttgval 26655 | Define a function to augme... |
| ttglem 26656 | Lemma for ~ ttgbas and ~ t... |
| ttgbas 26657 | The base set of a subcompl... |
| ttgplusg 26658 | The addition operation of ... |
| ttgsub 26659 | The subtraction operation ... |
| ttgvsca 26660 | The scalar product of a su... |
| ttgds 26661 | The metric of a subcomplex... |
| ttgitvval 26662 | Betweenness for a subcompl... |
| ttgelitv 26663 | Betweenness for a subcompl... |
| ttgbtwnid 26664 | Any subcomplex module equi... |
| ttgcontlem1 26665 | Lemma for % ttgcont . (Co... |
| xmstrkgc 26666 | Any metric space fulfills ... |
| cchhllem 26667 | Lemma for chlbas and chlvs... |
| elee 26674 | Membership in a Euclidean ... |
| mptelee 26675 | A condition for a mapping ... |
| eleenn 26676 | If ` A ` is in ` ( EE `` N... |
| eleei 26677 | The forward direction of ~... |
| eedimeq 26678 | A point belongs to at most... |
| brbtwn 26679 | The binary relation form o... |
| brcgr 26680 | The binary relation form o... |
| fveere 26681 | The function value of a po... |
| fveecn 26682 | The function value of a po... |
| eqeefv 26683 | Two points are equal iff t... |
| eqeelen 26684 | Two points are equal iff t... |
| brbtwn2 26685 | Alternate characterization... |
| colinearalglem1 26686 | Lemma for ~ colinearalg . ... |
| colinearalglem2 26687 | Lemma for ~ colinearalg . ... |
| colinearalglem3 26688 | Lemma for ~ colinearalg . ... |
| colinearalglem4 26689 | Lemma for ~ colinearalg . ... |
| colinearalg 26690 | An algebraic characterizat... |
| eleesub 26691 | Membership of a subtractio... |
| eleesubd 26692 | Membership of a subtractio... |
| axdimuniq 26693 | The unique dimension axiom... |
| axcgrrflx 26694 | ` A ` is as far from ` B `... |
| axcgrtr 26695 | Congruence is transitive. ... |
| axcgrid 26696 | If there is no distance be... |
| axsegconlem1 26697 | Lemma for ~ axsegcon . Ha... |
| axsegconlem2 26698 | Lemma for ~ axsegcon . Sh... |
| axsegconlem3 26699 | Lemma for ~ axsegcon . Sh... |
| axsegconlem4 26700 | Lemma for ~ axsegcon . Sh... |
| axsegconlem5 26701 | Lemma for ~ axsegcon . Sh... |
| axsegconlem6 26702 | Lemma for ~ axsegcon . Sh... |
| axsegconlem7 26703 | Lemma for ~ axsegcon . Sh... |
| axsegconlem8 26704 | Lemma for ~ axsegcon . Sh... |
| axsegconlem9 26705 | Lemma for ~ axsegcon . Sh... |
| axsegconlem10 26706 | Lemma for ~ axsegcon . Sh... |
| axsegcon 26707 | Any segment ` A B ` can be... |
| ax5seglem1 26708 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem2 26709 | Lemma for ~ ax5seg . Rexp... |
| ax5seglem3a 26710 | Lemma for ~ ax5seg . (Con... |
| ax5seglem3 26711 | Lemma for ~ ax5seg . Comb... |
| ax5seglem4 26712 | Lemma for ~ ax5seg . Give... |
| ax5seglem5 26713 | Lemma for ~ ax5seg . If `... |
| ax5seglem6 26714 | Lemma for ~ ax5seg . Give... |
| ax5seglem7 26715 | Lemma for ~ ax5seg . An a... |
| ax5seglem8 26716 | Lemma for ~ ax5seg . Use ... |
| ax5seglem9 26717 | Lemma for ~ ax5seg . Take... |
| ax5seg 26718 | The five segment axiom. T... |
| axbtwnid 26719 | Points are indivisible. T... |
| axpaschlem 26720 | Lemma for ~ axpasch . Set... |
| axpasch 26721 | The inner Pasch axiom. Ta... |
| axlowdimlem1 26722 | Lemma for ~ axlowdim . Es... |
| axlowdimlem2 26723 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem3 26724 | Lemma for ~ axlowdim . Se... |
| axlowdimlem4 26725 | Lemma for ~ axlowdim . Se... |
| axlowdimlem5 26726 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem6 26727 | Lemma for ~ axlowdim . Sh... |
| axlowdimlem7 26728 | Lemma for ~ axlowdim . Se... |
| axlowdimlem8 26729 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem9 26730 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem10 26731 | Lemma for ~ axlowdim . Se... |
| axlowdimlem11 26732 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem12 26733 | Lemma for ~ axlowdim . Ca... |
| axlowdimlem13 26734 | Lemma for ~ axlowdim . Es... |
| axlowdimlem14 26735 | Lemma for ~ axlowdim . Ta... |
| axlowdimlem15 26736 | Lemma for ~ axlowdim . Se... |
| axlowdimlem16 26737 | Lemma for ~ axlowdim . Se... |
| axlowdimlem17 26738 | Lemma for ~ axlowdim . Es... |
| axlowdim1 26739 | The lower dimension axiom ... |
| axlowdim2 26740 | The lower two-dimensional ... |
| axlowdim 26741 | The general lower dimensio... |
| axeuclidlem 26742 | Lemma for ~ axeuclid . Ha... |
| axeuclid 26743 | Euclid's axiom. Take an a... |
| axcontlem1 26744 | Lemma for ~ axcont . Chan... |
| axcontlem2 26745 | Lemma for ~ axcont . The ... |
| axcontlem3 26746 | Lemma for ~ axcont . Give... |
| axcontlem4 26747 | Lemma for ~ axcont . Give... |
| axcontlem5 26748 | Lemma for ~ axcont . Comp... |
| axcontlem6 26749 | Lemma for ~ axcont . Stat... |
| axcontlem7 26750 | Lemma for ~ axcont . Give... |
| axcontlem8 26751 | Lemma for ~ axcont . A po... |
| axcontlem9 26752 | Lemma for ~ axcont . Give... |
| axcontlem10 26753 | Lemma for ~ axcont . Give... |
| axcontlem11 26754 | Lemma for ~ axcont . Elim... |
| axcontlem12 26755 | Lemma for ~ axcont . Elim... |
| axcont 26756 | The axiom of continuity. ... |
| eengv 26759 | The value of the Euclidean... |
| eengstr 26760 | The Euclidean geometry as ... |
| eengbas 26761 | The Base of the Euclidean ... |
| ebtwntg 26762 | The betweenness relation u... |
| ecgrtg 26763 | The congruence relation us... |
| elntg 26764 | The line definition in the... |
| elntg2 26765 | The line definition in the... |
| eengtrkg 26766 | The geometry structure for... |
| eengtrkge 26767 | The geometry structure for... |
| edgfid 26770 | Utility theorem: index-ind... |
| edgfndxnn 26771 | The index value of the edg... |
| edgfndxid 26772 | The value of the edge func... |
| baseltedgf 26773 | The index value of the ` B... |
| slotsbaseefdif 26774 | The slots ` Base ` and ` .... |
| vtxval 26779 | The set of vertices of a g... |
| iedgval 26780 | The set of indexed edges o... |
| 1vgrex 26781 | A graph with at least one ... |
| opvtxval 26782 | The set of vertices of a g... |
| opvtxfv 26783 | The set of vertices of a g... |
| opvtxov 26784 | The set of vertices of a g... |
| opiedgval 26785 | The set of indexed edges o... |
| opiedgfv 26786 | The set of indexed edges o... |
| opiedgov 26787 | The set of indexed edges o... |
| opvtxfvi 26788 | The set of vertices of a g... |
| opiedgfvi 26789 | The set of indexed edges o... |
| funvtxdmge2val 26790 | The set of vertices of an ... |
| funiedgdmge2val 26791 | The set of indexed edges o... |
| funvtxdm2val 26792 | The set of vertices of an ... |
| funiedgdm2val 26793 | The set of indexed edges o... |
| funvtxval0 26794 | The set of vertices of an ... |
| basvtxval 26795 | The set of vertices of a g... |
| edgfiedgval 26796 | The set of indexed edges o... |
| funvtxval 26797 | The set of vertices of a g... |
| funiedgval 26798 | The set of indexed edges o... |
| structvtxvallem 26799 | Lemma for ~ structvtxval a... |
| structvtxval 26800 | The set of vertices of an ... |
| structiedg0val 26801 | The set of indexed edges o... |
| structgrssvtxlem 26802 | Lemma for ~ structgrssvtx ... |
| structgrssvtx 26803 | The set of vertices of a g... |
| structgrssiedg 26804 | The set of indexed edges o... |
| struct2grstr 26805 | A graph represented as an ... |
| struct2grvtx 26806 | The set of vertices of a g... |
| struct2griedg 26807 | The set of indexed edges o... |
| graop 26808 | Any representation of a gr... |
| grastruct 26809 | Any representation of a gr... |
| gropd 26810 | If any representation of a... |
| grstructd 26811 | If any representation of a... |
| gropeld 26812 | If any representation of a... |
| grstructeld 26813 | If any representation of a... |
| setsvtx 26814 | The vertices of a structur... |
| setsiedg 26815 | The (indexed) edges of a s... |
| snstrvtxval 26816 | The set of vertices of a g... |
| snstriedgval 26817 | The set of indexed edges o... |
| vtxval0 26818 | Degenerated case 1 for ver... |
| iedgval0 26819 | Degenerated case 1 for edg... |
| vtxvalsnop 26820 | Degenerated case 2 for ver... |
| iedgvalsnop 26821 | Degenerated case 2 for edg... |
| vtxval3sn 26822 | Degenerated case 3 for ver... |
| iedgval3sn 26823 | Degenerated case 3 for edg... |
| vtxvalprc 26824 | Degenerated case 4 for ver... |
| iedgvalprc 26825 | Degenerated case 4 for edg... |
| edgval 26828 | The edges of a graph. (Co... |
| iedgedg 26829 | An indexed edge is an edge... |
| edgopval 26830 | The edges of a graph repre... |
| edgov 26831 | The edges of a graph repre... |
| edgstruct 26832 | The edges of a graph repre... |
| edgiedgb 26833 | A set is an edge iff it is... |
| edg0iedg0 26834 | There is no edge in a grap... |
| isuhgr 26839 | The predicate "is an undir... |
| isushgr 26840 | The predicate "is an undir... |
| uhgrf 26841 | The edge function of an un... |
| ushgrf 26842 | The edge function of an un... |
| uhgrss 26843 | An edge is a subset of ver... |
| uhgreq12g 26844 | If two sets have the same ... |
| uhgrfun 26845 | The edge function of an un... |
| uhgrn0 26846 | An edge is a nonempty subs... |
| lpvtx 26847 | The endpoints of a loop (w... |
| ushgruhgr 26848 | An undirected simple hyper... |
| isuhgrop 26849 | The property of being an u... |
| uhgr0e 26850 | The empty graph, with vert... |
| uhgr0vb 26851 | The null graph, with no ve... |
| uhgr0 26852 | The null graph represented... |
| uhgrun 26853 | The union ` U ` of two (un... |
| uhgrunop 26854 | The union of two (undirect... |
| ushgrun 26855 | The union ` U ` of two (un... |
| ushgrunop 26856 | The union of two (undirect... |
| uhgrstrrepe 26857 | Replacing (or adding) the ... |
| incistruhgr 26858 | An _incidence structure_ `... |
| isupgr 26863 | The property of being an u... |
| wrdupgr 26864 | The property of being an u... |
| upgrf 26865 | The edge function of an un... |
| upgrfn 26866 | The edge function of an un... |
| upgrss 26867 | An edge is a subset of ver... |
| upgrn0 26868 | An edge is a nonempty subs... |
| upgrle 26869 | An edge of an undirected p... |
| upgrfi 26870 | An edge is a finite subset... |
| upgrex 26871 | An edge is an unordered pa... |
| upgrbi 26872 | Show that an unordered pai... |
| upgrop 26873 | A pseudograph represented ... |
| isumgr 26874 | The property of being an u... |
| isumgrs 26875 | The simplified property of... |
| wrdumgr 26876 | The property of being an u... |
| umgrf 26877 | The edge function of an un... |
| umgrfn 26878 | The edge function of an un... |
| umgredg2 26879 | An edge of a multigraph ha... |
| umgrbi 26880 | Show that an unordered pai... |
| upgruhgr 26881 | An undirected pseudograph ... |
| umgrupgr 26882 | An undirected multigraph i... |
| umgruhgr 26883 | An undirected multigraph i... |
| upgrle2 26884 | An edge of an undirected p... |
| umgrnloopv 26885 | In a multigraph, there is ... |
| umgredgprv 26886 | In a multigraph, an edge i... |
| umgrnloop 26887 | In a multigraph, there is ... |
| umgrnloop0 26888 | A multigraph has no loops.... |
| umgr0e 26889 | The empty graph, with vert... |
| upgr0e 26890 | The empty graph, with vert... |
| upgr1elem 26891 | Lemma for ~ upgr1e and ~ u... |
| upgr1e 26892 | A pseudograph with one edg... |
| upgr0eop 26893 | The empty graph, with vert... |
| upgr1eop 26894 | A pseudograph with one edg... |
| upgr0eopALT 26895 | Alternate proof of ~ upgr0... |
| upgr1eopALT 26896 | Alternate proof of ~ upgr1... |
| upgrun 26897 | The union ` U ` of two pse... |
| upgrunop 26898 | The union of two pseudogra... |
| umgrun 26899 | The union ` U ` of two mul... |
| umgrunop 26900 | The union of two multigrap... |
| umgrislfupgrlem 26901 | Lemma for ~ umgrislfupgr a... |
| umgrislfupgr 26902 | A multigraph is a loop-fre... |
| lfgredgge2 26903 | An edge of a loop-free gra... |
| lfgrnloop 26904 | A loop-free graph has no l... |
| uhgredgiedgb 26905 | In a hypergraph, a set is ... |
| uhgriedg0edg0 26906 | A hypergraph has no edges ... |
| uhgredgn0 26907 | An edge of a hypergraph is... |
| edguhgr 26908 | An edge of a hypergraph is... |
| uhgredgrnv 26909 | An edge of a hypergraph co... |
| uhgredgss 26910 | The set of edges of a hype... |
| upgredgss 26911 | The set of edges of a pseu... |
| umgredgss 26912 | The set of edges of a mult... |
| edgupgr 26913 | Properties of an edge of a... |
| edgumgr 26914 | Properties of an edge of a... |
| uhgrvtxedgiedgb 26915 | In a hypergraph, a vertex ... |
| upgredg 26916 | For each edge in a pseudog... |
| umgredg 26917 | For each edge in a multigr... |
| upgrpredgv 26918 | An edge of a pseudograph a... |
| umgrpredgv 26919 | An edge of a multigraph al... |
| upgredg2vtx 26920 | For a vertex incident to a... |
| upgredgpr 26921 | If a proper pair (of verti... |
| edglnl 26922 | The edges incident with a ... |
| numedglnl 26923 | The number of edges incide... |
| umgredgne 26924 | An edge of a multigraph al... |
| umgrnloop2 26925 | A multigraph has no loops.... |
| umgredgnlp 26926 | An edge of a multigraph is... |
| isuspgr 26931 | The property of being a si... |
| isusgr 26932 | The property of being a si... |
| uspgrf 26933 | The edge function of a sim... |
| usgrf 26934 | The edge function of a sim... |
| isusgrs 26935 | The property of being a si... |
| usgrfs 26936 | The edge function of a sim... |
| usgrfun 26937 | The edge function of a sim... |
| usgredgss 26938 | The set of edges of a simp... |
| edgusgr 26939 | An edge of a simple graph ... |
| isuspgrop 26940 | The property of being an u... |
| isusgrop 26941 | The property of being an u... |
| usgrop 26942 | A simple graph represented... |
| isausgr 26943 | The property of an unorder... |
| ausgrusgrb 26944 | The equivalence of the def... |
| usgrausgri 26945 | A simple graph represented... |
| ausgrumgri 26946 | If an alternatively define... |
| ausgrusgri 26947 | The equivalence of the def... |
| usgrausgrb 26948 | The equivalence of the def... |
| usgredgop 26949 | An edge of a simple graph ... |
| usgrf1o 26950 | The edge function of a sim... |
| usgrf1 26951 | The edge function of a sim... |
| uspgrf1oedg 26952 | The edge function of a sim... |
| usgrss 26953 | An edge is a subset of ver... |
| uspgrushgr 26954 | A simple pseudograph is an... |
| uspgrupgr 26955 | A simple pseudograph is an... |
| uspgrupgrushgr 26956 | A graph is a simple pseudo... |
| usgruspgr 26957 | A simple graph is a simple... |
| usgrumgr 26958 | A simple graph is an undir... |
| usgrumgruspgr 26959 | A graph is a simple graph ... |
| usgruspgrb 26960 | A class is a simple graph ... |
| usgrupgr 26961 | A simple graph is an undir... |
| usgruhgr 26962 | A simple graph is an undir... |
| usgrislfuspgr 26963 | A simple graph is a loop-f... |
| uspgrun 26964 | The union ` U ` of two sim... |
| uspgrunop 26965 | The union of two simple ps... |
| usgrun 26966 | The union ` U ` of two sim... |
| usgrunop 26967 | The union of two simple gr... |
| usgredg2 26968 | The value of the "edge fun... |
| usgredg2ALT 26969 | Alternate proof of ~ usgre... |
| usgredgprv 26970 | In a simple graph, an edge... |
| usgredgprvALT 26971 | Alternate proof of ~ usgre... |
| usgredgppr 26972 | An edge of a simple graph ... |
| usgrpredgv 26973 | An edge of a simple graph ... |
| edgssv2 26974 | An edge of a simple graph ... |
| usgredg 26975 | For each edge in a simple ... |
| usgrnloopv 26976 | In a simple graph, there i... |
| usgrnloopvALT 26977 | Alternate proof of ~ usgrn... |
| usgrnloop 26978 | In a simple graph, there i... |
| usgrnloopALT 26979 | Alternate proof of ~ usgrn... |
| usgrnloop0 26980 | A simple graph has no loop... |
| usgrnloop0ALT 26981 | Alternate proof of ~ usgrn... |
| usgredgne 26982 | An edge of a simple graph ... |
| usgrf1oedg 26983 | The edge function of a sim... |
| uhgr2edg 26984 | If a vertex is adjacent to... |
| umgr2edg 26985 | If a vertex is adjacent to... |
| usgr2edg 26986 | If a vertex is adjacent to... |
| umgr2edg1 26987 | If a vertex is adjacent to... |
| usgr2edg1 26988 | If a vertex is adjacent to... |
| umgrvad2edg 26989 | If a vertex is adjacent to... |
| umgr2edgneu 26990 | If a vertex is adjacent to... |
| usgrsizedg 26991 | In a simple graph, the siz... |
| usgredg3 26992 | The value of the "edge fun... |
| usgredg4 26993 | For a vertex incident to a... |
| usgredgreu 26994 | For a vertex incident to a... |
| usgredg2vtx 26995 | For a vertex incident to a... |
| uspgredg2vtxeu 26996 | For a vertex incident to a... |
| usgredg2vtxeu 26997 | For a vertex incident to a... |
| usgredg2vtxeuALT 26998 | Alternate proof of ~ usgre... |
| uspgredg2vlem 26999 | Lemma for ~ uspgredg2v . ... |
| uspgredg2v 27000 | In a simple pseudograph, t... |
| usgredg2vlem1 27001 | Lemma 1 for ~ usgredg2v . ... |
| usgredg2vlem2 27002 | Lemma 2 for ~ usgredg2v . ... |
| usgredg2v 27003 | In a simple graph, the map... |
| usgriedgleord 27004 | Alternate version of ~ usg... |
| ushgredgedg 27005 | In a simple hypergraph the... |
| usgredgedg 27006 | In a simple graph there is... |
| ushgredgedgloop 27007 | In a simple hypergraph the... |
| uspgredgleord 27008 | In a simple pseudograph th... |
| usgredgleord 27009 | In a simple graph the numb... |
| usgredgleordALT 27010 | Alternate proof for ~ usgr... |
| usgrstrrepe 27011 | Replacing (or adding) the ... |
| usgr0e 27012 | The empty graph, with vert... |
| usgr0vb 27013 | The null graph, with no ve... |
| uhgr0v0e 27014 | The null graph, with no ve... |
| uhgr0vsize0 27015 | The size of a hypergraph w... |
| uhgr0edgfi 27016 | A graph of order 0 (i.e. w... |
| usgr0v 27017 | The null graph, with no ve... |
| uhgr0vusgr 27018 | The null graph, with no ve... |
| usgr0 27019 | The null graph represented... |
| uspgr1e 27020 | A simple pseudograph with ... |
| usgr1e 27021 | A simple graph with one ed... |
| usgr0eop 27022 | The empty graph, with vert... |
| uspgr1eop 27023 | A simple pseudograph with ... |
| uspgr1ewop 27024 | A simple pseudograph with ... |
| uspgr1v1eop 27025 | A simple pseudograph with ... |
| usgr1eop 27026 | A simple graph with (at le... |
| uspgr2v1e2w 27027 | A simple pseudograph with ... |
| usgr2v1e2w 27028 | A simple graph with two ve... |
| edg0usgr 27029 | A class without edges is a... |
| lfuhgr1v0e 27030 | A loop-free hypergraph wit... |
| usgr1vr 27031 | A simple graph with one ve... |
| usgr1v 27032 | A class with one (or no) v... |
| usgr1v0edg 27033 | A class with one (or no) v... |
| usgrexmpldifpr 27034 | Lemma for ~ usgrexmpledg :... |
| usgrexmplef 27035 | Lemma for ~ usgrexmpl . (... |
| usgrexmpllem 27036 | Lemma for ~ usgrexmpl . (... |
| usgrexmplvtx 27037 | The vertices ` 0 , 1 , 2 ,... |
| usgrexmpledg 27038 | The edges ` { 0 , 1 } , { ... |
| usgrexmpl 27039 | ` G ` is a simple graph of... |
| griedg0prc 27040 | The class of empty graphs ... |
| griedg0ssusgr 27041 | The class of all simple gr... |
| usgrprc 27042 | The class of simple graphs... |
| relsubgr 27045 | The class of the subgraph ... |
| subgrv 27046 | If a class is a subgraph o... |
| issubgr 27047 | The property of a set to b... |
| issubgr2 27048 | The property of a set to b... |
| subgrprop 27049 | The properties of a subgra... |
| subgrprop2 27050 | The properties of a subgra... |
| uhgrissubgr 27051 | The property of a hypergra... |
| subgrprop3 27052 | The properties of a subgra... |
| egrsubgr 27053 | An empty graph consisting ... |
| 0grsubgr 27054 | The null graph (represente... |
| 0uhgrsubgr 27055 | The null graph (as hypergr... |
| uhgrsubgrself 27056 | A hypergraph is a subgraph... |
| subgrfun 27057 | The edge function of a sub... |
| subgruhgrfun 27058 | The edge function of a sub... |
| subgreldmiedg 27059 | An element of the domain o... |
| subgruhgredgd 27060 | An edge of a subgraph of a... |
| subumgredg2 27061 | An edge of a subgraph of a... |
| subuhgr 27062 | A subgraph of a hypergraph... |
| subupgr 27063 | A subgraph of a pseudograp... |
| subumgr 27064 | A subgraph of a multigraph... |
| subusgr 27065 | A subgraph of a simple gra... |
| uhgrspansubgrlem 27066 | Lemma for ~ uhgrspansubgr ... |
| uhgrspansubgr 27067 | A spanning subgraph ` S ` ... |
| uhgrspan 27068 | A spanning subgraph ` S ` ... |
| upgrspan 27069 | A spanning subgraph ` S ` ... |
| umgrspan 27070 | A spanning subgraph ` S ` ... |
| usgrspan 27071 | A spanning subgraph ` S ` ... |
| uhgrspanop 27072 | A spanning subgraph of a h... |
| upgrspanop 27073 | A spanning subgraph of a p... |
| umgrspanop 27074 | A spanning subgraph of a m... |
| usgrspanop 27075 | A spanning subgraph of a s... |
| uhgrspan1lem1 27076 | Lemma 1 for ~ uhgrspan1 . ... |
| uhgrspan1lem2 27077 | Lemma 2 for ~ uhgrspan1 . ... |
| uhgrspan1lem3 27078 | Lemma 3 for ~ uhgrspan1 . ... |
| uhgrspan1 27079 | The induced subgraph ` S `... |
| upgrreslem 27080 | Lemma for ~ upgrres . (Co... |
| umgrreslem 27081 | Lemma for ~ umgrres and ~ ... |
| upgrres 27082 | A subgraph obtained by rem... |
| umgrres 27083 | A subgraph obtained by rem... |
| usgrres 27084 | A subgraph obtained by rem... |
| upgrres1lem1 27085 | Lemma 1 for ~ upgrres1 . ... |
| umgrres1lem 27086 | Lemma for ~ umgrres1 . (C... |
| upgrres1lem2 27087 | Lemma 2 for ~ upgrres1 . ... |
| upgrres1lem3 27088 | Lemma 3 for ~ upgrres1 . ... |
| upgrres1 27089 | A pseudograph obtained by ... |
| umgrres1 27090 | A multigraph obtained by r... |
| usgrres1 27091 | Restricting a simple graph... |
| isfusgr 27094 | The property of being a fi... |
| fusgrvtxfi 27095 | A finite simple graph has ... |
| isfusgrf1 27096 | The property of being a fi... |
| isfusgrcl 27097 | The property of being a fi... |
| fusgrusgr 27098 | A finite simple graph is a... |
| opfusgr 27099 | A finite simple graph repr... |
| usgredgffibi 27100 | The number of edges in a s... |
| fusgredgfi 27101 | In a finite simple graph t... |
| usgr1v0e 27102 | The size of a (finite) sim... |
| usgrfilem 27103 | In a finite simple graph, ... |
| fusgrfisbase 27104 | Induction base for ~ fusgr... |
| fusgrfisstep 27105 | Induction step in ~ fusgrf... |
| fusgrfis 27106 | A finite simple graph is o... |
| fusgrfupgrfs 27107 | A finite simple graph is a... |
| nbgrprc0 27110 | The set of neighbors is em... |
| nbgrcl 27111 | If a class ` X ` has at le... |
| nbgrval 27112 | The set of neighbors of a ... |
| dfnbgr2 27113 | Alternate definition of th... |
| dfnbgr3 27114 | Alternate definition of th... |
| nbgrnvtx0 27115 | If a class ` X ` is not a ... |
| nbgrel 27116 | Characterization of a neig... |
| nbgrisvtx 27117 | Every neighbor ` N ` of a ... |
| nbgrssvtx 27118 | The neighbors of a vertex ... |
| nbuhgr 27119 | The set of neighbors of a ... |
| nbupgr 27120 | The set of neighbors of a ... |
| nbupgrel 27121 | A neighbor of a vertex in ... |
| nbumgrvtx 27122 | The set of neighbors of a ... |
| nbumgr 27123 | The set of neighbors of an... |
| nbusgrvtx 27124 | The set of neighbors of a ... |
| nbusgr 27125 | The set of neighbors of an... |
| nbgr2vtx1edg 27126 | If a graph has two vertice... |
| nbuhgr2vtx1edgblem 27127 | Lemma for ~ nbuhgr2vtx1edg... |
| nbuhgr2vtx1edgb 27128 | If a hypergraph has two ve... |
| nbusgreledg 27129 | A class/vertex is a neighb... |
| uhgrnbgr0nb 27130 | A vertex which is not endp... |
| nbgr0vtxlem 27131 | Lemma for ~ nbgr0vtx and ~... |
| nbgr0vtx 27132 | In a null graph (with no v... |
| nbgr0edg 27133 | In an empty graph (with no... |
| nbgr1vtx 27134 | In a graph with one vertex... |
| nbgrnself 27135 | A vertex in a graph is not... |
| nbgrnself2 27136 | A class ` X ` is not a nei... |
| nbgrssovtx 27137 | The neighbors of a vertex ... |
| nbgrssvwo2 27138 | The neighbors of a vertex ... |
| nbgrsym 27139 | In a graph, the neighborho... |
| nbupgrres 27140 | The neighborhood of a vert... |
| usgrnbcnvfv 27141 | Applying the edge function... |
| nbusgredgeu 27142 | For each neighbor of a ver... |
| edgnbusgreu 27143 | For each edge incident to ... |
| nbusgredgeu0 27144 | For each neighbor of a ver... |
| nbusgrf1o0 27145 | The mapping of neighbors o... |
| nbusgrf1o1 27146 | The set of neighbors of a ... |
| nbusgrf1o 27147 | The set of neighbors of a ... |
| nbedgusgr 27148 | The number of neighbors of... |
| edgusgrnbfin 27149 | The number of neighbors of... |
| nbusgrfi 27150 | The class of neighbors of ... |
| nbfiusgrfi 27151 | The class of neighbors of ... |
| hashnbusgrnn0 27152 | The number of neighbors of... |
| nbfusgrlevtxm1 27153 | The number of neighbors of... |
| nbfusgrlevtxm2 27154 | If there is a vertex which... |
| nbusgrvtxm1 27155 | If the number of neighbors... |
| nb3grprlem1 27156 | Lemma 1 for ~ nb3grpr . (... |
| nb3grprlem2 27157 | Lemma 2 for ~ nb3grpr . (... |
| nb3grpr 27158 | The neighbors of a vertex ... |
| nb3grpr2 27159 | The neighbors of a vertex ... |
| nb3gr2nb 27160 | If the neighbors of two ve... |
| uvtxval 27163 | The set of all universal v... |
| uvtxel 27164 | A universal vertex, i.e. a... |
| uvtxisvtx 27165 | A universal vertex is a ve... |
| uvtxssvtx 27166 | The set of the universal v... |
| vtxnbuvtx 27167 | A universal vertex has all... |
| uvtxnbgrss 27168 | A universal vertex has all... |
| uvtxnbgrvtx 27169 | A universal vertex is neig... |
| uvtx0 27170 | There is no universal vert... |
| isuvtx 27171 | The set of all universal v... |
| uvtxel1 27172 | Characterization of a univ... |
| uvtx01vtx 27173 | If a graph/class has no ed... |
| uvtx2vtx1edg 27174 | If a graph has two vertice... |
| uvtx2vtx1edgb 27175 | If a hypergraph has two ve... |
| uvtxnbgr 27176 | A universal vertex has all... |
| uvtxnbgrb 27177 | A vertex is universal iff ... |
| uvtxusgr 27178 | The set of all universal v... |
| uvtxusgrel 27179 | A universal vertex, i.e. a... |
| uvtxnm1nbgr 27180 | A universal vertex has ` n... |
| nbusgrvtxm1uvtx 27181 | If the number of neighbors... |
| uvtxnbvtxm1 27182 | A universal vertex has ` n... |
| nbupgruvtxres 27183 | The neighborhood of a univ... |
| uvtxupgrres 27184 | A universal vertex is univ... |
| cplgruvtxb 27189 | A graph ` G ` is complete ... |
| prcliscplgr 27190 | A proper class (representi... |
| iscplgr 27191 | The property of being a co... |
| iscplgrnb 27192 | A graph is complete iff al... |
| iscplgredg 27193 | A graph ` G ` is complete ... |
| iscusgr 27194 | The property of being a co... |
| cusgrusgr 27195 | A complete simple graph is... |
| cusgrcplgr 27196 | A complete simple graph is... |
| iscusgrvtx 27197 | A simple graph is complete... |
| cusgruvtxb 27198 | A simple graph is complete... |
| iscusgredg 27199 | A simple graph is complete... |
| cusgredg 27200 | In a complete simple graph... |
| cplgr0 27201 | The null graph (with no ve... |
| cusgr0 27202 | The null graph (with no ve... |
| cplgr0v 27203 | A null graph (with no vert... |
| cusgr0v 27204 | A graph with no vertices a... |
| cplgr1vlem 27205 | Lemma for ~ cplgr1v and ~ ... |
| cplgr1v 27206 | A graph with one vertex is... |
| cusgr1v 27207 | A graph with one vertex an... |
| cplgr2v 27208 | An undirected hypergraph w... |
| cplgr2vpr 27209 | An undirected hypergraph w... |
| nbcplgr 27210 | In a complete graph, each ... |
| cplgr3v 27211 | A pseudograph with three (... |
| cusgr3vnbpr 27212 | The neighbors of a vertex ... |
| cplgrop 27213 | A complete graph represent... |
| cusgrop 27214 | A complete simple graph re... |
| cusgrexilem1 27215 | Lemma 1 for ~ cusgrexi . ... |
| usgrexilem 27216 | Lemma for ~ usgrexi . (Co... |
| usgrexi 27217 | An arbitrary set regarded ... |
| cusgrexilem2 27218 | Lemma 2 for ~ cusgrexi . ... |
| cusgrexi 27219 | An arbitrary set ` V ` reg... |
| cusgrexg 27220 | For each set there is a se... |
| structtousgr 27221 | Any (extensible) structure... |
| structtocusgr 27222 | Any (extensible) structure... |
| cffldtocusgr 27223 | The field of complex numbe... |
| cusgrres 27224 | Restricting a complete sim... |
| cusgrsizeindb0 27225 | Base case of the induction... |
| cusgrsizeindb1 27226 | Base case of the induction... |
| cusgrsizeindslem 27227 | Lemma for ~ cusgrsizeinds ... |
| cusgrsizeinds 27228 | Part 1 of induction step i... |
| cusgrsize2inds 27229 | Induction step in ~ cusgrs... |
| cusgrsize 27230 | The size of a finite compl... |
| cusgrfilem1 27231 | Lemma 1 for ~ cusgrfi . (... |
| cusgrfilem2 27232 | Lemma 2 for ~ cusgrfi . (... |
| cusgrfilem3 27233 | Lemma 3 for ~ cusgrfi . (... |
| cusgrfi 27234 | If the size of a complete ... |
| usgredgsscusgredg 27235 | A simple graph is a subgra... |
| usgrsscusgr 27236 | A simple graph is a subgra... |
| sizusglecusglem1 27237 | Lemma 1 for ~ sizusglecusg... |
| sizusglecusglem2 27238 | Lemma 2 for ~ sizusglecusg... |
| sizusglecusg 27239 | The size of a simple graph... |
| fusgrmaxsize 27240 | The maximum size of a fini... |
| vtxdgfval 27243 | The value of the vertex de... |
| vtxdgval 27244 | The degree of a vertex. (... |
| vtxdgfival 27245 | The degree of a vertex for... |
| vtxdgop 27246 | The vertex degree expresse... |
| vtxdgf 27247 | The vertex degree function... |
| vtxdgelxnn0 27248 | The degree of a vertex is ... |
| vtxdg0v 27249 | The degree of a vertex in ... |
| vtxdg0e 27250 | The degree of a vertex in ... |
| vtxdgfisnn0 27251 | The degree of a vertex in ... |
| vtxdgfisf 27252 | The vertex degree function... |
| vtxdeqd 27253 | Equality theorem for the v... |
| vtxduhgr0e 27254 | The degree of a vertex in ... |
| vtxdlfuhgr1v 27255 | The degree of the vertex i... |
| vdumgr0 27256 | A vertex in a multigraph h... |
| vtxdun 27257 | The degree of a vertex in ... |
| vtxdfiun 27258 | The degree of a vertex in ... |
| vtxduhgrun 27259 | The degree of a vertex in ... |
| vtxduhgrfiun 27260 | The degree of a vertex in ... |
| vtxdlfgrval 27261 | The value of the vertex de... |
| vtxdumgrval 27262 | The value of the vertex de... |
| vtxdusgrval 27263 | The value of the vertex de... |
| vtxd0nedgb 27264 | A vertex has degree 0 iff ... |
| vtxdushgrfvedglem 27265 | Lemma for ~ vtxdushgrfvedg... |
| vtxdushgrfvedg 27266 | The value of the vertex de... |
| vtxdusgrfvedg 27267 | The value of the vertex de... |
| vtxduhgr0nedg 27268 | If a vertex in a hypergrap... |
| vtxdumgr0nedg 27269 | If a vertex in a multigrap... |
| vtxduhgr0edgnel 27270 | A vertex in a hypergraph h... |
| vtxdusgr0edgnel 27271 | A vertex in a simple graph... |
| vtxdusgr0edgnelALT 27272 | Alternate proof of ~ vtxdu... |
| vtxdgfusgrf 27273 | The vertex degree function... |
| vtxdgfusgr 27274 | In a finite simple graph, ... |
| fusgrn0degnn0 27275 | In a nonempty, finite grap... |
| 1loopgruspgr 27276 | A graph with one edge whic... |
| 1loopgredg 27277 | The set of edges in a grap... |
| 1loopgrnb0 27278 | In a graph (simple pseudog... |
| 1loopgrvd2 27279 | The vertex degree of a one... |
| 1loopgrvd0 27280 | The vertex degree of a one... |
| 1hevtxdg0 27281 | The vertex degree of verte... |
| 1hevtxdg1 27282 | The vertex degree of verte... |
| 1hegrvtxdg1 27283 | The vertex degree of a gra... |
| 1hegrvtxdg1r 27284 | The vertex degree of a gra... |
| 1egrvtxdg1 27285 | The vertex degree of a one... |
| 1egrvtxdg1r 27286 | The vertex degree of a one... |
| 1egrvtxdg0 27287 | The vertex degree of a one... |
| p1evtxdeqlem 27288 | Lemma for ~ p1evtxdeq and ... |
| p1evtxdeq 27289 | If an edge ` E ` which doe... |
| p1evtxdp1 27290 | If an edge ` E ` (not bein... |
| uspgrloopvtx 27291 | The set of vertices in a g... |
| uspgrloopvtxel 27292 | A vertex in a graph (simpl... |
| uspgrloopiedg 27293 | The set of edges in a grap... |
| uspgrloopedg 27294 | The set of edges in a grap... |
| uspgrloopnb0 27295 | In a graph (simple pseudog... |
| uspgrloopvd2 27296 | The vertex degree of a one... |
| umgr2v2evtx 27297 | The set of vertices in a m... |
| umgr2v2evtxel 27298 | A vertex in a multigraph w... |
| umgr2v2eiedg 27299 | The edge function in a mul... |
| umgr2v2eedg 27300 | The set of edges in a mult... |
| umgr2v2e 27301 | A multigraph with two edge... |
| umgr2v2enb1 27302 | In a multigraph with two e... |
| umgr2v2evd2 27303 | In a multigraph with two e... |
| hashnbusgrvd 27304 | In a simple graph, the num... |
| usgruvtxvdb 27305 | In a finite simple graph w... |
| vdiscusgrb 27306 | A finite simple graph with... |
| vdiscusgr 27307 | In a finite complete simpl... |
| vtxdusgradjvtx 27308 | The degree of a vertex in ... |
| usgrvd0nedg 27309 | If a vertex in a simple gr... |
| uhgrvd00 27310 | If every vertex in a hyper... |
| usgrvd00 27311 | If every vertex in a simpl... |
| vdegp1ai 27312 | The induction step for a v... |
| vdegp1bi 27313 | The induction step for a v... |
| vdegp1ci 27314 | The induction step for a v... |
| vtxdginducedm1lem1 27315 | Lemma 1 for ~ vtxdginduced... |
| vtxdginducedm1lem2 27316 | Lemma 2 for ~ vtxdginduced... |
| vtxdginducedm1lem3 27317 | Lemma 3 for ~ vtxdginduced... |
| vtxdginducedm1lem4 27318 | Lemma 4 for ~ vtxdginduced... |
| vtxdginducedm1 27319 | The degree of a vertex ` v... |
| vtxdginducedm1fi 27320 | The degree of a vertex ` v... |
| finsumvtxdg2ssteplem1 27321 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem2 27322 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem3 27323 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2ssteplem4 27324 | Lemma for ~ finsumvtxdg2ss... |
| finsumvtxdg2sstep 27325 | Induction step of ~ finsum... |
| finsumvtxdg2size 27326 | The sum of the degrees of ... |
| fusgr1th 27327 | The sum of the degrees of ... |
| finsumvtxdgeven 27328 | The sum of the degrees of ... |
| vtxdgoddnumeven 27329 | The number of vertices of ... |
| fusgrvtxdgonume 27330 | The number of vertices of ... |
| isrgr 27335 | The property of a class be... |
| rgrprop 27336 | The properties of a k-regu... |
| isrusgr 27337 | The property of being a k-... |
| rusgrprop 27338 | The properties of a k-regu... |
| rusgrrgr 27339 | A k-regular simple graph i... |
| rusgrusgr 27340 | A k-regular simple graph i... |
| finrusgrfusgr 27341 | A finite regular simple gr... |
| isrusgr0 27342 | The property of being a k-... |
| rusgrprop0 27343 | The properties of a k-regu... |
| usgreqdrusgr 27344 | If all vertices in a simpl... |
| fusgrregdegfi 27345 | In a nonempty finite simpl... |
| fusgrn0eqdrusgr 27346 | If all vertices in a nonem... |
| frusgrnn0 27347 | In a nonempty finite k-reg... |
| 0edg0rgr 27348 | A graph is 0-regular if it... |
| uhgr0edg0rgr 27349 | A hypergraph is 0-regular ... |
| uhgr0edg0rgrb 27350 | A hypergraph is 0-regular ... |
| usgr0edg0rusgr 27351 | A simple graph is 0-regula... |
| 0vtxrgr 27352 | A null graph (with no vert... |
| 0vtxrusgr 27353 | A graph with no vertices a... |
| 0uhgrrusgr 27354 | The null graph as hypergra... |
| 0grrusgr 27355 | The null graph represented... |
| 0grrgr 27356 | The null graph represented... |
| cusgrrusgr 27357 | A complete simple graph wi... |
| cusgrm1rusgr 27358 | A finite simple graph with... |
| rusgrpropnb 27359 | The properties of a k-regu... |
| rusgrpropedg 27360 | The properties of a k-regu... |
| rusgrpropadjvtx 27361 | The properties of a k-regu... |
| rusgrnumwrdl2 27362 | In a k-regular simple grap... |
| rusgr1vtxlem 27363 | Lemma for ~ rusgr1vtx . (... |
| rusgr1vtx 27364 | If a k-regular simple grap... |
| rgrusgrprc 27365 | The class of 0-regular sim... |
| rusgrprc 27366 | The class of 0-regular sim... |
| rgrprc 27367 | The class of 0-regular gra... |
| rgrprcx 27368 | The class of 0-regular gra... |
| rgrx0ndm 27369 | 0 is not in the domain of ... |
| rgrx0nd 27370 | The potentially alternativ... |
| ewlksfval 27377 | The set of s-walks of edge... |
| isewlk 27378 | Conditions for a function ... |
| ewlkprop 27379 | Properties of an s-walk of... |
| ewlkinedg 27380 | The intersection (common v... |
| ewlkle 27381 | An s-walk of edges is also... |
| upgrewlkle2 27382 | In a pseudograph, there is... |
| wkslem1 27383 | Lemma 1 for walks to subst... |
| wkslem2 27384 | Lemma 2 for walks to subst... |
| wksfval 27385 | The set of walks (in an un... |
| iswlk 27386 | Properties of a pair of fu... |
| wlkprop 27387 | Properties of a walk. (Co... |
| wlkv 27388 | The classes involved in a ... |
| iswlkg 27389 | Generalization of ~ iswlk ... |
| wlkf 27390 | The mapping enumerating th... |
| wlkcl 27391 | A walk has length ` # ( F ... |
| wlkp 27392 | The mapping enumerating th... |
| wlkpwrd 27393 | The sequence of vertices o... |
| wlklenvp1 27394 | The number of vertices of ... |
| wksv 27395 | The class of walks is a se... |
| wlkn0 27396 | The sequence of vertices o... |
| wlklenvm1 27397 | The number of edges of a w... |
| ifpsnprss 27398 | Lemma for ~ wlkvtxeledg : ... |
| wlkvtxeledg 27399 | Each pair of adjacent vert... |
| wlkvtxiedg 27400 | The vertices of a walk are... |
| relwlk 27401 | The set ` ( Walks `` G ) `... |
| wlkvv 27402 | If there is at least one w... |
| wlkop 27403 | A walk is an ordered pair.... |
| wlkcpr 27404 | A walk as class with two c... |
| wlk2f 27405 | If there is a walk ` W ` t... |
| wlkcomp 27406 | A walk expressed by proper... |
| wlkcompim 27407 | Implications for the prope... |
| wlkelwrd 27408 | The components of a walk a... |
| wlkeq 27409 | Conditions for two walks (... |
| edginwlk 27410 | The value of the edge func... |
| upgredginwlk 27411 | The value of the edge func... |
| iedginwlk 27412 | The value of the edge func... |
| wlkl1loop 27413 | A walk of length 1 from a ... |
| wlk1walk 27414 | A walk is a 1-walk "on the... |
| wlk1ewlk 27415 | A walk is an s-walk "on th... |
| upgriswlk 27416 | Properties of a pair of fu... |
| upgrwlkedg 27417 | The edges of a walk in a p... |
| upgrwlkcompim 27418 | Implications for the prope... |
| wlkvtxedg 27419 | The vertices of a walk are... |
| upgrwlkvtxedg 27420 | The pairs of connected ver... |
| uspgr2wlkeq 27421 | Conditions for two walks w... |
| uspgr2wlkeq2 27422 | Conditions for two walks w... |
| uspgr2wlkeqi 27423 | Conditions for two walks w... |
| umgrwlknloop 27424 | In a multigraph, each walk... |
| wlkRes 27425 | Restrictions of walks (i.e... |
| wlkv0 27426 | If there is a walk in the ... |
| g0wlk0 27427 | There is no walk in a null... |
| 0wlk0 27428 | There is no walk for the e... |
| wlk0prc 27429 | There is no walk in a null... |
| wlklenvclwlk 27430 | The number of vertices in ... |
| wlklenvclwlkOLD 27431 | Obsolete version of ~ wlkl... |
| wlkson 27432 | The set of walks between t... |
| iswlkon 27433 | Properties of a pair of fu... |
| wlkonprop 27434 | Properties of a walk betwe... |
| wlkpvtx 27435 | A walk connects vertices. ... |
| wlkepvtx 27436 | The endpoints of a walk ar... |
| wlkoniswlk 27437 | A walk between two vertice... |
| wlkonwlk 27438 | A walk is a walk between i... |
| wlkonwlk1l 27439 | A walk is a walk from its ... |
| wlksoneq1eq2 27440 | Two walks with identical s... |
| wlkonl1iedg 27441 | If there is a walk between... |
| wlkon2n0 27442 | The length of a walk betwe... |
| 2wlklem 27443 | Lemma for theorems for wal... |
| upgr2wlk 27444 | Properties of a pair of fu... |
| wlkreslem 27445 | Lemma for ~ wlkres . (Con... |
| wlkres 27446 | The restriction ` <. H , Q... |
| redwlklem 27447 | Lemma for ~ redwlk . (Con... |
| redwlk 27448 | A walk ending at the last ... |
| wlkp1lem1 27449 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem2 27450 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem3 27451 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem4 27452 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem5 27453 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem6 27454 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem7 27455 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1lem8 27456 | Lemma for ~ wlkp1 . (Cont... |
| wlkp1 27457 | Append one path segment (e... |
| wlkdlem1 27458 | Lemma 1 for ~ wlkd . (Con... |
| wlkdlem2 27459 | Lemma 2 for ~ wlkd . (Con... |
| wlkdlem3 27460 | Lemma 3 for ~ wlkd . (Con... |
| wlkdlem4 27461 | Lemma 4 for ~ wlkd . (Con... |
| wlkd 27462 | Two words representing a w... |
| lfgrwlkprop 27463 | Two adjacent vertices in a... |
| lfgriswlk 27464 | Conditions for a pair of f... |
| lfgrwlknloop 27465 | In a loop-free graph, each... |
| reltrls 27470 | The set ` ( Trails `` G ) ... |
| trlsfval 27471 | The set of trails (in an u... |
| istrl 27472 | Conditions for a pair of c... |
| trliswlk 27473 | A trail is a walk. (Contr... |
| trlf1 27474 | The enumeration ` F ` of a... |
| trlreslem 27475 | Lemma for ~ trlres . Form... |
| trlres 27476 | The restriction ` <. H , Q... |
| upgrtrls 27477 | The set of trails in a pse... |
| upgristrl 27478 | Properties of a pair of fu... |
| upgrf1istrl 27479 | Properties of a pair of a ... |
| wksonproplem 27480 | Lemma for theorems for pro... |
| trlsonfval 27481 | The set of trails between ... |
| istrlson 27482 | Properties of a pair of fu... |
| trlsonprop 27483 | Properties of a trail betw... |
| trlsonistrl 27484 | A trail between two vertic... |
| trlsonwlkon 27485 | A trail between two vertic... |
| trlontrl 27486 | A trail is a trail between... |
| relpths 27495 | The set ` ( Paths `` G ) `... |
| pthsfval 27496 | The set of paths (in an un... |
| spthsfval 27497 | The set of simple paths (i... |
| ispth 27498 | Conditions for a pair of c... |
| isspth 27499 | Conditions for a pair of c... |
| pthistrl 27500 | A path is a trail (in an u... |
| spthispth 27501 | A simple path is a path (i... |
| pthiswlk 27502 | A path is a walk (in an un... |
| spthiswlk 27503 | A simple path is a walk (i... |
| pthdivtx 27504 | The inner vertices of a pa... |
| pthdadjvtx 27505 | The adjacent vertices of a... |
| 2pthnloop 27506 | A path of length at least ... |
| upgr2pthnlp 27507 | A path of length at least ... |
| spthdifv 27508 | The vertices of a simple p... |
| spthdep 27509 | A simple path (at least of... |
| pthdepisspth 27510 | A path with different star... |
| upgrwlkdvdelem 27511 | Lemma for ~ upgrwlkdvde . ... |
| upgrwlkdvde 27512 | In a pseudograph, all edge... |
| upgrspthswlk 27513 | The set of simple paths in... |
| upgrwlkdvspth 27514 | A walk consisting of diffe... |
| pthsonfval 27515 | The set of paths between t... |
| spthson 27516 | The set of simple paths be... |
| ispthson 27517 | Properties of a pair of fu... |
| isspthson 27518 | Properties of a pair of fu... |
| pthsonprop 27519 | Properties of a path betwe... |
| spthonprop 27520 | Properties of a simple pat... |
| pthonispth 27521 | A path between two vertice... |
| pthontrlon 27522 | A path between two vertice... |
| pthonpth 27523 | A path is a path between i... |
| isspthonpth 27524 | A pair of functions is a s... |
| spthonisspth 27525 | A simple path between to v... |
| spthonpthon 27526 | A simple path between two ... |
| spthonepeq 27527 | The endpoints of a simple ... |
| uhgrwkspthlem1 27528 | Lemma 1 for ~ uhgrwkspth .... |
| uhgrwkspthlem2 27529 | Lemma 2 for ~ uhgrwkspth .... |
| uhgrwkspth 27530 | Any walk of length 1 betwe... |
| usgr2wlkneq 27531 | The vertices and edges are... |
| usgr2wlkspthlem1 27532 | Lemma 1 for ~ usgr2wlkspth... |
| usgr2wlkspthlem2 27533 | Lemma 2 for ~ usgr2wlkspth... |
| usgr2wlkspth 27534 | In a simple graph, any wal... |
| usgr2trlncl 27535 | In a simple graph, any tra... |
| usgr2trlspth 27536 | In a simple graph, any tra... |
| usgr2pthspth 27537 | In a simple graph, any pat... |
| usgr2pthlem 27538 | Lemma for ~ usgr2pth . (C... |
| usgr2pth 27539 | In a simple graph, there i... |
| usgr2pth0 27540 | In a simply graph, there i... |
| pthdlem1 27541 | Lemma 1 for ~ pthd . (Con... |
| pthdlem2lem 27542 | Lemma for ~ pthdlem2 . (C... |
| pthdlem2 27543 | Lemma 2 for ~ pthd . (Con... |
| pthd 27544 | Two words representing a t... |
| clwlks 27547 | The set of closed walks (i... |
| isclwlk 27548 | A pair of functions repres... |
| clwlkiswlk 27549 | A closed walk is a walk (i... |
| clwlkwlk 27550 | Closed walks are walks (in... |
| clwlkswks 27551 | Closed walks are walks (in... |
| isclwlke 27552 | Properties of a pair of fu... |
| isclwlkupgr 27553 | Properties of a pair of fu... |
| clwlkcomp 27554 | A closed walk expressed by... |
| clwlkcompim 27555 | Implications for the prope... |
| upgrclwlkcompim 27556 | Implications for the prope... |
| clwlkcompbp 27557 | Basic properties of the co... |
| clwlkl1loop 27558 | A closed walk of length 1 ... |
| crcts 27563 | The set of circuits (in an... |
| cycls 27564 | The set of cycles (in an u... |
| iscrct 27565 | Sufficient and necessary c... |
| iscycl 27566 | Sufficient and necessary c... |
| crctprop 27567 | The properties of a circui... |
| cyclprop 27568 | The properties of a cycle:... |
| crctisclwlk 27569 | A circuit is a closed walk... |
| crctistrl 27570 | A circuit is a trail. (Co... |
| crctiswlk 27571 | A circuit is a walk. (Con... |
| cyclispth 27572 | A cycle is a path. (Contr... |
| cycliswlk 27573 | A cycle is a walk. (Contr... |
| cycliscrct 27574 | A cycle is a circuit. (Co... |
| cyclnspth 27575 | A (non-trivial) cycle is n... |
| cyclispthon 27576 | A cycle is a path starting... |
| lfgrn1cycl 27577 | In a loop-free graph there... |
| usgr2trlncrct 27578 | In a simple graph, any tra... |
| umgrn1cycl 27579 | In a multigraph graph (wit... |
| uspgrn2crct 27580 | In a simple pseudograph th... |
| usgrn2cycl 27581 | In a simple graph there ar... |
| crctcshwlkn0lem1 27582 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem2 27583 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem3 27584 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem4 27585 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem5 27586 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem6 27587 | Lemma for ~ crctcshwlkn0 .... |
| crctcshwlkn0lem7 27588 | Lemma for ~ crctcshwlkn0 .... |
| crctcshlem1 27589 | Lemma for ~ crctcsh . (Co... |
| crctcshlem2 27590 | Lemma for ~ crctcsh . (Co... |
| crctcshlem3 27591 | Lemma for ~ crctcsh . (Co... |
| crctcshlem4 27592 | Lemma for ~ crctcsh . (Co... |
| crctcshwlkn0 27593 | Cyclically shifting the in... |
| crctcshwlk 27594 | Cyclically shifting the in... |
| crctcshtrl 27595 | Cyclically shifting the in... |
| crctcsh 27596 | Cyclically shifting the in... |
| wwlks 27607 | The set of walks (in an un... |
| iswwlks 27608 | A word over the set of ver... |
| wwlksn 27609 | The set of walks (in an un... |
| iswwlksn 27610 | A word over the set of ver... |
| wwlksnprcl 27611 | Derivation of the length o... |
| iswwlksnx 27612 | Properties of a word to re... |
| wwlkbp 27613 | Basic properties of a walk... |
| wwlknbp 27614 | Basic properties of a walk... |
| wwlknp 27615 | Properties of a set being ... |
| wwlknbp1 27616 | Other basic properties of ... |
| wwlknvtx 27617 | The symbols of a word ` W ... |
| wwlknllvtx 27618 | If a word ` W ` represents... |
| wwlknlsw 27619 | If a word represents a wal... |
| wspthsn 27620 | The set of simple paths of... |
| iswspthn 27621 | An element of the set of s... |
| wspthnp 27622 | Properties of a set being ... |
| wwlksnon 27623 | The set of walks of a fixe... |
| wspthsnon 27624 | The set of simple paths of... |
| iswwlksnon 27625 | The set of walks of a fixe... |
| wwlksnon0 27626 | Sufficient conditions for ... |
| wwlksonvtx 27627 | If a word ` W ` represents... |
| iswspthsnon 27628 | The set of simple paths of... |
| wwlknon 27629 | An element of the set of w... |
| wspthnon 27630 | An element of the set of s... |
| wspthnonp 27631 | Properties of a set being ... |
| wspthneq1eq2 27632 | Two simple paths with iden... |
| wwlksn0s 27633 | The set of all walks as wo... |
| wwlkssswrd 27634 | Walks (represented by word... |
| wwlksn0 27635 | A walk of length 0 is repr... |
| 0enwwlksnge1 27636 | In graphs without edges, t... |
| wwlkswwlksn 27637 | A walk of a fixed length a... |
| wwlkssswwlksn 27638 | The walks of a fixed lengt... |
| wlkiswwlks1 27639 | The sequence of vertices i... |
| wlklnwwlkln1 27640 | The sequence of vertices i... |
| wlkiswwlks2lem1 27641 | Lemma 1 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem2 27642 | Lemma 2 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem3 27643 | Lemma 3 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem4 27644 | Lemma 4 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem5 27645 | Lemma 5 for ~ wlkiswwlks2 ... |
| wlkiswwlks2lem6 27646 | Lemma 6 for ~ wlkiswwlks2 ... |
| wlkiswwlks2 27647 | A walk as word corresponds... |
| wlkiswwlks 27648 | A walk as word corresponds... |
| wlkiswwlksupgr2 27649 | A walk as word corresponds... |
| wlkiswwlkupgr 27650 | A walk as word corresponds... |
| wlkswwlksf1o 27651 | The mapping of (ordinary) ... |
| wlkswwlksen 27652 | The set of walks as words ... |
| wwlksm1edg 27653 | Removing the trailing edge... |
| wlklnwwlkln2lem 27654 | Lemma for ~ wlklnwwlkln2 a... |
| wlklnwwlkln2 27655 | A walk of length ` N ` as ... |
| wlklnwwlkn 27656 | A walk of length ` N ` as ... |
| wlklnwwlklnupgr2 27657 | A walk of length ` N ` as ... |
| wlklnwwlknupgr 27658 | A walk of length ` N ` as ... |
| wlknewwlksn 27659 | If a walk in a pseudograph... |
| wlknwwlksnbij 27660 | The mapping ` ( t e. T |->... |
| wlknwwlksnen 27661 | In a simple pseudograph, t... |
| wlknwwlksneqs 27662 | The set of walks of a fixe... |
| wwlkseq 27663 | Equality of two walks (as ... |
| wwlksnred 27664 | Reduction of a walk (as wo... |
| wwlksnext 27665 | Extension of a walk (as wo... |
| wwlksnextbi 27666 | Extension of a walk (as wo... |
| wwlksnredwwlkn 27667 | For each walk (as word) of... |
| wwlksnredwwlkn0 27668 | For each walk (as word) of... |
| wwlksnextwrd 27669 | Lemma for ~ wwlksnextbij .... |
| wwlksnextfun 27670 | Lemma for ~ wwlksnextbij .... |
| wwlksnextinj 27671 | Lemma for ~ wwlksnextbij .... |
| wwlksnextsurj 27672 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij0 27673 | Lemma for ~ wwlksnextbij .... |
| wwlksnextbij 27674 | There is a bijection betwe... |
| wwlksnexthasheq 27675 | The number of the extensio... |
| disjxwwlksn 27676 | Sets of walks (as words) e... |
| wwlksnndef 27677 | Conditions for ` WWalksN `... |
| wwlksnfi 27678 | The number of walks repres... |
| wwlksnfiOLD 27679 | Obsolete version of ~ wwlk... |
| wlksnfi 27680 | The number of walks of fix... |
| wlksnwwlknvbij 27681 | There is a bijection betwe... |
| wwlksnextproplem1 27682 | Lemma 1 for ~ wwlksnextpro... |
| wwlksnextproplem2 27683 | Lemma 2 for ~ wwlksnextpro... |
| wwlksnextproplem3 27684 | Lemma 3 for ~ wwlksnextpro... |
| wwlksnextprop 27685 | Adding additional properti... |
| disjxwwlkn 27686 | Sets of walks (as words) e... |
| hashwwlksnext 27687 | Number of walks (as words)... |
| wwlksnwwlksnon 27688 | A walk of fixed length is ... |
| wspthsnwspthsnon 27689 | A simple path of fixed len... |
| wspthsnonn0vne 27690 | If the set of simple paths... |
| wspthsswwlkn 27691 | The set of simple paths of... |
| wspthnfi 27692 | In a finite graph, the set... |
| wwlksnonfi 27693 | In a finite graph, the set... |
| wspthsswwlknon 27694 | The set of simple paths of... |
| wspthnonfi 27695 | In a finite graph, the set... |
| wspniunwspnon 27696 | The set of nonempty simple... |
| wspn0 27697 | If there are no vertices, ... |
| 2wlkdlem1 27698 | Lemma 1 for ~ 2wlkd . (Co... |
| 2wlkdlem2 27699 | Lemma 2 for ~ 2wlkd . (Co... |
| 2wlkdlem3 27700 | Lemma 3 for ~ 2wlkd . (Co... |
| 2wlkdlem4 27701 | Lemma 4 for ~ 2wlkd . (Co... |
| 2wlkdlem5 27702 | Lemma 5 for ~ 2wlkd . (Co... |
| 2pthdlem1 27703 | Lemma 1 for ~ 2pthd . (Co... |
| 2wlkdlem6 27704 | Lemma 6 for ~ 2wlkd . (Co... |
| 2wlkdlem7 27705 | Lemma 7 for ~ 2wlkd . (Co... |
| 2wlkdlem8 27706 | Lemma 8 for ~ 2wlkd . (Co... |
| 2wlkdlem9 27707 | Lemma 9 for ~ 2wlkd . (Co... |
| 2wlkdlem10 27708 | Lemma 10 for ~ 3wlkd . (C... |
| 2wlkd 27709 | Construction of a walk fro... |
| 2wlkond 27710 | A walk of length 2 from on... |
| 2trld 27711 | Construction of a trail fr... |
| 2trlond 27712 | A trail of length 2 from o... |
| 2pthd 27713 | A path of length 2 from on... |
| 2spthd 27714 | A simple path of length 2 ... |
| 2pthond 27715 | A simple path of length 2 ... |
| 2pthon3v 27716 | For a vertex adjacent to t... |
| umgr2adedgwlklem 27717 | Lemma for ~ umgr2adedgwlk ... |
| umgr2adedgwlk 27718 | In a multigraph, two adjac... |
| umgr2adedgwlkon 27719 | In a multigraph, two adjac... |
| umgr2adedgwlkonALT 27720 | Alternate proof for ~ umgr... |
| umgr2adedgspth 27721 | In a multigraph, two adjac... |
| umgr2wlk 27722 | In a multigraph, there is ... |
| umgr2wlkon 27723 | For each pair of adjacent ... |
| elwwlks2s3 27724 | A walk of length 2 as word... |
| midwwlks2s3 27725 | There is a vertex between ... |
| wwlks2onv 27726 | If a length 3 string repre... |
| elwwlks2ons3im 27727 | A walk as word of length 2... |
| elwwlks2ons3 27728 | For each walk of length 2 ... |
| s3wwlks2on 27729 | A length 3 string which re... |
| umgrwwlks2on 27730 | A walk of length 2 between... |
| wwlks2onsym 27731 | There is a walk of length ... |
| elwwlks2on 27732 | A walk of length 2 between... |
| elwspths2on 27733 | A simple path of length 2 ... |
| wpthswwlks2on 27734 | For two different vertices... |
| 2wspdisj 27735 | All simple paths of length... |
| 2wspiundisj 27736 | All simple paths of length... |
| usgr2wspthons3 27737 | A simple path of length 2 ... |
| usgr2wspthon 27738 | A simple path of length 2 ... |
| elwwlks2 27739 | A walk of length 2 between... |
| elwspths2spth 27740 | A simple path of length 2 ... |
| rusgrnumwwlkl1 27741 | In a k-regular graph, ther... |
| rusgrnumwwlkslem 27742 | Lemma for ~ rusgrnumwwlks ... |
| rusgrnumwwlklem 27743 | Lemma for ~ rusgrnumwwlk e... |
| rusgrnumwwlkb0 27744 | Induction base 0 for ~ rus... |
| rusgrnumwwlkb1 27745 | Induction base 1 for ~ rus... |
| rusgr0edg 27746 | Special case for graphs wi... |
| rusgrnumwwlks 27747 | Induction step for ~ rusgr... |
| rusgrnumwwlk 27748 | In a ` K `-regular graph, ... |
| rusgrnumwwlkg 27749 | In a ` K `-regular graph, ... |
| rusgrnumwlkg 27750 | In a k-regular graph, the ... |
| clwwlknclwwlkdif 27751 | The set ` A ` of walks of ... |
| clwwlknclwwlkdifnum 27752 | In a ` K `-regular graph, ... |
| clwwlk 27755 | The set of closed walks (i... |
| isclwwlk 27756 | Properties of a word to re... |
| clwwlkbp 27757 | Basic properties of a clos... |
| clwwlkgt0 27758 | There is no empty closed w... |
| clwwlksswrd 27759 | Closed walks (represented ... |
| clwwlk1loop 27760 | A closed walk of length 1 ... |
| clwwlkccatlem 27761 | Lemma for ~ clwwlkccat : i... |
| clwwlkccat 27762 | The concatenation of two w... |
| umgrclwwlkge2 27763 | A closed walk in a multigr... |
| clwlkclwwlklem2a1 27764 | Lemma 1 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a2 27765 | Lemma 2 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a3 27766 | Lemma 3 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2fv1 27767 | Lemma 4a for ~ clwlkclwwlk... |
| clwlkclwwlklem2fv2 27768 | Lemma 4b for ~ clwlkclwwlk... |
| clwlkclwwlklem2a4 27769 | Lemma 4 for ~ clwlkclwwlkl... |
| clwlkclwwlklem2a 27770 | Lemma for ~ clwlkclwwlklem... |
| clwlkclwwlklem1 27771 | Lemma 1 for ~ clwlkclwwlk ... |
| clwlkclwwlklem2 27772 | Lemma 2 for ~ clwlkclwwlk ... |
| clwlkclwwlklem3 27773 | Lemma 3 for ~ clwlkclwwlk ... |
| clwlkclwwlk 27774 | A closed walk as word of l... |
| clwlkclwwlk2 27775 | A closed walk corresponds ... |
| clwlkclwwlkflem 27776 | Lemma for ~ clwlkclwwlkf .... |
| clwlkclwwlkf1lem2 27777 | Lemma 2 for ~ clwlkclwwlkf... |
| clwlkclwwlkf1lem3 27778 | Lemma 3 for ~ clwlkclwwlkf... |
| clwlkclwwlkfolem 27779 | Lemma for ~ clwlkclwwlkfo ... |
| clwlkclwwlkf 27780 | ` F ` is a function from t... |
| clwlkclwwlkfo 27781 | ` F ` is a function from t... |
| clwlkclwwlkf1 27782 | ` F ` is a one-to-one func... |
| clwlkclwwlkf1o 27783 | ` F ` is a bijection betwe... |
| clwlkclwwlken 27784 | The set of the nonempty cl... |
| clwwisshclwwslemlem 27785 | Lemma for ~ clwwisshclwwsl... |
| clwwisshclwwslem 27786 | Lemma for ~ clwwisshclwws ... |
| clwwisshclwws 27787 | Cyclically shifting a clos... |
| clwwisshclwwsn 27788 | Cyclically shifting a clos... |
| erclwwlkrel 27789 | ` .~ ` is a relation. (Co... |
| erclwwlkeq 27790 | Two classes are equivalent... |
| erclwwlkeqlen 27791 | If two classes are equival... |
| erclwwlkref 27792 | ` .~ ` is a reflexive rela... |
| erclwwlksym 27793 | ` .~ ` is a symmetric rela... |
| erclwwlktr 27794 | ` .~ ` is a transitive rel... |
| erclwwlk 27795 | ` .~ ` is an equivalence r... |
| clwwlkn 27798 | The set of closed walks of... |
| isclwwlkn 27799 | A word over the set of ver... |
| clwwlkn0 27800 | There is no closed walk of... |
| clwwlkneq0 27801 | Sufficient conditions for ... |
| clwwlkclwwlkn 27802 | A closed walk of a fixed l... |
| clwwlksclwwlkn 27803 | The closed walks of a fixe... |
| clwwlknlen 27804 | The length of a word repre... |
| clwwlknnn 27805 | The length of a closed wal... |
| clwwlknwrd 27806 | A closed walk of a fixed l... |
| clwwlknbp 27807 | Basic properties of a clos... |
| isclwwlknx 27808 | Characterization of a word... |
| clwwlknp 27809 | Properties of a set being ... |
| clwwlknwwlksn 27810 | A word representing a clos... |
| clwwlknlbonbgr1 27811 | The last but one vertex in... |
| clwwlkinwwlk 27812 | If the initial vertex of a... |
| clwwlkn1 27813 | A closed walk of length 1 ... |
| loopclwwlkn1b 27814 | The singleton word consist... |
| clwwlkn1loopb 27815 | A word represents a closed... |
| clwwlkn2 27816 | A closed walk of length 2 ... |
| clwwlknfi 27817 | If there is only a finite ... |
| clwwlknfiOLD 27818 | Obsolete version of ~ clww... |
| clwwlkel 27819 | Obtaining a closed walk (a... |
| clwwlkf 27820 | Lemma 1 for ~ clwwlkf1o : ... |
| clwwlkfv 27821 | Lemma 2 for ~ clwwlkf1o : ... |
| clwwlkf1 27822 | Lemma 3 for ~ clwwlkf1o : ... |
| clwwlkfo 27823 | Lemma 4 for ~ clwwlkf1o : ... |
| clwwlkf1o 27824 | F is a 1-1 onto function, ... |
| clwwlken 27825 | The set of closed walks of... |
| clwwlknwwlkncl 27826 | Obtaining a closed walk (a... |
| clwwlkwwlksb 27827 | A nonempty word over verti... |
| clwwlknwwlksnb 27828 | A word over vertices repre... |
| clwwlkext2edg 27829 | If a word concatenated wit... |
| wwlksext2clwwlk 27830 | If a word represents a wal... |
| wwlksubclwwlk 27831 | Any prefix of a word repre... |
| clwwnisshclwwsn 27832 | Cyclically shifting a clos... |
| eleclclwwlknlem1 27833 | Lemma 1 for ~ eleclclwwlkn... |
| eleclclwwlknlem2 27834 | Lemma 2 for ~ eleclclwwlkn... |
| clwwlknscsh 27835 | The set of cyclical shifts... |
| clwwlknccat 27836 | The concatenation of two w... |
| umgr2cwwk2dif 27837 | If a word represents a clo... |
| umgr2cwwkdifex 27838 | If a word represents a clo... |
| erclwwlknrel 27839 | ` .~ ` is a relation. (Co... |
| erclwwlkneq 27840 | Two classes are equivalent... |
| erclwwlkneqlen 27841 | If two classes are equival... |
| erclwwlknref 27842 | ` .~ ` is a reflexive rela... |
| erclwwlknsym 27843 | ` .~ ` is a symmetric rela... |
| erclwwlkntr 27844 | ` .~ ` is a transitive rel... |
| erclwwlkn 27845 | ` .~ ` is an equivalence r... |
| qerclwwlknfi 27846 | The quotient set of the se... |
| hashclwwlkn0 27847 | The number of closed walks... |
| eclclwwlkn1 27848 | An equivalence class accor... |
| eleclclwwlkn 27849 | A member of an equivalence... |
| hashecclwwlkn1 27850 | The size of every equivale... |
| umgrhashecclwwlk 27851 | The size of every equivale... |
| fusgrhashclwwlkn 27852 | The size of the set of clo... |
| clwwlkndivn 27853 | The size of the set of clo... |
| clwlknf1oclwwlknlem1 27854 | Lemma 1 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem2 27855 | Lemma 2 for ~ clwlknf1oclw... |
| clwlknf1oclwwlknlem3 27856 | Lemma 3 for ~ clwlknf1oclw... |
| clwlknf1oclwwlkn 27857 | There is a one-to-one onto... |
| clwlkssizeeq 27858 | The size of the set of clo... |
| clwlksndivn 27859 | The size of the set of clo... |
| clwwlknonmpo 27862 | ` ( ClWWalksNOn `` G ) ` i... |
| clwwlknon 27863 | The set of closed walks on... |
| isclwwlknon 27864 | A word over the set of ver... |
| clwwlk0on0 27865 | There is no word over the ... |
| clwwlknon0 27866 | Sufficient conditions for ... |
| clwwlknonfin 27867 | In a finite graph ` G ` , ... |
| clwwlknonel 27868 | Characterization of a word... |
| clwwlknonccat 27869 | The concatenation of two w... |
| clwwlknon1 27870 | The set of closed walks on... |
| clwwlknon1loop 27871 | If there is a loop at vert... |
| clwwlknon1nloop 27872 | If there is no loop at ver... |
| clwwlknon1sn 27873 | The set of (closed) walks ... |
| clwwlknon1le1 27874 | There is at most one (clos... |
| clwwlknon2 27875 | The set of closed walks on... |
| clwwlknon2x 27876 | The set of closed walks on... |
| s2elclwwlknon2 27877 | Sufficient conditions of a... |
| clwwlknon2num 27878 | In a ` K `-regular graph `... |
| clwwlknonwwlknonb 27879 | A word over vertices repre... |
| clwwlknonex2lem1 27880 | Lemma 1 for ~ clwwlknonex2... |
| clwwlknonex2lem2 27881 | Lemma 2 for ~ clwwlknonex2... |
| clwwlknonex2 27882 | Extending a closed walk ` ... |
| clwwlknonex2e 27883 | Extending a closed walk ` ... |
| clwwlknondisj 27884 | The sets of closed walks o... |
| clwwlknun 27885 | The set of closed walks of... |
| clwwlkvbij 27886 | There is a bijection betwe... |
| 0ewlk 27887 | The empty set (empty seque... |
| 1ewlk 27888 | A sequence of 1 edge is an... |
| 0wlk 27889 | A pair of an empty set (of... |
| is0wlk 27890 | A pair of an empty set (of... |
| 0wlkonlem1 27891 | Lemma 1 for ~ 0wlkon and ~... |
| 0wlkonlem2 27892 | Lemma 2 for ~ 0wlkon and ~... |
| 0wlkon 27893 | A walk of length 0 from a ... |
| 0wlkons1 27894 | A walk of length 0 from a ... |
| 0trl 27895 | A pair of an empty set (of... |
| is0trl 27896 | A pair of an empty set (of... |
| 0trlon 27897 | A trail of length 0 from a... |
| 0pth 27898 | A pair of an empty set (of... |
| 0spth 27899 | A pair of an empty set (of... |
| 0pthon 27900 | A path of length 0 from a ... |
| 0pthon1 27901 | A path of length 0 from a ... |
| 0pthonv 27902 | For each vertex there is a... |
| 0clwlk 27903 | A pair of an empty set (of... |
| 0clwlkv 27904 | Any vertex (more precisely... |
| 0clwlk0 27905 | There is no closed walk in... |
| 0crct 27906 | A pair of an empty set (of... |
| 0cycl 27907 | A pair of an empty set (of... |
| 1pthdlem1 27908 | Lemma 1 for ~ 1pthd . (Co... |
| 1pthdlem2 27909 | Lemma 2 for ~ 1pthd . (Co... |
| 1wlkdlem1 27910 | Lemma 1 for ~ 1wlkd . (Co... |
| 1wlkdlem2 27911 | Lemma 2 for ~ 1wlkd . (Co... |
| 1wlkdlem3 27912 | Lemma 3 for ~ 1wlkd . (Co... |
| 1wlkdlem4 27913 | Lemma 4 for ~ 1wlkd . (Co... |
| 1wlkd 27914 | In a graph with two vertic... |
| 1trld 27915 | In a graph with two vertic... |
| 1pthd 27916 | In a graph with two vertic... |
| 1pthond 27917 | In a graph with two vertic... |
| upgr1wlkdlem1 27918 | Lemma 1 for ~ upgr1wlkd . ... |
| upgr1wlkdlem2 27919 | Lemma 2 for ~ upgr1wlkd . ... |
| upgr1wlkd 27920 | In a pseudograph with two ... |
| upgr1trld 27921 | In a pseudograph with two ... |
| upgr1pthd 27922 | In a pseudograph with two ... |
| upgr1pthond 27923 | In a pseudograph with two ... |
| lppthon 27924 | A loop (which is an edge a... |
| lp1cycl 27925 | A loop (which is an edge a... |
| 1pthon2v 27926 | For each pair of adjacent ... |
| 1pthon2ve 27927 | For each pair of adjacent ... |
| wlk2v2elem1 27928 | Lemma 1 for ~ wlk2v2e : ` ... |
| wlk2v2elem2 27929 | Lemma 2 for ~ wlk2v2e : T... |
| wlk2v2e 27930 | In a graph with two vertic... |
| ntrl2v2e 27931 | A walk which is not a trai... |
| 3wlkdlem1 27932 | Lemma 1 for ~ 3wlkd . (Co... |
| 3wlkdlem2 27933 | Lemma 2 for ~ 3wlkd . (Co... |
| 3wlkdlem3 27934 | Lemma 3 for ~ 3wlkd . (Co... |
| 3wlkdlem4 27935 | Lemma 4 for ~ 3wlkd . (Co... |
| 3wlkdlem5 27936 | Lemma 5 for ~ 3wlkd . (Co... |
| 3pthdlem1 27937 | Lemma 1 for ~ 3pthd . (Co... |
| 3wlkdlem6 27938 | Lemma 6 for ~ 3wlkd . (Co... |
| 3wlkdlem7 27939 | Lemma 7 for ~ 3wlkd . (Co... |
| 3wlkdlem8 27940 | Lemma 8 for ~ 3wlkd . (Co... |
| 3wlkdlem9 27941 | Lemma 9 for ~ 3wlkd . (Co... |
| 3wlkdlem10 27942 | Lemma 10 for ~ 3wlkd . (C... |
| 3wlkd 27943 | Construction of a walk fro... |
| 3wlkond 27944 | A walk of length 3 from on... |
| 3trld 27945 | Construction of a trail fr... |
| 3trlond 27946 | A trail of length 3 from o... |
| 3pthd 27947 | A path of length 3 from on... |
| 3pthond 27948 | A path of length 3 from on... |
| 3spthd 27949 | A simple path of length 3 ... |
| 3spthond 27950 | A simple path of length 3 ... |
| 3cycld 27951 | Construction of a 3-cycle ... |
| 3cyclpd 27952 | Construction of a 3-cycle ... |
| upgr3v3e3cycl 27953 | If there is a cycle of len... |
| uhgr3cyclexlem 27954 | Lemma for ~ uhgr3cyclex . ... |
| uhgr3cyclex 27955 | If there are three differe... |
| umgr3cyclex 27956 | If there are three (differ... |
| umgr3v3e3cycl 27957 | If and only if there is a ... |
| upgr4cycl4dv4e 27958 | If there is a cycle of len... |
| dfconngr1 27961 | Alternative definition of ... |
| isconngr 27962 | The property of being a co... |
| isconngr1 27963 | The property of being a co... |
| cusconngr 27964 | A complete hypergraph is c... |
| 0conngr 27965 | A graph without vertices i... |
| 0vconngr 27966 | A graph without vertices i... |
| 1conngr 27967 | A graph with (at most) one... |
| conngrv2edg 27968 | A vertex in a connected gr... |
| vdn0conngrumgrv2 27969 | A vertex in a connected mu... |
| releupth 27972 | The set ` ( EulerPaths `` ... |
| eupths 27973 | The Eulerian paths on the ... |
| iseupth 27974 | The property " ` <. F , P ... |
| iseupthf1o 27975 | The property " ` <. F , P ... |
| eupthi 27976 | Properties of an Eulerian ... |
| eupthf1o 27977 | The ` F ` function in an E... |
| eupthfi 27978 | Any graph with an Eulerian... |
| eupthseg 27979 | The ` N ` -th edge in an e... |
| upgriseupth 27980 | The property " ` <. F , P ... |
| upgreupthi 27981 | Properties of an Eulerian ... |
| upgreupthseg 27982 | The ` N ` -th edge in an e... |
| eupthcl 27983 | An Eulerian path has lengt... |
| eupthistrl 27984 | An Eulerian path is a trai... |
| eupthiswlk 27985 | An Eulerian path is a walk... |
| eupthpf 27986 | The ` P ` function in an E... |
| eupth0 27987 | There is an Eulerian path ... |
| eupthres 27988 | The restriction ` <. H , Q... |
| eupthp1 27989 | Append one path segment to... |
| eupth2eucrct 27990 | Append one path segment to... |
| eupth2lem1 27991 | Lemma for ~ eupth2 . (Con... |
| eupth2lem2 27992 | Lemma for ~ eupth2 . (Con... |
| trlsegvdeglem1 27993 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem2 27994 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem3 27995 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem4 27996 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem5 27997 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem6 27998 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeglem7 27999 | Lemma for ~ trlsegvdeg . ... |
| trlsegvdeg 28000 | Formerly part of proof of ... |
| eupth2lem3lem1 28001 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem2 28002 | Lemma for ~ eupth2lem3 . ... |
| eupth2lem3lem3 28003 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem4 28004 | Lemma for ~ eupth2lem3 , f... |
| eupth2lem3lem5 28005 | Lemma for ~ eupth2 . (Con... |
| eupth2lem3lem6 28006 | Formerly part of proof of ... |
| eupth2lem3lem7 28007 | Lemma for ~ eupth2lem3 : ... |
| eupthvdres 28008 | Formerly part of proof of ... |
| eupth2lem3 28009 | Lemma for ~ eupth2 . (Con... |
| eupth2lemb 28010 | Lemma for ~ eupth2 (induct... |
| eupth2lems 28011 | Lemma for ~ eupth2 (induct... |
| eupth2 28012 | The only vertices of odd d... |
| eulerpathpr 28013 | A graph with an Eulerian p... |
| eulerpath 28014 | A pseudograph with an Eule... |
| eulercrct 28015 | A pseudograph with an Eule... |
| eucrctshift 28016 | Cyclically shifting the in... |
| eucrct2eupth1 28017 | Removing one edge ` ( I ``... |
| eucrct2eupth 28018 | Removing one edge ` ( I ``... |
| konigsbergvtx 28019 | The set of vertices of the... |
| konigsbergiedg 28020 | The indexed edges of the K... |
| konigsbergiedgw 28021 | The indexed edges of the K... |
| konigsbergssiedgwpr 28022 | Each subset of the indexed... |
| konigsbergssiedgw 28023 | Each subset of the indexed... |
| konigsbergumgr 28024 | The Königsberg graph ... |
| konigsberglem1 28025 | Lemma 1 for ~ konigsberg :... |
| konigsberglem2 28026 | Lemma 2 for ~ konigsberg :... |
| konigsberglem3 28027 | Lemma 3 for ~ konigsberg :... |
| konigsberglem4 28028 | Lemma 4 for ~ konigsberg :... |
| konigsberglem5 28029 | Lemma 5 for ~ konigsberg :... |
| konigsberg 28030 | The Königsberg Bridge... |
| isfrgr 28033 | The property of being a fr... |
| frgrusgr 28034 | A friendship graph is a si... |
| frgr0v 28035 | Any null graph (set with n... |
| frgr0vb 28036 | Any null graph (without ve... |
| frgruhgr0v 28037 | Any null graph (without ve... |
| frgr0 28038 | The null graph (graph with... |
| frcond1 28039 | The friendship condition: ... |
| frcond2 28040 | The friendship condition: ... |
| frgreu 28041 | Variant of ~ frcond2 : An... |
| frcond3 28042 | The friendship condition, ... |
| frcond4 28043 | The friendship condition, ... |
| frgr1v 28044 | Any graph with (at most) o... |
| nfrgr2v 28045 | Any graph with two (differ... |
| frgr3vlem1 28046 | Lemma 1 for ~ frgr3v . (C... |
| frgr3vlem2 28047 | Lemma 2 for ~ frgr3v . (C... |
| frgr3v 28048 | Any graph with three verti... |
| 1vwmgr 28049 | Every graph with one verte... |
| 3vfriswmgrlem 28050 | Lemma for ~ 3vfriswmgr . ... |
| 3vfriswmgr 28051 | Every friendship graph wit... |
| 1to2vfriswmgr 28052 | Every friendship graph wit... |
| 1to3vfriswmgr 28053 | Every friendship graph wit... |
| 1to3vfriendship 28054 | The friendship theorem for... |
| 2pthfrgrrn 28055 | Between any two (different... |
| 2pthfrgrrn2 28056 | Between any two (different... |
| 2pthfrgr 28057 | Between any two (different... |
| 3cyclfrgrrn1 28058 | Every vertex in a friendsh... |
| 3cyclfrgrrn 28059 | Every vertex in a friendsh... |
| 3cyclfrgrrn2 28060 | Every vertex in a friendsh... |
| 3cyclfrgr 28061 | Every vertex in a friendsh... |
| 4cycl2v2nb 28062 | In a (maybe degenerate) 4-... |
| 4cycl2vnunb 28063 | In a 4-cycle, two distinct... |
| n4cyclfrgr 28064 | There is no 4-cycle in a f... |
| 4cyclusnfrgr 28065 | A graph with a 4-cycle is ... |
| frgrnbnb 28066 | If two neighbors ` U ` and... |
| frgrconngr 28067 | A friendship graph is conn... |
| vdgn0frgrv2 28068 | A vertex in a friendship g... |
| vdgn1frgrv2 28069 | Any vertex in a friendship... |
| vdgn1frgrv3 28070 | Any vertex in a friendship... |
| vdgfrgrgt2 28071 | Any vertex in a friendship... |
| frgrncvvdeqlem1 28072 | Lemma 1 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem2 28073 | Lemma 2 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem3 28074 | Lemma 3 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem4 28075 | Lemma 4 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem5 28076 | Lemma 5 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem6 28077 | Lemma 6 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem7 28078 | Lemma 7 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem8 28079 | Lemma 8 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem9 28080 | Lemma 9 for ~ frgrncvvdeq ... |
| frgrncvvdeqlem10 28081 | Lemma 10 for ~ frgrncvvdeq... |
| frgrncvvdeq 28082 | In a friendship graph, two... |
| frgrwopreglem4a 28083 | In a friendship graph any ... |
| frgrwopreglem5a 28084 | If a friendship graph has ... |
| frgrwopreglem1 28085 | Lemma 1 for ~ frgrwopreg :... |
| frgrwopreglem2 28086 | Lemma 2 for ~ frgrwopreg .... |
| frgrwopreglem3 28087 | Lemma 3 for ~ frgrwopreg .... |
| frgrwopreglem4 28088 | Lemma 4 for ~ frgrwopreg .... |
| frgrwopregasn 28089 | According to statement 5 i... |
| frgrwopregbsn 28090 | According to statement 5 i... |
| frgrwopreg1 28091 | According to statement 5 i... |
| frgrwopreg2 28092 | According to statement 5 i... |
| frgrwopreglem5lem 28093 | Lemma for ~ frgrwopreglem5... |
| frgrwopreglem5 28094 | Lemma 5 for ~ frgrwopreg .... |
| frgrwopreglem5ALT 28095 | Alternate direct proof of ... |
| frgrwopreg 28096 | In a friendship graph ther... |
| frgrregorufr0 28097 | In a friendship graph ther... |
| frgrregorufr 28098 | If there is a vertex havin... |
| frgrregorufrg 28099 | If there is a vertex havin... |
| frgr2wwlkeu 28100 | For two different vertices... |
| frgr2wwlkn0 28101 | In a friendship graph, the... |
| frgr2wwlk1 28102 | In a friendship graph, the... |
| frgr2wsp1 28103 | In a friendship graph, the... |
| frgr2wwlkeqm 28104 | If there is a (simple) pat... |
| frgrhash2wsp 28105 | The number of simple paths... |
| fusgreg2wsplem 28106 | Lemma for ~ fusgreg2wsp an... |
| fusgr2wsp2nb 28107 | The set of paths of length... |
| fusgreghash2wspv 28108 | According to statement 7 i... |
| fusgreg2wsp 28109 | In a finite simple graph, ... |
| 2wspmdisj 28110 | The sets of paths of lengt... |
| fusgreghash2wsp 28111 | In a finite k-regular grap... |
| frrusgrord0lem 28112 | Lemma for ~ frrusgrord0 . ... |
| frrusgrord0 28113 | If a nonempty finite frien... |
| frrusgrord 28114 | If a nonempty finite frien... |
| numclwwlk2lem1lem 28115 | Lemma for ~ numclwwlk2lem1... |
| 2clwwlklem 28116 | Lemma for ~ clwwnonrepclww... |
| clwwnrepclwwn 28117 | If the initial vertex of a... |
| clwwnonrepclwwnon 28118 | If the initial vertex of a... |
| 2clwwlk2clwwlklem 28119 | Lemma for ~ 2clwwlk2clwwlk... |
| 2clwwlk 28120 | Value of operation ` C ` ,... |
| 2clwwlk2 28121 | The set ` ( X C 2 ) ` of d... |
| 2clwwlkel 28122 | Characterization of an ele... |
| 2clwwlk2clwwlk 28123 | An element of the value of... |
| numclwwlk1lem2foalem 28124 | Lemma for ~ numclwwlk1lem2... |
| extwwlkfab 28125 | The set ` ( X C N ) ` of d... |
| extwwlkfabel 28126 | Characterization of an ele... |
| numclwwlk1lem2foa 28127 | Going forth and back from ... |
| numclwwlk1lem2f 28128 | ` T ` is a function, mappi... |
| numclwwlk1lem2fv 28129 | Value of the function ` T ... |
| numclwwlk1lem2f1 28130 | ` T ` is a 1-1 function. ... |
| numclwwlk1lem2fo 28131 | ` T ` is an onto function.... |
| numclwwlk1lem2f1o 28132 | ` T ` is a 1-1 onto functi... |
| numclwwlk1lem2 28133 | The set of double loops of... |
| numclwwlk1 28134 | Statement 9 in [Huneke] p.... |
| clwwlknonclwlknonf1o 28135 | ` F ` is a bijection betwe... |
| clwwlknonclwlknonen 28136 | The sets of the two repres... |
| dlwwlknondlwlknonf1olem1 28137 | Lemma 1 for ~ dlwwlknondlw... |
| dlwwlknondlwlknonf1o 28138 | ` F ` is a bijection betwe... |
| dlwwlknondlwlknonen 28139 | The sets of the two repres... |
| wlkl0 28140 | There is exactly one walk ... |
| clwlknon2num 28141 | There are k walks of lengt... |
| numclwlk1lem1 28142 | Lemma 1 for ~ numclwlk1 (S... |
| numclwlk1lem2 28143 | Lemma 2 for ~ numclwlk1 (S... |
| numclwlk1 28144 | Statement 9 in [Huneke] p.... |
| numclwwlkovh0 28145 | Value of operation ` H ` ,... |
| numclwwlkovh 28146 | Value of operation ` H ` ,... |
| numclwwlkovq 28147 | Value of operation ` Q ` ,... |
| numclwwlkqhash 28148 | In a ` K `-regular graph, ... |
| numclwwlk2lem1 28149 | In a friendship graph, for... |
| numclwlk2lem2f 28150 | ` R ` is a function mappin... |
| numclwlk2lem2fv 28151 | Value of the function ` R ... |
| numclwlk2lem2f1o 28152 | ` R ` is a 1-1 onto functi... |
| numclwwlk2lem3 28153 | In a friendship graph, the... |
| numclwwlk2 28154 | Statement 10 in [Huneke] p... |
| numclwwlk3lem1 28155 | Lemma 2 for ~ numclwwlk3 .... |
| numclwwlk3lem2lem 28156 | Lemma for ~ numclwwlk3lem2... |
| numclwwlk3lem2 28157 | Lemma 1 for ~ numclwwlk3 :... |
| numclwwlk3 28158 | Statement 12 in [Huneke] p... |
| numclwwlk4 28159 | The total number of closed... |
| numclwwlk5lem 28160 | Lemma for ~ numclwwlk5 . ... |
| numclwwlk5 28161 | Statement 13 in [Huneke] p... |
| numclwwlk7lem 28162 | Lemma for ~ numclwwlk7 , ~... |
| numclwwlk6 28163 | For a prime divisor ` P ` ... |
| numclwwlk7 28164 | Statement 14 in [Huneke] p... |
| numclwwlk8 28165 | The size of the set of clo... |
| frgrreggt1 28166 | If a finite nonempty frien... |
| frgrreg 28167 | If a finite nonempty frien... |
| frgrregord013 28168 | If a finite friendship gra... |
| frgrregord13 28169 | If a nonempty finite frien... |
| frgrogt3nreg 28170 | If a finite friendship gra... |
| friendshipgt3 28171 | The friendship theorem for... |
| friendship 28172 | The friendship theorem: I... |
| conventions 28173 |
... |
| conventions-labels 28174 |
... |
| conventions-comments 28175 |
... |
| natded 28176 | Here are typical n... |
| ex-natded5.2 28177 | Theorem 5.2 of [Clemente] ... |
| ex-natded5.2-2 28178 | A more efficient proof of ... |
| ex-natded5.2i 28179 | The same as ~ ex-natded5.2... |
| ex-natded5.3 28180 | Theorem 5.3 of [Clemente] ... |
| ex-natded5.3-2 28181 | A more efficient proof of ... |
| ex-natded5.3i 28182 | The same as ~ ex-natded5.3... |
| ex-natded5.5 28183 | Theorem 5.5 of [Clemente] ... |
| ex-natded5.7 28184 | Theorem 5.7 of [Clemente] ... |
| ex-natded5.7-2 28185 | A more efficient proof of ... |
| ex-natded5.8 28186 | Theorem 5.8 of [Clemente] ... |
| ex-natded5.8-2 28187 | A more efficient proof of ... |
| ex-natded5.13 28188 | Theorem 5.13 of [Clemente]... |
| ex-natded5.13-2 28189 | A more efficient proof of ... |
| ex-natded9.20 28190 | Theorem 9.20 of [Clemente]... |
| ex-natded9.20-2 28191 | A more efficient proof of ... |
| ex-natded9.26 28192 | Theorem 9.26 of [Clemente]... |
| ex-natded9.26-2 28193 | A more efficient proof of ... |
| ex-or 28194 | Example for ~ df-or . Exa... |
| ex-an 28195 | Example for ~ df-an . Exa... |
| ex-dif 28196 | Example for ~ df-dif . Ex... |
| ex-un 28197 | Example for ~ df-un . Exa... |
| ex-in 28198 | Example for ~ df-in . Exa... |
| ex-uni 28199 | Example for ~ df-uni . Ex... |
| ex-ss 28200 | Example for ~ df-ss . Exa... |
| ex-pss 28201 | Example for ~ df-pss . Ex... |
| ex-pw 28202 | Example for ~ df-pw . Exa... |
| ex-pr 28203 | Example for ~ df-pr . (Co... |
| ex-br 28204 | Example for ~ df-br . Exa... |
| ex-opab 28205 | Example for ~ df-opab . E... |
| ex-eprel 28206 | Example for ~ df-eprel . ... |
| ex-id 28207 | Example for ~ df-id . Exa... |
| ex-po 28208 | Example for ~ df-po . Exa... |
| ex-xp 28209 | Example for ~ df-xp . Exa... |
| ex-cnv 28210 | Example for ~ df-cnv . Ex... |
| ex-co 28211 | Example for ~ df-co . Exa... |
| ex-dm 28212 | Example for ~ df-dm . Exa... |
| ex-rn 28213 | Example for ~ df-rn . Exa... |
| ex-res 28214 | Example for ~ df-res . Ex... |
| ex-ima 28215 | Example for ~ df-ima . Ex... |
| ex-fv 28216 | Example for ~ df-fv . Exa... |
| ex-1st 28217 | Example for ~ df-1st . Ex... |
| ex-2nd 28218 | Example for ~ df-2nd . Ex... |
| 1kp2ke3k 28219 | Example for ~ df-dec , 100... |
| ex-fl 28220 | Example for ~ df-fl . Exa... |
| ex-ceil 28221 | Example for ~ df-ceil . (... |
| ex-mod 28222 | Example for ~ df-mod . (C... |
| ex-exp 28223 | Example for ~ df-exp . (C... |
| ex-fac 28224 | Example for ~ df-fac . (C... |
| ex-bc 28225 | Example for ~ df-bc . (Co... |
| ex-hash 28226 | Example for ~ df-hash . (... |
| ex-sqrt 28227 | Example for ~ df-sqrt . (... |
| ex-abs 28228 | Example for ~ df-abs . (C... |
| ex-dvds 28229 | Example for ~ df-dvds : 3 ... |
| ex-gcd 28230 | Example for ~ df-gcd . (C... |
| ex-lcm 28231 | Example for ~ df-lcm . (C... |
| ex-prmo 28232 | Example for ~ df-prmo : ` ... |
| aevdemo 28233 | Proof illustrating the com... |
| ex-ind-dvds 28234 | Example of a proof by indu... |
| ex-fpar 28235 | Formalized example provide... |
| avril1 28236 | Poisson d'Avril's Theorem.... |
| 2bornot2b 28237 | The law of excluded middle... |
| helloworld 28238 | The classic "Hello world" ... |
| 1p1e2apr1 28239 | One plus one equals two. ... |
| eqid1 28240 | Law of identity (reflexivi... |
| 1div0apr 28241 | Division by zero is forbid... |
| topnfbey 28242 | Nothing seems to be imposs... |
| 9p10ne21 28243 | 9 + 10 is not equal to 21.... |
| 9p10ne21fool 28244 | 9 + 10 equals 21. This as... |
| isplig 28247 | The predicate "is a planar... |
| ispligb 28248 | The predicate "is a planar... |
| tncp 28249 | In any planar incidence ge... |
| l2p 28250 | For any line in a planar i... |
| lpni 28251 | For any line in a planar i... |
| nsnlplig 28252 | There is no "one-point lin... |
| nsnlpligALT 28253 | Alternate version of ~ nsn... |
| n0lplig 28254 | There is no "empty line" i... |
| n0lpligALT 28255 | Alternate version of ~ n0l... |
| eulplig 28256 | Through two distinct point... |
| pliguhgr 28257 | Any planar incidence geome... |
| dummylink 28258 | Alias for ~ a1ii that may ... |
| id1 28259 | Alias for ~ idALT that may... |
| isgrpo 28268 | The predicate "is a group ... |
| isgrpoi 28269 | Properties that determine ... |
| grpofo 28270 | A group operation maps ont... |
| grpocl 28271 | Closure law for a group op... |
| grpolidinv 28272 | A group has a left identit... |
| grpon0 28273 | The base set of a group is... |
| grpoass 28274 | A group operation is assoc... |
| grpoidinvlem1 28275 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem2 28276 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem3 28277 | Lemma for ~ grpoidinv . (... |
| grpoidinvlem4 28278 | Lemma for ~ grpoidinv . (... |
| grpoidinv 28279 | A group has a left and rig... |
| grpoideu 28280 | The left identity element ... |
| grporndm 28281 | A group's range in terms o... |
| 0ngrp 28282 | The empty set is not a gro... |
| gidval 28283 | The value of the identity ... |
| grpoidval 28284 | Lemma for ~ grpoidcl and o... |
| grpoidcl 28285 | The identity element of a ... |
| grpoidinv2 28286 | A group's properties using... |
| grpolid 28287 | The identity element of a ... |
| grporid 28288 | The identity element of a ... |
| grporcan 28289 | Right cancellation law for... |
| grpoinveu 28290 | The left inverse element o... |
| grpoid 28291 | Two ways of saying that an... |
| grporn 28292 | The range of a group opera... |
| grpoinvfval 28293 | The inverse function of a ... |
| grpoinvval 28294 | The inverse of a group ele... |
| grpoinvcl 28295 | A group element's inverse ... |
| grpoinv 28296 | The properties of a group ... |
| grpolinv 28297 | The left inverse of a grou... |
| grporinv 28298 | The right inverse of a gro... |
| grpoinvid1 28299 | The inverse of a group ele... |
| grpoinvid2 28300 | The inverse of a group ele... |
| grpolcan 28301 | Left cancellation law for ... |
| grpo2inv 28302 | Double inverse law for gro... |
| grpoinvf 28303 | Mapping of the inverse fun... |
| grpoinvop 28304 | The inverse of the group o... |
| grpodivfval 28305 | Group division (or subtrac... |
| grpodivval 28306 | Group division (or subtrac... |
| grpodivinv 28307 | Group division by an inver... |
| grpoinvdiv 28308 | Inverse of a group divisio... |
| grpodivf 28309 | Mapping for group division... |
| grpodivcl 28310 | Closure of group division ... |
| grpodivdiv 28311 | Double group division. (C... |
| grpomuldivass 28312 | Associative-type law for m... |
| grpodivid 28313 | Division of a group member... |
| grponpcan 28314 | Cancellation law for group... |
| isablo 28317 | The predicate "is an Abeli... |
| ablogrpo 28318 | An Abelian group operation... |
| ablocom 28319 | An Abelian group operation... |
| ablo32 28320 | Commutative/associative la... |
| ablo4 28321 | Commutative/associative la... |
| isabloi 28322 | Properties that determine ... |
| ablomuldiv 28323 | Law for group multiplicati... |
| ablodivdiv 28324 | Law for double group divis... |
| ablodivdiv4 28325 | Law for double group divis... |
| ablodiv32 28326 | Swap the second and third ... |
| ablonncan 28327 | Cancellation law for group... |
| ablonnncan1 28328 | Cancellation law for group... |
| vcrel 28331 | The class of all complex v... |
| vciOLD 28332 | Obsolete version of ~ cvsi... |
| vcsm 28333 | Functionality of th scalar... |
| vccl 28334 | Closure of the scalar prod... |
| vcidOLD 28335 | Identity element for the s... |
| vcdi 28336 | Distributive law for the s... |
| vcdir 28337 | Distributive law for the s... |
| vcass 28338 | Associative law for the sc... |
| vc2OLD 28339 | A vector plus itself is tw... |
| vcablo 28340 | Vector addition is an Abel... |
| vcgrp 28341 | Vector addition is a group... |
| vclcan 28342 | Left cancellation law for ... |
| vczcl 28343 | The zero vector is a vecto... |
| vc0rid 28344 | The zero vector is a right... |
| vc0 28345 | Zero times a vector is the... |
| vcz 28346 | Anything times the zero ve... |
| vcm 28347 | Minus 1 times a vector is ... |
| isvclem 28348 | Lemma for ~ isvcOLD . (Co... |
| vcex 28349 | The components of a comple... |
| isvcOLD 28350 | The predicate "is a comple... |
| isvciOLD 28351 | Properties that determine ... |
| cnaddabloOLD 28352 | Obsolete version of ~ cnad... |
| cnidOLD 28353 | Obsolete version of ~ cnad... |
| cncvcOLD 28354 | Obsolete version of ~ cncv... |
| nvss 28364 | Structure of the class of ... |
| nvvcop 28365 | A normed complex vector sp... |
| nvrel 28373 | The class of all normed co... |
| vafval 28374 | Value of the function for ... |
| bafval 28375 | Value of the function for ... |
| smfval 28376 | Value of the function for ... |
| 0vfval 28377 | Value of the function for ... |
| nmcvfval 28378 | Value of the norm function... |
| nvop2 28379 | A normed complex vector sp... |
| nvvop 28380 | The vector space component... |
| isnvlem 28381 | Lemma for ~ isnv . (Contr... |
| nvex 28382 | The components of a normed... |
| isnv 28383 | The predicate "is a normed... |
| isnvi 28384 | Properties that determine ... |
| nvi 28385 | The properties of a normed... |
| nvvc 28386 | The vector space component... |
| nvablo 28387 | The vector addition operat... |
| nvgrp 28388 | The vector addition operat... |
| nvgf 28389 | Mapping for the vector add... |
| nvsf 28390 | Mapping for the scalar mul... |
| nvgcl 28391 | Closure law for the vector... |
| nvcom 28392 | The vector addition (group... |
| nvass 28393 | The vector addition (group... |
| nvadd32 28394 | Commutative/associative la... |
| nvrcan 28395 | Right cancellation law for... |
| nvadd4 28396 | Rearrangement of 4 terms i... |
| nvscl 28397 | Closure law for the scalar... |
| nvsid 28398 | Identity element for the s... |
| nvsass 28399 | Associative law for the sc... |
| nvscom 28400 | Commutative law for the sc... |
| nvdi 28401 | Distributive law for the s... |
| nvdir 28402 | Distributive law for the s... |
| nv2 28403 | A vector plus itself is tw... |
| vsfval 28404 | Value of the function for ... |
| nvzcl 28405 | Closure law for the zero v... |
| nv0rid 28406 | The zero vector is a right... |
| nv0lid 28407 | The zero vector is a left ... |
| nv0 28408 | Zero times a vector is the... |
| nvsz 28409 | Anything times the zero ve... |
| nvinv 28410 | Minus 1 times a vector is ... |
| nvinvfval 28411 | Function for the negative ... |
| nvm 28412 | Vector subtraction in term... |
| nvmval 28413 | Value of vector subtractio... |
| nvmval2 28414 | Value of vector subtractio... |
| nvmfval 28415 | Value of the function for ... |
| nvmf 28416 | Mapping for the vector sub... |
| nvmcl 28417 | Closure law for the vector... |
| nvnnncan1 28418 | Cancellation law for vecto... |
| nvmdi 28419 | Distributive law for scala... |
| nvnegneg 28420 | Double negative of a vecto... |
| nvmul0or 28421 | If a scalar product is zer... |
| nvrinv 28422 | A vector minus itself. (C... |
| nvlinv 28423 | Minus a vector plus itself... |
| nvpncan2 28424 | Cancellation law for vecto... |
| nvpncan 28425 | Cancellation law for vecto... |
| nvaddsub 28426 | Commutative/associative la... |
| nvnpcan 28427 | Cancellation law for a nor... |
| nvaddsub4 28428 | Rearrangement of 4 terms i... |
| nvmeq0 28429 | The difference between two... |
| nvmid 28430 | A vector minus itself is t... |
| nvf 28431 | Mapping for the norm funct... |
| nvcl 28432 | The norm of a normed compl... |
| nvcli 28433 | The norm of a normed compl... |
| nvs 28434 | Proportionality property o... |
| nvsge0 28435 | The norm of a scalar produ... |
| nvm1 28436 | The norm of the negative o... |
| nvdif 28437 | The norm of the difference... |
| nvpi 28438 | The norm of a vector plus ... |
| nvz0 28439 | The norm of a zero vector ... |
| nvz 28440 | The norm of a vector is ze... |
| nvtri 28441 | Triangle inequality for th... |
| nvmtri 28442 | Triangle inequality for th... |
| nvabs 28443 | Norm difference property o... |
| nvge0 28444 | The norm of a normed compl... |
| nvgt0 28445 | A nonzero norm is positive... |
| nv1 28446 | From any nonzero vector, c... |
| nvop 28447 | A complex inner product sp... |
| cnnv 28448 | The set of complex numbers... |
| cnnvg 28449 | The vector addition (group... |
| cnnvba 28450 | The base set of the normed... |
| cnnvs 28451 | The scalar product operati... |
| cnnvnm 28452 | The norm operation of the ... |
| cnnvm 28453 | The vector subtraction ope... |
| elimnv 28454 | Hypothesis elimination lem... |
| elimnvu 28455 | Hypothesis elimination lem... |
| imsval 28456 | Value of the induced metri... |
| imsdval 28457 | Value of the induced metri... |
| imsdval2 28458 | Value of the distance func... |
| nvnd 28459 | The norm of a normed compl... |
| imsdf 28460 | Mapping for the induced me... |
| imsmetlem 28461 | Lemma for ~ imsmet . (Con... |
| imsmet 28462 | The induced metric of a no... |
| imsxmet 28463 | The induced metric of a no... |
| cnims 28464 | The metric induced on the ... |
| vacn 28465 | Vector addition is jointly... |
| nmcvcn 28466 | The norm of a normed compl... |
| nmcnc 28467 | The norm of a normed compl... |
| smcnlem 28468 | Lemma for ~ smcn . (Contr... |
| smcn 28469 | Scalar multiplication is j... |
| vmcn 28470 | Vector subtraction is join... |
| dipfval 28473 | The inner product function... |
| ipval 28474 | Value of the inner product... |
| ipval2lem2 28475 | Lemma for ~ ipval3 . (Con... |
| ipval2lem3 28476 | Lemma for ~ ipval3 . (Con... |
| ipval2lem4 28477 | Lemma for ~ ipval3 . (Con... |
| ipval2 28478 | Expansion of the inner pro... |
| 4ipval2 28479 | Four times the inner produ... |
| ipval3 28480 | Expansion of the inner pro... |
| ipidsq 28481 | The inner product of a vec... |
| ipnm 28482 | Norm expressed in terms of... |
| dipcl 28483 | An inner product is a comp... |
| ipf 28484 | Mapping for the inner prod... |
| dipcj 28485 | The complex conjugate of a... |
| ipipcj 28486 | An inner product times its... |
| diporthcom 28487 | Orthogonality (meaning inn... |
| dip0r 28488 | Inner product with a zero ... |
| dip0l 28489 | Inner product with a zero ... |
| ipz 28490 | The inner product of a vec... |
| dipcn 28491 | Inner product is jointly c... |
| sspval 28494 | The set of all subspaces o... |
| isssp 28495 | The predicate "is a subspa... |
| sspid 28496 | A normed complex vector sp... |
| sspnv 28497 | A subspace is a normed com... |
| sspba 28498 | The base set of a subspace... |
| sspg 28499 | Vector addition on a subsp... |
| sspgval 28500 | Vector addition on a subsp... |
| ssps 28501 | Scalar multiplication on a... |
| sspsval 28502 | Scalar multiplication on a... |
| sspmlem 28503 | Lemma for ~ sspm and other... |
| sspmval 28504 | Vector addition on a subsp... |
| sspm 28505 | Vector subtraction on a su... |
| sspz 28506 | The zero vector of a subsp... |
| sspn 28507 | The norm on a subspace is ... |
| sspnval 28508 | The norm on a subspace in ... |
| sspimsval 28509 | The induced metric on a su... |
| sspims 28510 | The induced metric on a su... |
| lnoval 28523 | The set of linear operator... |
| islno 28524 | The predicate "is a linear... |
| lnolin 28525 | Basic linearity property o... |
| lnof 28526 | A linear operator is a map... |
| lno0 28527 | The value of a linear oper... |
| lnocoi 28528 | The composition of two lin... |
| lnoadd 28529 | Addition property of a lin... |
| lnosub 28530 | Subtraction property of a ... |
| lnomul 28531 | Scalar multiplication prop... |
| nvo00 28532 | Two ways to express a zero... |
| nmoofval 28533 | The operator norm function... |
| nmooval 28534 | The operator norm function... |
| nmosetre 28535 | The set in the supremum of... |
| nmosetn0 28536 | The set in the supremum of... |
| nmoxr 28537 | The norm of an operator is... |
| nmooge0 28538 | The norm of an operator is... |
| nmorepnf 28539 | The norm of an operator is... |
| nmoreltpnf 28540 | The norm of any operator i... |
| nmogtmnf 28541 | The norm of an operator is... |
| nmoolb 28542 | A lower bound for an opera... |
| nmoubi 28543 | An upper bound for an oper... |
| nmoub3i 28544 | An upper bound for an oper... |
| nmoub2i 28545 | An upper bound for an oper... |
| nmobndi 28546 | Two ways to express that a... |
| nmounbi 28547 | Two ways two express that ... |
| nmounbseqi 28548 | An unbounded operator dete... |
| nmounbseqiALT 28549 | Alternate shorter proof of... |
| nmobndseqi 28550 | A bounded sequence determi... |
| nmobndseqiALT 28551 | Alternate shorter proof of... |
| bloval 28552 | The class of bounded linea... |
| isblo 28553 | The predicate "is a bounde... |
| isblo2 28554 | The predicate "is a bounde... |
| bloln 28555 | A bounded operator is a li... |
| blof 28556 | A bounded operator is an o... |
| nmblore 28557 | The norm of a bounded oper... |
| 0ofval 28558 | The zero operator between ... |
| 0oval 28559 | Value of the zero operator... |
| 0oo 28560 | The zero operator is an op... |
| 0lno 28561 | The zero operator is linea... |
| nmoo0 28562 | The operator norm of the z... |
| 0blo 28563 | The zero operator is a bou... |
| nmlno0lem 28564 | Lemma for ~ nmlno0i . (Co... |
| nmlno0i 28565 | The norm of a linear opera... |
| nmlno0 28566 | The norm of a linear opera... |
| nmlnoubi 28567 | An upper bound for the ope... |
| nmlnogt0 28568 | The norm of a nonzero line... |
| lnon0 28569 | The domain of a nonzero li... |
| nmblolbii 28570 | A lower bound for the norm... |
| nmblolbi 28571 | A lower bound for the norm... |
| isblo3i 28572 | The predicate "is a bounde... |
| blo3i 28573 | Properties that determine ... |
| blometi 28574 | Upper bound for the distan... |
| blocnilem 28575 | Lemma for ~ blocni and ~ l... |
| blocni 28576 | A linear operator is conti... |
| lnocni 28577 | If a linear operator is co... |
| blocn 28578 | A linear operator is conti... |
| blocn2 28579 | A bounded linear operator ... |
| ajfval 28580 | The adjoint function. (Co... |
| hmoval 28581 | The set of Hermitian (self... |
| ishmo 28582 | The predicate "is a hermit... |
| phnv 28585 | Every complex inner produc... |
| phrel 28586 | The class of all complex i... |
| phnvi 28587 | Every complex inner produc... |
| isphg 28588 | The predicate "is a comple... |
| phop 28589 | A complex inner product sp... |
| cncph 28590 | The set of complex numbers... |
| elimph 28591 | Hypothesis elimination lem... |
| elimphu 28592 | Hypothesis elimination lem... |
| isph 28593 | The predicate "is an inner... |
| phpar2 28594 | The parallelogram law for ... |
| phpar 28595 | The parallelogram law for ... |
| ip0i 28596 | A slight variant of Equati... |
| ip1ilem 28597 | Lemma for ~ ip1i . (Contr... |
| ip1i 28598 | Equation 6.47 of [Ponnusam... |
| ip2i 28599 | Equation 6.48 of [Ponnusam... |
| ipdirilem 28600 | Lemma for ~ ipdiri . (Con... |
| ipdiri 28601 | Distributive law for inner... |
| ipasslem1 28602 | Lemma for ~ ipassi . Show... |
| ipasslem2 28603 | Lemma for ~ ipassi . Show... |
| ipasslem3 28604 | Lemma for ~ ipassi . Show... |
| ipasslem4 28605 | Lemma for ~ ipassi . Show... |
| ipasslem5 28606 | Lemma for ~ ipassi . Show... |
| ipasslem7 28607 | Lemma for ~ ipassi . Show... |
| ipasslem8 28608 | Lemma for ~ ipassi . By ~... |
| ipasslem9 28609 | Lemma for ~ ipassi . Conc... |
| ipasslem10 28610 | Lemma for ~ ipassi . Show... |
| ipasslem11 28611 | Lemma for ~ ipassi . Show... |
| ipassi 28612 | Associative law for inner ... |
| dipdir 28613 | Distributive law for inner... |
| dipdi 28614 | Distributive law for inner... |
| ip2dii 28615 | Inner product of two sums.... |
| dipass 28616 | Associative law for inner ... |
| dipassr 28617 | "Associative" law for seco... |
| dipassr2 28618 | "Associative" law for inne... |
| dipsubdir 28619 | Distributive law for inner... |
| dipsubdi 28620 | Distributive law for inner... |
| pythi 28621 | The Pythagorean theorem fo... |
| siilem1 28622 | Lemma for ~ sii . (Contri... |
| siilem2 28623 | Lemma for ~ sii . (Contri... |
| siii 28624 | Inference from ~ sii . (C... |
| sii 28625 | Schwarz inequality. Part ... |
| ipblnfi 28626 | A function ` F ` generated... |
| ip2eqi 28627 | Two vectors are equal iff ... |
| phoeqi 28628 | A condition implying that ... |
| ajmoi 28629 | Every operator has at most... |
| ajfuni 28630 | The adjoint function is a ... |
| ajfun 28631 | The adjoint function is a ... |
| ajval 28632 | Value of the adjoint funct... |
| iscbn 28635 | A complex Banach space is ... |
| cbncms 28636 | The induced metric on comp... |
| bnnv 28637 | Every complex Banach space... |
| bnrel 28638 | The class of all complex B... |
| bnsscmcl 28639 | A subspace of a Banach spa... |
| cnbn 28640 | The set of complex numbers... |
| ubthlem1 28641 | Lemma for ~ ubth . The fu... |
| ubthlem2 28642 | Lemma for ~ ubth . Given ... |
| ubthlem3 28643 | Lemma for ~ ubth . Prove ... |
| ubth 28644 | Uniform Boundedness Theore... |
| minvecolem1 28645 | Lemma for ~ minveco . The... |
| minvecolem2 28646 | Lemma for ~ minveco . Any... |
| minvecolem3 28647 | Lemma for ~ minveco . The... |
| minvecolem4a 28648 | Lemma for ~ minveco . ` F ... |
| minvecolem4b 28649 | Lemma for ~ minveco . The... |
| minvecolem4c 28650 | Lemma for ~ minveco . The... |
| minvecolem4 28651 | Lemma for ~ minveco . The... |
| minvecolem5 28652 | Lemma for ~ minveco . Dis... |
| minvecolem6 28653 | Lemma for ~ minveco . Any... |
| minvecolem7 28654 | Lemma for ~ minveco . Sin... |
| minveco 28655 | Minimizing vector theorem,... |
| ishlo 28658 | The predicate "is a comple... |
| hlobn 28659 | Every complex Hilbert spac... |
| hlph 28660 | Every complex Hilbert spac... |
| hlrel 28661 | The class of all complex H... |
| hlnv 28662 | Every complex Hilbert spac... |
| hlnvi 28663 | Every complex Hilbert spac... |
| hlvc 28664 | Every complex Hilbert spac... |
| hlcmet 28665 | The induced metric on a co... |
| hlmet 28666 | The induced metric on a co... |
| hlpar2 28667 | The parallelogram law sati... |
| hlpar 28668 | The parallelogram law sati... |
| hlex 28669 | The base set of a Hilbert ... |
| hladdf 28670 | Mapping for Hilbert space ... |
| hlcom 28671 | Hilbert space vector addit... |
| hlass 28672 | Hilbert space vector addit... |
| hl0cl 28673 | The Hilbert space zero vec... |
| hladdid 28674 | Hilbert space addition wit... |
| hlmulf 28675 | Mapping for Hilbert space ... |
| hlmulid 28676 | Hilbert space scalar multi... |
| hlmulass 28677 | Hilbert space scalar multi... |
| hldi 28678 | Hilbert space scalar multi... |
| hldir 28679 | Hilbert space scalar multi... |
| hlmul0 28680 | Hilbert space scalar multi... |
| hlipf 28681 | Mapping for Hilbert space ... |
| hlipcj 28682 | Conjugate law for Hilbert ... |
| hlipdir 28683 | Distributive law for Hilbe... |
| hlipass 28684 | Associative law for Hilber... |
| hlipgt0 28685 | The inner product of a Hil... |
| hlcompl 28686 | Completeness of a Hilbert ... |
| cnchl 28687 | The set of complex numbers... |
| htthlem 28688 | Lemma for ~ htth . The co... |
| htth 28689 | Hellinger-Toeplitz Theorem... |
| The list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here | |
| h2hva 28745 | The group (addition) opera... |
| h2hsm 28746 | The scalar product operati... |
| h2hnm 28747 | The norm function of Hilbe... |
| h2hvs 28748 | The vector subtraction ope... |
| h2hmetdval 28749 | Value of the distance func... |
| h2hcau 28750 | The Cauchy sequences of Hi... |
| h2hlm 28751 | The limit sequences of Hil... |
| axhilex-zf 28752 | Derive axiom ~ ax-hilex fr... |
| axhfvadd-zf 28753 | Derive axiom ~ ax-hfvadd f... |
| axhvcom-zf 28754 | Derive axiom ~ ax-hvcom fr... |
| axhvass-zf 28755 | Derive axiom ~ ax-hvass fr... |
| axhv0cl-zf 28756 | Derive axiom ~ ax-hv0cl fr... |
| axhvaddid-zf 28757 | Derive axiom ~ ax-hvaddid ... |
| axhfvmul-zf 28758 | Derive axiom ~ ax-hfvmul f... |
| axhvmulid-zf 28759 | Derive axiom ~ ax-hvmulid ... |
| axhvmulass-zf 28760 | Derive axiom ~ ax-hvmulass... |
| axhvdistr1-zf 28761 | Derive axiom ~ ax-hvdistr1... |
| axhvdistr2-zf 28762 | Derive axiom ~ ax-hvdistr2... |
| axhvmul0-zf 28763 | Derive axiom ~ ax-hvmul0 f... |
| axhfi-zf 28764 | Derive axiom ~ ax-hfi from... |
| axhis1-zf 28765 | Derive axiom ~ ax-his1 fro... |
| axhis2-zf 28766 | Derive axiom ~ ax-his2 fro... |
| axhis3-zf 28767 | Derive axiom ~ ax-his3 fro... |
| axhis4-zf 28768 | Derive axiom ~ ax-his4 fro... |
| axhcompl-zf 28769 | Derive axiom ~ ax-hcompl f... |
| hvmulex 28782 | The Hilbert space scalar p... |
| hvaddcl 28783 | Closure of vector addition... |
| hvmulcl 28784 | Closure of scalar multipli... |
| hvmulcli 28785 | Closure inference for scal... |
| hvsubf 28786 | Mapping domain and codomai... |
| hvsubval 28787 | Value of vector subtractio... |
| hvsubcl 28788 | Closure of vector subtract... |
| hvaddcli 28789 | Closure of vector addition... |
| hvcomi 28790 | Commutation of vector addi... |
| hvsubvali 28791 | Value of vector subtractio... |
| hvsubcli 28792 | Closure of vector subtract... |
| ifhvhv0 28793 | Prove ` if ( A e. ~H , A ,... |
| hvaddid2 28794 | Addition with the zero vec... |
| hvmul0 28795 | Scalar multiplication with... |
| hvmul0or 28796 | If a scalar product is zer... |
| hvsubid 28797 | Subtraction of a vector fr... |
| hvnegid 28798 | Addition of negative of a ... |
| hv2neg 28799 | Two ways to express the ne... |
| hvaddid2i 28800 | Addition with the zero vec... |
| hvnegidi 28801 | Addition of negative of a ... |
| hv2negi 28802 | Two ways to express the ne... |
| hvm1neg 28803 | Convert minus one times a ... |
| hvaddsubval 28804 | Value of vector addition i... |
| hvadd32 28805 | Commutative/associative la... |
| hvadd12 28806 | Commutative/associative la... |
| hvadd4 28807 | Hilbert vector space addit... |
| hvsub4 28808 | Hilbert vector space addit... |
| hvaddsub12 28809 | Commutative/associative la... |
| hvpncan 28810 | Addition/subtraction cance... |
| hvpncan2 28811 | Addition/subtraction cance... |
| hvaddsubass 28812 | Associativity of sum and d... |
| hvpncan3 28813 | Subtraction and addition o... |
| hvmulcom 28814 | Scalar multiplication comm... |
| hvsubass 28815 | Hilbert vector space assoc... |
| hvsub32 28816 | Hilbert vector space commu... |
| hvmulassi 28817 | Scalar multiplication asso... |
| hvmulcomi 28818 | Scalar multiplication comm... |
| hvmul2negi 28819 | Double negative in scalar ... |
| hvsubdistr1 28820 | Scalar multiplication dist... |
| hvsubdistr2 28821 | Scalar multiplication dist... |
| hvdistr1i 28822 | Scalar multiplication dist... |
| hvsubdistr1i 28823 | Scalar multiplication dist... |
| hvassi 28824 | Hilbert vector space assoc... |
| hvadd32i 28825 | Hilbert vector space commu... |
| hvsubassi 28826 | Hilbert vector space assoc... |
| hvsub32i 28827 | Hilbert vector space commu... |
| hvadd12i 28828 | Hilbert vector space commu... |
| hvadd4i 28829 | Hilbert vector space addit... |
| hvsubsub4i 28830 | Hilbert vector space addit... |
| hvsubsub4 28831 | Hilbert vector space addit... |
| hv2times 28832 | Two times a vector. (Cont... |
| hvnegdii 28833 | Distribution of negative o... |
| hvsubeq0i 28834 | If the difference between ... |
| hvsubcan2i 28835 | Vector cancellation law. ... |
| hvaddcani 28836 | Cancellation law for vecto... |
| hvsubaddi 28837 | Relationship between vecto... |
| hvnegdi 28838 | Distribution of negative o... |
| hvsubeq0 28839 | If the difference between ... |
| hvaddeq0 28840 | If the sum of two vectors ... |
| hvaddcan 28841 | Cancellation law for vecto... |
| hvaddcan2 28842 | Cancellation law for vecto... |
| hvmulcan 28843 | Cancellation law for scala... |
| hvmulcan2 28844 | Cancellation law for scala... |
| hvsubcan 28845 | Cancellation law for vecto... |
| hvsubcan2 28846 | Cancellation law for vecto... |
| hvsub0 28847 | Subtraction of a zero vect... |
| hvsubadd 28848 | Relationship between vecto... |
| hvaddsub4 28849 | Hilbert vector space addit... |
| hicl 28851 | Closure of inner product. ... |
| hicli 28852 | Closure inference for inne... |
| his5 28857 | Associative law for inner ... |
| his52 28858 | Associative law for inner ... |
| his35 28859 | Move scalar multiplication... |
| his35i 28860 | Move scalar multiplication... |
| his7 28861 | Distributive law for inner... |
| hiassdi 28862 | Distributive/associative l... |
| his2sub 28863 | Distributive law for inner... |
| his2sub2 28864 | Distributive law for inner... |
| hire 28865 | A necessary and sufficient... |
| hiidrcl 28866 | Real closure of inner prod... |
| hi01 28867 | Inner product with the 0 v... |
| hi02 28868 | Inner product with the 0 v... |
| hiidge0 28869 | Inner product with self is... |
| his6 28870 | Zero inner product with se... |
| his1i 28871 | Conjugate law for inner pr... |
| abshicom 28872 | Commuted inner products ha... |
| hial0 28873 | A vector whose inner produ... |
| hial02 28874 | A vector whose inner produ... |
| hisubcomi 28875 | Two vector subtractions si... |
| hi2eq 28876 | Lemma used to prove equali... |
| hial2eq 28877 | Two vectors whose inner pr... |
| hial2eq2 28878 | Two vectors whose inner pr... |
| orthcom 28879 | Orthogonality commutes. (... |
| normlem0 28880 | Lemma used to derive prope... |
| normlem1 28881 | Lemma used to derive prope... |
| normlem2 28882 | Lemma used to derive prope... |
| normlem3 28883 | Lemma used to derive prope... |
| normlem4 28884 | Lemma used to derive prope... |
| normlem5 28885 | Lemma used to derive prope... |
| normlem6 28886 | Lemma used to derive prope... |
| normlem7 28887 | Lemma used to derive prope... |
| normlem8 28888 | Lemma used to derive prope... |
| normlem9 28889 | Lemma used to derive prope... |
| normlem7tALT 28890 | Lemma used to derive prope... |
| bcseqi 28891 | Equality case of Bunjakova... |
| normlem9at 28892 | Lemma used to derive prope... |
| dfhnorm2 28893 | Alternate definition of th... |
| normf 28894 | The norm function maps fro... |
| normval 28895 | The value of the norm of a... |
| normcl 28896 | Real closure of the norm o... |
| normge0 28897 | The norm of a vector is no... |
| normgt0 28898 | The norm of nonzero vector... |
| norm0 28899 | The norm of a zero vector.... |
| norm-i 28900 | Theorem 3.3(i) of [Beran] ... |
| normne0 28901 | A norm is nonzero iff its ... |
| normcli 28902 | Real closure of the norm o... |
| normsqi 28903 | The square of a norm. (Co... |
| norm-i-i 28904 | Theorem 3.3(i) of [Beran] ... |
| normsq 28905 | The square of a norm. (Co... |
| normsub0i 28906 | Two vectors are equal iff ... |
| normsub0 28907 | Two vectors are equal iff ... |
| norm-ii-i 28908 | Triangle inequality for no... |
| norm-ii 28909 | Triangle inequality for no... |
| norm-iii-i 28910 | Theorem 3.3(iii) of [Beran... |
| norm-iii 28911 | Theorem 3.3(iii) of [Beran... |
| normsubi 28912 | Negative doesn't change th... |
| normpythi 28913 | Analogy to Pythagorean the... |
| normsub 28914 | Swapping order of subtract... |
| normneg 28915 | The norm of a vector equal... |
| normpyth 28916 | Analogy to Pythagorean the... |
| normpyc 28917 | Corollary to Pythagorean t... |
| norm3difi 28918 | Norm of differences around... |
| norm3adifii 28919 | Norm of differences around... |
| norm3lem 28920 | Lemma involving norm of di... |
| norm3dif 28921 | Norm of differences around... |
| norm3dif2 28922 | Norm of differences around... |
| norm3lemt 28923 | Lemma involving norm of di... |
| norm3adifi 28924 | Norm of differences around... |
| normpari 28925 | Parallelogram law for norm... |
| normpar 28926 | Parallelogram law for norm... |
| normpar2i 28927 | Corollary of parallelogram... |
| polid2i 28928 | Generalized polarization i... |
| polidi 28929 | Polarization identity. Re... |
| polid 28930 | Polarization identity. Re... |
| hilablo 28931 | Hilbert space vector addit... |
| hilid 28932 | The group identity element... |
| hilvc 28933 | Hilbert space is a complex... |
| hilnormi 28934 | Hilbert space norm in term... |
| hilhhi 28935 | Deduce the structure of Hi... |
| hhnv 28936 | Hilbert space is a normed ... |
| hhva 28937 | The group (addition) opera... |
| hhba 28938 | The base set of Hilbert sp... |
| hh0v 28939 | The zero vector of Hilbert... |
| hhsm 28940 | The scalar product operati... |
| hhvs 28941 | The vector subtraction ope... |
| hhnm 28942 | The norm function of Hilbe... |
| hhims 28943 | The induced metric of Hilb... |
| hhims2 28944 | Hilbert space distance met... |
| hhmet 28945 | The induced metric of Hilb... |
| hhxmet 28946 | The induced metric of Hilb... |
| hhmetdval 28947 | Value of the distance func... |
| hhip 28948 | The inner product operatio... |
| hhph 28949 | The Hilbert space of the H... |
| bcsiALT 28950 | Bunjakovaskij-Cauchy-Schwa... |
| bcsiHIL 28951 | Bunjakovaskij-Cauchy-Schwa... |
| bcs 28952 | Bunjakovaskij-Cauchy-Schwa... |
| bcs2 28953 | Corollary of the Bunjakova... |
| bcs3 28954 | Corollary of the Bunjakova... |
| hcau 28955 | Member of the set of Cauch... |
| hcauseq 28956 | A Cauchy sequences on a Hi... |
| hcaucvg 28957 | A Cauchy sequence on a Hil... |
| seq1hcau 28958 | A sequence on a Hilbert sp... |
| hlimi 28959 | Express the predicate: Th... |
| hlimseqi 28960 | A sequence with a limit on... |
| hlimveci 28961 | Closure of the limit of a ... |
| hlimconvi 28962 | Convergence of a sequence ... |
| hlim2 28963 | The limit of a sequence on... |
| hlimadd 28964 | Limit of the sum of two se... |
| hilmet 28965 | The Hilbert space norm det... |
| hilxmet 28966 | The Hilbert space norm det... |
| hilmetdval 28967 | Value of the distance func... |
| hilims 28968 | Hilbert space distance met... |
| hhcau 28969 | The Cauchy sequences of Hi... |
| hhlm 28970 | The limit sequences of Hil... |
| hhcmpl 28971 | Lemma used for derivation ... |
| hilcompl 28972 | Lemma used for derivation ... |
| hhcms 28974 | The Hilbert space induced ... |
| hhhl 28975 | The Hilbert space structur... |
| hilcms 28976 | The Hilbert space norm det... |
| hilhl 28977 | The Hilbert space of the H... |
| issh 28979 | Subspace ` H ` of a Hilber... |
| issh2 28980 | Subspace ` H ` of a Hilber... |
| shss 28981 | A subspace is a subset of ... |
| shel 28982 | A member of a subspace of ... |
| shex 28983 | The set of subspaces of a ... |
| shssii 28984 | A closed subspace of a Hil... |
| sheli 28985 | A member of a subspace of ... |
| shelii 28986 | A member of a subspace of ... |
| sh0 28987 | The zero vector belongs to... |
| shaddcl 28988 | Closure of vector addition... |
| shmulcl 28989 | Closure of vector scalar m... |
| issh3 28990 | Subspace ` H ` of a Hilber... |
| shsubcl 28991 | Closure of vector subtract... |
| isch 28993 | Closed subspace ` H ` of a... |
| isch2 28994 | Closed subspace ` H ` of a... |
| chsh 28995 | A closed subspace is a sub... |
| chsssh 28996 | Closed subspaces are subsp... |
| chex 28997 | The set of closed subspace... |
| chshii 28998 | A closed subspace is a sub... |
| ch0 28999 | The zero vector belongs to... |
| chss 29000 | A closed subspace of a Hil... |
| chel 29001 | A member of a closed subsp... |
| chssii 29002 | A closed subspace of a Hil... |
| cheli 29003 | A member of a closed subsp... |
| chelii 29004 | A member of a closed subsp... |
| chlimi 29005 | The limit property of a cl... |
| hlim0 29006 | The zero sequence in Hilbe... |
| hlimcaui 29007 | If a sequence in Hilbert s... |
| hlimf 29008 | Function-like behavior of ... |
| hlimuni 29009 | A Hilbert space sequence c... |
| hlimreui 29010 | The limit of a Hilbert spa... |
| hlimeui 29011 | The limit of a Hilbert spa... |
| isch3 29012 | A Hilbert subspace is clos... |
| chcompl 29013 | Completeness of a closed s... |
| helch 29014 | The unit Hilbert lattice e... |
| ifchhv 29015 | Prove ` if ( A e. CH , A ,... |
| helsh 29016 | Hilbert space is a subspac... |
| shsspwh 29017 | Subspaces are subsets of H... |
| chsspwh 29018 | Closed subspaces are subse... |
| hsn0elch 29019 | The zero subspace belongs ... |
| norm1 29020 | From any nonzero Hilbert s... |
| norm1exi 29021 | A normalized vector exists... |
| norm1hex 29022 | A normalized vector can ex... |
| elch0 29025 | Membership in zero for clo... |
| h0elch 29026 | The zero subspace is a clo... |
| h0elsh 29027 | The zero subspace is a sub... |
| hhssva 29028 | The vector addition operat... |
| hhsssm 29029 | The scalar multiplication ... |
| hhssnm 29030 | The norm operation on a su... |
| issubgoilem 29031 | Lemma for ~ hhssabloilem .... |
| hhssabloilem 29032 | Lemma for ~ hhssabloi . F... |
| hhssabloi 29033 | Abelian group property of ... |
| hhssablo 29034 | Abelian group property of ... |
| hhssnv 29035 | Normed complex vector spac... |
| hhssnvt 29036 | Normed complex vector spac... |
| hhsst 29037 | A member of ` SH ` is a su... |
| hhshsslem1 29038 | Lemma for ~ hhsssh . (Con... |
| hhshsslem2 29039 | Lemma for ~ hhsssh . (Con... |
| hhsssh 29040 | The predicate " ` H ` is a... |
| hhsssh2 29041 | The predicate " ` H ` is a... |
| hhssba 29042 | The base set of a subspace... |
| hhssvs 29043 | The vector subtraction ope... |
| hhssvsf 29044 | Mapping of the vector subt... |
| hhssims 29045 | Induced metric of a subspa... |
| hhssims2 29046 | Induced metric of a subspa... |
| hhssmet 29047 | Induced metric of a subspa... |
| hhssmetdval 29048 | Value of the distance func... |
| hhsscms 29049 | The induced metric of a cl... |
| hhssbnOLD 29050 | Obsolete version of ~ cssb... |
| ocval 29051 | Value of orthogonal comple... |
| ocel 29052 | Membership in orthogonal c... |
| shocel 29053 | Membership in orthogonal c... |
| ocsh 29054 | The orthogonal complement ... |
| shocsh 29055 | The orthogonal complement ... |
| ocss 29056 | An orthogonal complement i... |
| shocss 29057 | An orthogonal complement i... |
| occon 29058 | Contraposition law for ort... |
| occon2 29059 | Double contraposition for ... |
| occon2i 29060 | Double contraposition for ... |
| oc0 29061 | The zero vector belongs to... |
| ocorth 29062 | Members of a subset and it... |
| shocorth 29063 | Members of a subspace and ... |
| ococss 29064 | Inclusion in complement of... |
| shococss 29065 | Inclusion in complement of... |
| shorth 29066 | Members of orthogonal subs... |
| ocin 29067 | Intersection of a Hilbert ... |
| occon3 29068 | Hilbert lattice contraposi... |
| ocnel 29069 | A nonzero vector in the co... |
| chocvali 29070 | Value of the orthogonal co... |
| shuni 29071 | Two subspaces with trivial... |
| chocunii 29072 | Lemma for uniqueness part ... |
| pjhthmo 29073 | Projection Theorem, unique... |
| occllem 29074 | Lemma for ~ occl . (Contr... |
| occl 29075 | Closure of complement of H... |
| shoccl 29076 | Closure of complement of H... |
| choccl 29077 | Closure of complement of H... |
| choccli 29078 | Closure of ` CH ` orthocom... |
| shsval 29083 | Value of subspace sum of t... |
| shsss 29084 | The subspace sum is a subs... |
| shsel 29085 | Membership in the subspace... |
| shsel3 29086 | Membership in the subspace... |
| shseli 29087 | Membership in subspace sum... |
| shscli 29088 | Closure of subspace sum. ... |
| shscl 29089 | Closure of subspace sum. ... |
| shscom 29090 | Commutative law for subspa... |
| shsva 29091 | Vector sum belongs to subs... |
| shsel1 29092 | A subspace sum contains a ... |
| shsel2 29093 | A subspace sum contains a ... |
| shsvs 29094 | Vector subtraction belongs... |
| shsub1 29095 | Subspace sum is an upper b... |
| shsub2 29096 | Subspace sum is an upper b... |
| choc0 29097 | The orthocomplement of the... |
| choc1 29098 | The orthocomplement of the... |
| chocnul 29099 | Orthogonal complement of t... |
| shintcli 29100 | Closure of intersection of... |
| shintcl 29101 | The intersection of a none... |
| chintcli 29102 | The intersection of a none... |
| chintcl 29103 | The intersection (infimum)... |
| spanval 29104 | Value of the linear span o... |
| hsupval 29105 | Value of supremum of set o... |
| chsupval 29106 | The value of the supremum ... |
| spancl 29107 | The span of a subset of Hi... |
| elspancl 29108 | A member of a span is a ve... |
| shsupcl 29109 | Closure of the subspace su... |
| hsupcl 29110 | Closure of supremum of set... |
| chsupcl 29111 | Closure of supremum of sub... |
| hsupss 29112 | Subset relation for suprem... |
| chsupss 29113 | Subset relation for suprem... |
| hsupunss 29114 | The union of a set of Hilb... |
| chsupunss 29115 | The union of a set of clos... |
| spanss2 29116 | A subset of Hilbert space ... |
| shsupunss 29117 | The union of a set of subs... |
| spanid 29118 | A subspace of Hilbert spac... |
| spanss 29119 | Ordering relationship for ... |
| spanssoc 29120 | The span of a subset of Hi... |
| sshjval 29121 | Value of join for subsets ... |
| shjval 29122 | Value of join in ` SH ` . ... |
| chjval 29123 | Value of join in ` CH ` . ... |
| chjvali 29124 | Value of join in ` CH ` . ... |
| sshjval3 29125 | Value of join for subsets ... |
| sshjcl 29126 | Closure of join for subset... |
| shjcl 29127 | Closure of join in ` SH ` ... |
| chjcl 29128 | Closure of join in ` CH ` ... |
| shjcom 29129 | Commutative law for Hilber... |
| shless 29130 | Subset implies subset of s... |
| shlej1 29131 | Add disjunct to both sides... |
| shlej2 29132 | Add disjunct to both sides... |
| shincli 29133 | Closure of intersection of... |
| shscomi 29134 | Commutative law for subspa... |
| shsvai 29135 | Vector sum belongs to subs... |
| shsel1i 29136 | A subspace sum contains a ... |
| shsel2i 29137 | A subspace sum contains a ... |
| shsvsi 29138 | Vector subtraction belongs... |
| shunssi 29139 | Union is smaller than subs... |
| shunssji 29140 | Union is smaller than Hilb... |
| shsleji 29141 | Subspace sum is smaller th... |
| shjcomi 29142 | Commutative law for join i... |
| shsub1i 29143 | Subspace sum is an upper b... |
| shsub2i 29144 | Subspace sum is an upper b... |
| shub1i 29145 | Hilbert lattice join is an... |
| shjcli 29146 | Closure of ` CH ` join. (... |
| shjshcli 29147 | ` SH ` closure of join. (... |
| shlessi 29148 | Subset implies subset of s... |
| shlej1i 29149 | Add disjunct to both sides... |
| shlej2i 29150 | Add disjunct to both sides... |
| shslej 29151 | Subspace sum is smaller th... |
| shincl 29152 | Closure of intersection of... |
| shub1 29153 | Hilbert lattice join is an... |
| shub2 29154 | A subspace is a subset of ... |
| shsidmi 29155 | Idempotent law for Hilbert... |
| shslubi 29156 | The least upper bound law ... |
| shlesb1i 29157 | Hilbert lattice ordering i... |
| shsval2i 29158 | An alternate way to expres... |
| shsval3i 29159 | An alternate way to expres... |
| shmodsi 29160 | The modular law holds for ... |
| shmodi 29161 | The modular law is implied... |
| pjhthlem1 29162 | Lemma for ~ pjhth . (Cont... |
| pjhthlem2 29163 | Lemma for ~ pjhth . (Cont... |
| pjhth 29164 | Projection Theorem: Any H... |
| pjhtheu 29165 | Projection Theorem: Any H... |
| pjhfval 29167 | The value of the projectio... |
| pjhval 29168 | Value of a projection. (C... |
| pjpreeq 29169 | Equality with a projection... |
| pjeq 29170 | Equality with a projection... |
| axpjcl 29171 | Closure of a projection in... |
| pjhcl 29172 | Closure of a projection in... |
| omlsilem 29173 | Lemma for orthomodular law... |
| omlsii 29174 | Subspace inference form of... |
| omlsi 29175 | Subspace form of orthomodu... |
| ococi 29176 | Complement of complement o... |
| ococ 29177 | Complement of complement o... |
| dfch2 29178 | Alternate definition of th... |
| ococin 29179 | The double complement is t... |
| hsupval2 29180 | Alternate definition of su... |
| chsupval2 29181 | The value of the supremum ... |
| sshjval2 29182 | Value of join in the set o... |
| chsupid 29183 | A subspace is the supremum... |
| chsupsn 29184 | Value of supremum of subse... |
| shlub 29185 | Hilbert lattice join is th... |
| shlubi 29186 | Hilbert lattice join is th... |
| pjhtheu2 29187 | Uniqueness of ` y ` for th... |
| pjcli 29188 | Closure of a projection in... |
| pjhcli 29189 | Closure of a projection in... |
| pjpjpre 29190 | Decomposition of a vector ... |
| axpjpj 29191 | Decomposition of a vector ... |
| pjclii 29192 | Closure of a projection in... |
| pjhclii 29193 | Closure of a projection in... |
| pjpj0i 29194 | Decomposition of a vector ... |
| pjpji 29195 | Decomposition of a vector ... |
| pjpjhth 29196 | Projection Theorem: Any H... |
| pjpjhthi 29197 | Projection Theorem: Any H... |
| pjop 29198 | Orthocomplement projection... |
| pjpo 29199 | Projection in terms of ort... |
| pjopi 29200 | Orthocomplement projection... |
| pjpoi 29201 | Projection in terms of ort... |
| pjoc1i 29202 | Projection of a vector in ... |
| pjchi 29203 | Projection of a vector in ... |
| pjoccl 29204 | The part of a vector that ... |
| pjoc1 29205 | Projection of a vector in ... |
| pjomli 29206 | Subspace form of orthomodu... |
| pjoml 29207 | Subspace form of orthomodu... |
| pjococi 29208 | Proof of orthocomplement t... |
| pjoc2i 29209 | Projection of a vector in ... |
| pjoc2 29210 | Projection of a vector in ... |
| sh0le 29211 | The zero subspace is the s... |
| ch0le 29212 | The zero subspace is the s... |
| shle0 29213 | No subspace is smaller tha... |
| chle0 29214 | No Hilbert lattice element... |
| chnlen0 29215 | A Hilbert lattice element ... |
| ch0pss 29216 | The zero subspace is a pro... |
| orthin 29217 | The intersection of orthog... |
| ssjo 29218 | The lattice join of a subs... |
| shne0i 29219 | A nonzero subspace has a n... |
| shs0i 29220 | Hilbert subspace sum with ... |
| shs00i 29221 | Two subspaces are zero iff... |
| ch0lei 29222 | The closed subspace zero i... |
| chle0i 29223 | No Hilbert closed subspace... |
| chne0i 29224 | A nonzero closed subspace ... |
| chocini 29225 | Intersection of a closed s... |
| chj0i 29226 | Join with lattice zero in ... |
| chm1i 29227 | Meet with lattice one in `... |
| chjcli 29228 | Closure of ` CH ` join. (... |
| chsleji 29229 | Subspace sum is smaller th... |
| chseli 29230 | Membership in subspace sum... |
| chincli 29231 | Closure of Hilbert lattice... |
| chsscon3i 29232 | Hilbert lattice contraposi... |
| chsscon1i 29233 | Hilbert lattice contraposi... |
| chsscon2i 29234 | Hilbert lattice contraposi... |
| chcon2i 29235 | Hilbert lattice contraposi... |
| chcon1i 29236 | Hilbert lattice contraposi... |
| chcon3i 29237 | Hilbert lattice contraposi... |
| chunssji 29238 | Union is smaller than ` CH... |
| chjcomi 29239 | Commutative law for join i... |
| chub1i 29240 | ` CH ` join is an upper bo... |
| chub2i 29241 | ` CH ` join is an upper bo... |
| chlubi 29242 | Hilbert lattice join is th... |
| chlubii 29243 | Hilbert lattice join is th... |
| chlej1i 29244 | Add join to both sides of ... |
| chlej2i 29245 | Add join to both sides of ... |
| chlej12i 29246 | Add join to both sides of ... |
| chlejb1i 29247 | Hilbert lattice ordering i... |
| chdmm1i 29248 | De Morgan's law for meet i... |
| chdmm2i 29249 | De Morgan's law for meet i... |
| chdmm3i 29250 | De Morgan's law for meet i... |
| chdmm4i 29251 | De Morgan's law for meet i... |
| chdmj1i 29252 | De Morgan's law for join i... |
| chdmj2i 29253 | De Morgan's law for join i... |
| chdmj3i 29254 | De Morgan's law for join i... |
| chdmj4i 29255 | De Morgan's law for join i... |
| chnlei 29256 | Equivalent expressions for... |
| chjassi 29257 | Associative law for Hilber... |
| chj00i 29258 | Two Hilbert lattice elemen... |
| chjoi 29259 | The join of a closed subsp... |
| chj1i 29260 | Join with Hilbert lattice ... |
| chm0i 29261 | Meet with Hilbert lattice ... |
| chm0 29262 | Meet with Hilbert lattice ... |
| shjshsi 29263 | Hilbert lattice join equal... |
| shjshseli 29264 | A closed subspace sum equa... |
| chne0 29265 | A nonzero closed subspace ... |
| chocin 29266 | Intersection of a closed s... |
| chssoc 29267 | A closed subspace less tha... |
| chj0 29268 | Join with Hilbert lattice ... |
| chslej 29269 | Subspace sum is smaller th... |
| chincl 29270 | Closure of Hilbert lattice... |
| chsscon3 29271 | Hilbert lattice contraposi... |
| chsscon1 29272 | Hilbert lattice contraposi... |
| chsscon2 29273 | Hilbert lattice contraposi... |
| chpsscon3 29274 | Hilbert lattice contraposi... |
| chpsscon1 29275 | Hilbert lattice contraposi... |
| chpsscon2 29276 | Hilbert lattice contraposi... |
| chjcom 29277 | Commutative law for Hilber... |
| chub1 29278 | Hilbert lattice join is gr... |
| chub2 29279 | Hilbert lattice join is gr... |
| chlub 29280 | Hilbert lattice join is th... |
| chlej1 29281 | Add join to both sides of ... |
| chlej2 29282 | Add join to both sides of ... |
| chlejb1 29283 | Hilbert lattice ordering i... |
| chlejb2 29284 | Hilbert lattice ordering i... |
| chnle 29285 | Equivalent expressions for... |
| chjo 29286 | The join of a closed subsp... |
| chabs1 29287 | Hilbert lattice absorption... |
| chabs2 29288 | Hilbert lattice absorption... |
| chabs1i 29289 | Hilbert lattice absorption... |
| chabs2i 29290 | Hilbert lattice absorption... |
| chjidm 29291 | Idempotent law for Hilbert... |
| chjidmi 29292 | Idempotent law for Hilbert... |
| chj12i 29293 | A rearrangement of Hilbert... |
| chj4i 29294 | Rearrangement of the join ... |
| chjjdiri 29295 | Hilbert lattice join distr... |
| chdmm1 29296 | De Morgan's law for meet i... |
| chdmm2 29297 | De Morgan's law for meet i... |
| chdmm3 29298 | De Morgan's law for meet i... |
| chdmm4 29299 | De Morgan's law for meet i... |
| chdmj1 29300 | De Morgan's law for join i... |
| chdmj2 29301 | De Morgan's law for join i... |
| chdmj3 29302 | De Morgan's law for join i... |
| chdmj4 29303 | De Morgan's law for join i... |
| chjass 29304 | Associative law for Hilber... |
| chj12 29305 | A rearrangement of Hilbert... |
| chj4 29306 | Rearrangement of the join ... |
| ledii 29307 | An ortholattice is distrib... |
| lediri 29308 | An ortholattice is distrib... |
| lejdii 29309 | An ortholattice is distrib... |
| lejdiri 29310 | An ortholattice is distrib... |
| ledi 29311 | An ortholattice is distrib... |
| spansn0 29312 | The span of the singleton ... |
| span0 29313 | The span of the empty set ... |
| elspani 29314 | Membership in the span of ... |
| spanuni 29315 | The span of a union is the... |
| spanun 29316 | The span of a union is the... |
| sshhococi 29317 | The join of two Hilbert sp... |
| hne0 29318 | Hilbert space has a nonzer... |
| chsup0 29319 | The supremum of the empty ... |
| h1deoi 29320 | Membership in orthocomplem... |
| h1dei 29321 | Membership in 1-dimensiona... |
| h1did 29322 | A generating vector belong... |
| h1dn0 29323 | A nonzero vector generates... |
| h1de2i 29324 | Membership in 1-dimensiona... |
| h1de2bi 29325 | Membership in 1-dimensiona... |
| h1de2ctlem 29326 | Lemma for ~ h1de2ci . (Co... |
| h1de2ci 29327 | Membership in 1-dimensiona... |
| spansni 29328 | The span of a singleton in... |
| elspansni 29329 | Membership in the span of ... |
| spansn 29330 | The span of a singleton in... |
| spansnch 29331 | The span of a Hilbert spac... |
| spansnsh 29332 | The span of a Hilbert spac... |
| spansnchi 29333 | The span of a singleton in... |
| spansnid 29334 | A vector belongs to the sp... |
| spansnmul 29335 | A scalar product with a ve... |
| elspansncl 29336 | A member of a span of a si... |
| elspansn 29337 | Membership in the span of ... |
| elspansn2 29338 | Membership in the span of ... |
| spansncol 29339 | The singletons of collinea... |
| spansneleqi 29340 | Membership relation implie... |
| spansneleq 29341 | Membership relation that i... |
| spansnss 29342 | The span of the singleton ... |
| elspansn3 29343 | A member of the span of th... |
| elspansn4 29344 | A span membership conditio... |
| elspansn5 29345 | A vector belonging to both... |
| spansnss2 29346 | The span of the singleton ... |
| normcan 29347 | Cancellation-type law that... |
| pjspansn 29348 | A projection on the span o... |
| spansnpji 29349 | A subset of Hilbert space ... |
| spanunsni 29350 | The span of the union of a... |
| spanpr 29351 | The span of a pair of vect... |
| h1datomi 29352 | A 1-dimensional subspace i... |
| h1datom 29353 | A 1-dimensional subspace i... |
| cmbr 29355 | Binary relation expressing... |
| pjoml2i 29356 | Variation of orthomodular ... |
| pjoml3i 29357 | Variation of orthomodular ... |
| pjoml4i 29358 | Variation of orthomodular ... |
| pjoml5i 29359 | The orthomodular law. Rem... |
| pjoml6i 29360 | An equivalent of the ortho... |
| cmbri 29361 | Binary relation expressing... |
| cmcmlem 29362 | Commutation is symmetric. ... |
| cmcmi 29363 | Commutation is symmetric. ... |
| cmcm2i 29364 | Commutation with orthocomp... |
| cmcm3i 29365 | Commutation with orthocomp... |
| cmcm4i 29366 | Commutation with orthocomp... |
| cmbr2i 29367 | Alternate definition of th... |
| cmcmii 29368 | Commutation is symmetric. ... |
| cmcm2ii 29369 | Commutation with orthocomp... |
| cmcm3ii 29370 | Commutation with orthocomp... |
| cmbr3i 29371 | Alternate definition for t... |
| cmbr4i 29372 | Alternate definition for t... |
| lecmi 29373 | Comparable Hilbert lattice... |
| lecmii 29374 | Comparable Hilbert lattice... |
| cmj1i 29375 | A Hilbert lattice element ... |
| cmj2i 29376 | A Hilbert lattice element ... |
| cmm1i 29377 | A Hilbert lattice element ... |
| cmm2i 29378 | A Hilbert lattice element ... |
| cmbr3 29379 | Alternate definition for t... |
| cm0 29380 | The zero Hilbert lattice e... |
| cmidi 29381 | The commutes relation is r... |
| pjoml2 29382 | Variation of orthomodular ... |
| pjoml3 29383 | Variation of orthomodular ... |
| pjoml5 29384 | The orthomodular law. Rem... |
| cmcm 29385 | Commutation is symmetric. ... |
| cmcm3 29386 | Commutation with orthocomp... |
| cmcm2 29387 | Commutation with orthocomp... |
| lecm 29388 | Comparable Hilbert lattice... |
| fh1 29389 | Foulis-Holland Theorem. I... |
| fh2 29390 | Foulis-Holland Theorem. I... |
| cm2j 29391 | A lattice element that com... |
| fh1i 29392 | Foulis-Holland Theorem. I... |
| fh2i 29393 | Foulis-Holland Theorem. I... |
| fh3i 29394 | Variation of the Foulis-Ho... |
| fh4i 29395 | Variation of the Foulis-Ho... |
| cm2ji 29396 | A lattice element that com... |
| cm2mi 29397 | A lattice element that com... |
| qlax1i 29398 | One of the equations showi... |
| qlax2i 29399 | One of the equations showi... |
| qlax3i 29400 | One of the equations showi... |
| qlax4i 29401 | One of the equations showi... |
| qlax5i 29402 | One of the equations showi... |
| qlaxr1i 29403 | One of the conditions show... |
| qlaxr2i 29404 | One of the conditions show... |
| qlaxr4i 29405 | One of the conditions show... |
| qlaxr5i 29406 | One of the conditions show... |
| qlaxr3i 29407 | A variation of the orthomo... |
| chscllem1 29408 | Lemma for ~ chscl . (Cont... |
| chscllem2 29409 | Lemma for ~ chscl . (Cont... |
| chscllem3 29410 | Lemma for ~ chscl . (Cont... |
| chscllem4 29411 | Lemma for ~ chscl . (Cont... |
| chscl 29412 | The subspace sum of two cl... |
| osumi 29413 | If two closed subspaces of... |
| osumcori 29414 | Corollary of ~ osumi . (C... |
| osumcor2i 29415 | Corollary of ~ osumi , sho... |
| osum 29416 | If two closed subspaces of... |
| spansnji 29417 | The subspace sum of a clos... |
| spansnj 29418 | The subspace sum of a clos... |
| spansnscl 29419 | The subspace sum of a clos... |
| sumspansn 29420 | The sum of two vectors bel... |
| spansnm0i 29421 | The meet of different one-... |
| nonbooli 29422 | A Hilbert lattice with two... |
| spansncvi 29423 | Hilbert space has the cove... |
| spansncv 29424 | Hilbert space has the cove... |
| 5oalem1 29425 | Lemma for orthoarguesian l... |
| 5oalem2 29426 | Lemma for orthoarguesian l... |
| 5oalem3 29427 | Lemma for orthoarguesian l... |
| 5oalem4 29428 | Lemma for orthoarguesian l... |
| 5oalem5 29429 | Lemma for orthoarguesian l... |
| 5oalem6 29430 | Lemma for orthoarguesian l... |
| 5oalem7 29431 | Lemma for orthoarguesian l... |
| 5oai 29432 | Orthoarguesian law 5OA. Th... |
| 3oalem1 29433 | Lemma for 3OA (weak) ortho... |
| 3oalem2 29434 | Lemma for 3OA (weak) ortho... |
| 3oalem3 29435 | Lemma for 3OA (weak) ortho... |
| 3oalem4 29436 | Lemma for 3OA (weak) ortho... |
| 3oalem5 29437 | Lemma for 3OA (weak) ortho... |
| 3oalem6 29438 | Lemma for 3OA (weak) ortho... |
| 3oai 29439 | 3OA (weak) orthoarguesian ... |
| pjorthi 29440 | Projection components on o... |
| pjch1 29441 | Property of identity proje... |
| pjo 29442 | The orthogonal projection.... |
| pjcompi 29443 | Component of a projection.... |
| pjidmi 29444 | A projection is idempotent... |
| pjadjii 29445 | A projection is self-adjoi... |
| pjaddii 29446 | Projection of vector sum i... |
| pjinormii 29447 | The inner product of a pro... |
| pjmulii 29448 | Projection of (scalar) pro... |
| pjsubii 29449 | Projection of vector diffe... |
| pjsslem 29450 | Lemma for subset relations... |
| pjss2i 29451 | Subset relationship for pr... |
| pjssmii 29452 | Projection meet property. ... |
| pjssge0ii 29453 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormii 29454 | Theorem 4.5(v)<->(vi) of [... |
| pjcji 29455 | The projection on a subspa... |
| pjadji 29456 | A projection is self-adjoi... |
| pjaddi 29457 | Projection of vector sum i... |
| pjinormi 29458 | The inner product of a pro... |
| pjsubi 29459 | Projection of vector diffe... |
| pjmuli 29460 | Projection of scalar produ... |
| pjige0i 29461 | The inner product of a pro... |
| pjige0 29462 | The inner product of a pro... |
| pjcjt2 29463 | The projection on a subspa... |
| pj0i 29464 | The projection of the zero... |
| pjch 29465 | Projection of a vector in ... |
| pjid 29466 | The projection of a vector... |
| pjvec 29467 | The set of vectors belongi... |
| pjocvec 29468 | The set of vectors belongi... |
| pjocini 29469 | Membership of projection i... |
| pjini 29470 | Membership of projection i... |
| pjjsi 29471 | A sufficient condition for... |
| pjfni 29472 | Functionality of a project... |
| pjrni 29473 | The range of a projection.... |
| pjfoi 29474 | A projection maps onto its... |
| pjfi 29475 | The mapping of a projectio... |
| pjvi 29476 | The value of a projection ... |
| pjhfo 29477 | A projection maps onto its... |
| pjrn 29478 | The range of a projection.... |
| pjhf 29479 | The mapping of a projectio... |
| pjfn 29480 | Functionality of a project... |
| pjsumi 29481 | The projection on a subspa... |
| pj11i 29482 | One-to-one correspondence ... |
| pjdsi 29483 | Vector decomposition into ... |
| pjds3i 29484 | Vector decomposition into ... |
| pj11 29485 | One-to-one correspondence ... |
| pjmfn 29486 | Functionality of the proje... |
| pjmf1 29487 | The projector function map... |
| pjoi0 29488 | The inner product of proje... |
| pjoi0i 29489 | The inner product of proje... |
| pjopythi 29490 | Pythagorean theorem for pr... |
| pjopyth 29491 | Pythagorean theorem for pr... |
| pjnormi 29492 | The norm of the projection... |
| pjpythi 29493 | Pythagorean theorem for pr... |
| pjneli 29494 | If a vector does not belon... |
| pjnorm 29495 | The norm of the projection... |
| pjpyth 29496 | Pythagorean theorem for pr... |
| pjnel 29497 | If a vector does not belon... |
| pjnorm2 29498 | A vector belongs to the su... |
| mayete3i 29499 | Mayet's equation E_3. Par... |
| mayetes3i 29500 | Mayet's equation E^*_3, de... |
| hosmval 29506 | Value of the sum of two Hi... |
| hommval 29507 | Value of the scalar produc... |
| hodmval 29508 | Value of the difference of... |
| hfsmval 29509 | Value of the sum of two Hi... |
| hfmmval 29510 | Value of the scalar produc... |
| hosval 29511 | Value of the sum of two Hi... |
| homval 29512 | Value of the scalar produc... |
| hodval 29513 | Value of the difference of... |
| hfsval 29514 | Value of the sum of two Hi... |
| hfmval 29515 | Value of the scalar produc... |
| hoscl 29516 | Closure of the sum of two ... |
| homcl 29517 | Closure of the scalar prod... |
| hodcl 29518 | Closure of the difference ... |
| ho0val 29521 | Value of the zero Hilbert ... |
| ho0f 29522 | Functionality of the zero ... |
| df0op2 29523 | Alternate definition of Hi... |
| dfiop2 29524 | Alternate definition of Hi... |
| hoif 29525 | Functionality of the Hilbe... |
| hoival 29526 | The value of the Hilbert s... |
| hoico1 29527 | Composition with the Hilbe... |
| hoico2 29528 | Composition with the Hilbe... |
| hoaddcl 29529 | The sum of Hilbert space o... |
| homulcl 29530 | The scalar product of a Hi... |
| hoeq 29531 | Equality of Hilbert space ... |
| hoeqi 29532 | Equality of Hilbert space ... |
| hoscli 29533 | Closure of Hilbert space o... |
| hodcli 29534 | Closure of Hilbert space o... |
| hocoi 29535 | Composition of Hilbert spa... |
| hococli 29536 | Closure of composition of ... |
| hocofi 29537 | Mapping of composition of ... |
| hocofni 29538 | Functionality of compositi... |
| hoaddcli 29539 | Mapping of sum of Hilbert ... |
| hosubcli 29540 | Mapping of difference of H... |
| hoaddfni 29541 | Functionality of sum of Hi... |
| hosubfni 29542 | Functionality of differenc... |
| hoaddcomi 29543 | Commutativity of sum of Hi... |
| hosubcl 29544 | Mapping of difference of H... |
| hoaddcom 29545 | Commutativity of sum of Hi... |
| hodsi 29546 | Relationship between Hilbe... |
| hoaddassi 29547 | Associativity of sum of Hi... |
| hoadd12i 29548 | Commutative/associative la... |
| hoadd32i 29549 | Commutative/associative la... |
| hocadddiri 29550 | Distributive law for Hilbe... |
| hocsubdiri 29551 | Distributive law for Hilbe... |
| ho2coi 29552 | Double composition of Hilb... |
| hoaddass 29553 | Associativity of sum of Hi... |
| hoadd32 29554 | Commutative/associative la... |
| hoadd4 29555 | Rearrangement of 4 terms i... |
| hocsubdir 29556 | Distributive law for Hilbe... |
| hoaddid1i 29557 | Sum of a Hilbert space ope... |
| hodidi 29558 | Difference of a Hilbert sp... |
| ho0coi 29559 | Composition of the zero op... |
| hoid1i 29560 | Composition of Hilbert spa... |
| hoid1ri 29561 | Composition of Hilbert spa... |
| hoaddid1 29562 | Sum of a Hilbert space ope... |
| hodid 29563 | Difference of a Hilbert sp... |
| hon0 29564 | A Hilbert space operator i... |
| hodseqi 29565 | Subtraction and addition o... |
| ho0subi 29566 | Subtraction of Hilbert spa... |
| honegsubi 29567 | Relationship between Hilbe... |
| ho0sub 29568 | Subtraction of Hilbert spa... |
| hosubid1 29569 | The zero operator subtract... |
| honegsub 29570 | Relationship between Hilbe... |
| homulid2 29571 | An operator equals its sca... |
| homco1 29572 | Associative law for scalar... |
| homulass 29573 | Scalar product associative... |
| hoadddi 29574 | Scalar product distributiv... |
| hoadddir 29575 | Scalar product reverse dis... |
| homul12 29576 | Swap first and second fact... |
| honegneg 29577 | Double negative of a Hilbe... |
| hosubneg 29578 | Relationship between opera... |
| hosubdi 29579 | Scalar product distributiv... |
| honegdi 29580 | Distribution of negative o... |
| honegsubdi 29581 | Distribution of negative o... |
| honegsubdi2 29582 | Distribution of negative o... |
| hosubsub2 29583 | Law for double subtraction... |
| hosub4 29584 | Rearrangement of 4 terms i... |
| hosubadd4 29585 | Rearrangement of 4 terms i... |
| hoaddsubass 29586 | Associative-type law for a... |
| hoaddsub 29587 | Law for operator addition ... |
| hosubsub 29588 | Law for double subtraction... |
| hosubsub4 29589 | Law for double subtraction... |
| ho2times 29590 | Two times a Hilbert space ... |
| hoaddsubassi 29591 | Associativity of sum and d... |
| hoaddsubi 29592 | Law for sum and difference... |
| hosd1i 29593 | Hilbert space operator sum... |
| hosd2i 29594 | Hilbert space operator sum... |
| hopncani 29595 | Hilbert space operator can... |
| honpcani 29596 | Hilbert space operator can... |
| hosubeq0i 29597 | If the difference between ... |
| honpncani 29598 | Hilbert space operator can... |
| ho01i 29599 | A condition implying that ... |
| ho02i 29600 | A condition implying that ... |
| hoeq1 29601 | A condition implying that ... |
| hoeq2 29602 | A condition implying that ... |
| adjmo 29603 | Every Hilbert space operat... |
| adjsym 29604 | Symmetry property of an ad... |
| eigrei 29605 | A necessary and sufficient... |
| eigre 29606 | A necessary and sufficient... |
| eigposi 29607 | A sufficient condition (fi... |
| eigorthi 29608 | A necessary and sufficient... |
| eigorth 29609 | A necessary and sufficient... |
| nmopval 29627 | Value of the norm of a Hil... |
| elcnop 29628 | Property defining a contin... |
| ellnop 29629 | Property defining a linear... |
| lnopf 29630 | A linear Hilbert space ope... |
| elbdop 29631 | Property defining a bounde... |
| bdopln 29632 | A bounded linear Hilbert s... |
| bdopf 29633 | A bounded linear Hilbert s... |
| nmopsetretALT 29634 | The set in the supremum of... |
| nmopsetretHIL 29635 | The set in the supremum of... |
| nmopsetn0 29636 | The set in the supremum of... |
| nmopxr 29637 | The norm of a Hilbert spac... |
| nmoprepnf 29638 | The norm of a Hilbert spac... |
| nmopgtmnf 29639 | The norm of a Hilbert spac... |
| nmopreltpnf 29640 | The norm of a Hilbert spac... |
| nmopre 29641 | The norm of a bounded oper... |
| elbdop2 29642 | Property defining a bounde... |
| elunop 29643 | Property defining a unitar... |
| elhmop 29644 | Property defining a Hermit... |
| hmopf 29645 | A Hermitian operator is a ... |
| hmopex 29646 | The class of Hermitian ope... |
| nmfnval 29647 | Value of the norm of a Hil... |
| nmfnsetre 29648 | The set in the supremum of... |
| nmfnsetn0 29649 | The set in the supremum of... |
| nmfnxr 29650 | The norm of any Hilbert sp... |
| nmfnrepnf 29651 | The norm of a Hilbert spac... |
| nlfnval 29652 | Value of the null space of... |
| elcnfn 29653 | Property defining a contin... |
| ellnfn 29654 | Property defining a linear... |
| lnfnf 29655 | A linear Hilbert space fun... |
| dfadj2 29656 | Alternate definition of th... |
| funadj 29657 | Functionality of the adjoi... |
| dmadjss 29658 | The domain of the adjoint ... |
| dmadjop 29659 | A member of the domain of ... |
| adjeu 29660 | Elementhood in the domain ... |
| adjval 29661 | Value of the adjoint funct... |
| adjval2 29662 | Value of the adjoint funct... |
| cnvadj 29663 | The adjoint function equal... |
| funcnvadj 29664 | The converse of the adjoin... |
| adj1o 29665 | The adjoint function maps ... |
| dmadjrn 29666 | The adjoint of an operator... |
| eigvecval 29667 | The set of eigenvectors of... |
| eigvalfval 29668 | The eigenvalues of eigenve... |
| specval 29669 | The value of the spectrum ... |
| speccl 29670 | The spectrum of an operato... |
| hhlnoi 29671 | The linear operators of Hi... |
| hhnmoi 29672 | The norm of an operator in... |
| hhbloi 29673 | A bounded linear operator ... |
| hh0oi 29674 | The zero operator in Hilbe... |
| hhcno 29675 | The continuous operators o... |
| hhcnf 29676 | The continuous functionals... |
| dmadjrnb 29677 | The adjoint of an operator... |
| nmoplb 29678 | A lower bound for an opera... |
| nmopub 29679 | An upper bound for an oper... |
| nmopub2tALT 29680 | An upper bound for an oper... |
| nmopub2tHIL 29681 | An upper bound for an oper... |
| nmopge0 29682 | The norm of any Hilbert sp... |
| nmopgt0 29683 | A linear Hilbert space ope... |
| cnopc 29684 | Basic continuity property ... |
| lnopl 29685 | Basic linearity property o... |
| unop 29686 | Basic inner product proper... |
| unopf1o 29687 | A unitary operator in Hilb... |
| unopnorm 29688 | A unitary operator is idem... |
| cnvunop 29689 | The inverse (converse) of ... |
| unopadj 29690 | The inverse (converse) of ... |
| unoplin 29691 | A unitary operator is line... |
| counop 29692 | The composition of two uni... |
| hmop 29693 | Basic inner product proper... |
| hmopre 29694 | The inner product of the v... |
| nmfnlb 29695 | A lower bound for a functi... |
| nmfnleub 29696 | An upper bound for the nor... |
| nmfnleub2 29697 | An upper bound for the nor... |
| nmfnge0 29698 | The norm of any Hilbert sp... |
| elnlfn 29699 | Membership in the null spa... |
| elnlfn2 29700 | Membership in the null spa... |
| cnfnc 29701 | Basic continuity property ... |
| lnfnl 29702 | Basic linearity property o... |
| adjcl 29703 | Closure of the adjoint of ... |
| adj1 29704 | Property of an adjoint Hil... |
| adj2 29705 | Property of an adjoint Hil... |
| adjeq 29706 | A property that determines... |
| adjadj 29707 | Double adjoint. Theorem 3... |
| adjvalval 29708 | Value of the value of the ... |
| unopadj2 29709 | The adjoint of a unitary o... |
| hmopadj 29710 | A Hermitian operator is se... |
| hmdmadj 29711 | Every Hermitian operator h... |
| hmopadj2 29712 | An operator is Hermitian i... |
| hmoplin 29713 | A Hermitian operator is li... |
| brafval 29714 | The bra of a vector, expre... |
| braval 29715 | A bra-ket juxtaposition, e... |
| braadd 29716 | Linearity property of bra ... |
| bramul 29717 | Linearity property of bra ... |
| brafn 29718 | The bra function is a func... |
| bralnfn 29719 | The Dirac bra function is ... |
| bracl 29720 | Closure of the bra functio... |
| bra0 29721 | The Dirac bra of the zero ... |
| brafnmul 29722 | Anti-linearity property of... |
| kbfval 29723 | The outer product of two v... |
| kbop 29724 | The outer product of two v... |
| kbval 29725 | The value of the operator ... |
| kbmul 29726 | Multiplication property of... |
| kbpj 29727 | If a vector ` A ` has norm... |
| eleigvec 29728 | Membership in the set of e... |
| eleigvec2 29729 | Membership in the set of e... |
| eleigveccl 29730 | Closure of an eigenvector ... |
| eigvalval 29731 | The eigenvalue of an eigen... |
| eigvalcl 29732 | An eigenvalue is a complex... |
| eigvec1 29733 | Property of an eigenvector... |
| eighmre 29734 | The eigenvalues of a Hermi... |
| eighmorth 29735 | Eigenvectors of a Hermitia... |
| nmopnegi 29736 | Value of the norm of the n... |
| lnop0 29737 | The value of a linear Hilb... |
| lnopmul 29738 | Multiplicative property of... |
| lnopli 29739 | Basic scalar product prope... |
| lnopfi 29740 | A linear Hilbert space ope... |
| lnop0i 29741 | The value of a linear Hilb... |
| lnopaddi 29742 | Additive property of a lin... |
| lnopmuli 29743 | Multiplicative property of... |
| lnopaddmuli 29744 | Sum/product property of a ... |
| lnopsubi 29745 | Subtraction property for a... |
| lnopsubmuli 29746 | Subtraction/product proper... |
| lnopmulsubi 29747 | Product/subtraction proper... |
| homco2 29748 | Move a scalar product out ... |
| idunop 29749 | The identity function (res... |
| 0cnop 29750 | The identically zero funct... |
| 0cnfn 29751 | The identically zero funct... |
| idcnop 29752 | The identity function (res... |
| idhmop 29753 | The Hilbert space identity... |
| 0hmop 29754 | The identically zero funct... |
| 0lnop 29755 | The identically zero funct... |
| 0lnfn 29756 | The identically zero funct... |
| nmop0 29757 | The norm of the zero opera... |
| nmfn0 29758 | The norm of the identicall... |
| hmopbdoptHIL 29759 | A Hermitian operator is a ... |
| hoddii 29760 | Distributive law for Hilbe... |
| hoddi 29761 | Distributive law for Hilbe... |
| nmop0h 29762 | The norm of any operator o... |
| idlnop 29763 | The identity function (res... |
| 0bdop 29764 | The identically zero opera... |
| adj0 29765 | Adjoint of the zero operat... |
| nmlnop0iALT 29766 | A linear operator with a z... |
| nmlnop0iHIL 29767 | A linear operator with a z... |
| nmlnopgt0i 29768 | A linear Hilbert space ope... |
| nmlnop0 29769 | A linear operator with a z... |
| nmlnopne0 29770 | A linear operator with a n... |
| lnopmi 29771 | The scalar product of a li... |
| lnophsi 29772 | The sum of two linear oper... |
| lnophdi 29773 | The difference of two line... |
| lnopcoi 29774 | The composition of two lin... |
| lnopco0i 29775 | The composition of a linea... |
| lnopeq0lem1 29776 | Lemma for ~ lnopeq0i . Ap... |
| lnopeq0lem2 29777 | Lemma for ~ lnopeq0i . (C... |
| lnopeq0i 29778 | A condition implying that ... |
| lnopeqi 29779 | Two linear Hilbert space o... |
| lnopeq 29780 | Two linear Hilbert space o... |
| lnopunilem1 29781 | Lemma for ~ lnopunii . (C... |
| lnopunilem2 29782 | Lemma for ~ lnopunii . (C... |
| lnopunii 29783 | If a linear operator (whos... |
| elunop2 29784 | An operator is unitary iff... |
| nmopun 29785 | Norm of a unitary Hilbert ... |
| unopbd 29786 | A unitary operator is a bo... |
| lnophmlem1 29787 | Lemma for ~ lnophmi . (Co... |
| lnophmlem2 29788 | Lemma for ~ lnophmi . (Co... |
| lnophmi 29789 | A linear operator is Hermi... |
| lnophm 29790 | A linear operator is Hermi... |
| hmops 29791 | The sum of two Hermitian o... |
| hmopm 29792 | The scalar product of a He... |
| hmopd 29793 | The difference of two Herm... |
| hmopco 29794 | The composition of two com... |
| nmbdoplbi 29795 | A lower bound for the norm... |
| nmbdoplb 29796 | A lower bound for the norm... |
| nmcexi 29797 | Lemma for ~ nmcopexi and ~... |
| nmcopexi 29798 | The norm of a continuous l... |
| nmcoplbi 29799 | A lower bound for the norm... |
| nmcopex 29800 | The norm of a continuous l... |
| nmcoplb 29801 | A lower bound for the norm... |
| nmophmi 29802 | The norm of the scalar pro... |
| bdophmi 29803 | The scalar product of a bo... |
| lnconi 29804 | Lemma for ~ lnopconi and ~... |
| lnopconi 29805 | A condition equivalent to ... |
| lnopcon 29806 | A condition equivalent to ... |
| lnopcnbd 29807 | A linear operator is conti... |
| lncnopbd 29808 | A continuous linear operat... |
| lncnbd 29809 | A continuous linear operat... |
| lnopcnre 29810 | A linear operator is conti... |
| lnfnli 29811 | Basic property of a linear... |
| lnfnfi 29812 | A linear Hilbert space fun... |
| lnfn0i 29813 | The value of a linear Hilb... |
| lnfnaddi 29814 | Additive property of a lin... |
| lnfnmuli 29815 | Multiplicative property of... |
| lnfnaddmuli 29816 | Sum/product property of a ... |
| lnfnsubi 29817 | Subtraction property for a... |
| lnfn0 29818 | The value of a linear Hilb... |
| lnfnmul 29819 | Multiplicative property of... |
| nmbdfnlbi 29820 | A lower bound for the norm... |
| nmbdfnlb 29821 | A lower bound for the norm... |
| nmcfnexi 29822 | The norm of a continuous l... |
| nmcfnlbi 29823 | A lower bound for the norm... |
| nmcfnex 29824 | The norm of a continuous l... |
| nmcfnlb 29825 | A lower bound of the norm ... |
| lnfnconi 29826 | A condition equivalent to ... |
| lnfncon 29827 | A condition equivalent to ... |
| lnfncnbd 29828 | A linear functional is con... |
| imaelshi 29829 | The image of a subspace un... |
| rnelshi 29830 | The range of a linear oper... |
| nlelshi 29831 | The null space of a linear... |
| nlelchi 29832 | The null space of a contin... |
| riesz3i 29833 | A continuous linear functi... |
| riesz4i 29834 | A continuous linear functi... |
| riesz4 29835 | A continuous linear functi... |
| riesz1 29836 | Part 1 of the Riesz repres... |
| riesz2 29837 | Part 2 of the Riesz repres... |
| cnlnadjlem1 29838 | Lemma for ~ cnlnadji (Theo... |
| cnlnadjlem2 29839 | Lemma for ~ cnlnadji . ` G... |
| cnlnadjlem3 29840 | Lemma for ~ cnlnadji . By... |
| cnlnadjlem4 29841 | Lemma for ~ cnlnadji . Th... |
| cnlnadjlem5 29842 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem6 29843 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem7 29844 | Lemma for ~ cnlnadji . He... |
| cnlnadjlem8 29845 | Lemma for ~ cnlnadji . ` F... |
| cnlnadjlem9 29846 | Lemma for ~ cnlnadji . ` F... |
| cnlnadji 29847 | Every continuous linear op... |
| cnlnadjeui 29848 | Every continuous linear op... |
| cnlnadjeu 29849 | Every continuous linear op... |
| cnlnadj 29850 | Every continuous linear op... |
| cnlnssadj 29851 | Every continuous linear Hi... |
| bdopssadj 29852 | Every bounded linear Hilbe... |
| bdopadj 29853 | Every bounded linear Hilbe... |
| adjbdln 29854 | The adjoint of a bounded l... |
| adjbdlnb 29855 | An operator is bounded and... |
| adjbd1o 29856 | The mapping of adjoints of... |
| adjlnop 29857 | The adjoint of an operator... |
| adjsslnop 29858 | Every operator with an adj... |
| nmopadjlei 29859 | Property of the norm of an... |
| nmopadjlem 29860 | Lemma for ~ nmopadji . (C... |
| nmopadji 29861 | Property of the norm of an... |
| adjeq0 29862 | An operator is zero iff it... |
| adjmul 29863 | The adjoint of the scalar ... |
| adjadd 29864 | The adjoint of the sum of ... |
| nmoptrii 29865 | Triangle inequality for th... |
| nmopcoi 29866 | Upper bound for the norm o... |
| bdophsi 29867 | The sum of two bounded lin... |
| bdophdi 29868 | The difference between two... |
| bdopcoi 29869 | The composition of two bou... |
| nmoptri2i 29870 | Triangle-type inequality f... |
| adjcoi 29871 | The adjoint of a compositi... |
| nmopcoadji 29872 | The norm of an operator co... |
| nmopcoadj2i 29873 | The norm of an operator co... |
| nmopcoadj0i 29874 | An operator composed with ... |
| unierri 29875 | If we approximate a chain ... |
| branmfn 29876 | The norm of the bra functi... |
| brabn 29877 | The bra of a vector is a b... |
| rnbra 29878 | The set of bras equals the... |
| bra11 29879 | The bra function maps vect... |
| bracnln 29880 | A bra is a continuous line... |
| cnvbraval 29881 | Value of the converse of t... |
| cnvbracl 29882 | Closure of the converse of... |
| cnvbrabra 29883 | The converse bra of the br... |
| bracnvbra 29884 | The bra of the converse br... |
| bracnlnval 29885 | The vector that a continuo... |
| cnvbramul 29886 | Multiplication property of... |
| kbass1 29887 | Dirac bra-ket associative ... |
| kbass2 29888 | Dirac bra-ket associative ... |
| kbass3 29889 | Dirac bra-ket associative ... |
| kbass4 29890 | Dirac bra-ket associative ... |
| kbass5 29891 | Dirac bra-ket associative ... |
| kbass6 29892 | Dirac bra-ket associative ... |
| leopg 29893 | Ordering relation for posi... |
| leop 29894 | Ordering relation for oper... |
| leop2 29895 | Ordering relation for oper... |
| leop3 29896 | Operator ordering in terms... |
| leoppos 29897 | Binary relation defining a... |
| leoprf2 29898 | The ordering relation for ... |
| leoprf 29899 | The ordering relation for ... |
| leopsq 29900 | The square of a Hermitian ... |
| 0leop 29901 | The zero operator is a pos... |
| idleop 29902 | The identity operator is a... |
| leopadd 29903 | The sum of two positive op... |
| leopmuli 29904 | The scalar product of a no... |
| leopmul 29905 | The scalar product of a po... |
| leopmul2i 29906 | Scalar product applied to ... |
| leoptri 29907 | The positive operator orde... |
| leoptr 29908 | The positive operator orde... |
| leopnmid 29909 | A bounded Hermitian operat... |
| nmopleid 29910 | A nonzero, bounded Hermiti... |
| opsqrlem1 29911 | Lemma for opsqri . (Contr... |
| opsqrlem2 29912 | Lemma for opsqri . ` F `` ... |
| opsqrlem3 29913 | Lemma for opsqri . (Contr... |
| opsqrlem4 29914 | Lemma for opsqri . (Contr... |
| opsqrlem5 29915 | Lemma for opsqri . (Contr... |
| opsqrlem6 29916 | Lemma for opsqri . (Contr... |
| pjhmopi 29917 | A projector is a Hermitian... |
| pjlnopi 29918 | A projector is a linear op... |
| pjnmopi 29919 | The operator norm of a pro... |
| pjbdlni 29920 | A projector is a bounded l... |
| pjhmop 29921 | A projection is a Hermitia... |
| hmopidmchi 29922 | An idempotent Hermitian op... |
| hmopidmpji 29923 | An idempotent Hermitian op... |
| hmopidmch 29924 | An idempotent Hermitian op... |
| hmopidmpj 29925 | An idempotent Hermitian op... |
| pjsdii 29926 | Distributive law for Hilbe... |
| pjddii 29927 | Distributive law for Hilbe... |
| pjsdi2i 29928 | Chained distributive law f... |
| pjcoi 29929 | Composition of projections... |
| pjcocli 29930 | Closure of composition of ... |
| pjcohcli 29931 | Closure of composition of ... |
| pjadjcoi 29932 | Adjoint of composition of ... |
| pjcofni 29933 | Functionality of compositi... |
| pjss1coi 29934 | Subset relationship for pr... |
| pjss2coi 29935 | Subset relationship for pr... |
| pjssmi 29936 | Projection meet property. ... |
| pjssge0i 29937 | Theorem 4.5(iv)->(v) of [B... |
| pjdifnormi 29938 | Theorem 4.5(v)<->(vi) of [... |
| pjnormssi 29939 | Theorem 4.5(i)<->(vi) of [... |
| pjorthcoi 29940 | Composition of projections... |
| pjscji 29941 | The projection of orthogon... |
| pjssumi 29942 | The projection on a subspa... |
| pjssposi 29943 | Projector ordering can be ... |
| pjordi 29944 | The definition of projecto... |
| pjssdif2i 29945 | The projection subspace of... |
| pjssdif1i 29946 | A necessary and sufficient... |
| pjimai 29947 | The image of a projection.... |
| pjidmcoi 29948 | A projection is idempotent... |
| pjoccoi 29949 | Composition of projections... |
| pjtoi 29950 | Subspace sum of projection... |
| pjoci 29951 | Projection of orthocomplem... |
| pjidmco 29952 | A projection operator is i... |
| dfpjop 29953 | Definition of projection o... |
| pjhmopidm 29954 | Two ways to express the se... |
| elpjidm 29955 | A projection operator is i... |
| elpjhmop 29956 | A projection operator is H... |
| 0leopj 29957 | A projector is a positive ... |
| pjadj2 29958 | A projector is self-adjoin... |
| pjadj3 29959 | A projector is self-adjoin... |
| elpjch 29960 | Reconstruction of the subs... |
| elpjrn 29961 | Reconstruction of the subs... |
| pjinvari 29962 | A closed subspace ` H ` wi... |
| pjin1i 29963 | Lemma for Theorem 1.22 of ... |
| pjin2i 29964 | Lemma for Theorem 1.22 of ... |
| pjin3i 29965 | Lemma for Theorem 1.22 of ... |
| pjclem1 29966 | Lemma for projection commu... |
| pjclem2 29967 | Lemma for projection commu... |
| pjclem3 29968 | Lemma for projection commu... |
| pjclem4a 29969 | Lemma for projection commu... |
| pjclem4 29970 | Lemma for projection commu... |
| pjci 29971 | Two subspaces commute iff ... |
| pjcmul1i 29972 | A necessary and sufficient... |
| pjcmul2i 29973 | The projection subspace of... |
| pjcohocli 29974 | Closure of composition of ... |
| pjadj2coi 29975 | Adjoint of double composit... |
| pj2cocli 29976 | Closure of double composit... |
| pj3lem1 29977 | Lemma for projection tripl... |
| pj3si 29978 | Stronger projection triple... |
| pj3i 29979 | Projection triplet theorem... |
| pj3cor1i 29980 | Projection triplet corolla... |
| pjs14i 29981 | Theorem S-14 of Watanabe, ... |
| isst 29984 | Property of a state. (Con... |
| ishst 29985 | Property of a complex Hilb... |
| sticl 29986 | ` [ 0 , 1 ] ` closure of t... |
| stcl 29987 | Real closure of the value ... |
| hstcl 29988 | Closure of the value of a ... |
| hst1a 29989 | Unit value of a Hilbert-sp... |
| hstel2 29990 | Properties of a Hilbert-sp... |
| hstorth 29991 | Orthogonality property of ... |
| hstosum 29992 | Orthogonal sum property of... |
| hstoc 29993 | Sum of a Hilbert-space-val... |
| hstnmoc 29994 | Sum of norms of a Hilbert-... |
| stge0 29995 | The value of a state is no... |
| stle1 29996 | The value of a state is le... |
| hstle1 29997 | The norm of the value of a... |
| hst1h 29998 | The norm of a Hilbert-spac... |
| hst0h 29999 | The norm of a Hilbert-spac... |
| hstpyth 30000 | Pythagorean property of a ... |
| hstle 30001 | Ordering property of a Hil... |
| hstles 30002 | Ordering property of a Hil... |
| hstoh 30003 | A Hilbert-space-valued sta... |
| hst0 30004 | A Hilbert-space-valued sta... |
| sthil 30005 | The value of a state at th... |
| stj 30006 | The value of a state on a ... |
| sto1i 30007 | The state of a subspace pl... |
| sto2i 30008 | The state of the orthocomp... |
| stge1i 30009 | If a state is greater than... |
| stle0i 30010 | If a state is less than or... |
| stlei 30011 | Ordering law for states. ... |
| stlesi 30012 | Ordering law for states. ... |
| stji1i 30013 | Join of components of Sasa... |
| stm1i 30014 | State of component of unit... |
| stm1ri 30015 | State of component of unit... |
| stm1addi 30016 | Sum of states whose meet i... |
| staddi 30017 | If the sum of 2 states is ... |
| stm1add3i 30018 | Sum of states whose meet i... |
| stadd3i 30019 | If the sum of 3 states is ... |
| st0 30020 | The state of the zero subs... |
| strlem1 30021 | Lemma for strong state the... |
| strlem2 30022 | Lemma for strong state the... |
| strlem3a 30023 | Lemma for strong state the... |
| strlem3 30024 | Lemma for strong state the... |
| strlem4 30025 | Lemma for strong state the... |
| strlem5 30026 | Lemma for strong state the... |
| strlem6 30027 | Lemma for strong state the... |
| stri 30028 | Strong state theorem. The... |
| strb 30029 | Strong state theorem (bidi... |
| hstrlem2 30030 | Lemma for strong set of CH... |
| hstrlem3a 30031 | Lemma for strong set of CH... |
| hstrlem3 30032 | Lemma for strong set of CH... |
| hstrlem4 30033 | Lemma for strong set of CH... |
| hstrlem5 30034 | Lemma for strong set of CH... |
| hstrlem6 30035 | Lemma for strong set of CH... |
| hstri 30036 | Hilbert space admits a str... |
| hstrbi 30037 | Strong CH-state theorem (b... |
| largei 30038 | A Hilbert lattice admits a... |
| jplem1 30039 | Lemma for Jauch-Piron theo... |
| jplem2 30040 | Lemma for Jauch-Piron theo... |
| jpi 30041 | The function ` S ` , that ... |
| golem1 30042 | Lemma for Godowski's equat... |
| golem2 30043 | Lemma for Godowski's equat... |
| goeqi 30044 | Godowski's equation, shown... |
| stcltr1i 30045 | Property of a strong class... |
| stcltr2i 30046 | Property of a strong class... |
| stcltrlem1 30047 | Lemma for strong classical... |
| stcltrlem2 30048 | Lemma for strong classical... |
| stcltrthi 30049 | Theorem for classically st... |
| cvbr 30053 | Binary relation expressing... |
| cvbr2 30054 | Binary relation expressing... |
| cvcon3 30055 | Contraposition law for the... |
| cvpss 30056 | The covers relation implie... |
| cvnbtwn 30057 | The covers relation implie... |
| cvnbtwn2 30058 | The covers relation implie... |
| cvnbtwn3 30059 | The covers relation implie... |
| cvnbtwn4 30060 | The covers relation implie... |
| cvnsym 30061 | The covers relation is not... |
| cvnref 30062 | The covers relation is not... |
| cvntr 30063 | The covers relation is not... |
| spansncv2 30064 | Hilbert space has the cove... |
| mdbr 30065 | Binary relation expressing... |
| mdi 30066 | Consequence of the modular... |
| mdbr2 30067 | Binary relation expressing... |
| mdbr3 30068 | Binary relation expressing... |
| mdbr4 30069 | Binary relation expressing... |
| dmdbr 30070 | Binary relation expressing... |
| dmdmd 30071 | The dual modular pair prop... |
| mddmd 30072 | The modular pair property ... |
| dmdi 30073 | Consequence of the dual mo... |
| dmdbr2 30074 | Binary relation expressing... |
| dmdi2 30075 | Consequence of the dual mo... |
| dmdbr3 30076 | Binary relation expressing... |
| dmdbr4 30077 | Binary relation expressing... |
| dmdi4 30078 | Consequence of the dual mo... |
| dmdbr5 30079 | Binary relation expressing... |
| mddmd2 30080 | Relationship between modul... |
| mdsl0 30081 | A sublattice condition tha... |
| ssmd1 30082 | Ordering implies the modul... |
| ssmd2 30083 | Ordering implies the modul... |
| ssdmd1 30084 | Ordering implies the dual ... |
| ssdmd2 30085 | Ordering implies the dual ... |
| dmdsl3 30086 | Sublattice mapping for a d... |
| mdsl3 30087 | Sublattice mapping for a m... |
| mdslle1i 30088 | Order preservation of the ... |
| mdslle2i 30089 | Order preservation of the ... |
| mdslj1i 30090 | Join preservation of the o... |
| mdslj2i 30091 | Meet preservation of the r... |
| mdsl1i 30092 | If the modular pair proper... |
| mdsl2i 30093 | If the modular pair proper... |
| mdsl2bi 30094 | If the modular pair proper... |
| cvmdi 30095 | The covering property impl... |
| mdslmd1lem1 30096 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem2 30097 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem3 30098 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1lem4 30099 | Lemma for ~ mdslmd1i . (C... |
| mdslmd1i 30100 | Preservation of the modula... |
| mdslmd2i 30101 | Preservation of the modula... |
| mdsldmd1i 30102 | Preservation of the dual m... |
| mdslmd3i 30103 | Modular pair conditions th... |
| mdslmd4i 30104 | Modular pair condition tha... |
| csmdsymi 30105 | Cross-symmetry implies M-s... |
| mdexchi 30106 | An exchange lemma for modu... |
| cvmd 30107 | The covering property impl... |
| cvdmd 30108 | The covering property impl... |
| ela 30110 | Atoms in a Hilbert lattice... |
| elat2 30111 | Expanded membership relati... |
| elatcv0 30112 | A Hilbert lattice element ... |
| atcv0 30113 | An atom covers the zero su... |
| atssch 30114 | Atoms are a subset of the ... |
| atelch 30115 | An atom is a Hilbert latti... |
| atne0 30116 | An atom is not the Hilbert... |
| atss 30117 | A lattice element smaller ... |
| atsseq 30118 | Two atoms in a subset rela... |
| atcveq0 30119 | A Hilbert lattice element ... |
| h1da 30120 | A 1-dimensional subspace i... |
| spansna 30121 | The span of the singleton ... |
| sh1dle 30122 | A 1-dimensional subspace i... |
| ch1dle 30123 | A 1-dimensional subspace i... |
| atom1d 30124 | The 1-dimensional subspace... |
| superpos 30125 | Superposition Principle. ... |
| chcv1 30126 | The Hilbert lattice has th... |
| chcv2 30127 | The Hilbert lattice has th... |
| chjatom 30128 | The join of a closed subsp... |
| shatomici 30129 | The lattice of Hilbert sub... |
| hatomici 30130 | The Hilbert lattice is ato... |
| hatomic 30131 | A Hilbert lattice is atomi... |
| shatomistici 30132 | The lattice of Hilbert sub... |
| hatomistici 30133 | ` CH ` is atomistic, i.e. ... |
| chpssati 30134 | Two Hilbert lattice elemen... |
| chrelati 30135 | The Hilbert lattice is rel... |
| chrelat2i 30136 | A consequence of relative ... |
| cvati 30137 | If a Hilbert lattice eleme... |
| cvbr4i 30138 | An alternate way to expres... |
| cvexchlem 30139 | Lemma for ~ cvexchi . (Co... |
| cvexchi 30140 | The Hilbert lattice satisf... |
| chrelat2 30141 | A consequence of relative ... |
| chrelat3 30142 | A consequence of relative ... |
| chrelat3i 30143 | A consequence of the relat... |
| chrelat4i 30144 | A consequence of relative ... |
| cvexch 30145 | The Hilbert lattice satisf... |
| cvp 30146 | The Hilbert lattice satisf... |
| atnssm0 30147 | The meet of a Hilbert latt... |
| atnemeq0 30148 | The meet of distinct atoms... |
| atssma 30149 | The meet with an atom's su... |
| atcv0eq 30150 | Two atoms covering the zer... |
| atcv1 30151 | Two atoms covering the zer... |
| atexch 30152 | The Hilbert lattice satisf... |
| atomli 30153 | An assertion holding in at... |
| atoml2i 30154 | An assertion holding in at... |
| atordi 30155 | An ordering law for a Hilb... |
| atcvatlem 30156 | Lemma for ~ atcvati . (Co... |
| atcvati 30157 | A nonzero Hilbert lattice ... |
| atcvat2i 30158 | A Hilbert lattice element ... |
| atord 30159 | An ordering law for a Hilb... |
| atcvat2 30160 | A Hilbert lattice element ... |
| chirredlem1 30161 | Lemma for ~ chirredi . (C... |
| chirredlem2 30162 | Lemma for ~ chirredi . (C... |
| chirredlem3 30163 | Lemma for ~ chirredi . (C... |
| chirredlem4 30164 | Lemma for ~ chirredi . (C... |
| chirredi 30165 | The Hilbert lattice is irr... |
| chirred 30166 | The Hilbert lattice is irr... |
| atcvat3i 30167 | A condition implying that ... |
| atcvat4i 30168 | A condition implying exist... |
| atdmd 30169 | Two Hilbert lattice elemen... |
| atmd 30170 | Two Hilbert lattice elemen... |
| atmd2 30171 | Two Hilbert lattice elemen... |
| atabsi 30172 | Absorption of an incompara... |
| atabs2i 30173 | Absorption of an incompara... |
| mdsymlem1 30174 | Lemma for ~ mdsymi . (Con... |
| mdsymlem2 30175 | Lemma for ~ mdsymi . (Con... |
| mdsymlem3 30176 | Lemma for ~ mdsymi . (Con... |
| mdsymlem4 30177 | Lemma for ~ mdsymi . This... |
| mdsymlem5 30178 | Lemma for ~ mdsymi . (Con... |
| mdsymlem6 30179 | Lemma for ~ mdsymi . This... |
| mdsymlem7 30180 | Lemma for ~ mdsymi . Lemm... |
| mdsymlem8 30181 | Lemma for ~ mdsymi . Lemm... |
| mdsymi 30182 | M-symmetry of the Hilbert ... |
| mdsym 30183 | M-symmetry of the Hilbert ... |
| dmdsym 30184 | Dual M-symmetry of the Hil... |
| atdmd2 30185 | Two Hilbert lattice elemen... |
| sumdmdii 30186 | If the subspace sum of two... |
| cmmdi 30187 | Commuting subspaces form a... |
| cmdmdi 30188 | Commuting subspaces form a... |
| sumdmdlem 30189 | Lemma for ~ sumdmdi . The... |
| sumdmdlem2 30190 | Lemma for ~ sumdmdi . (Co... |
| sumdmdi 30191 | The subspace sum of two Hi... |
| dmdbr4ati 30192 | Dual modular pair property... |
| dmdbr5ati 30193 | Dual modular pair property... |
| dmdbr6ati 30194 | Dual modular pair property... |
| dmdbr7ati 30195 | Dual modular pair property... |
| mdoc1i 30196 | Orthocomplements form a mo... |
| mdoc2i 30197 | Orthocomplements form a mo... |
| dmdoc1i 30198 | Orthocomplements form a du... |
| dmdoc2i 30199 | Orthocomplements form a du... |
| mdcompli 30200 | A condition equivalent to ... |
| dmdcompli 30201 | A condition equivalent to ... |
| mddmdin0i 30202 | If dual modular implies mo... |
| cdjreui 30203 | A member of the sum of dis... |
| cdj1i 30204 | Two ways to express " ` A ... |
| cdj3lem1 30205 | A property of " ` A ` and ... |
| cdj3lem2 30206 | Lemma for ~ cdj3i . Value... |
| cdj3lem2a 30207 | Lemma for ~ cdj3i . Closu... |
| cdj3lem2b 30208 | Lemma for ~ cdj3i . The f... |
| cdj3lem3 30209 | Lemma for ~ cdj3i . Value... |
| cdj3lem3a 30210 | Lemma for ~ cdj3i . Closu... |
| cdj3lem3b 30211 | Lemma for ~ cdj3i . The s... |
| cdj3i 30212 | Two ways to express " ` A ... |
| The list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here | |
| mathbox 30213 | (_This theorem is a dummy ... |
| sa-abvi 30214 | A theorem about the univer... |
| xfree 30215 | A partial converse to ~ 19... |
| xfree2 30216 | A partial converse to ~ 19... |
| addltmulALT 30217 | A proof readability experi... |
| bian1d 30218 | Adding a superfluous conju... |
| or3di 30219 | Distributive law for disju... |
| or3dir 30220 | Distributive law for disju... |
| 3o1cs 30221 | Deduction eliminating disj... |
| 3o2cs 30222 | Deduction eliminating disj... |
| 3o3cs 30223 | Deduction eliminating disj... |
| sbc2iedf 30224 | Conversion of implicit sub... |
| rspc2daf 30225 | Double restricted speciali... |
| nelbOLD 30226 | Obsolete version of ~ nelb... |
| ralcom4f 30227 | Commutation of restricted ... |
| rexcom4f 30228 | Commutation of restricted ... |
| 19.9d2rf 30229 | A deduction version of one... |
| 19.9d2r 30230 | A deduction version of one... |
| r19.29ffa 30231 | A commonly used pattern ba... |
| eqtrb 30232 | A transposition of equalit... |
| opsbc2ie 30233 | Conversion of implicit sub... |
| opreu2reuALT 30234 | Correspondence between uni... |
| 2reucom 30237 | Double restricted existent... |
| 2reu2rex1 30238 | Double restricted existent... |
| 2reureurex 30239 | Double restricted existent... |
| 2reu2reu2 30240 | Double restricted existent... |
| opreu2reu1 30241 | Equivalent definition of t... |
| sq2reunnltb 30242 | There exists a unique deco... |
| addsqnot2reu 30243 | For each complex number ` ... |
| sbceqbidf 30244 | Equality theorem for class... |
| sbcies 30245 | A special version of class... |
| moel 30246 | "At most one" element in a... |
| mo5f 30247 | Alternate definition of "a... |
| nmo 30248 | Negation of "at most one".... |
| reuxfrdf 30249 | Transfer existential uniqu... |
| rexunirn 30250 | Restricted existential qua... |
| rmoxfrd 30251 | Transfer "at most one" res... |
| rmoun 30252 | "At most one" restricted e... |
| rmounid 30253 | Case where an "at most one... |
| dmrab 30254 | Domain of a restricted cla... |
| difrab2 30255 | Difference of two restrict... |
| rabexgfGS 30256 | Separation Scheme in terms... |
| rabsnel 30257 | Truth implied by equality ... |
| rabeqsnd 30258 | Conditions for a restricte... |
| foresf1o 30259 | From a surjective function... |
| rabfodom 30260 | Domination relation for re... |
| abrexdomjm 30261 | An indexed set is dominate... |
| abrexdom2jm 30262 | An indexed set is dominate... |
| abrexexd 30263 | Existence of a class abstr... |
| elabreximd 30264 | Class substitution in an i... |
| elabreximdv 30265 | Class substitution in an i... |
| abrexss 30266 | A necessary condition for ... |
| elunsn 30267 | Elementhood to a union wit... |
| nelun 30268 | Negated membership for a u... |
| disjdifr 30269 | A class and its relative c... |
| rabss3d 30270 | Subclass law for restricte... |
| inin 30271 | Intersection with an inter... |
| inindif 30272 | See ~ inundif . (Contribu... |
| difininv 30273 | Condition for the intersec... |
| difeq 30274 | Rewriting an equation with... |
| eqdif 30275 | If both set differences of... |
| difxp1ss 30276 | Difference law for Cartesi... |
| difxp2ss 30277 | Difference law for Cartesi... |
| undifr 30278 | Union of complementary par... |
| indifundif 30279 | A remarkable equation with... |
| elpwincl1 30280 | Closure of intersection wi... |
| elpwdifcl 30281 | Closure of class differenc... |
| elpwiuncl 30282 | Closure of indexed union w... |
| eqsnd 30283 | Deduce that a set is a sin... |
| elpreq 30284 | Equality wihin a pair. (C... |
| nelpr 30285 | A set ` A ` not in a pair ... |
| inpr0 30286 | Rewrite an empty intersect... |
| neldifpr1 30287 | The first element of a pai... |
| neldifpr2 30288 | The second element of a pa... |
| unidifsnel 30289 | The other element of a pai... |
| unidifsnne 30290 | The other element of a pai... |
| ifeqeqx 30291 | An equality theorem tailor... |
| elimifd 30292 | Elimination of a condition... |
| elim2if 30293 | Elimination of two conditi... |
| elim2ifim 30294 | Elimination of two conditi... |
| ifeq3da 30295 | Given an expression ` C ` ... |
| uniinn0 30296 | Sufficient and necessary c... |
| uniin1 30297 | Union of intersection. Ge... |
| uniin2 30298 | Union of intersection. Ge... |
| difuncomp 30299 | Express a class difference... |
| elpwunicl 30300 | Closure of a set union wit... |
| cbviunf 30301 | Rule used to change the bo... |
| iuneq12daf 30302 | Equality deduction for ind... |
| iunin1f 30303 | Indexed union of intersect... |
| ssiun3 30304 | Subset equivalence for an ... |
| ssiun2sf 30305 | Subset relationship for an... |
| iuninc 30306 | The union of an increasing... |
| iundifdifd 30307 | The intersection of a set ... |
| iundifdif 30308 | The intersection of a set ... |
| iunrdx 30309 | Re-index an indexed union.... |
| iunpreima 30310 | Preimage of an indexed uni... |
| iunrnmptss 30311 | A subset relation for an i... |
| iunxunsn 30312 | Appending a set to an inde... |
| iunxunpr 30313 | Appending two sets to an i... |
| disjnf 30314 | In case ` x ` is not free ... |
| cbvdisjf 30315 | Change bound variables in ... |
| disjss1f 30316 | A subset of a disjoint col... |
| disjeq1f 30317 | Equality theorem for disjo... |
| disjxun0 30318 | Simplify a disjoint union.... |
| disjdifprg 30319 | A trivial partition into a... |
| disjdifprg2 30320 | A trivial partition of a s... |
| disji2f 30321 | Property of a disjoint col... |
| disjif 30322 | Property of a disjoint col... |
| disjorf 30323 | Two ways to say that a col... |
| disjorsf 30324 | Two ways to say that a col... |
| disjif2 30325 | Property of a disjoint col... |
| disjabrex 30326 | Rewriting a disjoint colle... |
| disjabrexf 30327 | Rewriting a disjoint colle... |
| disjpreima 30328 | A preimage of a disjoint s... |
| disjrnmpt 30329 | Rewriting a disjoint colle... |
| disjin 30330 | If a collection is disjoin... |
| disjin2 30331 | If a collection is disjoin... |
| disjxpin 30332 | Derive a disjunction over ... |
| iundisjf 30333 | Rewrite a countable union ... |
| iundisj2f 30334 | A disjoint union is disjoi... |
| disjrdx 30335 | Re-index a disjunct collec... |
| disjex 30336 | Two ways to say that two c... |
| disjexc 30337 | A variant of ~ disjex , ap... |
| disjunsn 30338 | Append an element to a dis... |
| disjun0 30339 | Adding the empty element p... |
| disjiunel 30340 | A set of elements B of a d... |
| disjuniel 30341 | A set of elements B of a d... |
| xpdisjres 30342 | Restriction of a constant ... |
| opeldifid 30343 | Ordered pair elementhood o... |
| difres 30344 | Case when class difference... |
| imadifxp 30345 | Image of the difference wi... |
| relfi 30346 | A relation (set) is finite... |
| reldisjun 30347 | Split a relation into two ... |
| 0res 30348 | Restriction of the empty f... |
| funresdm1 30349 | Restriction of a disjoint ... |
| fnunres1 30350 | Restriction of a disjoint ... |
| fcoinver 30351 | Build an equivalence relat... |
| fcoinvbr 30352 | Binary relation for the eq... |
| brabgaf 30353 | The law of concretion for ... |
| brelg 30354 | Two things in a binary rel... |
| br8d 30355 | Substitution for an eight-... |
| opabdm 30356 | Domain of an ordered-pair ... |
| opabrn 30357 | Range of an ordered-pair c... |
| opabssi 30358 | Sufficient condition for a... |
| opabid2ss 30359 | One direction of ~ opabid2... |
| ssrelf 30360 | A subclass relationship de... |
| eqrelrd2 30361 | A version of ~ eqrelrdv2 w... |
| erbr3b 30362 | Biconditional for equivale... |
| iunsnima 30363 | Image of a singleton by an... |
| ac6sf2 30364 | Alternate version of ~ ac6... |
| fnresin 30365 | Restriction of a function ... |
| f1o3d 30366 | Describe an implicit one-t... |
| eldmne0 30367 | A function of nonempty dom... |
| f1rnen 30368 | Equinumerosity of the rang... |
| rinvf1o 30369 | Sufficient conditions for ... |
| fresf1o 30370 | Conditions for a restricti... |
| nfpconfp 30371 | The set of fixed points of... |
| fmptco1f1o 30372 | The action of composing (t... |
| cofmpt2 30373 | Express composition of a m... |
| f1mptrn 30374 | Express injection for a ma... |
| dfimafnf 30375 | Alternate definition of th... |
| funimass4f 30376 | Membership relation for th... |
| elimampt 30377 | Membership in the image of... |
| suppss2f 30378 | Show that the support of a... |
| fovcld 30379 | Closure law for an operati... |
| ofrn 30380 | The range of the function ... |
| ofrn2 30381 | The range of the function ... |
| off2 30382 | The function operation pro... |
| ofresid 30383 | Applying an operation rest... |
| fimarab 30384 | Expressing the image of a ... |
| unipreima 30385 | Preimage of a class union.... |
| sspreima 30386 | The preimage of a subset i... |
| opfv 30387 | Value of a function produc... |
| xppreima 30388 | The preimage of a Cartesia... |
| xppreima2 30389 | The preimage of a Cartesia... |
| elunirn2 30390 | Condition for the membersh... |
| abfmpunirn 30391 | Membership in a union of a... |
| rabfmpunirn 30392 | Membership in a union of a... |
| abfmpeld 30393 | Membership in an element o... |
| abfmpel 30394 | Membership in an element o... |
| fmptdF 30395 | Domain and codomain of the... |
| fmptcof2 30396 | Composition of two functio... |
| fcomptf 30397 | Express composition of two... |
| acunirnmpt 30398 | Axiom of choice for the un... |
| acunirnmpt2 30399 | Axiom of choice for the un... |
| acunirnmpt2f 30400 | Axiom of choice for the un... |
| aciunf1lem 30401 | Choice in an index union. ... |
| aciunf1 30402 | Choice in an index union. ... |
| ofoprabco 30403 | Function operation as a co... |
| ofpreima 30404 | Express the preimage of a ... |
| ofpreima2 30405 | Express the preimage of a ... |
| funcnvmpt 30406 | Condition for a function i... |
| funcnv5mpt 30407 | Two ways to say that a fun... |
| funcnv4mpt 30408 | Two ways to say that a fun... |
| preimane 30409 | Different elements have di... |
| fnpreimac 30410 | Choose a set ` x ` contain... |
| fgreu 30411 | Exactly one point of a fun... |
| fcnvgreu 30412 | If the converse of a relat... |
| rnmposs 30413 | The range of an operation ... |
| mptssALT 30414 | Deduce subset relation of ... |
| partfun 30415 | Rewrite a function defined... |
| dfcnv2 30416 | Alternative definition of ... |
| fnimatp 30417 | The image of a triplet und... |
| fnunres2 30418 | Restriction of a disjoint ... |
| mpomptxf 30419 | Express a two-argument fun... |
| suppovss 30420 | A bound for the support of... |
| fvdifsupp 30421 | Function value is zero out... |
| fmptssfisupp 30422 | The restriction of a mappi... |
| brsnop 30423 | Binary relation for an ord... |
| cosnopne 30424 | Composition of two ordered... |
| cosnop 30425 | Composition of two ordered... |
| cnvprop 30426 | Converse of a pair of orde... |
| brprop 30427 | Binary relation for a pair... |
| mptprop 30428 | Rewrite pairs of ordered p... |
| coprprop 30429 | Composition of two pairs o... |
| gtiso 30430 | Two ways to write a strict... |
| isoun 30431 | Infer an isomorphism from ... |
| disjdsct 30432 | A disjoint collection is d... |
| df1stres 30433 | Definition for a restricti... |
| df2ndres 30434 | Definition for a restricti... |
| 1stpreimas 30435 | The preimage of a singleto... |
| 1stpreima 30436 | The preimage by ` 1st ` is... |
| 2ndpreima 30437 | The preimage by ` 2nd ` is... |
| curry2ima 30438 | The image of a curried fun... |
| supssd 30439 | Inequality deduction for s... |
| infssd 30440 | Inequality deduction for i... |
| imafi2 30441 | The image by a finite set ... |
| unifi3 30442 | If a union is finite, then... |
| snct 30443 | A singleton is countable. ... |
| prct 30444 | An unordered pair is count... |
| mpocti 30445 | An operation is countable ... |
| abrexct 30446 | An image set of a countabl... |
| mptctf 30447 | A countable mapping set is... |
| abrexctf 30448 | An image set of a countabl... |
| padct 30449 | Index a countable set with... |
| cnvoprabOLD 30450 | The converse of a class ab... |
| f1od2 30451 | Sufficient condition for a... |
| fcobij 30452 | Composing functions with a... |
| fcobijfs 30453 | Composing finitely support... |
| suppss3 30454 | Deduce a function's suppor... |
| fsuppcurry1 30455 | Finite support of a currie... |
| fsuppcurry2 30456 | Finite support of a currie... |
| offinsupp1 30457 | Finite support for a funct... |
| ffs2 30458 | Rewrite a function's suppo... |
| ffsrn 30459 | The range of a finitely su... |
| resf1o 30460 | Restriction of functions t... |
| maprnin 30461 | Restricting the range of t... |
| fpwrelmapffslem 30462 | Lemma for ~ fpwrelmapffs .... |
| fpwrelmap 30463 | Define a canonical mapping... |
| fpwrelmapffs 30464 | Define a canonical mapping... |
| creq0 30465 | The real representation of... |
| 1nei 30466 | The imaginary unit ` _i ` ... |
| 1neg1t1neg1 30467 | An integer unit times itse... |
| nnmulge 30468 | Multiplying by a positive ... |
| lt2addrd 30469 | If the right-hand side of ... |
| xrlelttric 30470 | Trichotomy law for extende... |
| xaddeq0 30471 | Two extended reals which a... |
| xrinfm 30472 | The extended real numbers ... |
| le2halvesd 30473 | A sum is less than the who... |
| xraddge02 30474 | A number is less than or e... |
| xrge0addge 30475 | A number is less than or e... |
| xlt2addrd 30476 | If the right-hand side of ... |
| xrsupssd 30477 | Inequality deduction for s... |
| xrge0infss 30478 | Any subset of nonnegative ... |
| xrge0infssd 30479 | Inequality deduction for i... |
| xrge0addcld 30480 | Nonnegative extended reals... |
| xrge0subcld 30481 | Condition for closure of n... |
| infxrge0lb 30482 | A member of a set of nonne... |
| infxrge0glb 30483 | The infimum of a set of no... |
| infxrge0gelb 30484 | The infimum of a set of no... |
| dfrp2 30485 | Alternate definition of th... |
| xrofsup 30486 | The supremum is preserved ... |
| supxrnemnf 30487 | The supremum of a nonempty... |
| xnn0gt0 30488 | Nonzero extended nonnegati... |
| xnn01gt 30489 | An extended nonnegative in... |
| nn0xmulclb 30490 | Finite multiplication in t... |
| joiniooico 30491 | Disjoint joining an open i... |
| ubico 30492 | A right-open interval does... |
| xeqlelt 30493 | Equality in terms of 'less... |
| eliccelico 30494 | Relate elementhood to a cl... |
| elicoelioo 30495 | Relate elementhood to a cl... |
| iocinioc2 30496 | Intersection between two o... |
| xrdifh 30497 | Class difference of a half... |
| iocinif 30498 | Relate intersection of two... |
| difioo 30499 | The difference between two... |
| difico 30500 | The difference between two... |
| uzssico 30501 | Upper integer sets are a s... |
| fz2ssnn0 30502 | A finite set of sequential... |
| nndiffz1 30503 | Upper set of the positive ... |
| ssnnssfz 30504 | For any finite subset of `... |
| fzne1 30505 | Elementhood in a finite se... |
| fzm1ne1 30506 | Elementhood of an integer ... |
| fzspl 30507 | Split the last element of ... |
| fzdif2 30508 | Split the last element of ... |
| fzodif2 30509 | Split the last element of ... |
| fzodif1 30510 | Set difference of two half... |
| fzsplit3 30511 | Split a finite interval of... |
| bcm1n 30512 | The proportion of one bino... |
| iundisjfi 30513 | Rewrite a countable union ... |
| iundisj2fi 30514 | A disjoint union is disjoi... |
| iundisjcnt 30515 | Rewrite a countable union ... |
| iundisj2cnt 30516 | A countable disjoint union... |
| fzone1 30517 | Elementhood in a half-open... |
| fzom1ne1 30518 | Elementhood in a half-open... |
| f1ocnt 30519 | Given a countable set ` A ... |
| fz1nnct 30520 | NN and integer ranges star... |
| fz1nntr 30521 | NN and integer ranges star... |
| hashunif 30522 | The cardinality of a disjo... |
| hashxpe 30523 | The size of the Cartesian ... |
| hashgt1 30524 | Restate "set contains at l... |
| dvdszzq 30525 | Divisibility for an intege... |
| prmdvdsbc 30526 | Condition for a prime numb... |
| numdenneg 30527 | Numerator and denominator ... |
| divnumden2 30528 | Calculate the reduced form... |
| nnindf 30529 | Principle of Mathematical ... |
| nnindd 30530 | Principle of Mathematical ... |
| nn0min 30531 | Extracting the minimum pos... |
| subne0nn 30532 | A nonnegative difference i... |
| ltesubnnd 30533 | Subtracting an integer num... |
| fprodeq02 30534 | If one of the factors is z... |
| pr01ssre 30535 | The range of the indicator... |
| fprodex01 30536 | A product of factors equal... |
| prodpr 30537 | A product over a pair is t... |
| prodtp 30538 | A product over a triple is... |
| fsumub 30539 | An upper bound for a term ... |
| fsumiunle 30540 | Upper bound for a sum of n... |
| dfdec100 30541 | Split the hundreds from a ... |
| dp2eq1 30544 | Equality theorem for the d... |
| dp2eq2 30545 | Equality theorem for the d... |
| dp2eq1i 30546 | Equality theorem for the d... |
| dp2eq2i 30547 | Equality theorem for the d... |
| dp2eq12i 30548 | Equality theorem for the d... |
| dp20u 30549 | Add a zero in the tenths (... |
| dp20h 30550 | Add a zero in the unit pla... |
| dp2cl 30551 | Closure for the decimal fr... |
| dp2clq 30552 | Closure for a decimal frac... |
| rpdp2cl 30553 | Closure for a decimal frac... |
| rpdp2cl2 30554 | Closure for a decimal frac... |
| dp2lt10 30555 | Decimal fraction builds re... |
| dp2lt 30556 | Comparing two decimal frac... |
| dp2ltsuc 30557 | Comparing a decimal fracti... |
| dp2ltc 30558 | Comparing two decimal expa... |
| dpval 30561 | Define the value of the de... |
| dpcl 30562 | Prove that the closure of ... |
| dpfrac1 30563 | Prove a simple equivalence... |
| dpval2 30564 | Value of the decimal point... |
| dpval3 30565 | Value of the decimal point... |
| dpmul10 30566 | Multiply by 10 a decimal e... |
| decdiv10 30567 | Divide a decimal number by... |
| dpmul100 30568 | Multiply by 100 a decimal ... |
| dp3mul10 30569 | Multiply by 10 a decimal e... |
| dpmul1000 30570 | Multiply by 1000 a decimal... |
| dpval3rp 30571 | Value of the decimal point... |
| dp0u 30572 | Add a zero in the tenths p... |
| dp0h 30573 | Remove a zero in the units... |
| rpdpcl 30574 | Closure of the decimal poi... |
| dplt 30575 | Comparing two decimal expa... |
| dplti 30576 | Comparing a decimal expans... |
| dpgti 30577 | Comparing a decimal expans... |
| dpltc 30578 | Comparing two decimal inte... |
| dpexpp1 30579 | Add one zero to the mantis... |
| 0dp2dp 30580 | Multiply by 10 a decimal e... |
| dpadd2 30581 | Addition with one decimal,... |
| dpadd 30582 | Addition with one decimal.... |
| dpadd3 30583 | Addition with two decimals... |
| dpmul 30584 | Multiplication with one de... |
| dpmul4 30585 | An upper bound to multipli... |
| threehalves 30586 | Example theorem demonstrat... |
| 1mhdrd 30587 | Example theorem demonstrat... |
| xdivval 30590 | Value of division: the (un... |
| xrecex 30591 | Existence of reciprocal of... |
| xmulcand 30592 | Cancellation law for exten... |
| xreceu 30593 | Existential uniqueness of ... |
| xdivcld 30594 | Closure law for the extend... |
| xdivcl 30595 | Closure law for the extend... |
| xdivmul 30596 | Relationship between divis... |
| rexdiv 30597 | The extended real division... |
| xdivrec 30598 | Relationship between divis... |
| xdivid 30599 | A number divided by itself... |
| xdiv0 30600 | Division into zero is zero... |
| xdiv0rp 30601 | Division into zero is zero... |
| eliccioo 30602 | Membership in a closed int... |
| elxrge02 30603 | Elementhood in the set of ... |
| xdivpnfrp 30604 | Plus infinity divided by a... |
| rpxdivcld 30605 | Closure law for extended d... |
| xrpxdivcld 30606 | Closure law for extended d... |
| wrdfd 30607 | A word is a zero-based seq... |
| wrdres 30608 | Condition for the restrict... |
| wrdsplex 30609 | Existence of a split of a ... |
| pfx1s2 30610 | The prefix of length 1 of ... |
| pfxrn2 30611 | The range of a prefix of a... |
| pfxrn3 30612 | Express the range of a pre... |
| pfxf1 30613 | Condition for a prefix to ... |
| s1f1 30614 | Conditions for a length 1 ... |
| s2rn 30615 | Range of a length 2 string... |
| s2f1 30616 | Conditions for a length 2 ... |
| s3rn 30617 | Range of a length 3 string... |
| s3f1 30618 | Conditions for a length 3 ... |
| s3clhash 30619 | Closure of the words of le... |
| ccatf1 30620 | Conditions for a concatena... |
| pfxlsw2ccat 30621 | Reconstruct a word from it... |
| wrdt2ind 30622 | Perform an induction over ... |
| swrdrn2 30623 | The range of a subword is ... |
| swrdrn3 30624 | Express the range of a sub... |
| swrdf1 30625 | Condition for a subword to... |
| swrdrndisj 30626 | Condition for the range of... |
| splfv3 30627 | Symbols to the right of a ... |
| 1cshid 30628 | Cyclically shifting a sing... |
| cshw1s2 30629 | Cyclically shifting a leng... |
| cshwrnid 30630 | Cyclically shifting a word... |
| cshf1o 30631 | Condition for the cyclic s... |
| ressplusf 30632 | The group operation functi... |
| ressnm 30633 | The norm in a restricted s... |
| abvpropd2 30634 | Weaker version of ~ abvpro... |
| oppgle 30635 | less-than relation of an o... |
| oppglt 30636 | less-than relation of an o... |
| ressprs 30637 | The restriction of a prose... |
| oduprs 30638 | Being a proset is a self-d... |
| posrasymb 30639 | A poset ordering is asymet... |
| tospos 30640 | A Toset is a Poset. (Cont... |
| resspos 30641 | The restriction of a Poset... |
| resstos 30642 | The restriction of a Toset... |
| tleile 30643 | In a Toset, two elements m... |
| tltnle 30644 | In a Toset, less-than is e... |
| odutos 30645 | Being a toset is a self-du... |
| tlt2 30646 | In a Toset, two elements m... |
| tlt3 30647 | In a Toset, two elements m... |
| trleile 30648 | In a Toset, two elements m... |
| toslublem 30649 | Lemma for ~ toslub and ~ x... |
| toslub 30650 | In a toset, the lowest upp... |
| tosglblem 30651 | Lemma for ~ tosglb and ~ x... |
| tosglb 30652 | Same theorem as ~ toslub ,... |
| clatp0cl 30653 | The poset zero of a comple... |
| clatp1cl 30654 | The poset one of a complet... |
| xrs0 30657 | The zero of the extended r... |
| xrslt 30658 | The "strictly less than" r... |
| xrsinvgval 30659 | The inversion operation in... |
| xrsmulgzz 30660 | The "multiple" function in... |
| xrstos 30661 | The extended real numbers ... |
| xrsclat 30662 | The extended real numbers ... |
| xrsp0 30663 | The poset 0 of the extende... |
| xrsp1 30664 | The poset 1 of the extende... |
| ressmulgnn 30665 | Values for the group multi... |
| ressmulgnn0 30666 | Values for the group multi... |
| xrge0base 30667 | The base of the extended n... |
| xrge00 30668 | The zero of the extended n... |
| xrge0plusg 30669 | The additive law of the ex... |
| xrge0le 30670 | The "less than or equal to... |
| xrge0mulgnn0 30671 | The group multiple functio... |
| xrge0addass 30672 | Associativity of extended ... |
| xrge0addgt0 30673 | The sum of nonnegative and... |
| xrge0adddir 30674 | Right-distributivity of ex... |
| xrge0adddi 30675 | Left-distributivity of ext... |
| xrge0npcan 30676 | Extended nonnegative real ... |
| fsumrp0cl 30677 | Closure of a finite sum of... |
| abliso 30678 | The image of an Abelian gr... |
| gsumsubg 30679 | The group sum in a subgrou... |
| gsumsra 30680 | The group sum in a subring... |
| gsummpt2co 30681 | Split a finite sum into a ... |
| gsummpt2d 30682 | Express a finite sum over ... |
| lmodvslmhm 30683 | Scalar multiplication in a... |
| gsumvsmul1 30684 | Pull a scalar multiplicati... |
| gsummptres 30685 | Extend a finite group sum ... |
| gsumzresunsn 30686 | Append an element to a fin... |
| xrge0tsmsd 30687 | Any finite or infinite sum... |
| xrge0tsmsbi 30688 | Any limit of a finite or i... |
| xrge0tsmseq 30689 | Any limit of a finite or i... |
| cntzun 30690 | The centralizer of a union... |
| cntzsnid 30691 | The centralizer of the ide... |
| cntrcrng 30692 | The center of a ring is a ... |
| isomnd 30697 | A (left) ordered monoid is... |
| isogrp 30698 | A (left-)ordered group is ... |
| ogrpgrp 30699 | A left-ordered group is a ... |
| omndmnd 30700 | A left-ordered monoid is a... |
| omndtos 30701 | A left-ordered monoid is a... |
| omndadd 30702 | In an ordered monoid, the ... |
| omndaddr 30703 | In a right ordered monoid,... |
| omndadd2d 30704 | In a commutative left orde... |
| omndadd2rd 30705 | In a left- and right- orde... |
| submomnd 30706 | A submonoid of an ordered ... |
| xrge0omnd 30707 | The nonnegative extended r... |
| omndmul2 30708 | In an ordered monoid, the ... |
| omndmul3 30709 | In an ordered monoid, the ... |
| omndmul 30710 | In a commutative ordered m... |
| ogrpinv0le 30711 | In an ordered group, the o... |
| ogrpsub 30712 | In an ordered group, the o... |
| ogrpaddlt 30713 | In an ordered group, stric... |
| ogrpaddltbi 30714 | In a right ordered group, ... |
| ogrpaddltrd 30715 | In a right ordered group, ... |
| ogrpaddltrbid 30716 | In a right ordered group, ... |
| ogrpsublt 30717 | In an ordered group, stric... |
| ogrpinv0lt 30718 | In an ordered group, the o... |
| ogrpinvlt 30719 | In an ordered group, the o... |
| gsumle 30720 | A finite sum in an ordered... |
| symgfcoeu 30721 | Uniqueness property of per... |
| symgcom 30722 | Two permutations ` X ` and... |
| symgcom2 30723 | Two permutations ` X ` and... |
| symgcntz 30724 | All elements of a (finite)... |
| odpmco 30725 | The composition of two odd... |
| symgsubg 30726 | The value of the group sub... |
| pmtrprfv2 30727 | In a transposition of two ... |
| pmtrcnel 30728 | Composing a permutation ` ... |
| pmtrcnel2 30729 | Variation on ~ pmtrcnel . ... |
| pmtrcnelor 30730 | Composing a permutation ` ... |
| pmtridf1o 30731 | Transpositions of ` X ` an... |
| pmtridfv1 30732 | Value at X of the transpos... |
| pmtridfv2 30733 | Value at Y of the transpos... |
| psgnid 30734 | Permutation sign of the id... |
| psgndmfi 30735 | For a finite base set, the... |
| pmtrto1cl 30736 | Useful lemma for the follo... |
| psgnfzto1stlem 30737 | Lemma for ~ psgnfzto1st . ... |
| fzto1stfv1 30738 | Value of our permutation `... |
| fzto1st1 30739 | Special case where the per... |
| fzto1st 30740 | The function moving one el... |
| fzto1stinvn 30741 | Value of the inverse of ou... |
| psgnfzto1st 30742 | The permutation sign for m... |
| tocycval 30745 | Value of the cycle builder... |
| tocycfv 30746 | Function value of a permut... |
| tocycfvres1 30747 | A cyclic permutation is a ... |
| tocycfvres2 30748 | A cyclic permutation is th... |
| cycpmfvlem 30749 | Lemma for ~ cycpmfv1 and ~... |
| cycpmfv1 30750 | Value of a cycle function ... |
| cycpmfv2 30751 | Value of a cycle function ... |
| cycpmfv3 30752 | Values outside of the orbi... |
| cycpmcl 30753 | Cyclic permutations are pe... |
| tocycf 30754 | The permutation cycle buil... |
| tocyc01 30755 | Permutation cycles built f... |
| cycpm2tr 30756 | A cyclic permutation of 2 ... |
| cycpm2cl 30757 | Closure for the 2-cycles. ... |
| cyc2fv1 30758 | Function value of a 2-cycl... |
| cyc2fv2 30759 | Function value of a 2-cycl... |
| trsp2cyc 30760 | Exhibit the word a transpo... |
| cycpmco2f1 30761 | The word U used in ~ cycpm... |
| cycpmco2rn 30762 | The orbit of the compositi... |
| cycpmco2lem1 30763 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem2 30764 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem3 30765 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem4 30766 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem5 30767 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem6 30768 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2lem7 30769 | Lemma for ~ cycpmco2 . (C... |
| cycpmco2 30770 | The composition of a cycli... |
| cyc2fvx 30771 | Function value of a 2-cycl... |
| cycpm3cl 30772 | Closure of the 3-cycles in... |
| cycpm3cl2 30773 | Closure of the 3-cycles in... |
| cyc3fv1 30774 | Function value of a 3-cycl... |
| cyc3fv2 30775 | Function value of a 3-cycl... |
| cyc3fv3 30776 | Function value of a 3-cycl... |
| cyc3co2 30777 | Represent a 3-cycle as a c... |
| cycpmconjvlem 30778 | Lemma for ~ cycpmconjv (Co... |
| cycpmconjv 30779 | A formula for computing co... |
| cycpmrn 30780 | The range of the word used... |
| tocyccntz 30781 | All elements of a (finite)... |
| evpmval 30782 | Value of the set of even p... |
| cnmsgn0g 30783 | The neutral element of the... |
| evpmsubg 30784 | The alternating group is a... |
| evpmid 30785 | The identity is an even pe... |
| altgnsg 30786 | The alternating group ` ( ... |
| cyc3evpm 30787 | 3-Cycles are even permutat... |
| cyc3genpmlem 30788 | Lemma for ~ cyc3genpm . (... |
| cyc3genpm 30789 | The alternating group ` A ... |
| cycpmgcl 30790 | Cyclic permutations are pe... |
| cycpmconjslem1 30791 | Lemma for ~ cycpmconjs (Co... |
| cycpmconjslem2 30792 | Lemma for ~ cycpmconjs (Co... |
| cycpmconjs 30793 | All cycles of the same len... |
| cyc3conja 30794 | All 3-cycles are conjugate... |
| sgnsv 30797 | The sign mapping. (Contri... |
| sgnsval 30798 | The sign value. (Contribu... |
| sgnsf 30799 | The sign function. (Contr... |
| inftmrel 30804 | The infinitesimal relation... |
| isinftm 30805 | Express ` x ` is infinites... |
| isarchi 30806 | Express the predicate " ` ... |
| pnfinf 30807 | Plus infinity is an infini... |
| xrnarchi 30808 | The completed real line is... |
| isarchi2 30809 | Alternative way to express... |
| submarchi 30810 | A submonoid is archimedean... |
| isarchi3 30811 | This is the usual definiti... |
| archirng 30812 | Property of Archimedean or... |
| archirngz 30813 | Property of Archimedean le... |
| archiexdiv 30814 | In an Archimedean group, g... |
| archiabllem1a 30815 | Lemma for ~ archiabl : In... |
| archiabllem1b 30816 | Lemma for ~ archiabl . (C... |
| archiabllem1 30817 | Archimedean ordered groups... |
| archiabllem2a 30818 | Lemma for ~ archiabl , whi... |
| archiabllem2c 30819 | Lemma for ~ archiabl . (C... |
| archiabllem2b 30820 | Lemma for ~ archiabl . (C... |
| archiabllem2 30821 | Archimedean ordered groups... |
| archiabl 30822 | Archimedean left- and righ... |
| isslmd 30825 | The predicate "is a semimo... |
| slmdlema 30826 | Lemma for properties of a ... |
| lmodslmd 30827 | Left semimodules generaliz... |
| slmdcmn 30828 | A semimodule is a commutat... |
| slmdmnd 30829 | A semimodule is a monoid. ... |
| slmdsrg 30830 | The scalar component of a ... |
| slmdbn0 30831 | The base set of a semimodu... |
| slmdacl 30832 | Closure of ring addition f... |
| slmdmcl 30833 | Closure of ring multiplica... |
| slmdsn0 30834 | The set of scalars in a se... |
| slmdvacl 30835 | Closure of vector addition... |
| slmdass 30836 | Semiring left module vecto... |
| slmdvscl 30837 | Closure of scalar product ... |
| slmdvsdi 30838 | Distributive law for scala... |
| slmdvsdir 30839 | Distributive law for scala... |
| slmdvsass 30840 | Associative law for scalar... |
| slmd0cl 30841 | The ring zero in a semimod... |
| slmd1cl 30842 | The ring unit in a semirin... |
| slmdvs1 30843 | Scalar product with ring u... |
| slmd0vcl 30844 | The zero vector is a vecto... |
| slmd0vlid 30845 | Left identity law for the ... |
| slmd0vrid 30846 | Right identity law for the... |
| slmd0vs 30847 | Zero times a vector is the... |
| slmdvs0 30848 | Anything times the zero ve... |
| gsumvsca1 30849 | Scalar product of a finite... |
| gsumvsca2 30850 | Scalar product of a finite... |
| prmsimpcyc 30851 | A group of prime order is ... |
| rngurd 30852 | Deduce the unit of a ring ... |
| dvdschrmulg 30853 | In a ring, any multiple of... |
| freshmansdream 30854 | For a prime number ` P ` ,... |
| ress1r 30855 | ` 1r ` is unaffected by re... |
| dvrdir 30856 | Distributive law for the d... |
| rdivmuldivd 30857 | Multiplication of two rati... |
| ringinvval 30858 | The ring inverse expressed... |
| dvrcan5 30859 | Cancellation law for commo... |
| subrgchr 30860 | If ` A ` is a subring of `... |
| rmfsupp2 30861 | A mapping of a multiplicat... |
| primefldchr 30862 | The characteristic of a pr... |
| isorng 30867 | An ordered ring is a ring ... |
| orngring 30868 | An ordered ring is a ring.... |
| orngogrp 30869 | An ordered ring is an orde... |
| isofld 30870 | An ordered field is a fiel... |
| orngmul 30871 | In an ordered ring, the or... |
| orngsqr 30872 | In an ordered ring, all sq... |
| ornglmulle 30873 | In an ordered ring, multip... |
| orngrmulle 30874 | In an ordered ring, multip... |
| ornglmullt 30875 | In an ordered ring, multip... |
| orngrmullt 30876 | In an ordered ring, multip... |
| orngmullt 30877 | In an ordered ring, the st... |
| ofldfld 30878 | An ordered field is a fiel... |
| ofldtos 30879 | An ordered field is a tota... |
| orng0le1 30880 | In an ordered ring, the ri... |
| ofldlt1 30881 | In an ordered field, the r... |
| ofldchr 30882 | The characteristic of an o... |
| suborng 30883 | Every subring of an ordere... |
| subofld 30884 | Every subfield of an order... |
| isarchiofld 30885 | Axiom of Archimedes : a ch... |
| rhmdvdsr 30886 | A ring homomorphism preser... |
| rhmopp 30887 | A ring homomorphism is als... |
| elrhmunit 30888 | Ring homomorphisms preserv... |
| rhmdvd 30889 | A ring homomorphism preser... |
| rhmunitinv 30890 | Ring homomorphisms preserv... |
| kerunit 30891 | If a unit element lies in ... |
| reldmresv 30894 | The scalar restriction is ... |
| resvval 30895 | Value of structure restric... |
| resvid2 30896 | General behavior of trivia... |
| resvval2 30897 | Value of nontrivial struct... |
| resvsca 30898 | Base set of a structure re... |
| resvlem 30899 | Other elements of a struct... |
| resvbas 30900 | ` Base ` is unaffected by ... |
| resvplusg 30901 | ` +g ` is unaffected by sc... |
| resvvsca 30902 | ` .s ` is unaffected by sc... |
| resvmulr 30903 | ` .s ` is unaffected by sc... |
| resv0g 30904 | ` 0g ` is unaffected by sc... |
| resv1r 30905 | ` 1r ` is unaffected by sc... |
| resvcmn 30906 | Scalar restriction preserv... |
| gzcrng 30907 | The gaussian integers form... |
| reofld 30908 | The real numbers form an o... |
| nn0omnd 30909 | The nonnegative integers f... |
| rearchi 30910 | The field of the real numb... |
| nn0archi 30911 | The monoid of the nonnegat... |
| xrge0slmod 30912 | The extended nonnegative r... |
| qusker 30913 | The kernel of a quotient m... |
| eqgvscpbl 30914 | The left coset equivalence... |
| qusvscpbl 30915 | The quotient map distribut... |
| qusscaval 30916 | Value of the scalar multip... |
| imaslmod 30917 | The image structure of a l... |
| quslmod 30918 | If ` G ` is a submodule in... |
| quslmhm 30919 | If ` G ` is a submodule of... |
| ecxpid 30920 | The equivalence class of a... |
| eqg0el 30921 | Equivalence class of a quo... |
| qsxpid 30922 | The quotient set of a cart... |
| qusxpid 30923 | The Group quotient equival... |
| qustriv 30924 | The quotient of a group ` ... |
| qustrivr 30925 | Converse of ~ qustriv . (... |
| fply1 30926 | Conditions for a function ... |
| islinds5 30927 | A set is linearly independ... |
| ellspds 30928 | Variation on ~ ellspd . (... |
| 0ellsp 30929 | Zero is in all spans. (Co... |
| 0nellinds 30930 | The group identity cannot ... |
| rspsnel 30931 | Membership in a principal ... |
| rspsnid 30932 | A principal ideal contains... |
| lbslsp 30933 | Any element of a left modu... |
| lindssn 30934 | Any singleton of a nonzero... |
| lindflbs 30935 | Conditions for an independ... |
| linds2eq 30936 | Deduce equality of element... |
| lindfpropd 30937 | Property deduction for lin... |
| lindspropd 30938 | Property deduction for lin... |
| elgrplsmsn 30939 | Membership in a sumset wit... |
| lsmsnorb 30940 | The sumset of a group with... |
| elringlsm 30941 | Membership in a product of... |
| ringlsmss 30942 | Closure of the product of ... |
| lsmsnpridl 30943 | The product of the ring wi... |
| lsmsnidl 30944 | The product of the ring wi... |
| lsmidllsp 30945 | The sum of two ideals is t... |
| lsmidl 30946 | The sum of two ideals is a... |
| prmidlval 30949 | The class of prime ideals ... |
| isprmidl 30950 | The predicate "is a prime ... |
| prmidlnr 30951 | A prime ideal is a proper ... |
| prmidl 30952 | The main property of a pri... |
| prmidl2 30953 | A condition that shows an ... |
| pridln1 30954 | A proper ideal cannot cont... |
| prmidlidl 30955 | A prime ideal is an ideal.... |
| lidlnsg 30956 | An ideal is a normal subgr... |
| cringm4 30957 | Commutative/associative la... |
| isprmidlc 30958 | The predicate "is prime id... |
| prmidlc 30959 | Property of a prime ideal ... |
| qsidomlem1 30960 | If the quotient ring of a ... |
| qsidomlem2 30961 | A quotient by a prime idea... |
| qsidom 30962 | An ideal ` I ` in the comm... |
| mxidlval 30965 | The set of maximal ideals ... |
| ismxidl 30966 | The predicate "is a maxima... |
| mxidlidl 30967 | A maximal ideal is an idea... |
| mxidlnr 30968 | A maximal ideal is proper.... |
| mxidlmax 30969 | A maximal ideal is a maxim... |
| mxidln1 30970 | One is not contained in an... |
| mxidlnzr 30971 | A ring with a maximal idea... |
| mxidlprm 30972 | Every maximal ideal is pri... |
| ssmxidllem 30973 | The set ` P ` used in the ... |
| ssmxidl 30974 | Let ` R ` be a ring, and l... |
| krull 30975 | Krull's theorem: Any nonz... |
| mxidlnzrb 30976 | A ring is nonzero if and o... |
| sra1r 30981 | The multiplicative neutral... |
| sraring 30982 | Condition for a subring al... |
| sradrng 30983 | Condition for a subring al... |
| srasubrg 30984 | A subring of the original ... |
| sralvec 30985 | Given a sub division ring ... |
| srafldlvec 30986 | Given a subfield ` F ` of ... |
| drgext0g 30987 | The additive neutral eleme... |
| drgextvsca 30988 | The scalar multiplication ... |
| drgext0gsca 30989 | The additive neutral eleme... |
| drgextsubrg 30990 | The scalar field is a subr... |
| drgextlsp 30991 | The scalar field is a subs... |
| drgextgsum 30992 | Group sum in a division ri... |
| lvecdimfi 30993 | Finite version of ~ lvecdi... |
| dimval 30996 | The dimension of a vector ... |
| dimvalfi 30997 | The dimension of a vector ... |
| dimcl 30998 | Closure of the vector spac... |
| lvecdim0i 30999 | A vector space of dimensio... |
| lvecdim0 31000 | A vector space of dimensio... |
| lssdimle 31001 | The dimension of a linear ... |
| dimpropd 31002 | If two structures have the... |
| rgmoddim 31003 | The left vector space indu... |
| frlmdim 31004 | Dimension of a free left m... |
| tnglvec 31005 | Augmenting a structure wit... |
| tngdim 31006 | Dimension of a left vector... |
| rrxdim 31007 | Dimension of the generaliz... |
| matdim 31008 | Dimension of the space of ... |
| lbslsat 31009 | A nonzero vector ` X ` is ... |
| lsatdim 31010 | A line, spanned by a nonze... |
| drngdimgt0 31011 | The dimension of a vector ... |
| lmhmlvec2 31012 | A homomorphism of left vec... |
| kerlmhm 31013 | The kernel of a vector spa... |
| imlmhm 31014 | The image of a vector spac... |
| lindsunlem 31015 | Lemma for ~ lindsun . (Co... |
| lindsun 31016 | Condition for the union of... |
| lbsdiflsp0 31017 | The linear spans of two di... |
| dimkerim 31018 | Given a linear map ` F ` b... |
| qusdimsum 31019 | Let ` W ` be a vector spac... |
| fedgmullem1 31020 | Lemma for ~ fedgmul (Contr... |
| fedgmullem2 31021 | Lemma for ~ fedgmul (Contr... |
| fedgmul 31022 | The multiplicativity formu... |
| relfldext 31031 | The field extension is a r... |
| brfldext 31032 | The field extension relati... |
| ccfldextrr 31033 | The field of the complex n... |
| fldextfld1 31034 | A field extension is only ... |
| fldextfld2 31035 | A field extension is only ... |
| fldextsubrg 31036 | Field extension implies a ... |
| fldextress 31037 | Field extension implies a ... |
| brfinext 31038 | The finite field extension... |
| extdgval 31039 | Value of the field extensi... |
| fldextsralvec 31040 | The subring algebra associ... |
| extdgcl 31041 | Closure of the field exten... |
| extdggt0 31042 | Degrees of field extension... |
| fldexttr 31043 | Field extension is a trans... |
| fldextid 31044 | The field extension relati... |
| extdgid 31045 | A trivial field extension ... |
| extdgmul 31046 | The multiplicativity formu... |
| finexttrb 31047 | The extension ` E ` of ` K... |
| extdg1id 31048 | If the degree of the exten... |
| extdg1b 31049 | The degree of the extensio... |
| fldextchr 31050 | The characteristic of a su... |
| ccfldsrarelvec 31051 | The subring algebra of the... |
| ccfldextdgrr 31052 | The degree of the field ex... |
| smatfval 31055 | Value of the submatrix. (... |
| smatrcl 31056 | Closure of the rectangular... |
| smatlem 31057 | Lemma for the next theorem... |
| smattl 31058 | Entries of a submatrix, to... |
| smattr 31059 | Entries of a submatrix, to... |
| smatbl 31060 | Entries of a submatrix, bo... |
| smatbr 31061 | Entries of a submatrix, bo... |
| smatcl 31062 | Closure of the square subm... |
| matmpo 31063 | Write a square matrix as a... |
| 1smat1 31064 | The submatrix of the ident... |
| submat1n 31065 | One case where the submatr... |
| submatres 31066 | Special case where the sub... |
| submateqlem1 31067 | Lemma for ~ submateq . (C... |
| submateqlem2 31068 | Lemma for ~ submateq . (C... |
| submateq 31069 | Sufficient condition for t... |
| submatminr1 31070 | If we take a submatrix by ... |
| lmatval 31073 | Value of the literal matri... |
| lmatfval 31074 | Entries of a literal matri... |
| lmatfvlem 31075 | Useful lemma to extract li... |
| lmatcl 31076 | Closure of the literal mat... |
| lmat22lem 31077 | Lemma for ~ lmat22e11 and ... |
| lmat22e11 31078 | Entry of a 2x2 literal mat... |
| lmat22e12 31079 | Entry of a 2x2 literal mat... |
| lmat22e21 31080 | Entry of a 2x2 literal mat... |
| lmat22e22 31081 | Entry of a 2x2 literal mat... |
| lmat22det 31082 | The determinant of a liter... |
| mdetpmtr1 31083 | The determinant of a matri... |
| mdetpmtr2 31084 | The determinant of a matri... |
| mdetpmtr12 31085 | The determinant of a matri... |
| mdetlap1 31086 | A Laplace expansion of the... |
| madjusmdetlem1 31087 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem2 31088 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem3 31089 | Lemma for ~ madjusmdet . ... |
| madjusmdetlem4 31090 | Lemma for ~ madjusmdet . ... |
| madjusmdet 31091 | Express the cofactor of th... |
| mdetlap 31092 | Laplace expansion of the d... |
| txomap 31093 | Given two open maps ` F ` ... |
| qtopt1 31094 | If every equivalence class... |
| qtophaus 31095 | If an open map's graph in ... |
| circtopn 31096 | The topology of the unit c... |
| circcn 31097 | The function gluing the re... |
| reff 31098 | For any cover refinement, ... |
| locfinreflem 31099 | A locally finite refinemen... |
| locfinref 31100 | A locally finite refinemen... |
| iscref 31103 | The property that every op... |
| crefeq 31104 | Equality theorem for the "... |
| creftop 31105 | A space where every open c... |
| crefi 31106 | The property that every op... |
| crefdf 31107 | A formulation of ~ crefi e... |
| crefss 31108 | The "every open cover has ... |
| cmpcref 31109 | Equivalent definition of c... |
| cmpfiref 31110 | Every open cover of a Comp... |
| ldlfcntref 31113 | Every open cover of a Lind... |
| ispcmp 31116 | The predicate "is a paraco... |
| cmppcmp 31117 | Every compact space is par... |
| dispcmp 31118 | Every discrete space is pa... |
| pcmplfin 31119 | Given a paracompact topolo... |
| pcmplfinf 31120 | Given a paracompact topolo... |
| metidval 31125 | Value of the metric identi... |
| metidss 31126 | As a relation, the metric ... |
| metidv 31127 | ` A ` and ` B ` identify b... |
| metideq 31128 | Basic property of the metr... |
| metider 31129 | The metric identification ... |
| pstmval 31130 | Value of the metric induce... |
| pstmfval 31131 | Function value of the metr... |
| pstmxmet 31132 | The metric induced by a ps... |
| hauseqcn 31133 | In a Hausdorff topology, t... |
| unitsscn 31134 | The closed unit interval i... |
| elunitrn 31135 | The closed unit interval i... |
| elunitcn 31136 | The closed unit interval i... |
| elunitge0 31137 | An element of the closed u... |
| unitssxrge0 31138 | The closed unit interval i... |
| unitdivcld 31139 | Necessary conditions for a... |
| iistmd 31140 | The closed unit interval f... |
| unicls 31141 | The union of the closed se... |
| tpr2tp 31142 | The usual topology on ` ( ... |
| tpr2uni 31143 | The usual topology on ` ( ... |
| xpinpreima 31144 | Rewrite the cartesian prod... |
| xpinpreima2 31145 | Rewrite the cartesian prod... |
| sqsscirc1 31146 | The complex square of side... |
| sqsscirc2 31147 | The complex square of side... |
| cnre2csqlem 31148 | Lemma for ~ cnre2csqima . ... |
| cnre2csqima 31149 | Image of a centered square... |
| tpr2rico 31150 | For any point of an open s... |
| cnvordtrestixx 31151 | The restriction of the 'gr... |
| prsdm 31152 | Domain of the relation of ... |
| prsrn 31153 | Range of the relation of a... |
| prsss 31154 | Relation of a subproset. ... |
| prsssdm 31155 | Domain of a subproset rela... |
| ordtprsval 31156 | Value of the order topolog... |
| ordtprsuni 31157 | Value of the order topolog... |
| ordtcnvNEW 31158 | The order dual generates t... |
| ordtrestNEW 31159 | The subspace topology of a... |
| ordtrest2NEWlem 31160 | Lemma for ~ ordtrest2NEW .... |
| ordtrest2NEW 31161 | An interval-closed set ` A... |
| ordtconnlem1 31162 | Connectedness in the order... |
| ordtconn 31163 | Connectedness in the order... |
| mndpluscn 31164 | A mapping that is both a h... |
| mhmhmeotmd 31165 | Deduce a Topological Monoi... |
| rmulccn 31166 | Multiplication by a real c... |
| raddcn 31167 | Addition in the real numbe... |
| xrmulc1cn 31168 | The operation multiplying ... |
| fmcncfil 31169 | The image of a Cauchy filt... |
| xrge0hmph 31170 | The extended nonnegative r... |
| xrge0iifcnv 31171 | Define a bijection from ` ... |
| xrge0iifcv 31172 | The defined function's val... |
| xrge0iifiso 31173 | The defined bijection from... |
| xrge0iifhmeo 31174 | Expose a homeomorphism fro... |
| xrge0iifhom 31175 | The defined function from ... |
| xrge0iif1 31176 | Condition for the defined ... |
| xrge0iifmhm 31177 | The defined function from ... |
| xrge0pluscn 31178 | The addition operation of ... |
| xrge0mulc1cn 31179 | The operation multiplying ... |
| xrge0tps 31180 | The extended nonnegative r... |
| xrge0topn 31181 | The topology of the extend... |
| xrge0haus 31182 | The topology of the extend... |
| xrge0tmd 31183 | The extended nonnegative r... |
| xrge0tmdALT 31184 | Alternate proof of ~ xrge0... |
| lmlim 31185 | Relate a limit in a given ... |
| lmlimxrge0 31186 | Relate a limit in the nonn... |
| rge0scvg 31187 | Implication of convergence... |
| fsumcvg4 31188 | A serie with finite suppor... |
| pnfneige0 31189 | A neighborhood of ` +oo ` ... |
| lmxrge0 31190 | Express "sequence ` F ` co... |
| lmdvg 31191 | If a monotonic sequence of... |
| lmdvglim 31192 | If a monotonic real number... |
| pl1cn 31193 | A univariate polynomial is... |
| zringnm 31196 | The norm (function) for a ... |
| zzsnm 31197 | The norm of the ring of th... |
| zlm0 31198 | Zero of a ` ZZ ` -module. ... |
| zlm1 31199 | Unit of a ` ZZ ` -module (... |
| zlmds 31200 | Distance in a ` ZZ ` -modu... |
| zlmtset 31201 | Topology in a ` ZZ ` -modu... |
| zlmnm 31202 | Norm of a ` ZZ ` -module (... |
| zhmnrg 31203 | The ` ZZ ` -module built f... |
| nmmulg 31204 | The norm of a group produc... |
| zrhnm 31205 | The norm of the image by `... |
| cnzh 31206 | The ` ZZ ` -module of ` CC... |
| rezh 31207 | The ` ZZ ` -module of ` RR... |
| qqhval 31210 | Value of the canonical hom... |
| zrhf1ker 31211 | The kernel of the homomorp... |
| zrhchr 31212 | The kernel of the homomorp... |
| zrhker 31213 | The kernel of the homomorp... |
| zrhunitpreima 31214 | The preimage by ` ZRHom ` ... |
| elzrhunit 31215 | Condition for the image by... |
| elzdif0 31216 | Lemma for ~ qqhval2 . (Co... |
| qqhval2lem 31217 | Lemma for ~ qqhval2 . (Co... |
| qqhval2 31218 | Value of the canonical hom... |
| qqhvval 31219 | Value of the canonical hom... |
| qqh0 31220 | The image of ` 0 ` by the ... |
| qqh1 31221 | The image of ` 1 ` by the ... |
| qqhf 31222 | ` QQHom ` as a function. ... |
| qqhvq 31223 | The image of a quotient by... |
| qqhghm 31224 | The ` QQHom ` homomorphism... |
| qqhrhm 31225 | The ` QQHom ` homomorphism... |
| qqhnm 31226 | The norm of the image by `... |
| qqhcn 31227 | The ` QQHom ` homomorphism... |
| qqhucn 31228 | The ` QQHom ` homomorphism... |
| rrhval 31232 | Value of the canonical hom... |
| rrhcn 31233 | If the topology of ` R ` i... |
| rrhf 31234 | If the topology of ` R ` i... |
| isrrext 31236 | Express the property " ` R... |
| rrextnrg 31237 | An extension of ` RR ` is ... |
| rrextdrg 31238 | An extension of ` RR ` is ... |
| rrextnlm 31239 | The norm of an extension o... |
| rrextchr 31240 | The ring characteristic of... |
| rrextcusp 31241 | An extension of ` RR ` is ... |
| rrexttps 31242 | An extension of ` RR ` is ... |
| rrexthaus 31243 | The topology of an extensi... |
| rrextust 31244 | The uniformity of an exten... |
| rerrext 31245 | The field of the real numb... |
| cnrrext 31246 | The field of the complex n... |
| qqtopn 31247 | The topology of the field ... |
| rrhfe 31248 | If ` R ` is an extension o... |
| rrhcne 31249 | If ` R ` is an extension o... |
| rrhqima 31250 | The ` RRHom ` homomorphism... |
| rrh0 31251 | The image of ` 0 ` by the ... |
| xrhval 31254 | The value of the embedding... |
| zrhre 31255 | The ` ZRHom ` homomorphism... |
| qqhre 31256 | The ` QQHom ` homomorphism... |
| rrhre 31257 | The ` RRHom ` homomorphism... |
| relmntop 31260 | Manifold is a relation. (... |
| ismntoplly 31261 | Property of being a manifo... |
| ismntop 31262 | Property of being a manifo... |
| nexple 31263 | A lower bound for an expon... |
| indv 31266 | Value of the indicator fun... |
| indval 31267 | Value of the indicator fun... |
| indval2 31268 | Alternate value of the ind... |
| indf 31269 | An indicator function as a... |
| indfval 31270 | Value of the indicator fun... |
| ind1 31271 | Value of the indicator fun... |
| ind0 31272 | Value of the indicator fun... |
| ind1a 31273 | Value of the indicator fun... |
| indpi1 31274 | Preimage of the singleton ... |
| indsum 31275 | Finite sum of a product wi... |
| indsumin 31276 | Finite sum of a product wi... |
| prodindf 31277 | The product of indicators ... |
| indf1o 31278 | The bijection between a po... |
| indpreima 31279 | A function with range ` { ... |
| indf1ofs 31280 | The bijection between fini... |
| esumex 31283 | An extended sum is a set b... |
| esumcl 31284 | Closure for extended sum i... |
| esumeq12dvaf 31285 | Equality deduction for ext... |
| esumeq12dva 31286 | Equality deduction for ext... |
| esumeq12d 31287 | Equality deduction for ext... |
| esumeq1 31288 | Equality theorem for an ex... |
| esumeq1d 31289 | Equality theorem for an ex... |
| esumeq2 31290 | Equality theorem for exten... |
| esumeq2d 31291 | Equality deduction for ext... |
| esumeq2dv 31292 | Equality deduction for ext... |
| esumeq2sdv 31293 | Equality deduction for ext... |
| nfesum1 31294 | Bound-variable hypothesis ... |
| nfesum2 31295 | Bound-variable hypothesis ... |
| cbvesum 31296 | Change bound variable in a... |
| cbvesumv 31297 | Change bound variable in a... |
| esumid 31298 | Identify the extended sum ... |
| esumgsum 31299 | A finite extended sum is t... |
| esumval 31300 | Develop the value of the e... |
| esumel 31301 | The extended sum is a limi... |
| esumnul 31302 | Extended sum over the empt... |
| esum0 31303 | Extended sum of zero. (Co... |
| esumf1o 31304 | Re-index an extended sum u... |
| esumc 31305 | Convert from the collectio... |
| esumrnmpt 31306 | Rewrite an extended sum in... |
| esumsplit 31307 | Split an extended sum into... |
| esummono 31308 | Extended sum is monotonic.... |
| esumpad 31309 | Extend an extended sum by ... |
| esumpad2 31310 | Remove zeroes from an exte... |
| esumadd 31311 | Addition of infinite sums.... |
| esumle 31312 | If all of the terms of an ... |
| gsumesum 31313 | Relate a group sum on ` ( ... |
| esumlub 31314 | The extended sum is the lo... |
| esumaddf 31315 | Addition of infinite sums.... |
| esumlef 31316 | If all of the terms of an ... |
| esumcst 31317 | The extended sum of a cons... |
| esumsnf 31318 | The extended sum of a sing... |
| esumsn 31319 | The extended sum of a sing... |
| esumpr 31320 | Extended sum over a pair. ... |
| esumpr2 31321 | Extended sum over a pair, ... |
| esumrnmpt2 31322 | Rewrite an extended sum in... |
| esumfzf 31323 | Formulating a partial exte... |
| esumfsup 31324 | Formulating an extended su... |
| esumfsupre 31325 | Formulating an extended su... |
| esumss 31326 | Change the index set to a ... |
| esumpinfval 31327 | The value of the extended ... |
| esumpfinvallem 31328 | Lemma for ~ esumpfinval . ... |
| esumpfinval 31329 | The value of the extended ... |
| esumpfinvalf 31330 | Same as ~ esumpfinval , mi... |
| esumpinfsum 31331 | The value of the extended ... |
| esumpcvgval 31332 | The value of the extended ... |
| esumpmono 31333 | The partial sums in an ext... |
| esumcocn 31334 | Lemma for ~ esummulc2 and ... |
| esummulc1 31335 | An extended sum multiplied... |
| esummulc2 31336 | An extended sum multiplied... |
| esumdivc 31337 | An extended sum divided by... |
| hashf2 31338 | Lemma for ~ hasheuni . (C... |
| hasheuni 31339 | The cardinality of a disjo... |
| esumcvg 31340 | The sequence of partial su... |
| esumcvg2 31341 | Simpler version of ~ esumc... |
| esumcvgsum 31342 | The value of the extended ... |
| esumsup 31343 | Express an extended sum as... |
| esumgect 31344 | "Send ` n ` to ` +oo ` " i... |
| esumcvgre 31345 | All terms of a converging ... |
| esum2dlem 31346 | Lemma for ~ esum2d (finite... |
| esum2d 31347 | Write a double extended su... |
| esumiun 31348 | Sum over a nonnecessarily ... |
| ofceq 31351 | Equality theorem for funct... |
| ofcfval 31352 | Value of an operation appl... |
| ofcval 31353 | Evaluate a function/consta... |
| ofcfn 31354 | The function operation pro... |
| ofcfeqd2 31355 | Equality theorem for funct... |
| ofcfval3 31356 | General value of ` ( F oFC... |
| ofcf 31357 | The function/constant oper... |
| ofcfval2 31358 | The function operation exp... |
| ofcfval4 31359 | The function/constant oper... |
| ofcc 31360 | Left operation by a consta... |
| ofcof 31361 | Relate function operation ... |
| sigaex 31364 | Lemma for ~ issiga and ~ i... |
| sigaval 31365 | The set of sigma-algebra w... |
| issiga 31366 | An alternative definition ... |
| isrnsiga 31367 | The property of being a si... |
| 0elsiga 31368 | A sigma-algebra contains t... |
| baselsiga 31369 | A sigma-algebra contains i... |
| sigasspw 31370 | A sigma-algebra is a set o... |
| sigaclcu 31371 | A sigma-algebra is closed ... |
| sigaclcuni 31372 | A sigma-algebra is closed ... |
| sigaclfu 31373 | A sigma-algebra is closed ... |
| sigaclcu2 31374 | A sigma-algebra is closed ... |
| sigaclfu2 31375 | A sigma-algebra is closed ... |
| sigaclcu3 31376 | A sigma-algebra is closed ... |
| issgon 31377 | Property of being a sigma-... |
| sgon 31378 | A sigma-algebra is a sigma... |
| elsigass 31379 | An element of a sigma-alge... |
| elrnsiga 31380 | Dropping the base informat... |
| isrnsigau 31381 | The property of being a si... |
| unielsiga 31382 | A sigma-algebra contains i... |
| dmvlsiga 31383 | Lebesgue-measurable subset... |
| pwsiga 31384 | Any power set forms a sigm... |
| prsiga 31385 | The smallest possible sigm... |
| sigaclci 31386 | A sigma-algebra is closed ... |
| difelsiga 31387 | A sigma-algebra is closed ... |
| unelsiga 31388 | A sigma-algebra is closed ... |
| inelsiga 31389 | A sigma-algebra is closed ... |
| sigainb 31390 | Building a sigma-algebra f... |
| insiga 31391 | The intersection of a coll... |
| sigagenval 31394 | Value of the generated sig... |
| sigagensiga 31395 | A generated sigma-algebra ... |
| sgsiga 31396 | A generated sigma-algebra ... |
| unisg 31397 | The sigma-algebra generate... |
| dmsigagen 31398 | A sigma-algebra can be gen... |
| sssigagen 31399 | A set is a subset of the s... |
| sssigagen2 31400 | A subset of the generating... |
| elsigagen 31401 | Any element of a set is al... |
| elsigagen2 31402 | Any countable union of ele... |
| sigagenss 31403 | The generated sigma-algebr... |
| sigagenss2 31404 | Sufficient condition for i... |
| sigagenid 31405 | The sigma-algebra generate... |
| ispisys 31406 | The property of being a pi... |
| ispisys2 31407 | The property of being a pi... |
| inelpisys 31408 | Pi-systems are closed unde... |
| sigapisys 31409 | All sigma-algebras are pi-... |
| isldsys 31410 | The property of being a la... |
| pwldsys 31411 | The power set of the unive... |
| unelldsys 31412 | Lambda-systems are closed ... |
| sigaldsys 31413 | All sigma-algebras are lam... |
| ldsysgenld 31414 | The intersection of all la... |
| sigapildsyslem 31415 | Lemma for ~ sigapildsys . ... |
| sigapildsys 31416 | Sigma-algebra are exactly ... |
| ldgenpisyslem1 31417 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem2 31418 | Lemma for ~ ldgenpisys . ... |
| ldgenpisyslem3 31419 | Lemma for ~ ldgenpisys . ... |
| ldgenpisys 31420 | The lambda system ` E ` ge... |
| dynkin 31421 | Dynkin's lambda-pi theorem... |
| isros 31422 | The property of being a ri... |
| rossspw 31423 | A ring of sets is a collec... |
| 0elros 31424 | A ring of sets contains th... |
| unelros 31425 | A ring of sets is closed u... |
| difelros 31426 | A ring of sets is closed u... |
| inelros 31427 | A ring of sets is closed u... |
| fiunelros 31428 | A ring of sets is closed u... |
| issros 31429 | The property of being a se... |
| srossspw 31430 | A semiring of sets is a co... |
| 0elsros 31431 | A semiring of sets contain... |
| inelsros 31432 | A semiring of sets is clos... |
| diffiunisros 31433 | In semiring of sets, compl... |
| rossros 31434 | Rings of sets are semiring... |
| brsiga 31437 | The Borel Algebra on real ... |
| brsigarn 31438 | The Borel Algebra is a sig... |
| brsigasspwrn 31439 | The Borel Algebra is a set... |
| unibrsiga 31440 | The union of the Borel Alg... |
| cldssbrsiga 31441 | A Borel Algebra contains a... |
| sxval 31444 | Value of the product sigma... |
| sxsiga 31445 | A product sigma-algebra is... |
| sxsigon 31446 | A product sigma-algebra is... |
| sxuni 31447 | The base set of a product ... |
| elsx 31448 | The cartesian product of t... |
| measbase 31451 | The base set of a measure ... |
| measval 31452 | The value of the ` measure... |
| ismeas 31453 | The property of being a me... |
| isrnmeas 31454 | The property of being a me... |
| dmmeas 31455 | The domain of a measure is... |
| measbasedom 31456 | The base set of a measure ... |
| measfrge0 31457 | A measure is a function ov... |
| measfn 31458 | A measure is a function on... |
| measvxrge0 31459 | The values of a measure ar... |
| measvnul 31460 | The measure of the empty s... |
| measge0 31461 | A measure is nonnegative. ... |
| measle0 31462 | If the measure of a given ... |
| measvun 31463 | The measure of a countable... |
| measxun2 31464 | The measure the union of t... |
| measun 31465 | The measure the union of t... |
| measvunilem 31466 | Lemma for ~ measvuni . (C... |
| measvunilem0 31467 | Lemma for ~ measvuni . (C... |
| measvuni 31468 | The measure of a countable... |
| measssd 31469 | A measure is monotone with... |
| measunl 31470 | A measure is sub-additive ... |
| measiuns 31471 | The measure of the union o... |
| measiun 31472 | A measure is sub-additive.... |
| meascnbl 31473 | A measure is continuous fr... |
| measinblem 31474 | Lemma for ~ measinb . (Co... |
| measinb 31475 | Building a measure restric... |
| measres 31476 | Building a measure restric... |
| measinb2 31477 | Building a measure restric... |
| measdivcst 31478 | Division of a measure by a... |
| measdivcstALTV 31479 | Alternate version of ~ mea... |
| cntmeas 31480 | The Counting measure is a ... |
| pwcntmeas 31481 | The counting measure is a ... |
| cntnevol 31482 | Counting and Lebesgue meas... |
| voliune 31483 | The Lebesgue measure funct... |
| volfiniune 31484 | The Lebesgue measure funct... |
| volmeas 31485 | The Lebesgue measure is a ... |
| ddeval1 31488 | Value of the delta measure... |
| ddeval0 31489 | Value of the delta measure... |
| ddemeas 31490 | The Dirac delta measure is... |
| relae 31494 | 'almost everywhere' is a r... |
| brae 31495 | 'almost everywhere' relati... |
| braew 31496 | 'almost everywhere' relati... |
| truae 31497 | A truth holds almost every... |
| aean 31498 | A conjunction holds almost... |
| faeval 31500 | Value of the 'almost every... |
| relfae 31501 | The 'almost everywhere' bu... |
| brfae 31502 | 'almost everywhere' relati... |
| ismbfm 31505 | The predicate " ` F ` is a... |
| elunirnmbfm 31506 | The property of being a me... |
| mbfmfun 31507 | A measurable function is a... |
| mbfmf 31508 | A measurable function as a... |
| isanmbfm 31509 | The predicate to be a meas... |
| mbfmcnvima 31510 | The preimage by a measurab... |
| mbfmbfm 31511 | A measurable function to a... |
| mbfmcst 31512 | A constant function is mea... |
| 1stmbfm 31513 | The first projection map i... |
| 2ndmbfm 31514 | The second projection map ... |
| imambfm 31515 | If the sigma-algebra in th... |
| cnmbfm 31516 | A continuous function is m... |
| mbfmco 31517 | The composition of two mea... |
| mbfmco2 31518 | The pair building of two m... |
| mbfmvolf 31519 | Measurable functions with ... |
| elmbfmvol2 31520 | Measurable functions with ... |
| mbfmcnt 31521 | All functions are measurab... |
| br2base 31522 | The base set for the gener... |
| dya2ub 31523 | An upper bound for a dyadi... |
| sxbrsigalem0 31524 | The closed half-spaces of ... |
| sxbrsigalem3 31525 | The sigma-algebra generate... |
| dya2iocival 31526 | The function ` I ` returns... |
| dya2iocress 31527 | Dyadic intervals are subse... |
| dya2iocbrsiga 31528 | Dyadic intervals are Borel... |
| dya2icobrsiga 31529 | Dyadic intervals are Borel... |
| dya2icoseg 31530 | For any point and any clos... |
| dya2icoseg2 31531 | For any point and any open... |
| dya2iocrfn 31532 | The function returning dya... |
| dya2iocct 31533 | The dyadic rectangle set i... |
| dya2iocnrect 31534 | For any point of an open r... |
| dya2iocnei 31535 | For any point of an open s... |
| dya2iocuni 31536 | Every open set of ` ( RR X... |
| dya2iocucvr 31537 | The dyadic rectangular set... |
| sxbrsigalem1 31538 | The Borel algebra on ` ( R... |
| sxbrsigalem2 31539 | The sigma-algebra generate... |
| sxbrsigalem4 31540 | The Borel algebra on ` ( R... |
| sxbrsigalem5 31541 | First direction for ~ sxbr... |
| sxbrsigalem6 31542 | First direction for ~ sxbr... |
| sxbrsiga 31543 | The product sigma-algebra ... |
| omsval 31546 | Value of the function mapp... |
| omsfval 31547 | Value of the outer measure... |
| omscl 31548 | A closure lemma for the co... |
| omsf 31549 | A constructed outer measur... |
| oms0 31550 | A constructed outer measur... |
| omsmon 31551 | A constructed outer measur... |
| omssubaddlem 31552 | For any small margin ` E `... |
| omssubadd 31553 | A constructed outer measur... |
| carsgval 31556 | Value of the Caratheodory ... |
| carsgcl 31557 | Closure of the Caratheodor... |
| elcarsg 31558 | Property of being a Carath... |
| baselcarsg 31559 | The universe set, ` O ` , ... |
| 0elcarsg 31560 | The empty set is Caratheod... |
| carsguni 31561 | The union of all Caratheod... |
| elcarsgss 31562 | Caratheodory measurable se... |
| difelcarsg 31563 | The Caratheodory measurabl... |
| inelcarsg 31564 | The Caratheodory measurabl... |
| unelcarsg 31565 | The Caratheodory-measurabl... |
| difelcarsg2 31566 | The Caratheodory-measurabl... |
| carsgmon 31567 | Utility lemma: Apply mono... |
| carsgsigalem 31568 | Lemma for the following th... |
| fiunelcarsg 31569 | The Caratheodory measurabl... |
| carsgclctunlem1 31570 | Lemma for ~ carsgclctun . ... |
| carsggect 31571 | The outer measure is count... |
| carsgclctunlem2 31572 | Lemma for ~ carsgclctun . ... |
| carsgclctunlem3 31573 | Lemma for ~ carsgclctun . ... |
| carsgclctun 31574 | The Caratheodory measurabl... |
| carsgsiga 31575 | The Caratheodory measurabl... |
| omsmeas 31576 | The restriction of a const... |
| pmeasmono 31577 | This theorem's hypotheses ... |
| pmeasadd 31578 | A premeasure on a ring of ... |
| itgeq12dv 31579 | Equality theorem for an in... |
| sitgval 31585 | Value of the simple functi... |
| issibf 31586 | The predicate " ` F ` is a... |
| sibf0 31587 | The constant zero function... |
| sibfmbl 31588 | A simple function is measu... |
| sibff 31589 | A simple function is a fun... |
| sibfrn 31590 | A simple function has fini... |
| sibfima 31591 | Any preimage of a singleto... |
| sibfinima 31592 | The measure of the interse... |
| sibfof 31593 | Applying function operatio... |
| sitgfval 31594 | Value of the Bochner integ... |
| sitgclg 31595 | Closure of the Bochner int... |
| sitgclbn 31596 | Closure of the Bochner int... |
| sitgclcn 31597 | Closure of the Bochner int... |
| sitgclre 31598 | Closure of the Bochner int... |
| sitg0 31599 | The integral of the consta... |
| sitgf 31600 | The integral for simple fu... |
| sitgaddlemb 31601 | Lemma for * sitgadd . (Co... |
| sitmval 31602 | Value of the simple functi... |
| sitmfval 31603 | Value of the integral dist... |
| sitmcl 31604 | Closure of the integral di... |
| sitmf 31605 | The integral metric as a f... |
| oddpwdc 31607 | Lemma for ~ eulerpart . T... |
| oddpwdcv 31608 | Lemma for ~ eulerpart : va... |
| eulerpartlemsv1 31609 | Lemma for ~ eulerpart . V... |
| eulerpartlemelr 31610 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv2 31611 | Lemma for ~ eulerpart . V... |
| eulerpartlemsf 31612 | Lemma for ~ eulerpart . (... |
| eulerpartlems 31613 | Lemma for ~ eulerpart . (... |
| eulerpartlemsv3 31614 | Lemma for ~ eulerpart . V... |
| eulerpartlemgc 31615 | Lemma for ~ eulerpart . (... |
| eulerpartleme 31616 | Lemma for ~ eulerpart . (... |
| eulerpartlemv 31617 | Lemma for ~ eulerpart . (... |
| eulerpartlemo 31618 | Lemma for ~ eulerpart : ` ... |
| eulerpartlemd 31619 | Lemma for ~ eulerpart : ` ... |
| eulerpartlem1 31620 | Lemma for ~ eulerpart . (... |
| eulerpartlemb 31621 | Lemma for ~ eulerpart . T... |
| eulerpartlemt0 31622 | Lemma for ~ eulerpart . (... |
| eulerpartlemf 31623 | Lemma for ~ eulerpart : O... |
| eulerpartlemt 31624 | Lemma for ~ eulerpart . (... |
| eulerpartgbij 31625 | Lemma for ~ eulerpart : T... |
| eulerpartlemgv 31626 | Lemma for ~ eulerpart : va... |
| eulerpartlemr 31627 | Lemma for ~ eulerpart . (... |
| eulerpartlemmf 31628 | Lemma for ~ eulerpart . (... |
| eulerpartlemgvv 31629 | Lemma for ~ eulerpart : va... |
| eulerpartlemgu 31630 | Lemma for ~ eulerpart : R... |
| eulerpartlemgh 31631 | Lemma for ~ eulerpart : T... |
| eulerpartlemgf 31632 | Lemma for ~ eulerpart : I... |
| eulerpartlemgs2 31633 | Lemma for ~ eulerpart : T... |
| eulerpartlemn 31634 | Lemma for ~ eulerpart . (... |
| eulerpart 31635 | Euler's theorem on partiti... |
| subiwrd 31638 | Lemma for ~ sseqp1 . (Con... |
| subiwrdlen 31639 | Length of a subword of an ... |
| iwrdsplit 31640 | Lemma for ~ sseqp1 . (Con... |
| sseqval 31641 | Value of the strong sequen... |
| sseqfv1 31642 | Value of the strong sequen... |
| sseqfn 31643 | A strong recursive sequenc... |
| sseqmw 31644 | Lemma for ~ sseqf amd ~ ss... |
| sseqf 31645 | A strong recursive sequenc... |
| sseqfres 31646 | The first elements in the ... |
| sseqfv2 31647 | Value of the strong sequen... |
| sseqp1 31648 | Value of the strong sequen... |
| fiblem 31651 | Lemma for ~ fib0 , ~ fib1 ... |
| fib0 31652 | Value of the Fibonacci seq... |
| fib1 31653 | Value of the Fibonacci seq... |
| fibp1 31654 | Value of the Fibonacci seq... |
| fib2 31655 | Value of the Fibonacci seq... |
| fib3 31656 | Value of the Fibonacci seq... |
| fib4 31657 | Value of the Fibonacci seq... |
| fib5 31658 | Value of the Fibonacci seq... |
| fib6 31659 | Value of the Fibonacci seq... |
| elprob 31662 | The property of being a pr... |
| domprobmeas 31663 | A probability measure is a... |
| domprobsiga 31664 | The domain of a probabilit... |
| probtot 31665 | The probability of the uni... |
| prob01 31666 | A probability is an elemen... |
| probnul 31667 | The probability of the emp... |
| unveldomd 31668 | The universe is an element... |
| unveldom 31669 | The universe is an element... |
| nuleldmp 31670 | The empty set is an elemen... |
| probcun 31671 | The probability of the uni... |
| probun 31672 | The probability of the uni... |
| probdif 31673 | The probability of the dif... |
| probinc 31674 | A probability law is incre... |
| probdsb 31675 | The probability of the com... |
| probmeasd 31676 | A probability measure is a... |
| probvalrnd 31677 | The value of a probability... |
| probtotrnd 31678 | The probability of the uni... |
| totprobd 31679 | Law of total probability, ... |
| totprob 31680 | Law of total probability. ... |
| probfinmeasb 31681 | Build a probability measur... |
| probfinmeasbALTV 31682 | Alternate version of ~ pro... |
| probmeasb 31683 | Build a probability from a... |
| cndprobval 31686 | The value of the condition... |
| cndprobin 31687 | An identity linking condit... |
| cndprob01 31688 | The conditional probabilit... |
| cndprobtot 31689 | The conditional probabilit... |
| cndprobnul 31690 | The conditional probabilit... |
| cndprobprob 31691 | The conditional probabilit... |
| bayesth 31692 | Bayes Theorem. (Contribut... |
| rrvmbfm 31695 | A real-valued random varia... |
| isrrvv 31696 | Elementhood to the set of ... |
| rrvvf 31697 | A real-valued random varia... |
| rrvfn 31698 | A real-valued random varia... |
| rrvdm 31699 | The domain of a random var... |
| rrvrnss 31700 | The range of a random vari... |
| rrvf2 31701 | A real-valued random varia... |
| rrvdmss 31702 | The domain of a random var... |
| rrvfinvima 31703 | For a real-value random va... |
| 0rrv 31704 | The constant function equa... |
| rrvadd 31705 | The sum of two random vari... |
| rrvmulc 31706 | A random variable multipli... |
| rrvsum 31707 | An indexed sum of random v... |
| orvcval 31710 | Value of the preimage mapp... |
| orvcval2 31711 | Another way to express the... |
| elorvc 31712 | Elementhood of a preimage.... |
| orvcval4 31713 | The value of the preimage ... |
| orvcoel 31714 | If the relation produces o... |
| orvccel 31715 | If the relation produces c... |
| elorrvc 31716 | Elementhood of a preimage ... |
| orrvcval4 31717 | The value of the preimage ... |
| orrvcoel 31718 | If the relation produces o... |
| orrvccel 31719 | If the relation produces c... |
| orvcgteel 31720 | Preimage maps produced by ... |
| orvcelval 31721 | Preimage maps produced by ... |
| orvcelel 31722 | Preimage maps produced by ... |
| dstrvval 31723 | The value of the distribut... |
| dstrvprob 31724 | The distribution of a rand... |
| orvclteel 31725 | Preimage maps produced by ... |
| dstfrvel 31726 | Elementhood of preimage ma... |
| dstfrvunirn 31727 | The limit of all preimage ... |
| orvclteinc 31728 | Preimage maps produced by ... |
| dstfrvinc 31729 | A cumulative distribution ... |
| dstfrvclim1 31730 | The limit of the cumulativ... |
| coinfliplem 31731 | Division in the extended r... |
| coinflipprob 31732 | The ` P ` we defined for c... |
| coinflipspace 31733 | The space of our coin-flip... |
| coinflipuniv 31734 | The universe of our coin-f... |
| coinfliprv 31735 | The ` X ` we defined for c... |
| coinflippv 31736 | The probability of heads i... |
| coinflippvt 31737 | The probability of tails i... |
| ballotlemoex 31738 | ` O ` is a set. (Contribu... |
| ballotlem1 31739 | The size of the universe i... |
| ballotlemelo 31740 | Elementhood in ` O ` . (C... |
| ballotlem2 31741 | The probability that the f... |
| ballotlemfval 31742 | The value of F. (Contribut... |
| ballotlemfelz 31743 | ` ( F `` C ) ` has values ... |
| ballotlemfp1 31744 | If the ` J ` th ballot is ... |
| ballotlemfc0 31745 | ` F ` takes value 0 betwee... |
| ballotlemfcc 31746 | ` F ` takes value 0 betwee... |
| ballotlemfmpn 31747 | ` ( F `` C ) ` finishes co... |
| ballotlemfval0 31748 | ` ( F `` C ) ` always star... |
| ballotleme 31749 | Elements of ` E ` . (Cont... |
| ballotlemodife 31750 | Elements of ` ( O \ E ) ` ... |
| ballotlem4 31751 | If the first pick is a vot... |
| ballotlem5 31752 | If A is not ahead througho... |
| ballotlemi 31753 | Value of ` I ` for a given... |
| ballotlemiex 31754 | Properties of ` ( I `` C )... |
| ballotlemi1 31755 | The first tie cannot be re... |
| ballotlemii 31756 | The first tie cannot be re... |
| ballotlemsup 31757 | The set of zeroes of ` F `... |
| ballotlemimin 31758 | ` ( I `` C ) ` is the firs... |
| ballotlemic 31759 | If the first vote is for B... |
| ballotlem1c 31760 | If the first vote is for A... |
| ballotlemsval 31761 | Value of ` S ` . (Contrib... |
| ballotlemsv 31762 | Value of ` S ` evaluated a... |
| ballotlemsgt1 31763 | ` S ` maps values less tha... |
| ballotlemsdom 31764 | Domain of ` S ` for a give... |
| ballotlemsel1i 31765 | The range ` ( 1 ... ( I ``... |
| ballotlemsf1o 31766 | The defined ` S ` is a bij... |
| ballotlemsi 31767 | The image by ` S ` of the ... |
| ballotlemsima 31768 | The image by ` S ` of an i... |
| ballotlemieq 31769 | If two countings share the... |
| ballotlemrval 31770 | Value of ` R ` . (Contrib... |
| ballotlemscr 31771 | The image of ` ( R `` C ) ... |
| ballotlemrv 31772 | Value of ` R ` evaluated a... |
| ballotlemrv1 31773 | Value of ` R ` before the ... |
| ballotlemrv2 31774 | Value of ` R ` after the t... |
| ballotlemro 31775 | Range of ` R ` is included... |
| ballotlemgval 31776 | Expand the value of ` .^ `... |
| ballotlemgun 31777 | A property of the defined ... |
| ballotlemfg 31778 | Express the value of ` ( F... |
| ballotlemfrc 31779 | Express the value of ` ( F... |
| ballotlemfrci 31780 | Reverse counting preserves... |
| ballotlemfrceq 31781 | Value of ` F ` for a rever... |
| ballotlemfrcn0 31782 | Value of ` F ` for a rever... |
| ballotlemrc 31783 | Range of ` R ` . (Contrib... |
| ballotlemirc 31784 | Applying ` R ` does not ch... |
| ballotlemrinv0 31785 | Lemma for ~ ballotlemrinv ... |
| ballotlemrinv 31786 | ` R ` is its own inverse :... |
| ballotlem1ri 31787 | When the vote on the first... |
| ballotlem7 31788 | ` R ` is a bijection betwe... |
| ballotlem8 31789 | There are as many counting... |
| ballotth 31790 | Bertrand's ballot problem ... |
| sgncl 31791 | Closure of the signum. (C... |
| sgnclre 31792 | Closure of the signum. (C... |
| sgnneg 31793 | Negation of the signum. (... |
| sgn3da 31794 | A conditional containing a... |
| sgnmul 31795 | Signum of a product. (Con... |
| sgnmulrp2 31796 | Multiplication by a positi... |
| sgnsub 31797 | Subtraction of a number of... |
| sgnnbi 31798 | Negative signum. (Contrib... |
| sgnpbi 31799 | Positive signum. (Contrib... |
| sgn0bi 31800 | Zero signum. (Contributed... |
| sgnsgn 31801 | Signum is idempotent. (Co... |
| sgnmulsgn 31802 | If two real numbers are of... |
| sgnmulsgp 31803 | If two real numbers are of... |
| fzssfzo 31804 | Condition for an integer i... |
| gsumncl 31805 | Closure of a group sum in ... |
| gsumnunsn 31806 | Closure of a group sum in ... |
| ccatmulgnn0dir 31807 | Concatenation of words fol... |
| ofcccat 31808 | Letterwise operations on w... |
| ofcs1 31809 | Letterwise operations on a... |
| ofcs2 31810 | Letterwise operations on a... |
| plymul02 31811 | Product of a polynomial wi... |
| plymulx0 31812 | Coefficients of a polynomi... |
| plymulx 31813 | Coefficients of a polynomi... |
| plyrecld 31814 | Closure of a polynomial wi... |
| signsplypnf 31815 | The quotient of a polynomi... |
| signsply0 31816 | Lemma for the rule of sign... |
| signspval 31817 | The value of the skipping ... |
| signsw0glem 31818 | Neutral element property o... |
| signswbase 31819 | The base of ` W ` is the t... |
| signswplusg 31820 | The operation of ` W ` . ... |
| signsw0g 31821 | The neutral element of ` W... |
| signswmnd 31822 | ` W ` is a monoid structur... |
| signswrid 31823 | The zero-skipping operatio... |
| signswlid 31824 | The zero-skipping operatio... |
| signswn0 31825 | The zero-skipping operatio... |
| signswch 31826 | The zero-skipping operatio... |
| signslema 31827 | Computational part of sign... |
| signstfv 31828 | Value of the zero-skipping... |
| signstfval 31829 | Value of the zero-skipping... |
| signstcl 31830 | Closure of the zero skippi... |
| signstf 31831 | The zero skipping sign wor... |
| signstlen 31832 | Length of the zero skippin... |
| signstf0 31833 | Sign of a single letter wo... |
| signstfvn 31834 | Zero-skipping sign in a wo... |
| signsvtn0 31835 | If the last letter is nonz... |
| signstfvp 31836 | Zero-skipping sign in a wo... |
| signstfvneq0 31837 | In case the first letter i... |
| signstfvcl 31838 | Closure of the zero skippi... |
| signstfvc 31839 | Zero-skipping sign in a wo... |
| signstres 31840 | Restriction of a zero skip... |
| signstfveq0a 31841 | Lemma for ~ signstfveq0 . ... |
| signstfveq0 31842 | In case the last letter is... |
| signsvvfval 31843 | The value of ` V ` , which... |
| signsvvf 31844 | ` V ` is a function. (Con... |
| signsvf0 31845 | There is no change of sign... |
| signsvf1 31846 | In a single-letter word, w... |
| signsvfn 31847 | Number of changes in a wor... |
| signsvtp 31848 | Adding a letter of the sam... |
| signsvtn 31849 | Adding a letter of a diffe... |
| signsvfpn 31850 | Adding a letter of the sam... |
| signsvfnn 31851 | Adding a letter of a diffe... |
| signlem0 31852 | Adding a zero as the highe... |
| signshf 31853 | ` H ` , corresponding to t... |
| signshwrd 31854 | ` H ` , corresponding to t... |
| signshlen 31855 | Length of ` H ` , correspo... |
| signshnz 31856 | ` H ` is not the empty wor... |
| efcld 31857 | Closure law for the expone... |
| iblidicc 31858 | The identity function is i... |
| rpsqrtcn 31859 | Continuity of the real pos... |
| divsqrtid 31860 | A real number divided by i... |
| cxpcncf1 31861 | The power function on comp... |
| efmul2picn 31862 | Multiplying by ` ( _i x. (... |
| fct2relem 31863 | Lemma for ~ ftc2re . (Con... |
| ftc2re 31864 | The Fundamental Theorem of... |
| fdvposlt 31865 | Functions with a positive ... |
| fdvneggt 31866 | Functions with a negative ... |
| fdvposle 31867 | Functions with a nonnegati... |
| fdvnegge 31868 | Functions with a nonpositi... |
| prodfzo03 31869 | A product of three factors... |
| actfunsnf1o 31870 | The action ` F ` of extend... |
| actfunsnrndisj 31871 | The action ` F ` of extend... |
| itgexpif 31872 | The basis for the circle m... |
| fsum2dsub 31873 | Lemma for ~ breprexp - Re-... |
| reprval 31876 | Value of the representatio... |
| repr0 31877 | There is exactly one repre... |
| reprf 31878 | Members of the representat... |
| reprsum 31879 | Sums of values of the memb... |
| reprle 31880 | Upper bound to the terms i... |
| reprsuc 31881 | Express the representation... |
| reprfi 31882 | Bounded representations ar... |
| reprss 31883 | Representations with terms... |
| reprinrn 31884 | Representations with term ... |
| reprlt 31885 | There are no representatio... |
| hashreprin 31886 | Express a sum of represent... |
| reprgt 31887 | There are no representatio... |
| reprinfz1 31888 | For the representation of ... |
| reprfi2 31889 | Corollary of ~ reprinfz1 .... |
| reprfz1 31890 | Corollary of ~ reprinfz1 .... |
| hashrepr 31891 | Develop the number of repr... |
| reprpmtf1o 31892 | Transposing ` 0 ` and ` X ... |
| reprdifc 31893 | Express the representation... |
| chpvalz 31894 | Value of the second Chebys... |
| chtvalz 31895 | Value of the Chebyshev fun... |
| breprexplema 31896 | Lemma for ~ breprexp (indu... |
| breprexplemb 31897 | Lemma for ~ breprexp (clos... |
| breprexplemc 31898 | Lemma for ~ breprexp (indu... |
| breprexp 31899 | Express the ` S ` th power... |
| breprexpnat 31900 | Express the ` S ` th power... |
| vtsval 31903 | Value of the Vinogradov tr... |
| vtscl 31904 | Closure of the Vinogradov ... |
| vtsprod 31905 | Express the Vinogradov tri... |
| circlemeth 31906 | The Hardy, Littlewood and ... |
| circlemethnat 31907 | The Hardy, Littlewood and ... |
| circlevma 31908 | The Circle Method, where t... |
| circlemethhgt 31909 | The circle method, where t... |
| hgt750lemc 31913 | An upper bound to the summ... |
| hgt750lemd 31914 | An upper bound to the summ... |
| hgt749d 31915 | A deduction version of ~ a... |
| logdivsqrle 31916 | Conditions for ` ( ( log `... |
| hgt750lem 31917 | Lemma for ~ tgoldbachgtd .... |
| hgt750lem2 31918 | Decimal multiplication gal... |
| hgt750lemf 31919 | Lemma for the statement 7.... |
| hgt750lemg 31920 | Lemma for the statement 7.... |
| oddprm2 31921 | Two ways to write the set ... |
| hgt750lemb 31922 | An upper bound on the cont... |
| hgt750lema 31923 | An upper bound on the cont... |
| hgt750leme 31924 | An upper bound on the cont... |
| tgoldbachgnn 31925 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtde 31926 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtda 31927 | Lemma for ~ tgoldbachgtd .... |
| tgoldbachgtd 31928 | Odd integers greater than ... |
| tgoldbachgt 31929 | Odd integers greater than ... |
| istrkg2d 31932 | Property of fulfilling dim... |
| axtglowdim2ALTV 31933 | Alternate version of ~ axt... |
| axtgupdim2ALTV 31934 | Alternate version of ~ axt... |
| afsval 31937 | Value of the AFS relation ... |
| brafs 31938 | Binary relation form of th... |
| tg5segofs 31939 | Rephrase ~ axtg5seg using ... |
| lpadval 31942 | Value of the ` leftpad ` f... |
| lpadlem1 31943 | Lemma for the ` leftpad ` ... |
| lpadlem3 31944 | Lemma for ~ lpadlen1 (Cont... |
| lpadlen1 31945 | Length of a left-padded wo... |
| lpadlem2 31946 | Lemma for the ` leftpad ` ... |
| lpadlen2 31947 | Length of a left-padded wo... |
| lpadmax 31948 | Length of a left-padded wo... |
| lpadleft 31949 | The contents of prefix of ... |
| lpadright 31950 | The suffix of a left-padde... |
| bnj170 31963 | ` /\ ` -manipulation. (Co... |
| bnj240 31964 | ` /\ ` -manipulation. (Co... |
| bnj248 31965 | ` /\ ` -manipulation. (Co... |
| bnj250 31966 | ` /\ ` -manipulation. (Co... |
| bnj251 31967 | ` /\ ` -manipulation. (Co... |
| bnj252 31968 | ` /\ ` -manipulation. (Co... |
| bnj253 31969 | ` /\ ` -manipulation. (Co... |
| bnj255 31970 | ` /\ ` -manipulation. (Co... |
| bnj256 31971 | ` /\ ` -manipulation. (Co... |
| bnj257 31972 | ` /\ ` -manipulation. (Co... |
| bnj258 31973 | ` /\ ` -manipulation. (Co... |
| bnj268 31974 | ` /\ ` -manipulation. (Co... |
| bnj290 31975 | ` /\ ` -manipulation. (Co... |
| bnj291 31976 | ` /\ ` -manipulation. (Co... |
| bnj312 31977 | ` /\ ` -manipulation. (Co... |
| bnj334 31978 | ` /\ ` -manipulation. (Co... |
| bnj345 31979 | ` /\ ` -manipulation. (Co... |
| bnj422 31980 | ` /\ ` -manipulation. (Co... |
| bnj432 31981 | ` /\ ` -manipulation. (Co... |
| bnj446 31982 | ` /\ ` -manipulation. (Co... |
| bnj23 31983 | First-order logic and set ... |
| bnj31 31984 | First-order logic and set ... |
| bnj62 31985 | First-order logic and set ... |
| bnj89 31986 | First-order logic and set ... |
| bnj90 31987 | First-order logic and set ... |
| bnj101 31988 | First-order logic and set ... |
| bnj105 31989 | First-order logic and set ... |
| bnj115 31990 | First-order logic and set ... |
| bnj132 31991 | First-order logic and set ... |
| bnj133 31992 | First-order logic and set ... |
| bnj156 31993 | First-order logic and set ... |
| bnj158 31994 | First-order logic and set ... |
| bnj168 31995 | First-order logic and set ... |
| bnj206 31996 | First-order logic and set ... |
| bnj216 31997 | First-order logic and set ... |
| bnj219 31998 | First-order logic and set ... |
| bnj226 31999 | First-order logic and set ... |
| bnj228 32000 | First-order logic and set ... |
| bnj519 32001 | First-order logic and set ... |
| bnj521 32002 | First-order logic and set ... |
| bnj524 32003 | First-order logic and set ... |
| bnj525 32004 | First-order logic and set ... |
| bnj534 32005 | First-order logic and set ... |
| bnj538 32006 | First-order logic and set ... |
| bnj529 32007 | First-order logic and set ... |
| bnj551 32008 | First-order logic and set ... |
| bnj563 32009 | First-order logic and set ... |
| bnj564 32010 | First-order logic and set ... |
| bnj593 32011 | First-order logic and set ... |
| bnj596 32012 | First-order logic and set ... |
| bnj610 32013 | Pass from equality ( ` x =... |
| bnj642 32014 | ` /\ ` -manipulation. (Co... |
| bnj643 32015 | ` /\ ` -manipulation. (Co... |
| bnj645 32016 | ` /\ ` -manipulation. (Co... |
| bnj658 32017 | ` /\ ` -manipulation. (Co... |
| bnj667 32018 | ` /\ ` -manipulation. (Co... |
| bnj705 32019 | ` /\ ` -manipulation. (Co... |
| bnj706 32020 | ` /\ ` -manipulation. (Co... |
| bnj707 32021 | ` /\ ` -manipulation. (Co... |
| bnj708 32022 | ` /\ ` -manipulation. (Co... |
| bnj721 32023 | ` /\ ` -manipulation. (Co... |
| bnj832 32024 | ` /\ ` -manipulation. (Co... |
| bnj835 32025 | ` /\ ` -manipulation. (Co... |
| bnj836 32026 | ` /\ ` -manipulation. (Co... |
| bnj837 32027 | ` /\ ` -manipulation. (Co... |
| bnj769 32028 | ` /\ ` -manipulation. (Co... |
| bnj770 32029 | ` /\ ` -manipulation. (Co... |
| bnj771 32030 | ` /\ ` -manipulation. (Co... |
| bnj887 32031 | ` /\ ` -manipulation. (Co... |
| bnj918 32032 | First-order logic and set ... |
| bnj919 32033 | First-order logic and set ... |
| bnj923 32034 | First-order logic and set ... |
| bnj927 32035 | First-order logic and set ... |
| bnj930 32036 | First-order logic and set ... |
| bnj931 32037 | First-order logic and set ... |
| bnj937 32038 | First-order logic and set ... |
| bnj941 32039 | First-order logic and set ... |
| bnj945 32040 | Technical lemma for ~ bnj6... |
| bnj946 32041 | First-order logic and set ... |
| bnj951 32042 | ` /\ ` -manipulation. (Co... |
| bnj956 32043 | First-order logic and set ... |
| bnj976 32044 | First-order logic and set ... |
| bnj982 32045 | First-order logic and set ... |
| bnj1019 32046 | First-order logic and set ... |
| bnj1023 32047 | First-order logic and set ... |
| bnj1095 32048 | First-order logic and set ... |
| bnj1096 32049 | First-order logic and set ... |
| bnj1098 32050 | First-order logic and set ... |
| bnj1101 32051 | First-order logic and set ... |
| bnj1113 32052 | First-order logic and set ... |
| bnj1109 32053 | First-order logic and set ... |
| bnj1131 32054 | First-order logic and set ... |
| bnj1138 32055 | First-order logic and set ... |
| bnj1142 32056 | First-order logic and set ... |
| bnj1143 32057 | First-order logic and set ... |
| bnj1146 32058 | First-order logic and set ... |
| bnj1149 32059 | First-order logic and set ... |
| bnj1185 32060 | First-order logic and set ... |
| bnj1196 32061 | First-order logic and set ... |
| bnj1198 32062 | First-order logic and set ... |
| bnj1209 32063 | First-order logic and set ... |
| bnj1211 32064 | First-order logic and set ... |
| bnj1213 32065 | First-order logic and set ... |
| bnj1212 32066 | First-order logic and set ... |
| bnj1219 32067 | First-order logic and set ... |
| bnj1224 32068 | First-order logic and set ... |
| bnj1230 32069 | First-order logic and set ... |
| bnj1232 32070 | First-order logic and set ... |
| bnj1235 32071 | First-order logic and set ... |
| bnj1239 32072 | First-order logic and set ... |
| bnj1238 32073 | First-order logic and set ... |
| bnj1241 32074 | First-order logic and set ... |
| bnj1247 32075 | First-order logic and set ... |
| bnj1254 32076 | First-order logic and set ... |
| bnj1262 32077 | First-order logic and set ... |
| bnj1266 32078 | First-order logic and set ... |
| bnj1265 32079 | First-order logic and set ... |
| bnj1275 32080 | First-order logic and set ... |
| bnj1276 32081 | First-order logic and set ... |
| bnj1292 32082 | First-order logic and set ... |
| bnj1293 32083 | First-order logic and set ... |
| bnj1294 32084 | First-order logic and set ... |
| bnj1299 32085 | First-order logic and set ... |
| bnj1304 32086 | First-order logic and set ... |
| bnj1316 32087 | First-order logic and set ... |
| bnj1317 32088 | First-order logic and set ... |
| bnj1322 32089 | First-order logic and set ... |
| bnj1340 32090 | First-order logic and set ... |
| bnj1345 32091 | First-order logic and set ... |
| bnj1350 32092 | First-order logic and set ... |
| bnj1351 32093 | First-order logic and set ... |
| bnj1352 32094 | First-order logic and set ... |
| bnj1361 32095 | First-order logic and set ... |
| bnj1366 32096 | First-order logic and set ... |
| bnj1379 32097 | First-order logic and set ... |
| bnj1383 32098 | First-order logic and set ... |
| bnj1385 32099 | First-order logic and set ... |
| bnj1386 32100 | First-order logic and set ... |
| bnj1397 32101 | First-order logic and set ... |
| bnj1400 32102 | First-order logic and set ... |
| bnj1405 32103 | First-order logic and set ... |
| bnj1422 32104 | First-order logic and set ... |
| bnj1424 32105 | First-order logic and set ... |
| bnj1436 32106 | First-order logic and set ... |
| bnj1441 32107 | First-order logic and set ... |
| bnj1441g 32108 | First-order logic and set ... |
| bnj1454 32109 | First-order logic and set ... |
| bnj1459 32110 | First-order logic and set ... |
| bnj1464 32111 | Conversion of implicit sub... |
| bnj1465 32112 | First-order logic and set ... |
| bnj1468 32113 | Conversion of implicit sub... |
| bnj1476 32114 | First-order logic and set ... |
| bnj1502 32115 | First-order logic and set ... |
| bnj1503 32116 | First-order logic and set ... |
| bnj1517 32117 | First-order logic and set ... |
| bnj1521 32118 | First-order logic and set ... |
| bnj1533 32119 | First-order logic and set ... |
| bnj1534 32120 | First-order logic and set ... |
| bnj1536 32121 | First-order logic and set ... |
| bnj1538 32122 | First-order logic and set ... |
| bnj1541 32123 | First-order logic and set ... |
| bnj1542 32124 | First-order logic and set ... |
| bnj110 32125 | Well-founded induction res... |
| bnj157 32126 | Well-founded induction res... |
| bnj66 32127 | Technical lemma for ~ bnj6... |
| bnj91 32128 | First-order logic and set ... |
| bnj92 32129 | First-order logic and set ... |
| bnj93 32130 | Technical lemma for ~ bnj9... |
| bnj95 32131 | Technical lemma for ~ bnj1... |
| bnj96 32132 | Technical lemma for ~ bnj1... |
| bnj97 32133 | Technical lemma for ~ bnj1... |
| bnj98 32134 | Technical lemma for ~ bnj1... |
| bnj106 32135 | First-order logic and set ... |
| bnj118 32136 | First-order logic and set ... |
| bnj121 32137 | First-order logic and set ... |
| bnj124 32138 | Technical lemma for ~ bnj1... |
| bnj125 32139 | Technical lemma for ~ bnj1... |
| bnj126 32140 | Technical lemma for ~ bnj1... |
| bnj130 32141 | Technical lemma for ~ bnj1... |
| bnj149 32142 | Technical lemma for ~ bnj1... |
| bnj150 32143 | Technical lemma for ~ bnj1... |
| bnj151 32144 | Technical lemma for ~ bnj1... |
| bnj154 32145 | Technical lemma for ~ bnj1... |
| bnj155 32146 | Technical lemma for ~ bnj1... |
| bnj153 32147 | Technical lemma for ~ bnj8... |
| bnj207 32148 | Technical lemma for ~ bnj8... |
| bnj213 32149 | First-order logic and set ... |
| bnj222 32150 | Technical lemma for ~ bnj2... |
| bnj229 32151 | Technical lemma for ~ bnj5... |
| bnj517 32152 | Technical lemma for ~ bnj5... |
| bnj518 32153 | Technical lemma for ~ bnj8... |
| bnj523 32154 | Technical lemma for ~ bnj8... |
| bnj526 32155 | Technical lemma for ~ bnj8... |
| bnj528 32156 | Technical lemma for ~ bnj8... |
| bnj535 32157 | Technical lemma for ~ bnj8... |
| bnj539 32158 | Technical lemma for ~ bnj8... |
| bnj540 32159 | Technical lemma for ~ bnj8... |
| bnj543 32160 | Technical lemma for ~ bnj8... |
| bnj544 32161 | Technical lemma for ~ bnj8... |
| bnj545 32162 | Technical lemma for ~ bnj8... |
| bnj546 32163 | Technical lemma for ~ bnj8... |
| bnj548 32164 | Technical lemma for ~ bnj8... |
| bnj553 32165 | Technical lemma for ~ bnj8... |
| bnj554 32166 | Technical lemma for ~ bnj8... |
| bnj556 32167 | Technical lemma for ~ bnj8... |
| bnj557 32168 | Technical lemma for ~ bnj8... |
| bnj558 32169 | Technical lemma for ~ bnj8... |
| bnj561 32170 | Technical lemma for ~ bnj8... |
| bnj562 32171 | Technical lemma for ~ bnj8... |
| bnj570 32172 | Technical lemma for ~ bnj8... |
| bnj571 32173 | Technical lemma for ~ bnj8... |
| bnj605 32174 | Technical lemma. This lem... |
| bnj581 32175 | Technical lemma for ~ bnj5... |
| bnj589 32176 | Technical lemma for ~ bnj8... |
| bnj590 32177 | Technical lemma for ~ bnj8... |
| bnj591 32178 | Technical lemma for ~ bnj8... |
| bnj594 32179 | Technical lemma for ~ bnj8... |
| bnj580 32180 | Technical lemma for ~ bnj5... |
| bnj579 32181 | Technical lemma for ~ bnj8... |
| bnj602 32182 | Equality theorem for the `... |
| bnj607 32183 | Technical lemma for ~ bnj8... |
| bnj609 32184 | Technical lemma for ~ bnj8... |
| bnj611 32185 | Technical lemma for ~ bnj8... |
| bnj600 32186 | Technical lemma for ~ bnj8... |
| bnj601 32187 | Technical lemma for ~ bnj8... |
| bnj852 32188 | Technical lemma for ~ bnj6... |
| bnj864 32189 | Technical lemma for ~ bnj6... |
| bnj865 32190 | Technical lemma for ~ bnj6... |
| bnj873 32191 | Technical lemma for ~ bnj6... |
| bnj849 32192 | Technical lemma for ~ bnj6... |
| bnj882 32193 | Definition (using hypothes... |
| bnj18eq1 32194 | Equality theorem for trans... |
| bnj893 32195 | Property of ` _trCl ` . U... |
| bnj900 32196 | Technical lemma for ~ bnj6... |
| bnj906 32197 | Property of ` _trCl ` . (... |
| bnj908 32198 | Technical lemma for ~ bnj6... |
| bnj911 32199 | Technical lemma for ~ bnj6... |
| bnj916 32200 | Technical lemma for ~ bnj6... |
| bnj917 32201 | Technical lemma for ~ bnj6... |
| bnj934 32202 | Technical lemma for ~ bnj6... |
| bnj929 32203 | Technical lemma for ~ bnj6... |
| bnj938 32204 | Technical lemma for ~ bnj6... |
| bnj944 32205 | Technical lemma for ~ bnj6... |
| bnj953 32206 | Technical lemma for ~ bnj6... |
| bnj958 32207 | Technical lemma for ~ bnj6... |
| bnj1000 32208 | Technical lemma for ~ bnj8... |
| bnj965 32209 | Technical lemma for ~ bnj8... |
| bnj964 32210 | Technical lemma for ~ bnj6... |
| bnj966 32211 | Technical lemma for ~ bnj6... |
| bnj967 32212 | Technical lemma for ~ bnj6... |
| bnj969 32213 | Technical lemma for ~ bnj6... |
| bnj970 32214 | Technical lemma for ~ bnj6... |
| bnj910 32215 | Technical lemma for ~ bnj6... |
| bnj978 32216 | Technical lemma for ~ bnj6... |
| bnj981 32217 | Technical lemma for ~ bnj6... |
| bnj983 32218 | Technical lemma for ~ bnj6... |
| bnj984 32219 | Technical lemma for ~ bnj6... |
| bnj985v 32220 | Version of ~ bnj985 with a... |
| bnj985 32221 | Technical lemma for ~ bnj6... |
| bnj986 32222 | Technical lemma for ~ bnj6... |
| bnj996 32223 | Technical lemma for ~ bnj6... |
| bnj998 32224 | Technical lemma for ~ bnj6... |
| bnj999 32225 | Technical lemma for ~ bnj6... |
| bnj1001 32226 | Technical lemma for ~ bnj6... |
| bnj1006 32227 | Technical lemma for ~ bnj6... |
| bnj1014 32228 | Technical lemma for ~ bnj6... |
| bnj1015 32229 | Technical lemma for ~ bnj6... |
| bnj1018g 32230 | Version of ~ bnj1018 with ... |
| bnj1018 32231 | Technical lemma for ~ bnj6... |
| bnj1020 32232 | Technical lemma for ~ bnj6... |
| bnj1021 32233 | Technical lemma for ~ bnj6... |
| bnj907 32234 | Technical lemma for ~ bnj6... |
| bnj1029 32235 | Property of ` _trCl ` . (... |
| bnj1033 32236 | Technical lemma for ~ bnj6... |
| bnj1034 32237 | Technical lemma for ~ bnj6... |
| bnj1039 32238 | Technical lemma for ~ bnj6... |
| bnj1040 32239 | Technical lemma for ~ bnj6... |
| bnj1047 32240 | Technical lemma for ~ bnj6... |
| bnj1049 32241 | Technical lemma for ~ bnj6... |
| bnj1052 32242 | Technical lemma for ~ bnj6... |
| bnj1053 32243 | Technical lemma for ~ bnj6... |
| bnj1071 32244 | Technical lemma for ~ bnj6... |
| bnj1083 32245 | Technical lemma for ~ bnj6... |
| bnj1090 32246 | Technical lemma for ~ bnj6... |
| bnj1093 32247 | Technical lemma for ~ bnj6... |
| bnj1097 32248 | Technical lemma for ~ bnj6... |
| bnj1110 32249 | Technical lemma for ~ bnj6... |
| bnj1112 32250 | Technical lemma for ~ bnj6... |
| bnj1118 32251 | Technical lemma for ~ bnj6... |
| bnj1121 32252 | Technical lemma for ~ bnj6... |
| bnj1123 32253 | Technical lemma for ~ bnj6... |
| bnj1030 32254 | Technical lemma for ~ bnj6... |
| bnj1124 32255 | Property of ` _trCl ` . (... |
| bnj1133 32256 | Technical lemma for ~ bnj6... |
| bnj1128 32257 | Technical lemma for ~ bnj6... |
| bnj1127 32258 | Property of ` _trCl ` . (... |
| bnj1125 32259 | Property of ` _trCl ` . (... |
| bnj1145 32260 | Technical lemma for ~ bnj6... |
| bnj1147 32261 | Property of ` _trCl ` . (... |
| bnj1137 32262 | Property of ` _trCl ` . (... |
| bnj1148 32263 | Property of ` _pred ` . (... |
| bnj1136 32264 | Technical lemma for ~ bnj6... |
| bnj1152 32265 | Technical lemma for ~ bnj6... |
| bnj1154 32266 | Property of ` Fr ` . (Con... |
| bnj1171 32267 | Technical lemma for ~ bnj6... |
| bnj1172 32268 | Technical lemma for ~ bnj6... |
| bnj1173 32269 | Technical lemma for ~ bnj6... |
| bnj1174 32270 | Technical lemma for ~ bnj6... |
| bnj1175 32271 | Technical lemma for ~ bnj6... |
| bnj1176 32272 | Technical lemma for ~ bnj6... |
| bnj1177 32273 | Technical lemma for ~ bnj6... |
| bnj1186 32274 | Technical lemma for ~ bnj6... |
| bnj1190 32275 | Technical lemma for ~ bnj6... |
| bnj1189 32276 | Technical lemma for ~ bnj6... |
| bnj69 32277 | Existence of a minimal ele... |
| bnj1228 32278 | Existence of a minimal ele... |
| bnj1204 32279 | Well-founded induction. T... |
| bnj1234 32280 | Technical lemma for ~ bnj6... |
| bnj1245 32281 | Technical lemma for ~ bnj6... |
| bnj1256 32282 | Technical lemma for ~ bnj6... |
| bnj1259 32283 | Technical lemma for ~ bnj6... |
| bnj1253 32284 | Technical lemma for ~ bnj6... |
| bnj1279 32285 | Technical lemma for ~ bnj6... |
| bnj1286 32286 | Technical lemma for ~ bnj6... |
| bnj1280 32287 | Technical lemma for ~ bnj6... |
| bnj1296 32288 | Technical lemma for ~ bnj6... |
| bnj1309 32289 | Technical lemma for ~ bnj6... |
| bnj1307 32290 | Technical lemma for ~ bnj6... |
| bnj1311 32291 | Technical lemma for ~ bnj6... |
| bnj1318 32292 | Technical lemma for ~ bnj6... |
| bnj1326 32293 | Technical lemma for ~ bnj6... |
| bnj1321 32294 | Technical lemma for ~ bnj6... |
| bnj1364 32295 | Property of ` _FrSe ` . (... |
| bnj1371 32296 | Technical lemma for ~ bnj6... |
| bnj1373 32297 | Technical lemma for ~ bnj6... |
| bnj1374 32298 | Technical lemma for ~ bnj6... |
| bnj1384 32299 | Technical lemma for ~ bnj6... |
| bnj1388 32300 | Technical lemma for ~ bnj6... |
| bnj1398 32301 | Technical lemma for ~ bnj6... |
| bnj1413 32302 | Property of ` _trCl ` . (... |
| bnj1408 32303 | Technical lemma for ~ bnj1... |
| bnj1414 32304 | Property of ` _trCl ` . (... |
| bnj1415 32305 | Technical lemma for ~ bnj6... |
| bnj1416 32306 | Technical lemma for ~ bnj6... |
| bnj1418 32307 | Property of ` _pred ` . (... |
| bnj1417 32308 | Technical lemma for ~ bnj6... |
| bnj1421 32309 | Technical lemma for ~ bnj6... |
| bnj1444 32310 | Technical lemma for ~ bnj6... |
| bnj1445 32311 | Technical lemma for ~ bnj6... |
| bnj1446 32312 | Technical lemma for ~ bnj6... |
| bnj1447 32313 | Technical lemma for ~ bnj6... |
| bnj1448 32314 | Technical lemma for ~ bnj6... |
| bnj1449 32315 | Technical lemma for ~ bnj6... |
| bnj1442 32316 | Technical lemma for ~ bnj6... |
| bnj1450 32317 | Technical lemma for ~ bnj6... |
| bnj1423 32318 | Technical lemma for ~ bnj6... |
| bnj1452 32319 | Technical lemma for ~ bnj6... |
| bnj1466 32320 | Technical lemma for ~ bnj6... |
| bnj1467 32321 | Technical lemma for ~ bnj6... |
| bnj1463 32322 | Technical lemma for ~ bnj6... |
| bnj1489 32323 | Technical lemma for ~ bnj6... |
| bnj1491 32324 | Technical lemma for ~ bnj6... |
| bnj1312 32325 | Technical lemma for ~ bnj6... |
| bnj1493 32326 | Technical lemma for ~ bnj6... |
| bnj1497 32327 | Technical lemma for ~ bnj6... |
| bnj1498 32328 | Technical lemma for ~ bnj6... |
| bnj60 32329 | Well-founded recursion, pa... |
| bnj1514 32330 | Technical lemma for ~ bnj1... |
| bnj1518 32331 | Technical lemma for ~ bnj1... |
| bnj1519 32332 | Technical lemma for ~ bnj1... |
| bnj1520 32333 | Technical lemma for ~ bnj1... |
| bnj1501 32334 | Technical lemma for ~ bnj1... |
| bnj1500 32335 | Well-founded recursion, pa... |
| bnj1525 32336 | Technical lemma for ~ bnj1... |
| bnj1529 32337 | Technical lemma for ~ bnj1... |
| bnj1523 32338 | Technical lemma for ~ bnj1... |
| bnj1522 32339 | Well-founded recursion, pa... |
| exdifsn 32340 | There exists an element in... |
| srcmpltd 32341 | If a statement is true for... |
| prsrcmpltd 32342 | If a statement is true for... |
| zltp1ne 32343 | Integer ordering relation.... |
| nnltp1ne 32344 | Positive integer ordering ... |
| nn0ltp1ne 32345 | Nonnegative integer orderi... |
| 0nn0m1nnn0 32346 | A number is zero if and on... |
| fisshasheq 32347 | A finite set is equal to i... |
| dff15 32348 | A one-to-one function in t... |
| hashfundm 32349 | The size of a set function... |
| hashf1dmrn 32350 | The size of the domain of ... |
| hashf1dmcdm 32351 | The size of the domain of ... |
| funen1cnv 32352 | If a function is equinumer... |
| f1resveqaeq 32353 | If a function restricted t... |
| f1resrcmplf1dlem 32354 | Lemma for ~ f1resrcmplf1d ... |
| f1resrcmplf1d 32355 | If a function's restrictio... |
| f1resfz0f1d 32356 | If a function with a seque... |
| revpfxsfxrev 32357 | The reverse of a prefix of... |
| swrdrevpfx 32358 | A subword expressed in ter... |
| lfuhgr 32359 | A hypergraph is loop-free ... |
| lfuhgr2 32360 | A hypergraph is loop-free ... |
| lfuhgr3 32361 | A hypergraph is loop-free ... |
| cplgredgex 32362 | Any two (distinct) vertice... |
| cusgredgex 32363 | Any two (distinct) vertice... |
| cusgredgex2 32364 | Any two distinct vertices ... |
| pfxwlk 32365 | A prefix of a walk is a wa... |
| revwlk 32366 | The reverse of a walk is a... |
| revwlkb 32367 | Two words represent a walk... |
| swrdwlk 32368 | Two matching subwords of a... |
| pthhashvtx 32369 | A graph containing a path ... |
| pthisspthorcycl 32370 | A path is either a simple ... |
| spthcycl 32371 | A walk is a trivial path i... |
| usgrgt2cycl 32372 | A non-trivial cycle in a s... |
| usgrcyclgt2v 32373 | A simple graph with a non-... |
| subgrwlk 32374 | If a walk exists in a subg... |
| subgrtrl 32375 | If a trail exists in a sub... |
| subgrpth 32376 | If a path exists in a subg... |
| subgrcycl 32377 | If a cycle exists in a sub... |
| cusgr3cyclex 32378 | Every complete simple grap... |
| loop1cycl 32379 | A hypergraph has a cycle o... |
| 2cycld 32380 | Construction of a 2-cycle ... |
| 2cycl2d 32381 | Construction of a 2-cycle ... |
| umgr2cycllem 32382 | Lemma for ~ umgr2cycl . (... |
| umgr2cycl 32383 | A multigraph with two dist... |
| dfacycgr1 32386 | An alternate definition of... |
| isacycgr 32387 | The property of being an a... |
| isacycgr1 32388 | The property of being an a... |
| acycgrcycl 32389 | Any cycle in an acyclic gr... |
| acycgr0v 32390 | A null graph (with no vert... |
| acycgr1v 32391 | A multigraph with one vert... |
| acycgr2v 32392 | A simple graph with two ve... |
| prclisacycgr 32393 | A proper class (representi... |
| acycgrislfgr 32394 | An acyclic hypergraph is a... |
| upgracycumgr 32395 | An acyclic pseudograph is ... |
| umgracycusgr 32396 | An acyclic multigraph is a... |
| upgracycusgr 32397 | An acyclic pseudograph is ... |
| cusgracyclt3v 32398 | A complete simple graph is... |
| pthacycspth 32399 | A path in an acyclic graph... |
| acycgrsubgr 32400 | The subgraph of an acyclic... |
| quartfull 32407 | The quartic equation, writ... |
| deranglem 32408 | Lemma for derangements. (... |
| derangval 32409 | Define the derangement fun... |
| derangf 32410 | The derangement number is ... |
| derang0 32411 | The derangement number of ... |
| derangsn 32412 | The derangement number of ... |
| derangenlem 32413 | One half of ~ derangen . ... |
| derangen 32414 | The derangement number is ... |
| subfacval 32415 | The subfactorial is define... |
| derangen2 32416 | Write the derangement numb... |
| subfacf 32417 | The subfactorial is a func... |
| subfaclefac 32418 | The subfactorial is less t... |
| subfac0 32419 | The subfactorial at zero. ... |
| subfac1 32420 | The subfactorial at one. ... |
| subfacp1lem1 32421 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem2a 32422 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem2b 32423 | Lemma for ~ subfacp1 . Pr... |
| subfacp1lem3 32424 | Lemma for ~ subfacp1 . In... |
| subfacp1lem4 32425 | Lemma for ~ subfacp1 . Th... |
| subfacp1lem5 32426 | Lemma for ~ subfacp1 . In... |
| subfacp1lem6 32427 | Lemma for ~ subfacp1 . By... |
| subfacp1 32428 | A two-term recurrence for ... |
| subfacval2 32429 | A closed-form expression f... |
| subfaclim 32430 | The subfactorial converges... |
| subfacval3 32431 | Another closed form expres... |
| derangfmla 32432 | The derangements formula, ... |
| erdszelem1 32433 | Lemma for ~ erdsze . (Con... |
| erdszelem2 32434 | Lemma for ~ erdsze . (Con... |
| erdszelem3 32435 | Lemma for ~ erdsze . (Con... |
| erdszelem4 32436 | Lemma for ~ erdsze . (Con... |
| erdszelem5 32437 | Lemma for ~ erdsze . (Con... |
| erdszelem6 32438 | Lemma for ~ erdsze . (Con... |
| erdszelem7 32439 | Lemma for ~ erdsze . (Con... |
| erdszelem8 32440 | Lemma for ~ erdsze . (Con... |
| erdszelem9 32441 | Lemma for ~ erdsze . (Con... |
| erdszelem10 32442 | Lemma for ~ erdsze . (Con... |
| erdszelem11 32443 | Lemma for ~ erdsze . (Con... |
| erdsze 32444 | The Erdős-Szekeres th... |
| erdsze2lem1 32445 | Lemma for ~ erdsze2 . (Co... |
| erdsze2lem2 32446 | Lemma for ~ erdsze2 . (Co... |
| erdsze2 32447 | Generalize the statement o... |
| kur14lem1 32448 | Lemma for ~ kur14 . (Cont... |
| kur14lem2 32449 | Lemma for ~ kur14 . Write... |
| kur14lem3 32450 | Lemma for ~ kur14 . A clo... |
| kur14lem4 32451 | Lemma for ~ kur14 . Compl... |
| kur14lem5 32452 | Lemma for ~ kur14 . Closu... |
| kur14lem6 32453 | Lemma for ~ kur14 . If ` ... |
| kur14lem7 32454 | Lemma for ~ kur14 : main p... |
| kur14lem8 32455 | Lemma for ~ kur14 . Show ... |
| kur14lem9 32456 | Lemma for ~ kur14 . Since... |
| kur14lem10 32457 | Lemma for ~ kur14 . Disch... |
| kur14 32458 | Kuratowski's closure-compl... |
| ispconn 32465 | The property of being a pa... |
| pconncn 32466 | The property of being a pa... |
| pconntop 32467 | A simply connected space i... |
| issconn 32468 | The property of being a si... |
| sconnpconn 32469 | A simply connected space i... |
| sconntop 32470 | A simply connected space i... |
| sconnpht 32471 | A closed path in a simply ... |
| cnpconn 32472 | An image of a path-connect... |
| pconnconn 32473 | A path-connected space is ... |
| txpconn 32474 | The topological product of... |
| ptpconn 32475 | The topological product of... |
| indispconn 32476 | The indiscrete topology (o... |
| connpconn 32477 | A connected and locally pa... |
| qtoppconn 32478 | A quotient of a path-conne... |
| pconnpi1 32479 | All fundamental groups in ... |
| sconnpht2 32480 | Any two paths in a simply ... |
| sconnpi1 32481 | A path-connected topologic... |
| txsconnlem 32482 | Lemma for ~ txsconn . (Co... |
| txsconn 32483 | The topological product of... |
| cvxpconn 32484 | A convex subset of the com... |
| cvxsconn 32485 | A convex subset of the com... |
| blsconn 32486 | An open ball in the comple... |
| cnllysconn 32487 | The topology of the comple... |
| resconn 32488 | A subset of ` RR ` is simp... |
| ioosconn 32489 | An open interval is simply... |
| iccsconn 32490 | A closed interval is simpl... |
| retopsconn 32491 | The real numbers are simpl... |
| iccllysconn 32492 | A closed interval is local... |
| rellysconn 32493 | The real numbers are local... |
| iisconn 32494 | The unit interval is simpl... |
| iillysconn 32495 | The unit interval is local... |
| iinllyconn 32496 | The unit interval is local... |
| fncvm 32499 | Lemma for covering maps. ... |
| cvmscbv 32500 | Change bound variables in ... |
| iscvm 32501 | The property of being a co... |
| cvmtop1 32502 | Reverse closure for a cove... |
| cvmtop2 32503 | Reverse closure for a cove... |
| cvmcn 32504 | A covering map is a contin... |
| cvmcov 32505 | Property of a covering map... |
| cvmsrcl 32506 | Reverse closure for an eve... |
| cvmsi 32507 | One direction of ~ cvmsval... |
| cvmsval 32508 | Elementhood in the set ` S... |
| cvmsss 32509 | An even covering is a subs... |
| cvmsn0 32510 | An even covering is nonemp... |
| cvmsuni 32511 | An even covering of ` U ` ... |
| cvmsdisj 32512 | An even covering of ` U ` ... |
| cvmshmeo 32513 | Every element of an even c... |
| cvmsf1o 32514 | ` F ` , localized to an el... |
| cvmscld 32515 | The sets of an even coveri... |
| cvmsss2 32516 | An open subset of an evenl... |
| cvmcov2 32517 | The covering map property ... |
| cvmseu 32518 | Every element in ` U. T ` ... |
| cvmsiota 32519 | Identify the unique elemen... |
| cvmopnlem 32520 | Lemma for ~ cvmopn . (Con... |
| cvmfolem 32521 | Lemma for ~ cvmfo . (Cont... |
| cvmopn 32522 | A covering map is an open ... |
| cvmliftmolem1 32523 | Lemma for ~ cvmliftmo . (... |
| cvmliftmolem2 32524 | Lemma for ~ cvmliftmo . (... |
| cvmliftmoi 32525 | A lift of a continuous fun... |
| cvmliftmo 32526 | A lift of a continuous fun... |
| cvmliftlem1 32527 | Lemma for ~ cvmlift . In ... |
| cvmliftlem2 32528 | Lemma for ~ cvmlift . ` W ... |
| cvmliftlem3 32529 | Lemma for ~ cvmlift . Sin... |
| cvmliftlem4 32530 | Lemma for ~ cvmlift . The... |
| cvmliftlem5 32531 | Lemma for ~ cvmlift . Def... |
| cvmliftlem6 32532 | Lemma for ~ cvmlift . Ind... |
| cvmliftlem7 32533 | Lemma for ~ cvmlift . Pro... |
| cvmliftlem8 32534 | Lemma for ~ cvmlift . The... |
| cvmliftlem9 32535 | Lemma for ~ cvmlift . The... |
| cvmliftlem10 32536 | Lemma for ~ cvmlift . The... |
| cvmliftlem11 32537 | Lemma for ~ cvmlift . (Co... |
| cvmliftlem13 32538 | Lemma for ~ cvmlift . The... |
| cvmliftlem14 32539 | Lemma for ~ cvmlift . Put... |
| cvmliftlem15 32540 | Lemma for ~ cvmlift . Dis... |
| cvmlift 32541 | One of the important prope... |
| cvmfo 32542 | A covering map is an onto ... |
| cvmliftiota 32543 | Write out a function ` H `... |
| cvmlift2lem1 32544 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9a 32545 | Lemma for ~ cvmlift2 and ~... |
| cvmlift2lem2 32546 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem3 32547 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem4 32548 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem5 32549 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem6 32550 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem7 32551 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem8 32552 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem9 32553 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem10 32554 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem11 32555 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem12 32556 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2lem13 32557 | Lemma for ~ cvmlift2 . (C... |
| cvmlift2 32558 | A two-dimensional version ... |
| cvmliftphtlem 32559 | Lemma for ~ cvmliftpht . ... |
| cvmliftpht 32560 | If ` G ` and ` H ` are pat... |
| cvmlift3lem1 32561 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem2 32562 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem3 32563 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem4 32564 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem5 32565 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem6 32566 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem7 32567 | Lemma for ~ cvmlift3 . (C... |
| cvmlift3lem8 32568 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3lem9 32569 | Lemma for ~ cvmlift2 . (C... |
| cvmlift3 32570 | A general version of ~ cvm... |
| snmlff 32571 | The function ` F ` from ~ ... |
| snmlfval 32572 | The function ` F ` from ~ ... |
| snmlval 32573 | The property " ` A ` is si... |
| snmlflim 32574 | If ` A ` is simply normal,... |
| goel 32589 | A "Godel-set of membership... |
| goelel3xp 32590 | A "Godel-set of membership... |
| goeleq12bg 32591 | Two "Godel-set of membersh... |
| gonafv 32592 | The "Godel-set for the She... |
| goaleq12d 32593 | Equality of the "Godel-set... |
| gonanegoal 32594 | The Godel-set for the Shef... |
| satf 32595 | The satisfaction predicate... |
| satfsucom 32596 | The satisfaction predicate... |
| satfn 32597 | The satisfaction predicate... |
| satom 32598 | The satisfaction predicate... |
| satfvsucom 32599 | The satisfaction predicate... |
| satfv0 32600 | The value of the satisfact... |
| satfvsuclem1 32601 | Lemma 1 for ~ satfvsuc . ... |
| satfvsuclem2 32602 | Lemma 2 for ~ satfvsuc . ... |
| satfvsuc 32603 | The value of the satisfact... |
| satfv1lem 32604 | Lemma for ~ satfv1 . (Con... |
| satfv1 32605 | The value of the satisfact... |
| satfsschain 32606 | The binary relation of a s... |
| satfvsucsuc 32607 | The satisfaction predicate... |
| satfbrsuc 32608 | The binary relation of a s... |
| satfrel 32609 | The value of the satisfact... |
| satfdmlem 32610 | Lemma for ~ satfdm . (Con... |
| satfdm 32611 | The domain of the satisfac... |
| satfrnmapom 32612 | The range of the satisfact... |
| satfv0fun 32613 | The value of the satisfact... |
| satf0 32614 | The satisfaction predicate... |
| satf0sucom 32615 | The satisfaction predicate... |
| satf00 32616 | The value of the satisfact... |
| satf0suclem 32617 | Lemma for ~ satf0suc , ~ s... |
| satf0suc 32618 | The value of the satisfact... |
| satf0op 32619 | An element of a value of t... |
| satf0n0 32620 | The value of the satisfact... |
| sat1el2xp 32621 | The first component of an ... |
| fmlafv 32622 | The valid Godel formulas o... |
| fmla 32623 | The set of all valid Godel... |
| fmla0 32624 | The valid Godel formulas o... |
| fmla0xp 32625 | The valid Godel formulas o... |
| fmlasuc0 32626 | The valid Godel formulas o... |
| fmlafvel 32627 | A class is a valid Godel f... |
| fmlasuc 32628 | The valid Godel formulas o... |
| fmla1 32629 | The valid Godel formulas o... |
| isfmlasuc 32630 | The characterization of a ... |
| fmlasssuc 32631 | The Godel formulas of heig... |
| fmlaomn0 32632 | The empty set is not a God... |
| fmlan0 32633 | The empty set is not a God... |
| gonan0 32634 | The "Godel-set of NAND" is... |
| goaln0 32635 | The "Godel-set of universa... |
| gonarlem 32636 | Lemma for ~ gonar (inducti... |
| gonar 32637 | If the "Godel-set of NAND"... |
| goalrlem 32638 | Lemma for ~ goalr (inducti... |
| goalr 32639 | If the "Godel-set of unive... |
| fmla0disjsuc 32640 | The set of valid Godel for... |
| fmlasucdisj 32641 | The valid Godel formulas o... |
| satfdmfmla 32642 | The domain of the satisfac... |
| satffunlem 32643 | Lemma for ~ satffunlem1lem... |
| satffunlem1lem1 32644 | Lemma for ~ satffunlem1 . ... |
| satffunlem1lem2 32645 | Lemma 2 for ~ satffunlem1 ... |
| satffunlem2lem1 32646 | Lemma 1 for ~ satffunlem2 ... |
| dmopab3rexdif 32647 | The domain of an ordered p... |
| satffunlem2lem2 32648 | Lemma 2 for ~ satffunlem2 ... |
| satffunlem1 32649 | Lemma 1 for ~ satffun : in... |
| satffunlem2 32650 | Lemma 2 for ~ satffun : in... |
| satffun 32651 | The value of the satisfact... |
| satff 32652 | The satisfaction predicate... |
| satfun 32653 | The satisfaction predicate... |
| satfvel 32654 | An element of the value of... |
| satfv0fvfmla0 32655 | The value of the satisfact... |
| satefv 32656 | The simplified satisfactio... |
| sate0 32657 | The simplified satisfactio... |
| satef 32658 | The simplified satisfactio... |
| sate0fv0 32659 | A simplified satisfaction ... |
| satefvfmla0 32660 | The simplified satisfactio... |
| sategoelfvb 32661 | Characterization of a valu... |
| sategoelfv 32662 | Condition of a valuation `... |
| ex-sategoelel 32663 | Example of a valuation of ... |
| ex-sategoel 32664 | Instance of ~ sategoelfv f... |
| satfv1fvfmla1 32665 | The value of the satisfact... |
| 2goelgoanfmla1 32666 | Two Godel-sets of membersh... |
| satefvfmla1 32667 | The simplified satisfactio... |
| ex-sategoelelomsuc 32668 | Example of a valuation of ... |
| ex-sategoelel12 32669 | Example of a valuation of ... |
| prv 32670 | The "proves" relation on a... |
| elnanelprv 32671 | The wff ` ( A e. B -/\ B e... |
| prv0 32672 | Every wff encoded as ` U `... |
| prv1n 32673 | No wff encoded as a Godel-... |
| mvtval 32742 | The set of variable typeco... |
| mrexval 32743 | The set of "raw expression... |
| mexval 32744 | The set of expressions, wh... |
| mexval2 32745 | The set of expressions, wh... |
| mdvval 32746 | The set of disjoint variab... |
| mvrsval 32747 | The set of variables in an... |
| mvrsfpw 32748 | The set of variables in an... |
| mrsubffval 32749 | The substitution of some v... |
| mrsubfval 32750 | The substitution of some v... |
| mrsubval 32751 | The substitution of some v... |
| mrsubcv 32752 | The value of a substituted... |
| mrsubvr 32753 | The value of a substituted... |
| mrsubff 32754 | A substitution is a functi... |
| mrsubrn 32755 | Although it is defined for... |
| mrsubff1 32756 | When restricted to complet... |
| mrsubff1o 32757 | When restricted to complet... |
| mrsub0 32758 | The value of the substitut... |
| mrsubf 32759 | A substitution is a functi... |
| mrsubccat 32760 | Substitution distributes o... |
| mrsubcn 32761 | A substitution does not ch... |
| elmrsubrn 32762 | Characterization of the su... |
| mrsubco 32763 | The composition of two sub... |
| mrsubvrs 32764 | The set of variables in a ... |
| msubffval 32765 | A substitution applied to ... |
| msubfval 32766 | A substitution applied to ... |
| msubval 32767 | A substitution applied to ... |
| msubrsub 32768 | A substitution applied to ... |
| msubty 32769 | The type of a substituted ... |
| elmsubrn 32770 | Characterization of substi... |
| msubrn 32771 | Although it is defined for... |
| msubff 32772 | A substitution is a functi... |
| msubco 32773 | The composition of two sub... |
| msubf 32774 | A substitution is a functi... |
| mvhfval 32775 | Value of the function mapp... |
| mvhval 32776 | Value of the function mapp... |
| mpstval 32777 | A pre-statement is an orde... |
| elmpst 32778 | Property of being a pre-st... |
| msrfval 32779 | Value of the reduct of a p... |
| msrval 32780 | Value of the reduct of a p... |
| mpstssv 32781 | A pre-statement is an orde... |
| mpst123 32782 | Decompose a pre-statement ... |
| mpstrcl 32783 | The elements of a pre-stat... |
| msrf 32784 | The reduct of a pre-statem... |
| msrrcl 32785 | If ` X ` and ` Y ` have th... |
| mstaval 32786 | Value of the set of statem... |
| msrid 32787 | The reduct of a statement ... |
| msrfo 32788 | The reduct of a pre-statem... |
| mstapst 32789 | A statement is a pre-state... |
| elmsta 32790 | Property of being a statem... |
| ismfs 32791 | A formal system is a tuple... |
| mfsdisj 32792 | The constants and variable... |
| mtyf2 32793 | The type function maps var... |
| mtyf 32794 | The type function maps var... |
| mvtss 32795 | The set of variable typeco... |
| maxsta 32796 | An axiom is a statement. ... |
| mvtinf 32797 | Each variable typecode has... |
| msubff1 32798 | When restricted to complet... |
| msubff1o 32799 | When restricted to complet... |
| mvhf 32800 | The function mapping varia... |
| mvhf1 32801 | The function mapping varia... |
| msubvrs 32802 | The set of variables in a ... |
| mclsrcl 32803 | Reverse closure for the cl... |
| mclsssvlem 32804 | Lemma for ~ mclsssv . (Co... |
| mclsval 32805 | The function mapping varia... |
| mclsssv 32806 | The closure of a set of ex... |
| ssmclslem 32807 | Lemma for ~ ssmcls . (Con... |
| vhmcls 32808 | All variable hypotheses ar... |
| ssmcls 32809 | The original expressions a... |
| ss2mcls 32810 | The closure is monotonic u... |
| mclsax 32811 | The closure is closed unde... |
| mclsind 32812 | Induction theorem for clos... |
| mppspstlem 32813 | Lemma for ~ mppspst . (Co... |
| mppsval 32814 | Definition of a provable p... |
| elmpps 32815 | Definition of a provable p... |
| mppspst 32816 | A provable pre-statement i... |
| mthmval 32817 | A theorem is a pre-stateme... |
| elmthm 32818 | A theorem is a pre-stateme... |
| mthmi 32819 | A statement whose reduct i... |
| mthmsta 32820 | A theorem is a pre-stateme... |
| mppsthm 32821 | A provable pre-statement i... |
| mthmblem 32822 | Lemma for ~ mthmb . (Cont... |
| mthmb 32823 | If two statements have the... |
| mthmpps 32824 | Given a theorem, there is ... |
| mclsppslem 32825 | The closure is closed unde... |
| mclspps 32826 | The closure is closed unde... |
| problem1 32903 | Practice problem 1. Clues... |
| problem2 32904 | Practice problem 2. Clues... |
| problem3 32905 | Practice problem 3. Clues... |
| problem4 32906 | Practice problem 4. Clues... |
| problem5 32907 | Practice problem 5. Clues... |
| quad3 32908 | Variant of quadratic equat... |
| climuzcnv 32909 | Utility lemma to convert b... |
| sinccvglem 32910 | ` ( ( sin `` x ) / x ) ~~>... |
| sinccvg 32911 | ` ( ( sin `` x ) / x ) ~~>... |
| circum 32912 | The circumference of a cir... |
| elfzm12 32913 | Membership in a curtailed ... |
| nn0seqcvg 32914 | A strictly-decreasing nonn... |
| lediv2aALT 32915 | Division of both sides of ... |
| abs2sqlei 32916 | The absolute values of two... |
| abs2sqlti 32917 | The absolute values of two... |
| abs2sqle 32918 | The absolute values of two... |
| abs2sqlt 32919 | The absolute values of two... |
| abs2difi 32920 | Difference of absolute val... |
| abs2difabsi 32921 | Absolute value of differen... |
| axextprim 32922 | ~ ax-ext without distinct ... |
| axrepprim 32923 | ~ ax-rep without distinct ... |
| axunprim 32924 | ~ ax-un without distinct v... |
| axpowprim 32925 | ~ ax-pow without distinct ... |
| axregprim 32926 | ~ ax-reg without distinct ... |
| axinfprim 32927 | ~ ax-inf without distinct ... |
| axacprim 32928 | ~ ax-ac without distinct v... |
| untelirr 32929 | We call a class "untanged"... |
| untuni 32930 | The union of a class is un... |
| untsucf 32931 | If a class is untangled, t... |
| unt0 32932 | The null set is untangled.... |
| untint 32933 | If there is an untangled e... |
| efrunt 32934 | If ` A ` is well-founded b... |
| untangtr 32935 | A transitive class is unta... |
| 3orel2 32936 | Partial elimination of a t... |
| 3orel3 32937 | Partial elimination of a t... |
| 3pm3.2ni 32938 | Triple negated disjunction... |
| 3jaodd 32939 | Double deduction form of ~... |
| 3orit 32940 | Closed form of ~ 3ori . (... |
| biimpexp 32941 | A biconditional in the ant... |
| 3orel13 32942 | Elimination of two disjunc... |
| nepss 32943 | Two classes are unequal if... |
| 3ccased 32944 | Triple disjunction form of... |
| dfso3 32945 | Expansion of the definitio... |
| brtpid1 32946 | A binary relation involvin... |
| brtpid2 32947 | A binary relation involvin... |
| brtpid3 32948 | A binary relation involvin... |
| ceqsrexv2 32949 | Alternate elimitation of a... |
| iota5f 32950 | A method for computing iot... |
| ceqsralv2 32951 | Alternate elimination of a... |
| dford5 32952 | A class is ordinal iff it ... |
| jath 32953 | Closed form of ~ ja . Pro... |
| sqdivzi 32954 | Distribution of square ove... |
| supfz 32955 | The supremum of a finite s... |
| inffz 32956 | The infimum of a finite se... |
| fz0n 32957 | The sequence ` ( 0 ... ( N... |
| shftvalg 32958 | Value of a sequence shifte... |
| divcnvlin 32959 | Limit of the ratio of two ... |
| climlec3 32960 | Comparison of a constant t... |
| logi 32961 | Calculate the logarithm of... |
| iexpire 32962 | ` _i ` raised to itself is... |
| bcneg1 32963 | The binomial coefficent ov... |
| bcm1nt 32964 | The proportion of one bion... |
| bcprod 32965 | A product identity for bin... |
| bccolsum 32966 | A column-sum rule for bino... |
| iprodefisumlem 32967 | Lemma for ~ iprodefisum . ... |
| iprodefisum 32968 | Applying the exponential f... |
| iprodgam 32969 | An infinite product versio... |
| faclimlem1 32970 | Lemma for ~ faclim . Clos... |
| faclimlem2 32971 | Lemma for ~ faclim . Show... |
| faclimlem3 32972 | Lemma for ~ faclim . Alge... |
| faclim 32973 | An infinite product expres... |
| iprodfac 32974 | An infinite product expres... |
| faclim2 32975 | Another factorial limit du... |
| pdivsq 32976 | Condition for a prime divi... |
| dvdspw 32977 | Exponentiation law for div... |
| gcd32 32978 | Swap the second and third ... |
| gcdabsorb 32979 | Absorption law for gcd. (... |
| brtp 32980 | A condition for a binary r... |
| dftr6 32981 | A potential definition of ... |
| coep 32982 | Composition with the membe... |
| coepr 32983 | Composition with the conve... |
| dffr5 32984 | A quantifier free definiti... |
| dfso2 32985 | Quantifier free definition... |
| dfpo2 32986 | Quantifier free definition... |
| br8 32987 | Substitution for an eight-... |
| br6 32988 | Substitution for a six-pla... |
| br4 32989 | Substitution for a four-pl... |
| cnvco1 32990 | Another distributive law o... |
| cnvco2 32991 | Another distributive law o... |
| eldm3 32992 | Quantifier-free definition... |
| elrn3 32993 | Quantifier-free definition... |
| pocnv 32994 | The converse of a partial ... |
| socnv 32995 | The converse of a strict o... |
| sotrd 32996 | Transitivity law for stric... |
| sotr3 32997 | Transitivity law for stric... |
| sotrine 32998 | Trichotomy law for strict ... |
| eqfunresadj 32999 | Law for adjoining an eleme... |
| eqfunressuc 33000 | Law for equality of restri... |
| funeldmb 33001 | If ` (/) ` is not part of ... |
| elintfv 33002 | Membership in an intersect... |
| funpsstri 33003 | A condition for subset tri... |
| fundmpss 33004 | If a class ` F ` is a prop... |
| fvresval 33005 | The value of a function at... |
| funsseq 33006 | Given two functions with e... |
| fununiq 33007 | The uniqueness condition o... |
| funbreq 33008 | An equality condition for ... |
| br1steq 33009 | Uniqueness condition for t... |
| br2ndeq 33010 | Uniqueness condition for t... |
| dfdm5 33011 | Definition of domain in te... |
| dfrn5 33012 | Definition of range in ter... |
| opelco3 33013 | Alternate way of saying th... |
| elima4 33014 | Quantifier-free expression... |
| fv1stcnv 33015 | The value of the converse ... |
| fv2ndcnv 33016 | The value of the converse ... |
| imaindm 33017 | The image is unaffected by... |
| setinds 33018 | Principle of set induction... |
| setinds2f 33019 | ` _E ` induction schema, u... |
| setinds2 33020 | ` _E ` induction schema, u... |
| elpotr 33021 | A class of transitive sets... |
| dford5reg 33022 | Given ~ ax-reg , an ordina... |
| dfon2lem1 33023 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem2 33024 | Lemma for ~ dfon2 . (Cont... |
| dfon2lem3 33025 | Lemma for ~ dfon2 . All s... |
| dfon2lem4 33026 | Lemma for ~ dfon2 . If tw... |
| dfon2lem5 33027 | Lemma for ~ dfon2 . Two s... |
| dfon2lem6 33028 | Lemma for ~ dfon2 . A tra... |
| dfon2lem7 33029 | Lemma for ~ dfon2 . All e... |
| dfon2lem8 33030 | Lemma for ~ dfon2 . The i... |
| dfon2lem9 33031 | Lemma for ~ dfon2 . A cla... |
| dfon2 33032 | ` On ` consists of all set... |
| rdgprc0 33033 | The value of the recursive... |
| rdgprc 33034 | The value of the recursive... |
| dfrdg2 33035 | Alternate definition of th... |
| dfrdg3 33036 | Generalization of ~ dfrdg2... |
| axextdfeq 33037 | A version of ~ ax-ext for ... |
| ax8dfeq 33038 | A version of ~ ax-8 for us... |
| axextdist 33039 | ~ ax-ext with distinctors ... |
| axextbdist 33040 | ~ axextb with distinctors ... |
| 19.12b 33041 | Version of ~ 19.12vv with ... |
| exnel 33042 | There is always a set not ... |
| distel 33043 | Distinctors in terms of me... |
| axextndbi 33044 | ~ axextnd as a bicondition... |
| hbntg 33045 | A more general form of ~ h... |
| hbimtg 33046 | A more general and closed ... |
| hbaltg 33047 | A more general and closed ... |
| hbng 33048 | A more general form of ~ h... |
| hbimg 33049 | A more general form of ~ h... |
| tfisg 33050 | A closed form of ~ tfis . ... |
| dftrpred2 33053 | A definition of the transi... |
| trpredeq1 33054 | Equality theorem for trans... |
| trpredeq2 33055 | Equality theorem for trans... |
| trpredeq3 33056 | Equality theorem for trans... |
| trpredeq1d 33057 | Equality deduction for tra... |
| trpredeq2d 33058 | Equality deduction for tra... |
| trpredeq3d 33059 | Equality deduction for tra... |
| eltrpred 33060 | A class is a transitive pr... |
| trpredlem1 33061 | Technical lemma for transi... |
| trpredpred 33062 | Assuming it exists, the pr... |
| trpredss 33063 | The transitive predecessor... |
| trpredtr 33064 | The transitive predecessor... |
| trpredmintr 33065 | The transitive predecessor... |
| trpredelss 33066 | Given a transitive predece... |
| dftrpred3g 33067 | The transitive predecessor... |
| dftrpred4g 33068 | Another recursive expressi... |
| trpredpo 33069 | If ` R ` partially orders ... |
| trpred0 33070 | The class of transitive pr... |
| trpredex 33071 | The transitive predecessor... |
| trpredrec 33072 | If ` Y ` is an ` R ` , ` A... |
| frpomin 33073 | Every (possibly proper) su... |
| frpomin2 33074 | Every (possibly proper) su... |
| frpoind 33075 | The principle of founded i... |
| frpoinsg 33076 | Founded, Partial-Ordering ... |
| frpoins2fg 33077 | Founded Partial Induction ... |
| frpoins2g 33078 | Founded Partial Induction ... |
| frmin 33079 | Every (possibly proper) su... |
| frind 33080 | The principle of founded i... |
| frindi 33081 | The principle of founded i... |
| frinsg 33082 | Founded Induction Schema. ... |
| frins 33083 | Founded Induction Schema. ... |
| frins2fg 33084 | Founded Induction schema, ... |
| frins2f 33085 | Founded Induction schema, ... |
| frins2g 33086 | Founded Induction schema, ... |
| frins2 33087 | Founded Induction schema, ... |
| frins3 33088 | Founded Induction schema, ... |
| orderseqlem 33089 | Lemma for ~ poseq and ~ so... |
| poseq 33090 | A partial ordering of sequ... |
| soseq 33091 | A linear ordering of seque... |
| wsuceq123 33096 | Equality theorem for well-... |
| wsuceq1 33097 | Equality theorem for well-... |
| wsuceq2 33098 | Equality theorem for well-... |
| wsuceq3 33099 | Equality theorem for well-... |
| nfwsuc 33100 | Bound-variable hypothesis ... |
| wlimeq12 33101 | Equality theorem for the l... |
| wlimeq1 33102 | Equality theorem for the l... |
| wlimeq2 33103 | Equality theorem for the l... |
| nfwlim 33104 | Bound-variable hypothesis ... |
| elwlim 33105 | Membership in the limit cl... |
| wzel 33106 | The zero of a well-founded... |
| wsuclem 33107 | Lemma for the supremum pro... |
| wsucex 33108 | Existence theorem for well... |
| wsuccl 33109 | If ` X ` is a set with an ... |
| wsuclb 33110 | A well-founded successor i... |
| wlimss 33111 | The class of limit points ... |
| frecseq123 33114 | Equality theorem for found... |
| nffrecs 33115 | Bound-variable hypothesis ... |
| frr3g 33116 | Functions defined by found... |
| fpr3g 33117 | Functions defined by found... |
| frrlem1 33118 | Lemma for founded recursio... |
| frrlem2 33119 | Lemma for founded recursio... |
| frrlem3 33120 | Lemma for founded recursio... |
| frrlem4 33121 | Lemma for founded recursio... |
| frrlem5 33122 | Lemma for founded recursio... |
| frrlem6 33123 | Lemma for founded recursio... |
| frrlem7 33124 | Lemma for founded recursio... |
| frrlem8 33125 | Lemma for founded recursio... |
| frrlem9 33126 | Lemma for founded recursio... |
| frrlem10 33127 | Lemma for founded recursio... |
| frrlem11 33128 | Lemma for founded recursio... |
| frrlem12 33129 | Lemma for founded recursio... |
| frrlem13 33130 | Lemma for founded recursio... |
| frrlem14 33131 | Lemma for founded recursio... |
| fprlem1 33132 | Lemma for founded partial ... |
| fprlem2 33133 | Lemma for founded partial ... |
| fpr1 33134 | Law of founded partial rec... |
| fpr2 33135 | Law of founded partial rec... |
| fpr3 33136 | Law of founded partial rec... |
| frrlem15 33137 | Lemma for general founded ... |
| frrlem16 33138 | Lemma for general founded ... |
| frr1 33139 | Law of general founded rec... |
| frr2 33140 | Law of general founded rec... |
| frr3 33141 | Law of general founded rec... |
| elno 33148 | Membership in the surreals... |
| sltval 33149 | The value of the surreal l... |
| bdayval 33150 | The value of the birthday ... |
| nofun 33151 | A surreal is a function. ... |
| nodmon 33152 | The domain of a surreal is... |
| norn 33153 | The range of a surreal is ... |
| nofnbday 33154 | A surreal is a function ov... |
| nodmord 33155 | The domain of a surreal ha... |
| elno2 33156 | An alternative condition f... |
| elno3 33157 | Another condition for memb... |
| sltval2 33158 | Alternate expression for s... |
| nofv 33159 | The function value of a su... |
| nosgnn0 33160 | ` (/) ` is not a surreal s... |
| nosgnn0i 33161 | If ` X ` is a surreal sign... |
| noreson 33162 | The restriction of a surre... |
| sltintdifex 33163 |
If ` A |
| sltres 33164 | If the restrictions of two... |
| noxp1o 33165 | The Cartesian product of a... |
| noseponlem 33166 | Lemma for ~ nosepon . Con... |
| nosepon 33167 | Given two unequal surreals... |
| noextend 33168 | Extending a surreal by one... |
| noextendseq 33169 | Extend a surreal by a sequ... |
| noextenddif 33170 | Calculate the place where ... |
| noextendlt 33171 | Extending a surreal with a... |
| noextendgt 33172 | Extending a surreal with a... |
| nolesgn2o 33173 | Given ` A ` less than or e... |
| nolesgn2ores 33174 | Given ` A ` less than or e... |
| sltsolem1 33175 | Lemma for ~ sltso . The s... |
| sltso 33176 | Surreal less than totally ... |
| bdayfo 33177 | The birthday function maps... |
| fvnobday 33178 | The value of a surreal at ... |
| nosepnelem 33179 | Lemma for ~ nosepne . (Co... |
| nosepne 33180 | The value of two non-equal... |
| nosep1o 33181 | If the value of a surreal ... |
| nosepdmlem 33182 | Lemma for ~ nosepdm . (Co... |
| nosepdm 33183 | The first place two surrea... |
| nosepeq 33184 | The values of two surreals... |
| nosepssdm 33185 | Given two non-equal surrea... |
| nodenselem4 33186 | Lemma for ~ nodense . Sho... |
| nodenselem5 33187 | Lemma for ~ nodense . If ... |
| nodenselem6 33188 | The restriction of a surre... |
| nodenselem7 33189 | Lemma for ~ nodense . ` A ... |
| nodenselem8 33190 | Lemma for ~ nodense . Giv... |
| nodense 33191 | Given two distinct surreal... |
| bdayimaon 33192 | Lemma for full-eta propert... |
| nolt02olem 33193 | Lemma for ~ nolt02o . If ... |
| nolt02o 33194 | Given ` A ` less than ` B ... |
| noresle 33195 | Restriction law for surrea... |
| nomaxmo 33196 | A class of surreals has at... |
| noprefixmo 33197 | In any class of surreals, ... |
| nosupno 33198 | The next several theorems ... |
| nosupdm 33199 | The domain of the surreal ... |
| nosupbday 33200 | Birthday bounding law for ... |
| nosupfv 33201 | The value of surreal supre... |
| nosupres 33202 | A restriction law for surr... |
| nosupbnd1lem1 33203 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1lem2 33204 | Lemma for ~ nosupbnd1 . W... |
| nosupbnd1lem3 33205 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem4 33206 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem5 33207 | Lemma for ~ nosupbnd1 . I... |
| nosupbnd1lem6 33208 | Lemma for ~ nosupbnd1 . E... |
| nosupbnd1 33209 | Bounding law from below fo... |
| nosupbnd2lem1 33210 | Bounding law from above wh... |
| nosupbnd2 33211 | Bounding law from above fo... |
| noetalem1 33212 | Lemma for ~ noeta . Estab... |
| noetalem2 33213 | Lemma for ~ noeta . ` Z ` ... |
| noetalem3 33214 | Lemma for ~ noeta . When ... |
| noetalem4 33215 | Lemma for ~ noeta . Bound... |
| noetalem5 33216 | Lemma for ~ noeta . The f... |
| noeta 33217 | The full-eta axiom for the... |
| sltirr 33220 | Surreal less than is irref... |
| slttr 33221 | Surreal less than is trans... |
| sltasym 33222 | Surreal less than is asymm... |
| sltlin 33223 | Surreal less than obeys tr... |
| slttrieq2 33224 | Trichotomy law for surreal... |
| slttrine 33225 | Trichotomy law for surreal... |
| slenlt 33226 | Surreal less than or equal... |
| sltnle 33227 | Surreal less than in terms... |
| sleloe 33228 | Surreal less than or equal... |
| sletri3 33229 | Trichotomy law for surreal... |
| sltletr 33230 | Surreal transitive law. (... |
| slelttr 33231 | Surreal transitive law. (... |
| sletr 33232 | Surreal transitive law. (... |
| slttrd 33233 | Surreal less than is trans... |
| sltletrd 33234 | Surreal less than is trans... |
| slelttrd 33235 | Surreal less than is trans... |
| sletrd 33236 | Surreal less than or equal... |
| bdayfun 33237 | The birthday function is a... |
| bdayfn 33238 | The birthday function is a... |
| bdaydm 33239 | The birthday function's do... |
| bdayrn 33240 | The birthday function's ra... |
| bdayelon 33241 | The value of the birthday ... |
| nocvxminlem 33242 | Lemma for ~ nocvxmin . Gi... |
| nocvxmin 33243 | Given a nonempty convex cl... |
| noprc 33244 | The surreal numbers are a ... |
| brsslt 33249 | Binary relation form of th... |
| ssltex1 33250 | The first argument of surr... |
| ssltex2 33251 | The second argument of sur... |
| ssltss1 33252 | The first argument of surr... |
| ssltss2 33253 | The second argument of sur... |
| ssltsep 33254 | The separation property of... |
| sssslt1 33255 | Relationship between surre... |
| sssslt2 33256 | Relationship between surre... |
| nulsslt 33257 | The empty set is less than... |
| nulssgt 33258 | The empty set is greater t... |
| conway 33259 | Conway's Simplicity Theore... |
| scutval 33260 | The value of the surreal c... |
| scutcut 33261 | Cut properties of the surr... |
| scutbday 33262 | The birthday of the surrea... |
| sslttr 33263 | Transitive law for surreal... |
| ssltun1 33264 | Union law for surreal set ... |
| ssltun2 33265 | Union law for surreal set ... |
| scutun12 33266 | Union law for surreal cuts... |
| dmscut 33267 | The domain of the surreal ... |
| scutf 33268 | Functionhood statement for... |
| etasslt 33269 | A restatement of ~ noeta u... |
| scutbdaybnd 33270 | An upper bound on the birt... |
| scutbdaylt 33271 | If a surreal lies in a gap... |
| slerec 33272 | A comparison law for surre... |
| sltrec 33273 | A comparison law for surre... |
| madeval 33284 | The value of the made by f... |
| madeval2 33285 | Alternative characterizati... |
| txpss3v 33334 | A tail Cartesian product i... |
| txprel 33335 | A tail Cartesian product i... |
| brtxp 33336 | Characterize a ternary rel... |
| brtxp2 33337 | The binary relation over a... |
| dfpprod2 33338 | Expanded definition of par... |
| pprodcnveq 33339 | A converse law for paralle... |
| pprodss4v 33340 | The parallel product is a ... |
| brpprod 33341 | Characterize a quaternary ... |
| brpprod3a 33342 | Condition for parallel pro... |
| brpprod3b 33343 | Condition for parallel pro... |
| relsset 33344 | The subset class is a bina... |
| brsset 33345 | For sets, the ` SSet ` bin... |
| idsset 33346 | ` _I ` is equal to the int... |
| eltrans 33347 | Membership in the class of... |
| dfon3 33348 | A quantifier-free definiti... |
| dfon4 33349 | Another quantifier-free de... |
| brtxpsd 33350 | Expansion of a common form... |
| brtxpsd2 33351 | Another common abbreviatio... |
| brtxpsd3 33352 | A third common abbreviatio... |
| relbigcup 33353 | The ` Bigcup ` relationshi... |
| brbigcup 33354 | Binary relation over ` Big... |
| dfbigcup2 33355 | ` Bigcup ` using maps-to n... |
| fobigcup 33356 | ` Bigcup ` maps the univer... |
| fnbigcup 33357 | ` Bigcup ` is a function o... |
| fvbigcup 33358 | For sets, ` Bigcup ` yield... |
| elfix 33359 | Membership in the fixpoint... |
| elfix2 33360 | Alternative membership in ... |
| dffix2 33361 | The fixpoints of a class i... |
| fixssdm 33362 | The fixpoints of a class a... |
| fixssrn 33363 | The fixpoints of a class a... |
| fixcnv 33364 | The fixpoints of a class a... |
| fixun 33365 | The fixpoint operator dist... |
| ellimits 33366 | Membership in the class of... |
| limitssson 33367 | The class of all limit ord... |
| dfom5b 33368 | A quantifier-free definiti... |
| sscoid 33369 | A condition for subset and... |
| dffun10 33370 | Another potential definiti... |
| elfuns 33371 | Membership in the class of... |
| elfunsg 33372 | Closed form of ~ elfuns . ... |
| brsingle 33373 | The binary relation form o... |
| elsingles 33374 | Membership in the class of... |
| fnsingle 33375 | The singleton relationship... |
| fvsingle 33376 | The value of the singleton... |
| dfsingles2 33377 | Alternate definition of th... |
| snelsingles 33378 | A singleton is a member of... |
| dfiota3 33379 | A definition of iota using... |
| dffv5 33380 | Another quantifier free de... |
| unisnif 33381 | Express union of singleton... |
| brimage 33382 | Binary relation form of th... |
| brimageg 33383 | Closed form of ~ brimage .... |
| funimage 33384 | ` Image A ` is a function.... |
| fnimage 33385 | ` Image R ` is a function ... |
| imageval 33386 | The image functor in maps-... |
| fvimage 33387 | Value of the image functor... |
| brcart 33388 | Binary relation form of th... |
| brdomain 33389 | Binary relation form of th... |
| brrange 33390 | Binary relation form of th... |
| brdomaing 33391 | Closed form of ~ brdomain ... |
| brrangeg 33392 | Closed form of ~ brrange .... |
| brimg 33393 | Binary relation form of th... |
| brapply 33394 | Binary relation form of th... |
| brcup 33395 | Binary relation form of th... |
| brcap 33396 | Binary relation form of th... |
| brsuccf 33397 | Binary relation form of th... |
| funpartlem 33398 | Lemma for ~ funpartfun . ... |
| funpartfun 33399 | The functional part of ` F... |
| funpartss 33400 | The functional part of ` F... |
| funpartfv 33401 | The function value of the ... |
| fullfunfnv 33402 | The full functional part o... |
| fullfunfv 33403 | The function value of the ... |
| brfullfun 33404 | A binary relation form con... |
| brrestrict 33405 | Binary relation form of th... |
| dfrecs2 33406 | A quantifier-free definiti... |
| dfrdg4 33407 | A quantifier-free definiti... |
| dfint3 33408 | Quantifier-free definition... |
| imagesset 33409 | The Image functor applied ... |
| brub 33410 | Binary relation form of th... |
| brlb 33411 | Binary relation form of th... |
| altopex 33416 | Alternative ordered pairs ... |
| altopthsn 33417 | Two alternate ordered pair... |
| altopeq12 33418 | Equality for alternate ord... |
| altopeq1 33419 | Equality for alternate ord... |
| altopeq2 33420 | Equality for alternate ord... |
| altopth1 33421 | Equality of the first memb... |
| altopth2 33422 | Equality of the second mem... |
| altopthg 33423 | Alternate ordered pair the... |
| altopthbg 33424 | Alternate ordered pair the... |
| altopth 33425 | The alternate ordered pair... |
| altopthb 33426 | Alternate ordered pair the... |
| altopthc 33427 | Alternate ordered pair the... |
| altopthd 33428 | Alternate ordered pair the... |
| altxpeq1 33429 | Equality for alternate Car... |
| altxpeq2 33430 | Equality for alternate Car... |
| elaltxp 33431 | Membership in alternate Ca... |
| altopelaltxp 33432 | Alternate ordered pair mem... |
| altxpsspw 33433 | An inclusion rule for alte... |
| altxpexg 33434 | The alternate Cartesian pr... |
| rankaltopb 33435 | Compute the rank of an alt... |
| nfaltop 33436 | Bound-variable hypothesis ... |
| sbcaltop 33437 | Distribution of class subs... |
| cgrrflx2d 33440 | Deduction form of ~ axcgrr... |
| cgrtr4d 33441 | Deduction form of ~ axcgrt... |
| cgrtr4and 33442 | Deduction form of ~ axcgrt... |
| cgrrflx 33443 | Reflexivity law for congru... |
| cgrrflxd 33444 | Deduction form of ~ cgrrfl... |
| cgrcomim 33445 | Congruence commutes on the... |
| cgrcom 33446 | Congruence commutes betwee... |
| cgrcomand 33447 | Deduction form of ~ cgrcom... |
| cgrtr 33448 | Transitivity law for congr... |
| cgrtrand 33449 | Deduction form of ~ cgrtr ... |
| cgrtr3 33450 | Transitivity law for congr... |
| cgrtr3and 33451 | Deduction form of ~ cgrtr3... |
| cgrcoml 33452 | Congruence commutes on the... |
| cgrcomr 33453 | Congruence commutes on the... |
| cgrcomlr 33454 | Congruence commutes on bot... |
| cgrcomland 33455 | Deduction form of ~ cgrcom... |
| cgrcomrand 33456 | Deduction form of ~ cgrcom... |
| cgrcomlrand 33457 | Deduction form of ~ cgrcom... |
| cgrtriv 33458 | Degenerate segments are co... |
| cgrid2 33459 | Identity law for congruenc... |
| cgrdegen 33460 | Two congruent segments are... |
| brofs 33461 | Binary relation form of th... |
| 5segofs 33462 | Rephrase ~ ax5seg using th... |
| ofscom 33463 | The outer five segment pre... |
| cgrextend 33464 | Link congruence over a pai... |
| cgrextendand 33465 | Deduction form of ~ cgrext... |
| segconeq 33466 | Two points that satisfy th... |
| segconeu 33467 | Existential uniqueness ver... |
| btwntriv2 33468 | Betweenness always holds f... |
| btwncomim 33469 | Betweenness commutes. Imp... |
| btwncom 33470 | Betweenness commutes. (Co... |
| btwncomand 33471 | Deduction form of ~ btwnco... |
| btwntriv1 33472 | Betweenness always holds f... |
| btwnswapid 33473 | If you can swap the first ... |
| btwnswapid2 33474 | If you can swap arguments ... |
| btwnintr 33475 | Inner transitivity law for... |
| btwnexch3 33476 | Exchange the first endpoin... |
| btwnexch3and 33477 | Deduction form of ~ btwnex... |
| btwnouttr2 33478 | Outer transitivity law for... |
| btwnexch2 33479 | Exchange the outer point o... |
| btwnouttr 33480 | Outer transitivity law for... |
| btwnexch 33481 | Outer transitivity law for... |
| btwnexchand 33482 | Deduction form of ~ btwnex... |
| btwndiff 33483 | There is always a ` c ` di... |
| trisegint 33484 | A line segment between two... |
| funtransport 33487 | The ` TransportTo ` relati... |
| fvtransport 33488 | Calculate the value of the... |
| transportcl 33489 | Closure law for segment tr... |
| transportprops 33490 | Calculate the defining pro... |
| brifs 33499 | Binary relation form of th... |
| ifscgr 33500 | Inner five segment congrue... |
| cgrsub 33501 | Removing identical parts f... |
| brcgr3 33502 | Binary relation form of th... |
| cgr3permute3 33503 | Permutation law for three-... |
| cgr3permute1 33504 | Permutation law for three-... |
| cgr3permute2 33505 | Permutation law for three-... |
| cgr3permute4 33506 | Permutation law for three-... |
| cgr3permute5 33507 | Permutation law for three-... |
| cgr3tr4 33508 | Transitivity law for three... |
| cgr3com 33509 | Commutativity law for thre... |
| cgr3rflx 33510 | Identity law for three-pla... |
| cgrxfr 33511 | A line segment can be divi... |
| btwnxfr 33512 | A condition for extending ... |
| colinrel 33513 | Colinearity is a relations... |
| brcolinear2 33514 | Alternate colinearity bina... |
| brcolinear 33515 | The binary relation form o... |
| colinearex 33516 | The colinear predicate exi... |
| colineardim1 33517 | If ` A ` is colinear with ... |
| colinearperm1 33518 | Permutation law for coline... |
| colinearperm3 33519 | Permutation law for coline... |
| colinearperm2 33520 | Permutation law for coline... |
| colinearperm4 33521 | Permutation law for coline... |
| colinearperm5 33522 | Permutation law for coline... |
| colineartriv1 33523 | Trivial case of colinearit... |
| colineartriv2 33524 | Trivial case of colinearit... |
| btwncolinear1 33525 | Betweenness implies coline... |
| btwncolinear2 33526 | Betweenness implies coline... |
| btwncolinear3 33527 | Betweenness implies coline... |
| btwncolinear4 33528 | Betweenness implies coline... |
| btwncolinear5 33529 | Betweenness implies coline... |
| btwncolinear6 33530 | Betweenness implies coline... |
| colinearxfr 33531 | Transfer law for colineari... |
| lineext 33532 | Extend a line with a missi... |
| brofs2 33533 | Change some conditions for... |
| brifs2 33534 | Change some conditions for... |
| brfs 33535 | Binary relation form of th... |
| fscgr 33536 | Congruence law for the gen... |
| linecgr 33537 | Congruence rule for lines.... |
| linecgrand 33538 | Deduction form of ~ linecg... |
| lineid 33539 | Identity law for points on... |
| idinside 33540 | Law for finding a point in... |
| endofsegid 33541 | If ` A ` , ` B ` , and ` C... |
| endofsegidand 33542 | Deduction form of ~ endofs... |
| btwnconn1lem1 33543 | Lemma for ~ btwnconn1 . T... |
| btwnconn1lem2 33544 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem3 33545 | Lemma for ~ btwnconn1 . E... |
| btwnconn1lem4 33546 | Lemma for ~ btwnconn1 . A... |
| btwnconn1lem5 33547 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem6 33548 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem7 33549 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem8 33550 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem9 33551 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem10 33552 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem11 33553 | Lemma for ~ btwnconn1 . N... |
| btwnconn1lem12 33554 | Lemma for ~ btwnconn1 . U... |
| btwnconn1lem13 33555 | Lemma for ~ btwnconn1 . B... |
| btwnconn1lem14 33556 | Lemma for ~ btwnconn1 . F... |
| btwnconn1 33557 | Connectitivy law for betwe... |
| btwnconn2 33558 | Another connectivity law f... |
| btwnconn3 33559 | Inner connectivity law for... |
| midofsegid 33560 | If two points fall in the ... |
| segcon2 33561 | Generalization of ~ axsegc... |
| brsegle 33564 | Binary relation form of th... |
| brsegle2 33565 | Alternate characterization... |
| seglecgr12im 33566 | Substitution law for segme... |
| seglecgr12 33567 | Substitution law for segme... |
| seglerflx 33568 | Segment comparison is refl... |
| seglemin 33569 | Any segment is at least as... |
| segletr 33570 | Segment less than is trans... |
| segleantisym 33571 | Antisymmetry law for segme... |
| seglelin 33572 | Linearity law for segment ... |
| btwnsegle 33573 | If ` B ` falls between ` A... |
| colinbtwnle 33574 | Given three colinear point... |
| broutsideof 33577 | Binary relation form of ` ... |
| broutsideof2 33578 | Alternate form of ` Outsid... |
| outsidene1 33579 | Outsideness implies inequa... |
| outsidene2 33580 | Outsideness implies inequa... |
| btwnoutside 33581 | A principle linking outsid... |
| broutsideof3 33582 | Characterization of outsid... |
| outsideofrflx 33583 | Reflexitivity of outsidene... |
| outsideofcom 33584 | Commutitivity law for outs... |
| outsideoftr 33585 | Transitivity law for outsi... |
| outsideofeq 33586 | Uniqueness law for ` Outsi... |
| outsideofeu 33587 | Given a nondegenerate ray,... |
| outsidele 33588 | Relate ` OutsideOf ` to ` ... |
| outsideofcol 33589 | Outside of implies colinea... |
| funray 33596 | Show that the ` Ray ` rela... |
| fvray 33597 | Calculate the value of the... |
| funline 33598 | Show that the ` Line ` rel... |
| linedegen 33599 | When ` Line ` is applied w... |
| fvline 33600 | Calculate the value of the... |
| liness 33601 | A line is a subset of the ... |
| fvline2 33602 | Alternate definition of a ... |
| lineunray 33603 | A line is composed of a po... |
| lineelsb2 33604 | If ` S ` lies on ` P Q ` ,... |
| linerflx1 33605 | Reflexivity law for line m... |
| linecom 33606 | Commutativity law for line... |
| linerflx2 33607 | Reflexivity law for line m... |
| ellines 33608 | Membership in the set of a... |
| linethru 33609 | If ` A ` is a line contain... |
| hilbert1.1 33610 | There is a line through an... |
| hilbert1.2 33611 | There is at most one line ... |
| linethrueu 33612 | There is a unique line goi... |
| lineintmo 33613 | Two distinct lines interse... |
| fwddifval 33618 | Calculate the value of the... |
| fwddifnval 33619 | The value of the forward d... |
| fwddifn0 33620 | The value of the n-iterate... |
| fwddifnp1 33621 | The value of the n-iterate... |
| rankung 33622 | The rank of the union of t... |
| ranksng 33623 | The rank of a singleton. ... |
| rankelg 33624 | The membership relation is... |
| rankpwg 33625 | The rank of a power set. ... |
| rank0 33626 | The rank of the empty set ... |
| rankeq1o 33627 | The only set with rank ` 1... |
| elhf 33630 | Membership in the heredita... |
| elhf2 33631 | Alternate form of membersh... |
| elhf2g 33632 | Hereditarily finiteness vi... |
| 0hf 33633 | The empty set is a heredit... |
| hfun 33634 | The union of two HF sets i... |
| hfsn 33635 | The singleton of an HF set... |
| hfadj 33636 | Adjoining one HF element t... |
| hfelhf 33637 | Any member of an HF set is... |
| hftr 33638 | The class of all hereditar... |
| hfext 33639 | Extensionality for HF sets... |
| hfuni 33640 | The union of an HF set is ... |
| hfpw 33641 | The power class of an HF s... |
| hfninf 33642 | ` _om ` is not hereditaril... |
| a1i14 33643 | Add two antecedents to a w... |
| a1i24 33644 | Add two antecedents to a w... |
| exp5d 33645 | An exportation inference. ... |
| exp5g 33646 | An exportation inference. ... |
| exp5k 33647 | An exportation inference. ... |
| exp56 33648 | An exportation inference. ... |
| exp58 33649 | An exportation inference. ... |
| exp510 33650 | An exportation inference. ... |
| exp511 33651 | An exportation inference. ... |
| exp512 33652 | An exportation inference. ... |
| 3com12d 33653 | Commutation in consequent.... |
| imp5p 33654 | A triple importation infer... |
| imp5q 33655 | A triple importation infer... |
| ecase13d 33656 | Deduction for elimination ... |
| subtr 33657 | Transitivity of implicit s... |
| subtr2 33658 | Transitivity of implicit s... |
| trer 33659 | A relation intersected wit... |
| elicc3 33660 | An equivalent membership c... |
| finminlem 33661 | A useful lemma about finit... |
| gtinf 33662 | Any number greater than an... |
| opnrebl 33663 | A set is open in the stand... |
| opnrebl2 33664 | A set is open in the stand... |
| nn0prpwlem 33665 | Lemma for ~ nn0prpw . Use... |
| nn0prpw 33666 | Two nonnegative integers a... |
| topbnd 33667 | Two equivalent expressions... |
| opnbnd 33668 | A set is open iff it is di... |
| cldbnd 33669 | A set is closed iff it con... |
| ntruni 33670 | A union of interiors is a ... |
| clsun 33671 | A pairwise union of closur... |
| clsint2 33672 | The closure of an intersec... |
| opnregcld 33673 | A set is regularly closed ... |
| cldregopn 33674 | A set if regularly open if... |
| neiin 33675 | Two neighborhoods intersec... |
| hmeoclda 33676 | Homeomorphisms preserve cl... |
| hmeocldb 33677 | Homeomorphisms preserve cl... |
| ivthALT 33678 | An alternate proof of the ... |
| fnerel 33681 | Fineness is a relation. (... |
| isfne 33682 | The predicate " ` B ` is f... |
| isfne4 33683 | The predicate " ` B ` is f... |
| isfne4b 33684 | A condition for a topology... |
| isfne2 33685 | The predicate " ` B ` is f... |
| isfne3 33686 | The predicate " ` B ` is f... |
| fnebas 33687 | A finer cover covers the s... |
| fnetg 33688 | A finer cover generates a ... |
| fnessex 33689 | If ` B ` is finer than ` A... |
| fneuni 33690 | If ` B ` is finer than ` A... |
| fneint 33691 | If a cover is finer than a... |
| fness 33692 | A cover is finer than its ... |
| fneref 33693 | Reflexivity of the finenes... |
| fnetr 33694 | Transitivity of the finene... |
| fneval 33695 | Two covers are finer than ... |
| fneer 33696 | Fineness intersected with ... |
| topfne 33697 | Fineness for covers corres... |
| topfneec 33698 | A cover is equivalent to a... |
| topfneec2 33699 | A topology is precisely id... |
| fnessref 33700 | A cover is finer iff it ha... |
| refssfne 33701 | A cover is a refinement if... |
| neibastop1 33702 | A collection of neighborho... |
| neibastop2lem 33703 | Lemma for ~ neibastop2 . ... |
| neibastop2 33704 | In the topology generated ... |
| neibastop3 33705 | The topology generated by ... |
| topmtcl 33706 | The meet of a collection o... |
| topmeet 33707 | Two equivalent formulation... |
| topjoin 33708 | Two equivalent formulation... |
| fnemeet1 33709 | The meet of a collection o... |
| fnemeet2 33710 | The meet of equivalence cl... |
| fnejoin1 33711 | Join of equivalence classe... |
| fnejoin2 33712 | Join of equivalence classe... |
| fgmin 33713 | Minimality property of a g... |
| neifg 33714 | The neighborhood filter of... |
| tailfval 33715 | The tail function for a di... |
| tailval 33716 | The tail of an element in ... |
| eltail 33717 | An element of a tail. (Co... |
| tailf 33718 | The tail function of a dir... |
| tailini 33719 | A tail contains its initia... |
| tailfb 33720 | The collection of tails of... |
| filnetlem1 33721 | Lemma for ~ filnet . Chan... |
| filnetlem2 33722 | Lemma for ~ filnet . The ... |
| filnetlem3 33723 | Lemma for ~ filnet . (Con... |
| filnetlem4 33724 | Lemma for ~ filnet . (Con... |
| filnet 33725 | A filter has the same conv... |
| tb-ax1 33726 | The first of three axioms ... |
| tb-ax2 33727 | The second of three axioms... |
| tb-ax3 33728 | The third of three axioms ... |
| tbsyl 33729 | The weak syllogism from Ta... |
| re1ax2lem 33730 | Lemma for ~ re1ax2 . (Con... |
| re1ax2 33731 | ~ ax-2 rederived from the ... |
| naim1 33732 | Constructor theorem for ` ... |
| naim2 33733 | Constructor theorem for ` ... |
| naim1i 33734 | Constructor rule for ` -/\... |
| naim2i 33735 | Constructor rule for ` -/\... |
| naim12i 33736 | Constructor rule for ` -/\... |
| nabi1i 33737 | Constructor rule for ` -/\... |
| nabi2i 33738 | Constructor rule for ` -/\... |
| nabi12i 33739 | Constructor rule for ` -/\... |
| df3nandALT1 33742 | The double nand expressed ... |
| df3nandALT2 33743 | The double nand expressed ... |
| andnand1 33744 | Double and in terms of dou... |
| imnand2 33745 | An ` -> ` nand relation. ... |
| nalfal 33746 | Not all sets hold ` F. ` a... |
| nexntru 33747 | There does not exist a set... |
| nexfal 33748 | There does not exist a set... |
| neufal 33749 | There does not exist exact... |
| neutru 33750 | There does not exist exact... |
| nmotru 33751 | There does not exist at mo... |
| mofal 33752 | There exist at most one se... |
| nrmo 33753 | "At most one" restricted e... |
| meran1 33754 | A single axiom for proposi... |
| meran2 33755 | A single axiom for proposi... |
| meran3 33756 | A single axiom for proposi... |
| waj-ax 33757 | A single axiom for proposi... |
| lukshef-ax2 33758 | A single axiom for proposi... |
| arg-ax 33759 | A single axiom for proposi... |
| negsym1 33760 | In the paper "On Variable ... |
| imsym1 33761 | A symmetry with ` -> ` . ... |
| bisym1 33762 | A symmetry with ` <-> ` . ... |
| consym1 33763 | A symmetry with ` /\ ` . ... |
| dissym1 33764 | A symmetry with ` \/ ` . ... |
| nandsym1 33765 | A symmetry with ` -/\ ` . ... |
| unisym1 33766 | A symmetry with ` A. ` . ... |
| exisym1 33767 | A symmetry with ` E. ` . ... |
| unqsym1 33768 | A symmetry with ` E! ` . ... |
| amosym1 33769 | A symmetry with ` E* ` . ... |
| subsym1 33770 | A symmetry with ` [ x / y ... |
| ontopbas 33771 | An ordinal number is a top... |
| onsstopbas 33772 | The class of ordinal numbe... |
| onpsstopbas 33773 | The class of ordinal numbe... |
| ontgval 33774 | The topology generated fro... |
| ontgsucval 33775 | The topology generated fro... |
| onsuctop 33776 | A successor ordinal number... |
| onsuctopon 33777 | One of the topologies on a... |
| ordtoplem 33778 | Membership of the class of... |
| ordtop 33779 | An ordinal is a topology i... |
| onsucconni 33780 | A successor ordinal number... |
| onsucconn 33781 | A successor ordinal number... |
| ordtopconn 33782 | An ordinal topology is con... |
| onintopssconn 33783 | An ordinal topology is con... |
| onsuct0 33784 | A successor ordinal number... |
| ordtopt0 33785 | An ordinal topology is T_0... |
| onsucsuccmpi 33786 | The successor of a success... |
| onsucsuccmp 33787 | The successor of a success... |
| limsucncmpi 33788 | The successor of a limit o... |
| limsucncmp 33789 | The successor of a limit o... |
| ordcmp 33790 | An ordinal topology is com... |
| ssoninhaus 33791 | The ordinal topologies ` 1... |
| onint1 33792 | The ordinal T_1 spaces are... |
| oninhaus 33793 | The ordinal Hausdorff spac... |
| fveleq 33794 | Please add description her... |
| findfvcl 33795 | Please add description her... |
| findreccl 33796 | Please add description her... |
| findabrcl 33797 | Please add description her... |
| nnssi2 33798 | Convert a theorem for real... |
| nnssi3 33799 | Convert a theorem for real... |
| nndivsub 33800 | Please add description her... |
| nndivlub 33801 | A factor of a positive int... |
| ee7.2aOLD 33804 | Lemma for Euclid's Element... |
| dnival 33805 | Value of the "distance to ... |
| dnicld1 33806 | Closure theorem for the "d... |
| dnicld2 33807 | Closure theorem for the "d... |
| dnif 33808 | The "distance to nearest i... |
| dnizeq0 33809 | The distance to nearest in... |
| dnizphlfeqhlf 33810 | The distance to nearest in... |
| rddif2 33811 | Variant of ~ rddif . (Con... |
| dnibndlem1 33812 | Lemma for ~ dnibnd . (Con... |
| dnibndlem2 33813 | Lemma for ~ dnibnd . (Con... |
| dnibndlem3 33814 | Lemma for ~ dnibnd . (Con... |
| dnibndlem4 33815 | Lemma for ~ dnibnd . (Con... |
| dnibndlem5 33816 | Lemma for ~ dnibnd . (Con... |
| dnibndlem6 33817 | Lemma for ~ dnibnd . (Con... |
| dnibndlem7 33818 | Lemma for ~ dnibnd . (Con... |
| dnibndlem8 33819 | Lemma for ~ dnibnd . (Con... |
| dnibndlem9 33820 | Lemma for ~ dnibnd . (Con... |
| dnibndlem10 33821 | Lemma for ~ dnibnd . (Con... |
| dnibndlem11 33822 | Lemma for ~ dnibnd . (Con... |
| dnibndlem12 33823 | Lemma for ~ dnibnd . (Con... |
| dnibndlem13 33824 | Lemma for ~ dnibnd . (Con... |
| dnibnd 33825 | The "distance to nearest i... |
| dnicn 33826 | The "distance to nearest i... |
| knoppcnlem1 33827 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem2 33828 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem3 33829 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem4 33830 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem5 33831 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem6 33832 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem7 33833 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem8 33834 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem9 33835 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem10 33836 | Lemma for ~ knoppcn . (Co... |
| knoppcnlem11 33837 | Lemma for ~ knoppcn . (Co... |
| knoppcn 33838 | The continuous nowhere dif... |
| knoppcld 33839 | Closure theorem for Knopp'... |
| unblimceq0lem 33840 | Lemma for ~ unblimceq0 . ... |
| unblimceq0 33841 | If ` F ` is unbounded near... |
| unbdqndv1 33842 | If the difference quotient... |
| unbdqndv2lem1 33843 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2lem2 33844 | Lemma for ~ unbdqndv2 . (... |
| unbdqndv2 33845 | Variant of ~ unbdqndv1 wit... |
| knoppndvlem1 33846 | Lemma for ~ knoppndv . (C... |
| knoppndvlem2 33847 | Lemma for ~ knoppndv . (C... |
| knoppndvlem3 33848 | Lemma for ~ knoppndv . (C... |
| knoppndvlem4 33849 | Lemma for ~ knoppndv . (C... |
| knoppndvlem5 33850 | Lemma for ~ knoppndv . (C... |
| knoppndvlem6 33851 | Lemma for ~ knoppndv . (C... |
| knoppndvlem7 33852 | Lemma for ~ knoppndv . (C... |
| knoppndvlem8 33853 | Lemma for ~ knoppndv . (C... |
| knoppndvlem9 33854 | Lemma for ~ knoppndv . (C... |
| knoppndvlem10 33855 | Lemma for ~ knoppndv . (C... |
| knoppndvlem11 33856 | Lemma for ~ knoppndv . (C... |
| knoppndvlem12 33857 | Lemma for ~ knoppndv . (C... |
| knoppndvlem13 33858 | Lemma for ~ knoppndv . (C... |
| knoppndvlem14 33859 | Lemma for ~ knoppndv . (C... |
| knoppndvlem15 33860 | Lemma for ~ knoppndv . (C... |
| knoppndvlem16 33861 | Lemma for ~ knoppndv . (C... |
| knoppndvlem17 33862 | Lemma for ~ knoppndv . (C... |
| knoppndvlem18 33863 | Lemma for ~ knoppndv . (C... |
| knoppndvlem19 33864 | Lemma for ~ knoppndv . (C... |
| knoppndvlem20 33865 | Lemma for ~ knoppndv . (C... |
| knoppndvlem21 33866 | Lemma for ~ knoppndv . (C... |
| knoppndvlem22 33867 | Lemma for ~ knoppndv . (C... |
| knoppndv 33868 | The continuous nowhere dif... |
| knoppf 33869 | Knopp's function is a func... |
| knoppcn2 33870 | Variant of ~ knoppcn with ... |
| cnndvlem1 33871 | Lemma for ~ cnndv . (Cont... |
| cnndvlem2 33872 | Lemma for ~ cnndv . (Cont... |
| cnndv 33873 | There exists a continuous ... |
| bj-mp2c 33874 | A double modus ponens infe... |
| bj-mp2d 33875 | A double modus ponens infe... |
| bj-0 33876 | A syntactic theorem. See ... |
| bj-1 33877 | In this proof, the use of ... |
| bj-a1k 33878 | Weakening of ~ ax-1 . Thi... |
| bj-nnclav 33879 | When ` F. ` is substituted... |
| bj-jarrii 33880 | Inference associated with ... |
| bj-imim21 33881 | The propositional function... |
| bj-imim21i 33882 | Inference associated with ... |
| bj-peircestab 33883 | Over minimal implicational... |
| bj-stabpeirce 33884 | Over minimal implicational... |
| bj-syl66ib 33885 | A mixed syllogism inferenc... |
| bj-orim2 33886 | Proof of ~ orim2 from the ... |
| bj-currypeirce 33887 | Curry's axiom ~ curryax (a... |
| bj-peircecurry 33888 | Peirce's axiom ~ peirce im... |
| bj-animbi 33889 | Conjunction in terms of im... |
| bj-currypara 33890 | Curry's paradox. Note tha... |
| bj-con2com 33891 | A commuted form of the con... |
| bj-con2comi 33892 | Inference associated with ... |
| bj-pm2.01i 33893 | Inference associated with ... |
| bj-nimn 33894 | If a formula is true, then... |
| bj-nimni 33895 | Inference associated with ... |
| bj-peircei 33896 | Inference associated with ... |
| bj-looinvi 33897 | Inference associated with ... |
| bj-looinvii 33898 | Inference associated with ... |
| bj-jaoi1 33899 | Shortens ~ orfa2 (58>53), ... |
| bj-jaoi2 33900 | Shortens ~ consensus (110>... |
| bj-dfbi4 33901 | Alternate definition of th... |
| bj-dfbi5 33902 | Alternate definition of th... |
| bj-dfbi6 33903 | Alternate definition of th... |
| bj-bijust0ALT 33904 | Alternate proof of ~ bijus... |
| bj-bijust00 33905 | A self-implication does no... |
| bj-consensus 33906 | Version of ~ consensus exp... |
| bj-consensusALT 33907 | Alternate proof of ~ bj-co... |
| bj-df-ifc 33908 | Candidate definition for t... |
| bj-dfif 33909 | Alternate definition of th... |
| bj-ififc 33910 | A biconditional connecting... |
| bj-imbi12 33911 | Uncurried (imported) form ... |
| bj-biorfi 33912 | This should be labeled "bi... |
| bj-falor 33913 | Dual of ~ truan (which has... |
| bj-falor2 33914 | Dual of ~ truan . (Contri... |
| bj-bibibi 33915 | A property of the bicondit... |
| bj-imn3ani 33916 | Duplication of ~ bnj1224 .... |
| bj-andnotim 33917 | Two ways of expressing a c... |
| bj-bi3ant 33918 | This used to be in the mai... |
| bj-bisym 33919 | This used to be in the mai... |
| bj-bixor 33920 | Equivalence of two ternary... |
| bj-axdd2 33921 | This implication, proved u... |
| bj-axd2d 33922 | This implication, proved u... |
| bj-axtd 33923 | This implication, proved f... |
| bj-gl4 33924 | In a normal modal logic, t... |
| bj-axc4 33925 | Over minimal calculus, the... |
| prvlem1 33930 | An elementary property of ... |
| prvlem2 33931 | An elementary property of ... |
| bj-babygodel 33932 | See the section header com... |
| bj-babylob 33933 | See the section header com... |
| bj-godellob 33934 | Proof of Gödel's theo... |
| bj-genr 33935 | Generalization rule on the... |
| bj-genl 33936 | Generalization rule on the... |
| bj-genan 33937 | Generalization rule on a c... |
| bj-mpgs 33938 | From a closed form theorem... |
| bj-2alim 33939 | Closed form of ~ 2alimi . ... |
| bj-2exim 33940 | Closed form of ~ 2eximi . ... |
| bj-alanim 33941 | Closed form of ~ alanimi .... |
| bj-2albi 33942 | Closed form of ~ 2albii . ... |
| bj-notalbii 33943 | Equivalence of universal q... |
| bj-2exbi 33944 | Closed form of ~ 2exbii . ... |
| bj-3exbi 33945 | Closed form of ~ 3exbii . ... |
| bj-sylgt2 33946 | Uncurried (imported) form ... |
| bj-alrimg 33947 | The general form of the *a... |
| bj-alrimd 33948 | A slightly more general ~ ... |
| bj-sylget 33949 | Dual statement of ~ sylgt ... |
| bj-sylget2 33950 | Uncurried (imported) form ... |
| bj-exlimg 33951 | The general form of the *e... |
| bj-sylge 33952 | Dual statement of ~ sylg (... |
| bj-exlimd 33953 | A slightly more general ~ ... |
| bj-nfimexal 33954 | A weak from of nonfreeness... |
| bj-alexim 33955 | Closed form of ~ aleximi .... |
| bj-nexdh 33956 | Closed form of ~ nexdh (ac... |
| bj-nexdh2 33957 | Uncurried (imported) form ... |
| bj-hbxfrbi 33958 | Closed form of ~ hbxfrbi .... |
| bj-hbyfrbi 33959 | Version of ~ bj-hbxfrbi wi... |
| bj-exalim 33960 | Distribute quantifiers ove... |
| bj-exalimi 33961 | An inference for distribut... |
| bj-exalims 33962 | Distributing quantifiers o... |
| bj-exalimsi 33963 | An inference for distribut... |
| bj-ax12ig 33964 | A lemma used to prove a we... |
| bj-ax12i 33965 | A weakening of ~ bj-ax12ig... |
| bj-nfimt 33966 | Closed form of ~ nfim and ... |
| bj-cbvalimt 33967 | A lemma in closed form use... |
| bj-cbveximt 33968 | A lemma in closed form use... |
| bj-eximALT 33969 | Alternate proof of ~ exim ... |
| bj-aleximiALT 33970 | Alternate proof of ~ alexi... |
| bj-eximcom 33971 | A commuted form of ~ exim ... |
| bj-ax12wlem 33972 | A lemma used to prove a we... |
| bj-cbvalim 33973 | A lemma used to prove ~ bj... |
| bj-cbvexim 33974 | A lemma used to prove ~ bj... |
| bj-cbvalimi 33975 | An equality-free general i... |
| bj-cbveximi 33976 | An equality-free general i... |
| bj-cbval 33977 | Changing a bound variable ... |
| bj-cbvex 33978 | Changing a bound variable ... |
| bj-ssbeq 33981 | Substitution in an equalit... |
| bj-ssblem1 33982 | A lemma for the definiens ... |
| bj-ssblem2 33983 | An instance of ~ ax-11 pro... |
| bj-ax12v 33984 | A weaker form of ~ ax-12 a... |
| bj-ax12 33985 | Remove a DV condition from... |
| bj-ax12ssb 33986 | The axiom ~ bj-ax12 expres... |
| bj-19.41al 33987 | Special case of ~ 19.41 pr... |
| bj-equsexval 33988 | Special case of ~ equsexv ... |
| bj-sb56 33989 | Proof of ~ sb56 from Tarsk... |
| bj-ssbid2 33990 | A special case of ~ sbequ2... |
| bj-ssbid2ALT 33991 | Alternate proof of ~ bj-ss... |
| bj-ssbid1 33992 | A special case of ~ sbequ1... |
| bj-ssbid1ALT 33993 | Alternate proof of ~ bj-ss... |
| bj-ax6elem1 33994 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6elem2 33995 | Lemma for ~ bj-ax6e . (Co... |
| bj-ax6e 33996 | Proof of ~ ax6e (hence ~ a... |
| bj-spimvwt 33997 | Closed form of ~ spimvw . ... |
| bj-spnfw 33998 | Theorem close to a closed ... |
| bj-cbvexiw 33999 | Change bound variable. Th... |
| bj-cbvexivw 34000 | Change bound variable. Th... |
| bj-modald 34001 | A short form of the axiom ... |
| bj-denot 34002 | A weakening of ~ ax-6 and ... |
| bj-eqs 34003 | A lemma for substitutions,... |
| bj-cbvexw 34004 | Change bound variable. Th... |
| bj-ax12w 34005 | The general statement that... |
| bj-ax89 34006 | A theorem which could be u... |
| bj-elequ12 34007 | An identity law for the no... |
| bj-cleljusti 34008 | One direction of ~ cleljus... |
| bj-alcomexcom 34009 | Commutation of universal q... |
| bj-hbalt 34010 | Closed form of ~ hbal . W... |
| axc11n11 34011 | Proof of ~ axc11n from { ~... |
| axc11n11r 34012 | Proof of ~ axc11n from { ~... |
| bj-axc16g16 34013 | Proof of ~ axc16g from { ~... |
| bj-ax12v3 34014 | A weak version of ~ ax-12 ... |
| bj-ax12v3ALT 34015 | Alternate proof of ~ bj-ax... |
| bj-sb 34016 | A weak variant of ~ sbid2 ... |
| bj-modalbe 34017 | The predicate-calculus ver... |
| bj-spst 34018 | Closed form of ~ sps . On... |
| bj-19.21bit 34019 | Closed form of ~ 19.21bi .... |
| bj-19.23bit 34020 | Closed form of ~ 19.23bi .... |
| bj-nexrt 34021 | Closed form of ~ nexr . C... |
| bj-alrim 34022 | Closed form of ~ alrimi . ... |
| bj-alrim2 34023 | Uncurried (imported) form ... |
| bj-nfdt0 34024 | A theorem close to a close... |
| bj-nfdt 34025 | Closed form of ~ nf5d and ... |
| bj-nexdt 34026 | Closed form of ~ nexd . (... |
| bj-nexdvt 34027 | Closed form of ~ nexdv . ... |
| bj-alexbiex 34028 | Adding a second quantifier... |
| bj-exexbiex 34029 | Adding a second quantifier... |
| bj-alalbial 34030 | Adding a second quantifier... |
| bj-exalbial 34031 | Adding a second quantifier... |
| bj-19.9htbi 34032 | Strengthening ~ 19.9ht by ... |
| bj-hbntbi 34033 | Strengthening ~ hbnt by re... |
| bj-biexal1 34034 | A general FOL biconditiona... |
| bj-biexal2 34035 | When ` ph ` is substituted... |
| bj-biexal3 34036 | When ` ph ` is substituted... |
| bj-bialal 34037 | When ` ph ` is substituted... |
| bj-biexex 34038 | When ` ph ` is substituted... |
| bj-hbext 34039 | Closed form of ~ hbex . (... |
| bj-nfalt 34040 | Closed form of ~ nfal . (... |
| bj-nfext 34041 | Closed form of ~ nfex . (... |
| bj-eeanvw 34042 | Version of ~ exdistrv with... |
| bj-modal4 34043 | First-order logic form of ... |
| bj-modal4e 34044 | First-order logic form of ... |
| bj-modalb 34045 | A short form of the axiom ... |
| bj-wnf1 34046 | When ` ph ` is substituted... |
| bj-wnf2 34047 | When ` ph ` is substituted... |
| bj-wnfanf 34048 | When ` ph ` is substituted... |
| bj-wnfenf 34049 | When ` ph ` is substituted... |
| bj-nnfbi 34052 | If two formulas are equiva... |
| bj-nnfbd 34053 | If two formulas are equiva... |
| bj-nnfbii 34054 | If two formulas are equiva... |
| bj-nnfa 34055 | Nonfreeness implies the eq... |
| bj-nnfad 34056 | Nonfreeness implies the eq... |
| bj-nnfe 34057 | Nonfreeness implies the eq... |
| bj-nnfed 34058 | Nonfreeness implies the eq... |
| bj-nnfea 34059 | Nonfreeness implies the eq... |
| bj-nnfead 34060 | Nonfreeness implies the eq... |
| bj-dfnnf2 34061 | Alternate definition of ~ ... |
| bj-nnfnfTEMP 34062 | New nonfreeness implies ol... |
| bj-wnfnf 34063 | When ` ph ` is substituted... |
| bj-nnfnt 34064 | A variable is nonfree in a... |
| bj-nnftht 34065 | A variable is nonfree in a... |
| bj-nnfth 34066 | A variable is nonfree in a... |
| bj-nnfnth 34067 | A variable is nonfree in t... |
| bj-nnfim1 34068 | A consequence of nonfreene... |
| bj-nnfim2 34069 | A consequence of nonfreene... |
| bj-nnfim 34070 | Nonfreeness in the anteced... |
| bj-nnfimd 34071 | Nonfreeness in the anteced... |
| bj-nnfan 34072 | Nonfreeness in both conjun... |
| bj-nnfand 34073 | Nonfreeness in both conjun... |
| bj-nnfor 34074 | Nonfreeness in both disjun... |
| bj-nnford 34075 | Nonfreeness in both disjun... |
| bj-nnfbit 34076 | Nonfreeness in both sides ... |
| bj-nnfbid 34077 | Nonfreeness in both sides ... |
| bj-nnfv 34078 | A non-occurring variable i... |
| bj-nnf-alrim 34079 | Proof of the closed form o... |
| bj-nnf-exlim 34080 | Proof of the closed form o... |
| bj-dfnnf3 34081 | Alternate definition of no... |
| bj-nfnnfTEMP 34082 | New nonfreeness is equival... |
| bj-nnfa1 34083 | See ~ nfa1 . (Contributed... |
| bj-nnfe1 34084 | See ~ nfe1 . (Contributed... |
| bj-19.12 34085 | See ~ 19.12 . Could be la... |
| bj-nnflemaa 34086 | One of four lemmas for non... |
| bj-nnflemee 34087 | One of four lemmas for non... |
| bj-nnflemae 34088 | One of four lemmas for non... |
| bj-nnflemea 34089 | One of four lemmas for non... |
| bj-nnfalt 34090 | See ~ nfal and ~ bj-nfalt ... |
| bj-nnfext 34091 | See ~ nfex and ~ bj-nfext ... |
| bj-stdpc5t 34092 | Alias of ~ bj-nnf-alrim fo... |
| bj-19.21t 34093 | Statement ~ 19.21t proved ... |
| bj-19.23t 34094 | Statement ~ 19.23t proved ... |
| bj-19.36im 34095 | One direction of ~ 19.36 f... |
| bj-19.37im 34096 | One direction of ~ 19.37 f... |
| bj-19.42t 34097 | Closed form of ~ 19.42 fro... |
| bj-19.41t 34098 | Closed form of ~ 19.41 fro... |
| bj-sbft 34099 | Version of ~ sbft using ` ... |
| bj-axc10 34100 | Alternate (shorter) proof ... |
| bj-alequex 34101 | A fol lemma. See ~ aleque... |
| bj-spimt2 34102 | A step in the proof of ~ s... |
| bj-cbv3ta 34103 | Closed form of ~ cbv3 . (... |
| bj-cbv3tb 34104 | Closed form of ~ cbv3 . (... |
| bj-hbsb3t 34105 | A theorem close to a close... |
| bj-hbsb3 34106 | Shorter proof of ~ hbsb3 .... |
| bj-nfs1t 34107 | A theorem close to a close... |
| bj-nfs1t2 34108 | A theorem close to a close... |
| bj-nfs1 34109 | Shorter proof of ~ nfs1 (t... |
| bj-axc10v 34110 | Version of ~ axc10 with a ... |
| bj-spimtv 34111 | Version of ~ spimt with a ... |
| bj-cbv3hv2 34112 | Version of ~ cbv3h with tw... |
| bj-cbv1hv 34113 | Version of ~ cbv1h with a ... |
| bj-cbv2hv 34114 | Version of ~ cbv2h with a ... |
| bj-cbv2v 34115 | Version of ~ cbv2 with a d... |
| bj-cbvaldv 34116 | Version of ~ cbvald with a... |
| bj-cbvexdv 34117 | Version of ~ cbvexd with a... |
| bj-cbval2vv 34118 | Version of ~ cbval2vv with... |
| bj-cbvex2vv 34119 | Version of ~ cbvex2vv with... |
| bj-cbvaldvav 34120 | Version of ~ cbvaldva with... |
| bj-cbvexdvav 34121 | Version of ~ cbvexdva with... |
| bj-cbvex4vv 34122 | Version of ~ cbvex4v with ... |
| bj-equsalhv 34123 | Version of ~ equsalh with ... |
| bj-axc11nv 34124 | Version of ~ axc11n with a... |
| bj-aecomsv 34125 | Version of ~ aecoms with a... |
| bj-axc11v 34126 | Version of ~ axc11 with a ... |
| bj-drnf2v 34127 | Version of ~ drnf2 with a ... |
| bj-equs45fv 34128 | Version of ~ equs45f with ... |
| bj-hbs1 34129 | Version of ~ hbsb2 with a ... |
| bj-nfs1v 34130 | Version of ~ nfsb2 with a ... |
| bj-hbsb2av 34131 | Version of ~ hbsb2a with a... |
| bj-hbsb3v 34132 | Version of ~ hbsb3 with a ... |
| bj-nfsab1 34133 | Remove dependency on ~ ax-... |
| bj-dtru 34134 | Remove dependency on ~ ax-... |
| bj-dtrucor2v 34135 | Version of ~ dtrucor2 with... |
| bj-hbaeb2 34136 | Biconditional version of a... |
| bj-hbaeb 34137 | Biconditional version of ~... |
| bj-hbnaeb 34138 | Biconditional version of ~... |
| bj-dvv 34139 | A special instance of ~ bj... |
| bj-equsal1t 34140 | Duplication of ~ wl-equsal... |
| bj-equsal1ti 34141 | Inference associated with ... |
| bj-equsal1 34142 | One direction of ~ equsal ... |
| bj-equsal2 34143 | One direction of ~ equsal ... |
| bj-equsal 34144 | Shorter proof of ~ equsal ... |
| stdpc5t 34145 | Closed form of ~ stdpc5 . ... |
| bj-stdpc5 34146 | More direct proof of ~ std... |
| 2stdpc5 34147 | A double ~ stdpc5 (one dir... |
| bj-19.21t0 34148 | Proof of ~ 19.21t from ~ s... |
| exlimii 34149 | Inference associated with ... |
| ax11-pm 34150 | Proof of ~ ax-11 similar t... |
| ax6er 34151 | Commuted form of ~ ax6e . ... |
| exlimiieq1 34152 | Inferring a theorem when i... |
| exlimiieq2 34153 | Inferring a theorem when i... |
| ax11-pm2 34154 | Proof of ~ ax-11 from the ... |
| bj-sbsb 34155 | Biconditional showing two ... |
| bj-dfsb2 34156 | Alternate (dual) definitio... |
| bj-sbf3 34157 | Substitution has no effect... |
| bj-sbf4 34158 | Substitution has no effect... |
| bj-sbnf 34159 | Move nonfree predicate in ... |
| bj-eu3f 34160 | Version of ~ eu3v where th... |
| bj-sblem1 34161 | Lemma for substitution. (... |
| bj-sblem2 34162 | Lemma for substitution. (... |
| bj-sblem 34163 | Lemma for substitution. (... |
| bj-sbievw1 34164 | Lemma for substitution. (... |
| bj-sbievw2 34165 | Lemma for substitution. (... |
| bj-sbievw 34166 | Lemma for substitution. C... |
| bj-sbievv 34167 | Version of ~ sbie with a s... |
| bj-moeub 34168 | Uniqueness is equivalent t... |
| bj-sbidmOLD 34169 | Obsolete proof of ~ sbidm ... |
| bj-dvelimdv 34170 | Deduction form of ~ dvelim... |
| bj-dvelimdv1 34171 | Curried (exported) form of... |
| bj-dvelimv 34172 | A version of ~ dvelim usin... |
| bj-nfeel2 34173 | Nonfreeness in a membershi... |
| bj-axc14nf 34174 | Proof of a version of ~ ax... |
| bj-axc14 34175 | Alternate proof of ~ axc14... |
| mobidvALT 34176 | Alternate proof of ~ mobid... |
| eliminable1 34177 | A theorem used to prove th... |
| eliminable2a 34178 | A theorem used to prove th... |
| eliminable2b 34179 | A theorem used to prove th... |
| eliminable2c 34180 | A theorem used to prove th... |
| eliminable3a 34181 | A theorem used to prove th... |
| eliminable3b 34182 | A theorem used to prove th... |
| bj-denotes 34183 | This would be the justific... |
| bj-issetwt 34184 | Closed form of ~ bj-issetw... |
| bj-issetw 34185 | The closest one can get to... |
| bj-elissetv 34186 | Version of ~ bj-elisset wi... |
| bj-elisset 34187 | Remove from ~ elisset depe... |
| bj-issetiv 34188 | Version of ~ bj-isseti wit... |
| bj-isseti 34189 | Remove from ~ isseti depen... |
| bj-ralvw 34190 | A weak version of ~ ralv n... |
| bj-rexvw 34191 | A weak version of ~ rexv n... |
| bj-rababw 34192 | A weak version of ~ rabab ... |
| bj-rexcom4bv 34193 | Version of ~ rexcom4b and ... |
| bj-rexcom4b 34194 | Remove from ~ rexcom4b dep... |
| bj-ceqsalt0 34195 | The FOL content of ~ ceqsa... |
| bj-ceqsalt1 34196 | The FOL content of ~ ceqsa... |
| bj-ceqsalt 34197 | Remove from ~ ceqsalt depe... |
| bj-ceqsaltv 34198 | Version of ~ bj-ceqsalt wi... |
| bj-ceqsalg0 34199 | The FOL content of ~ ceqsa... |
| bj-ceqsalg 34200 | Remove from ~ ceqsalg depe... |
| bj-ceqsalgALT 34201 | Alternate proof of ~ bj-ce... |
| bj-ceqsalgv 34202 | Version of ~ bj-ceqsalg wi... |
| bj-ceqsalgvALT 34203 | Alternate proof of ~ bj-ce... |
| bj-ceqsal 34204 | Remove from ~ ceqsal depen... |
| bj-ceqsalv 34205 | Remove from ~ ceqsalv depe... |
| bj-spcimdv 34206 | Remove from ~ spcimdv depe... |
| bj-spcimdvv 34207 | Remove from ~ spcimdv depe... |
| elelb 34208 | Equivalence between two co... |
| bj-pwvrelb 34209 | Characterization of the el... |
| bj-nfcsym 34210 | The nonfreeness quantifier... |
| bj-ax9 34211 | Proof of ~ ax-9 from Tarsk... |
| bj-sbeqALT 34212 | Substitution in an equalit... |
| bj-sbeq 34213 | Distribute proper substitu... |
| bj-sbceqgALT 34214 | Distribute proper substitu... |
| bj-csbsnlem 34215 | Lemma for ~ bj-csbsn (in t... |
| bj-csbsn 34216 | Substitution in a singleto... |
| bj-sbel1 34217 | Version of ~ sbcel1g when ... |
| bj-abv 34218 | The class of sets verifyin... |
| bj-ab0 34219 | The class of sets verifyin... |
| bj-abf 34220 | Shorter proof of ~ abf (wh... |
| bj-csbprc 34221 | More direct proof of ~ csb... |
| bj-exlimvmpi 34222 | A Fol lemma ( ~ exlimiv fo... |
| bj-exlimmpi 34223 | Lemma for ~ bj-vtoclg1f1 (... |
| bj-exlimmpbi 34224 | Lemma for theorems of the ... |
| bj-exlimmpbir 34225 | Lemma for theorems of the ... |
| bj-vtoclf 34226 | Remove dependency on ~ ax-... |
| bj-vtocl 34227 | Remove dependency on ~ ax-... |
| bj-vtoclg1f1 34228 | The FOL content of ~ vtocl... |
| bj-vtoclg1f 34229 | Reprove ~ vtoclg1f from ~ ... |
| bj-vtoclg1fv 34230 | Version of ~ bj-vtoclg1f w... |
| bj-vtoclg 34231 | A version of ~ vtoclg with... |
| bj-rabbida2 34232 | Version of ~ rabbidva2 wit... |
| bj-rabeqd 34233 | Deduction form of ~ rabeq ... |
| bj-rabeqbid 34234 | Version of ~ rabeqbidv wit... |
| bj-rabeqbida 34235 | Version of ~ rabeqbidva wi... |
| bj-seex 34236 | Version of ~ seex with a d... |
| bj-nfcf 34237 | Version of ~ df-nfc with a... |
| bj-zfauscl 34238 | General version of ~ zfaus... |
| bj-unrab 34239 | Generalization of ~ unrab ... |
| bj-inrab 34240 | Generalization of ~ inrab ... |
| bj-inrab2 34241 | Shorter proof of ~ inrab .... |
| bj-inrab3 34242 | Generalization of ~ dfrab3... |
| bj-rabtr 34243 | Restricted class abstracti... |
| bj-rabtrALT 34244 | Alternate proof of ~ bj-ra... |
| bj-rabtrAUTO 34245 | Proof of ~ bj-rabtr found ... |
| bj-ru0 34248 | The FOL part of Russell's ... |
| bj-ru1 34249 | A version of Russell's par... |
| bj-ru 34250 | Remove dependency on ~ ax-... |
| currysetlem 34251 | Lemma for ~ currysetlem , ... |
| curryset 34252 | Curry's paradox in set the... |
| currysetlem1 34253 | Lemma for ~ currysetALT . ... |
| currysetlem2 34254 | Lemma for ~ currysetALT . ... |
| currysetlem3 34255 | Lemma for ~ currysetALT . ... |
| currysetALT 34256 | Alternate proof of ~ curry... |
| bj-n0i 34257 | Inference associated with ... |
| bj-disjcsn 34258 | A class is disjoint from i... |
| bj-disjsn01 34259 | Disjointness of the single... |
| bj-0nel1 34260 | The empty set does not bel... |
| bj-1nel0 34261 | ` 1o ` does not belong to ... |
| bj-xpimasn 34262 | The image of a singleton, ... |
| bj-xpima1sn 34263 | The image of a singleton b... |
| bj-xpima1snALT 34264 | Alternate proof of ~ bj-xp... |
| bj-xpima2sn 34265 | The image of a singleton b... |
| bj-xpnzex 34266 | If the first factor of a p... |
| bj-xpexg2 34267 | Curried (exported) form of... |
| bj-xpnzexb 34268 | If the first factor of a p... |
| bj-cleq 34269 | Substitution property for ... |
| bj-snsetex 34270 | The class of sets "whose s... |
| bj-clex 34271 | Sethood of certain classes... |
| bj-sngleq 34274 | Substitution property for ... |
| bj-elsngl 34275 | Characterization of the el... |
| bj-snglc 34276 | Characterization of the el... |
| bj-snglss 34277 | The singletonization of a ... |
| bj-0nelsngl 34278 | The empty set is not a mem... |
| bj-snglinv 34279 | Inverse of singletonizatio... |
| bj-snglex 34280 | A class is a set if and on... |
| bj-tageq 34283 | Substitution property for ... |
| bj-eltag 34284 | Characterization of the el... |
| bj-0eltag 34285 | The empty set belongs to t... |
| bj-tagn0 34286 | The tagging of a class is ... |
| bj-tagss 34287 | The tagging of a class is ... |
| bj-snglsstag 34288 | The singletonization is in... |
| bj-sngltagi 34289 | The singletonization is in... |
| bj-sngltag 34290 | The singletonization and t... |
| bj-tagci 34291 | Characterization of the el... |
| bj-tagcg 34292 | Characterization of the el... |
| bj-taginv 34293 | Inverse of tagging. (Cont... |
| bj-tagex 34294 | A class is a set if and on... |
| bj-xtageq 34295 | The products of a given cl... |
| bj-xtagex 34296 | The product of a set and t... |
| bj-projeq 34299 | Substitution property for ... |
| bj-projeq2 34300 | Substitution property for ... |
| bj-projun 34301 | The class projection on a ... |
| bj-projex 34302 | Sethood of the class proje... |
| bj-projval 34303 | Value of the class project... |
| bj-1upleq 34306 | Substitution property for ... |
| bj-pr1eq 34309 | Substitution property for ... |
| bj-pr1un 34310 | The first projection prese... |
| bj-pr1val 34311 | Value of the first project... |
| bj-pr11val 34312 | Value of the first project... |
| bj-pr1ex 34313 | Sethood of the first proje... |
| bj-1uplth 34314 | The characteristic propert... |
| bj-1uplex 34315 | A monuple is a set if and ... |
| bj-1upln0 34316 | A monuple is nonempty. (C... |
| bj-2upleq 34319 | Substitution property for ... |
| bj-pr21val 34320 | Value of the first project... |
| bj-pr2eq 34323 | Substitution property for ... |
| bj-pr2un 34324 | The second projection pres... |
| bj-pr2val 34325 | Value of the second projec... |
| bj-pr22val 34326 | Value of the second projec... |
| bj-pr2ex 34327 | Sethood of the second proj... |
| bj-2uplth 34328 | The characteristic propert... |
| bj-2uplex 34329 | A couple is a set if and o... |
| bj-2upln0 34330 | A couple is nonempty. (Co... |
| bj-2upln1upl 34331 | A couple is never equal to... |
| bj-rcleqf 34332 | Relative version of ~ cleq... |
| bj-rcleq 34333 | Relative version of ~ dfcl... |
| bj-reabeq 34334 | Relative form of ~ abeq2 .... |
| bj-disj2r 34335 | Relative version of ~ ssdi... |
| bj-sscon 34336 | Contraposition law for rel... |
| bj-sselpwuni 34337 | Quantitative version of ~ ... |
| bj-unirel 34338 | Quantitative version of ~ ... |
| bj-elpwg 34339 | If the intersection of two... |
| bj-vjust 34340 | Justification theorem for ... |
| bj-df-v 34341 | Alternate definition of th... |
| bj-df-nul 34342 | Alternate definition of th... |
| bj-nul 34343 | Two formulations of the ax... |
| bj-nuliota 34344 | Definition of the empty se... |
| bj-nuliotaALT 34345 | Alternate proof of ~ bj-nu... |
| bj-vtoclgfALT 34346 | Alternate proof of ~ vtocl... |
| bj-elsn12g 34347 | Join of ~ elsng and ~ elsn... |
| bj-elsnb 34348 | Biconditional version of ~... |
| bj-pwcfsdom 34349 | Remove hypothesis from ~ p... |
| bj-grur1 34350 | Remove hypothesis from ~ g... |
| bj-bm1.3ii 34351 | The extension of a predica... |
| bj-0nelopab 34352 | The empty set is never an ... |
| bj-brrelex12ALT 34353 | Two classes related by a b... |
| bj-epelg 34354 | The membership relation an... |
| bj-epelb 34355 | Two classes are related by... |
| bj-nsnid 34356 | A set does not contain the... |
| bj-evaleq 34357 | Equality theorem for the `... |
| bj-evalfun 34358 | The evaluation at a class ... |
| bj-evalfn 34359 | The evaluation at a class ... |
| bj-evalval 34360 | Value of the evaluation at... |
| bj-evalid 34361 | The evaluation at a set of... |
| bj-ndxarg 34362 | Proof of ~ ndxarg from ~ b... |
| bj-evalidval 34363 | Closed general form of ~ s... |
| bj-rest00 34366 | An elementwise intersectio... |
| bj-restsn 34367 | An elementwise intersectio... |
| bj-restsnss 34368 | Special case of ~ bj-rests... |
| bj-restsnss2 34369 | Special case of ~ bj-rests... |
| bj-restsn0 34370 | An elementwise intersectio... |
| bj-restsn10 34371 | Special case of ~ bj-rests... |
| bj-restsnid 34372 | The elementwise intersecti... |
| bj-rest10 34373 | An elementwise intersectio... |
| bj-rest10b 34374 | Alternate version of ~ bj-... |
| bj-restn0 34375 | An elementwise intersectio... |
| bj-restn0b 34376 | Alternate version of ~ bj-... |
| bj-restpw 34377 | The elementwise intersecti... |
| bj-rest0 34378 | An elementwise intersectio... |
| bj-restb 34379 | An elementwise intersectio... |
| bj-restv 34380 | An elementwise intersectio... |
| bj-resta 34381 | An elementwise intersectio... |
| bj-restuni 34382 | The union of an elementwis... |
| bj-restuni2 34383 | The union of an elementwis... |
| bj-restreg 34384 | A reformulation of the axi... |
| bj-intss 34385 | A nonempty intersection of... |
| bj-raldifsn 34386 | All elements in a set sati... |
| bj-0int 34387 | If ` A ` is a collection o... |
| bj-mooreset 34388 | A Moore collection is a se... |
| bj-ismoore 34391 | Characterization of Moore ... |
| bj-ismoored0 34392 | Necessary condition to be ... |
| bj-ismoored 34393 | Necessary condition to be ... |
| bj-ismoored2 34394 | Necessary condition to be ... |
| bj-ismooredr 34395 | Sufficient condition to be... |
| bj-ismooredr2 34396 | Sufficient condition to be... |
| bj-discrmoore 34397 | The powerclass ` ~P A ` is... |
| bj-0nmoore 34398 | The empty set is not a Moo... |
| bj-snmoore 34399 | A singleton is a Moore col... |
| bj-snmooreb 34400 | A singleton is a Moore col... |
| bj-prmoore 34401 | A pair formed of two neste... |
| bj-0nelmpt 34402 | The empty set is not an el... |
| bj-mptval 34403 | Value of a function given ... |
| bj-dfmpoa 34404 | An equivalent definition o... |
| bj-mpomptALT 34405 | Alternate proof of ~ mpomp... |
| setsstrset 34422 | Relation between ~ df-sets... |
| bj-nfald 34423 | Variant of ~ nfald . (Con... |
| bj-nfexd 34424 | Variant of ~ nfexd . (Con... |
| copsex2d 34425 | Implicit substitution dedu... |
| copsex2b 34426 | Biconditional form of ~ co... |
| opelopabd 34427 | Membership of an ordere pa... |
| opelopabb 34428 | Membership of an ordered p... |
| opelopabbv 34429 | Membership of an ordered p... |
| bj-opelrelex 34430 | The coordinates of an orde... |
| bj-opelresdm 34431 | If an ordered pair is in a... |
| bj-brresdm 34432 | If two classes are related... |
| brabd0 34433 | Expressing that two sets a... |
| brabd 34434 | Expressing that two sets a... |
| bj-brab2a1 34435 | "Unbounded" version of ~ b... |
| bj-opabssvv 34436 | A variant of ~ relopabiv (... |
| bj-funidres 34437 | The restricted identity re... |
| bj-opelidb 34438 | Characterization of the or... |
| bj-opelidb1 34439 | Characterization of the or... |
| bj-inexeqex 34440 | Lemma for ~ bj-opelid (but... |
| bj-elsn0 34441 | If the intersection of two... |
| bj-opelid 34442 | Characterization of the or... |
| bj-ideqg 34443 | Characterization of the cl... |
| bj-ideqgALT 34444 | Alternate proof of ~ bj-id... |
| bj-ideqb 34445 | Characterization of classe... |
| bj-idres 34446 | Alternate expression for t... |
| bj-opelidres 34447 | Characterization of the or... |
| bj-idreseq 34448 | Sufficient condition for t... |
| bj-idreseqb 34449 | Characterization for two c... |
| bj-ideqg1 34450 | For sets, the identity rel... |
| bj-ideqg1ALT 34451 | Alternate proof of bj-ideq... |
| bj-opelidb1ALT 34452 | Characterization of the co... |
| bj-elid3 34453 | Characterization of the co... |
| bj-elid4 34454 | Characterization of the el... |
| bj-elid5 34455 | Characterization of the el... |
| bj-elid6 34456 | Characterization of the el... |
| bj-elid7 34457 | Characterization of the el... |
| bj-diagval 34460 | Value of the funtionalized... |
| bj-diagval2 34461 | Value of the funtionalized... |
| bj-eldiag 34462 | Characterization of the el... |
| bj-eldiag2 34463 | Characterization of the el... |
| bj-imdirval 34466 | Value of the functionalize... |
| bj-imdirval2 34467 | Value of the functionalize... |
| bj-imdirval3 34468 | Value of the functionalize... |
| bj-imdirid 34469 | Functorial property of the... |
| bj-inftyexpitaufo 34478 | The function ` inftyexpita... |
| bj-inftyexpitaudisj 34481 | An element of the circle a... |
| bj-inftyexpiinv 34484 | Utility theorem for the in... |
| bj-inftyexpiinj 34485 | Injectivity of the paramet... |
| bj-inftyexpidisj 34486 | An element of the circle a... |
| bj-ccinftydisj 34489 | The circle at infinity is ... |
| bj-elccinfty 34490 | A lemma for infinite exten... |
| bj-ccssccbar 34493 | Complex numbers are extend... |
| bj-ccinftyssccbar 34494 | Infinite extended complex ... |
| bj-pinftyccb 34497 | The class ` pinfty ` is an... |
| bj-pinftynrr 34498 | The extended complex numbe... |
| bj-minftyccb 34501 | The class ` minfty ` is an... |
| bj-minftynrr 34502 | The extended complex numbe... |
| bj-pinftynminfty 34503 | The extended complex numbe... |
| bj-rrhatsscchat 34512 | The real projective line i... |
| bj-imafv 34527 | If the direct image of a s... |
| bj-funun 34528 | Value of a function expres... |
| bj-fununsn1 34529 | Value of a function expres... |
| bj-fununsn2 34530 | Value of a function expres... |
| bj-fvsnun1 34531 | The value of a function wi... |
| bj-fvsnun2 34532 | The value of a function wi... |
| bj-fvmptunsn1 34533 | Value of a function expres... |
| bj-fvmptunsn2 34534 | Value of a function expres... |
| bj-iomnnom 34535 | The canonical bijection fr... |
| bj-smgrpssmgm 34544 | Semigroups are magmas. (C... |
| bj-smgrpssmgmel 34545 | Semigroups are magmas (ele... |
| bj-mndsssmgrp 34546 | Monoids are semigroups. (... |
| bj-mndsssmgrpel 34547 | Monoids are semigroups (el... |
| bj-cmnssmnd 34548 | Commutative monoids are mo... |
| bj-cmnssmndel 34549 | Commutative monoids are mo... |
| bj-grpssmnd 34550 | Groups are monoids. (Cont... |
| bj-grpssmndel 34551 | Groups are monoids (elemen... |
| bj-ablssgrp 34552 | Abelian groups are groups.... |
| bj-ablssgrpel 34553 | Abelian groups are groups ... |
| bj-ablsscmn 34554 | Abelian groups are commuta... |
| bj-ablsscmnel 34555 | Abelian groups are commuta... |
| bj-modssabl 34556 | (The additive groups of) m... |
| bj-vecssmod 34557 | Vector spaces are modules.... |
| bj-vecssmodel 34558 | Vector spaces are modules ... |
| bj-finsumval0 34561 | Value of a finite sum. (C... |
| bj-fvimacnv0 34562 | Variant of ~ fvimacnv wher... |
| bj-isvec 34563 | The predicate "is a vector... |
| bj-flddrng 34564 | Fields are division rings.... |
| bj-rrdrg 34565 | The field of real numbers ... |
| bj-isclm 34566 | The predicate "is a subcom... |
| bj-isrvec 34569 | The predicate "is a real v... |
| bj-rvecmod 34570 | Real vector spaces are mod... |
| bj-rvecssmod 34571 | Real vector spaces are mod... |
| bj-rvecrr 34572 | The field of scalars of a ... |
| bj-isrvecd 34573 | The predicate "is a real v... |
| bj-rvecvec 34574 | Real vector spaces are vec... |
| bj-isrvec2 34575 | The predicate "is a real v... |
| bj-rvecssvec 34576 | Real vector spaces are vec... |
| bj-rveccmod 34577 | Real vector spaces are sub... |
| bj-rvecsscmod 34578 | Real vector spaces are sub... |
| bj-rvecsscvec 34579 | Real vector spaces are sub... |
| bj-rveccvec 34580 | Real vector spaces are sub... |
| bj-rvecssabl 34581 | (The additive groups of) r... |
| bj-rvecabl 34582 | (The additive groups of) r... |
| bj-subcom 34583 | A consequence of commutati... |
| bj-lineqi 34584 | Solution of a (scalar) lin... |
| bj-bary1lem 34585 | Lemma for ~ bj-bary1 : exp... |
| bj-bary1lem1 34586 | Lemma for bj-bary1: comput... |
| bj-bary1 34587 | Barycentric coordinates in... |
| bj-endval 34590 | Value of the monoid of end... |
| bj-endbase 34591 | Base set of the monoid of ... |
| bj-endcomp 34592 | Composition law of the mon... |
| bj-endmnd 34593 | The monoid of endomorphism... |
| taupilem3 34594 | Lemma for tau-related theo... |
| taupilemrplb 34595 | A set of positive reals ha... |
| taupilem1 34596 | Lemma for ~ taupi . A pos... |
| taupilem2 34597 | Lemma for ~ taupi . The s... |
| taupi 34598 | Relationship between ` _ta... |
| dfgcd3 34599 | Alternate definition of th... |
| csbdif 34600 | Distribution of class subs... |
| csbpredg 34601 | Move class substitution in... |
| csbwrecsg 34602 | Move class substitution in... |
| csbrecsg 34603 | Move class substitution in... |
| csbrdgg 34604 | Move class substitution in... |
| csboprabg 34605 | Move class substitution in... |
| csbmpo123 34606 | Move class substitution in... |
| con1bii2 34607 | A contraposition inference... |
| con2bii2 34608 | A contraposition inference... |
| vtoclefex 34609 | Implicit substitution of a... |
| rnmptsn 34610 | The range of a function ma... |
| f1omptsnlem 34611 | This is the core of the pr... |
| f1omptsn 34612 | A function mapping to sing... |
| mptsnunlem 34613 | This is the core of the pr... |
| mptsnun 34614 | A class ` B ` is equal to ... |
| dissneqlem 34615 | This is the core of the pr... |
| dissneq 34616 | Any topology that contains... |
| exlimim 34617 | Closed form of ~ exlimimd ... |
| exlimimd 34618 | Existential elimination ru... |
| exellim 34619 | Closed form of ~ exellimdd... |
| exellimddv 34620 | Eliminate an antecedent wh... |
| topdifinfindis 34621 | Part of Exercise 3 of [Mun... |
| topdifinffinlem 34622 | This is the core of the pr... |
| topdifinffin 34623 | Part of Exercise 3 of [Mun... |
| topdifinf 34624 | Part of Exercise 3 of [Mun... |
| topdifinfeq 34625 | Two different ways of defi... |
| icorempo 34626 | Closed-below, open-above i... |
| icoreresf 34627 | Closed-below, open-above i... |
| icoreval 34628 | Value of the closed-below,... |
| icoreelrnab 34629 | Elementhood in the set of ... |
| isbasisrelowllem1 34630 | Lemma for ~ isbasisrelowl ... |
| isbasisrelowllem2 34631 | Lemma for ~ isbasisrelowl ... |
| icoreclin 34632 | The set of closed-below, o... |
| isbasisrelowl 34633 | The set of all closed-belo... |
| icoreunrn 34634 | The union of all closed-be... |
| istoprelowl 34635 | The set of all closed-belo... |
| icoreelrn 34636 | A class abstraction which ... |
| iooelexlt 34637 | An element of an open inte... |
| relowlssretop 34638 | The lower limit topology o... |
| relowlpssretop 34639 | The lower limit topology o... |
| sucneqond 34640 | Inequality of an ordinal s... |
| sucneqoni 34641 | Inequality of an ordinal s... |
| onsucuni3 34642 | If an ordinal number has a... |
| 1oequni2o 34643 | The ordinal number ` 1o ` ... |
| rdgsucuni 34644 | If an ordinal number has a... |
| rdgeqoa 34645 | If a recursive function wi... |
| elxp8 34646 | Membership in a Cartesian ... |
| cbveud 34647 | Deduction used to change b... |
| cbvreud 34648 | Deduction used to change b... |
| difunieq 34649 | The difference of unions i... |
| inunissunidif 34650 | Theorem about subsets of t... |
| rdgellim 34651 | Elementhood in a recursive... |
| rdglimss 34652 | A recursive definition at ... |
| rdgssun 34653 | In a recursive definition ... |
| exrecfnlem 34654 | Lemma for ~ exrecfn . (Co... |
| exrecfn 34655 | Theorem about the existenc... |
| exrecfnpw 34656 | For any base set, a set wh... |
| finorwe 34657 | If the Axiom of Infinity i... |
| dffinxpf 34660 | This theorem is the same a... |
| finxpeq1 34661 | Equality theorem for Carte... |
| finxpeq2 34662 | Equality theorem for Carte... |
| csbfinxpg 34663 | Distribute proper substitu... |
| finxpreclem1 34664 | Lemma for ` ^^ ` recursion... |
| finxpreclem2 34665 | Lemma for ` ^^ ` recursion... |
| finxp0 34666 | The value of Cartesian exp... |
| finxp1o 34667 | The value of Cartesian exp... |
| finxpreclem3 34668 | Lemma for ` ^^ ` recursion... |
| finxpreclem4 34669 | Lemma for ` ^^ ` recursion... |
| finxpreclem5 34670 | Lemma for ` ^^ ` recursion... |
| finxpreclem6 34671 | Lemma for ` ^^ ` recursion... |
| finxpsuclem 34672 | Lemma for ~ finxpsuc . (C... |
| finxpsuc 34673 | The value of Cartesian exp... |
| finxp2o 34674 | The value of Cartesian exp... |
| finxp3o 34675 | The value of Cartesian exp... |
| finxpnom 34676 | Cartesian exponentiation w... |
| finxp00 34677 | Cartesian exponentiation o... |
| iunctb2 34678 | Using the axiom of countab... |
| domalom 34679 | A class which dominates ev... |
| isinf2 34680 | The converse of ~ isinf . ... |
| ctbssinf 34681 | Using the axiom of choice,... |
| ralssiun 34682 | The index set of an indexe... |
| nlpineqsn 34683 | For every point ` p ` of a... |
| nlpfvineqsn 34684 | Given a subset ` A ` of ` ... |
| fvineqsnf1 34685 | A theorem about functions ... |
| fvineqsneu 34686 | A theorem about functions ... |
| fvineqsneq 34687 | A theorem about functions ... |
| pibp16 34688 | Property P000016 of pi-bas... |
| pibp19 34689 | Property P000019 of pi-bas... |
| pibp21 34690 | Property P000021 of pi-bas... |
| pibt1 34691 | Theorem T000001 of pi-base... |
| pibt2 34692 | Theorem T000002 of pi-base... |
| wl-section-prop 34693 | Intuitionistic logic is no... |
| wl-section-boot 34697 | In this section, I provide... |
| wl-luk-imim1i 34698 | Inference adding common co... |
| wl-luk-syl 34699 | An inference version of th... |
| wl-luk-imtrid 34700 | A syllogism rule of infere... |
| wl-luk-pm2.18d 34701 | Deduction based on reducti... |
| wl-luk-con4i 34702 | Inference rule. Copy of ~... |
| wl-luk-pm2.24i 34703 | Inference rule. Copy of ~... |
| wl-luk-a1i 34704 | Inference rule. Copy of ~... |
| wl-luk-mpi 34705 | A nested modus ponens infe... |
| wl-luk-imim2i 34706 | Inference adding common an... |
| wl-luk-imtrdi 34707 | A syllogism rule of infere... |
| wl-luk-ax3 34708 | ~ ax-3 proved from Lukasie... |
| wl-luk-ax1 34709 | ~ ax-1 proved from Lukasie... |
| wl-luk-pm2.27 34710 | This theorem, called "Asse... |
| wl-luk-com12 34711 | Inference that swaps (comm... |
| wl-luk-pm2.21 34712 | From a wff and its negatio... |
| wl-luk-con1i 34713 | A contraposition inference... |
| wl-luk-ja 34714 | Inference joining the ante... |
| wl-luk-imim2 34715 | A closed form of syllogism... |
| wl-luk-a1d 34716 | Deduction introducing an e... |
| wl-luk-ax2 34717 | ~ ax-2 proved from Lukasie... |
| wl-luk-id 34718 | Principle of identity. Th... |
| wl-luk-notnotr 34719 | Converse of double negatio... |
| wl-luk-pm2.04 34720 | Swap antecedents. Theorem... |
| wl-section-impchain 34721 | An implication like ` ( ps... |
| wl-impchain-mp-x 34722 | This series of theorems pr... |
| wl-impchain-mp-0 34723 | This theorem is the start ... |
| wl-impchain-mp-1 34724 | This theorem is in fact a ... |
| wl-impchain-mp-2 34725 | This theorem is in fact a ... |
| wl-impchain-com-1.x 34726 | It is often convenient to ... |
| wl-impchain-com-1.1 34727 | A degenerate form of antec... |
| wl-impchain-com-1.2 34728 | This theorem is in fact a ... |
| wl-impchain-com-1.3 34729 | This theorem is in fact a ... |
| wl-impchain-com-1.4 34730 | This theorem is in fact a ... |
| wl-impchain-com-n.m 34731 | This series of theorems al... |
| wl-impchain-com-2.3 34732 | This theorem is in fact a ... |
| wl-impchain-com-2.4 34733 | This theorem is in fact a ... |
| wl-impchain-com-3.2.1 34734 | This theorem is in fact a ... |
| wl-impchain-a1-x 34735 | If an implication chain is... |
| wl-impchain-a1-1 34736 | Inference rule, a copy of ... |
| wl-impchain-a1-2 34737 | Inference rule, a copy of ... |
| wl-impchain-a1-3 34738 | Inference rule, a copy of ... |
| wl-ax13lem1 34740 | A version of ~ ax-wl-13v w... |
| wl-mps 34741 | Replacing a nested consequ... |
| wl-syls1 34742 | Replacing a nested consequ... |
| wl-syls2 34743 | Replacing a nested anteced... |
| wl-embant 34744 | A true wff can always be a... |
| wl-orel12 34745 | In a conjunctive normal fo... |
| wl-cases2-dnf 34746 | A particular instance of ~... |
| wl-cbvmotv 34747 | Change bound variable. Us... |
| wl-moteq 34748 | Change bound variable. Us... |
| wl-motae 34749 | Change bound variable. Us... |
| wl-moae 34750 | Two ways to express "at mo... |
| wl-euae 34751 | Two ways to express "exact... |
| wl-nax6im 34752 | The following series of th... |
| wl-hbae1 34753 | This specialization of ~ h... |
| wl-naevhba1v 34754 | An instance of ~ hbn1w app... |
| wl-spae 34755 | Prove an instance of ~ sp ... |
| wl-speqv 34756 | Under the assumption ` -. ... |
| wl-19.8eqv 34757 | Under the assumption ` -. ... |
| wl-19.2reqv 34758 | Under the assumption ` -. ... |
| wl-nfalv 34759 | If ` x ` is not present in... |
| wl-nfimf1 34760 | An antecedent is irrelevan... |
| wl-nfae1 34761 | Unlike ~ nfae , this speci... |
| wl-nfnae1 34762 | Unlike ~ nfnae , this spec... |
| wl-aetr 34763 | A transitive law for varia... |
| wl-axc11r 34764 | Same as ~ axc11r , but usi... |
| wl-dral1d 34765 | A version of ~ dral1 with ... |
| wl-cbvalnaed 34766 | ~ wl-cbvalnae with a conte... |
| wl-cbvalnae 34767 | A more general version of ... |
| wl-exeq 34768 | The semantics of ` E. x y ... |
| wl-aleq 34769 | The semantics of ` A. x y ... |
| wl-nfeqfb 34770 | Extend ~ nfeqf to an equiv... |
| wl-nfs1t 34771 | If ` y ` is not free in ` ... |
| wl-equsalvw 34772 | Version of ~ equsalv with ... |
| wl-equsald 34773 | Deduction version of ~ equ... |
| wl-equsal 34774 | A useful equivalence relat... |
| wl-equsal1t 34775 | The expression ` x = y ` i... |
| wl-equsalcom 34776 | This simple equivalence ea... |
| wl-equsal1i 34777 | The antecedent ` x = y ` i... |
| wl-sb6rft 34778 | A specialization of ~ wl-e... |
| wl-cbvalsbi 34779 | Change bounded variables i... |
| wl-sbrimt 34780 | Substitution with a variab... |
| wl-sblimt 34781 | Substitution with a variab... |
| wl-sb8t 34782 | Substitution of variable i... |
| wl-sb8et 34783 | Substitution of variable i... |
| wl-sbhbt 34784 | Closed form of ~ sbhb . C... |
| wl-sbnf1 34785 | Two ways expressing that `... |
| wl-equsb3 34786 | ~ equsb3 with a distinctor... |
| wl-equsb4 34787 | Substitution applied to an... |
| wl-2sb6d 34788 | Version of ~ 2sb6 with a c... |
| wl-sbcom2d-lem1 34789 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d-lem2 34790 | Lemma used to prove ~ wl-s... |
| wl-sbcom2d 34791 | Version of ~ sbcom2 with a... |
| wl-sbalnae 34792 | A theorem used in eliminat... |
| wl-sbal1 34793 | A theorem used in eliminat... |
| wl-sbal2 34794 | Move quantifier in and out... |
| wl-2spsbbi 34795 | ~ spsbbi applied twice. (... |
| wl-lem-exsb 34796 | This theorem provides a ba... |
| wl-lem-nexmo 34797 | This theorem provides a ba... |
| wl-lem-moexsb 34798 | The antecedent ` A. x ( ph... |
| wl-alanbii 34799 | This theorem extends ~ ala... |
| wl-mo2df 34800 | Version of ~ mof with a co... |
| wl-mo2tf 34801 | Closed form of ~ mof with ... |
| wl-eudf 34802 | Version of ~ eu6 with a co... |
| wl-eutf 34803 | Closed form of ~ eu6 with ... |
| wl-euequf 34804 | ~ euequ proved with a dist... |
| wl-mo2t 34805 | Closed form of ~ mof . (C... |
| wl-mo3t 34806 | Closed form of ~ mo3 . (C... |
| wl-sb8eut 34807 | Substitution of variable i... |
| wl-sb8mot 34808 | Substitution of variable i... |
| wl-axc11rc11 34809 | Proving ~ axc11r from ~ ax... |
| wl-ax11-lem1 34811 | A transitive law for varia... |
| wl-ax11-lem2 34812 | Lemma. (Contributed by Wo... |
| wl-ax11-lem3 34813 | Lemma. (Contributed by Wo... |
| wl-ax11-lem4 34814 | Lemma. (Contributed by Wo... |
| wl-ax11-lem5 34815 | Lemma. (Contributed by Wo... |
| wl-ax11-lem6 34816 | Lemma. (Contributed by Wo... |
| wl-ax11-lem7 34817 | Lemma. (Contributed by Wo... |
| wl-ax11-lem8 34818 | Lemma. (Contributed by Wo... |
| wl-ax11-lem9 34819 | The easy part when ` x ` c... |
| wl-ax11-lem10 34820 | We now have prepared every... |
| wl-clabv 34821 | Variant of ~ df-clab , whe... |
| wl-dfclab 34822 | Rederive ~ df-clab from ~ ... |
| wl-clabtv 34823 | Using class abstraction in... |
| wl-clabt 34824 | Using class abstraction in... |
| wl-dfralsb 34831 | An alternate definition of... |
| wl-dfralflem 34832 | Lemma for ~ wl-dfralf and ... |
| wl-dfralf 34833 | Restricted universal quant... |
| wl-dfralfi 34834 | Restricted universal quant... |
| wl-dfralv 34835 | Alternate definition of re... |
| wl-rgen 34836 | Generalization rule for re... |
| wl-rgenw 34837 | Generalization rule for re... |
| wl-rgen2w 34838 | Generalization rule for re... |
| wl-ralel 34839 | All elements of a class ar... |
| wl-dfrexf 34841 | Restricted existential qua... |
| wl-dfrexfi 34842 | Restricted universal quant... |
| wl-dfrexv 34843 | Alternate definition of re... |
| wl-dfrexex 34844 | Alternate definition of th... |
| wl-dfrexsb 34845 | An alternate definition of... |
| wl-dfrmosb 34847 | An alternate definition of... |
| wl-dfrmov 34848 | Alternate definition of re... |
| wl-dfrmof 34849 | Restricted "at most one" (... |
| wl-dfreusb 34851 | An alternate definition of... |
| wl-dfreuv 34852 | Alternate definition of re... |
| wl-dfreuf 34853 | Restricted existential uni... |
| wl-dfrabsb 34855 | Alternate definition of re... |
| wl-dfrabv 34856 | Alternate definition of re... |
| wl-clelsb3df 34857 | Deduction version of ~ cle... |
| wl-dfrabf 34858 | Alternate definition of re... |
| rabiun 34859 | Abstraction restricted to ... |
| iundif1 34860 | Indexed union of class dif... |
| imadifss 34861 | The difference of images i... |
| cureq 34862 | Equality theorem for curry... |
| unceq 34863 | Equality theorem for uncur... |
| curf 34864 | Functional property of cur... |
| uncf 34865 | Functional property of unc... |
| curfv 34866 | Value of currying. (Contr... |
| uncov 34867 | Value of uncurrying. (Con... |
| curunc 34868 | Currying of uncurrying. (... |
| unccur 34869 | Uncurrying of currying. (... |
| phpreu 34870 | Theorem related to pigeonh... |
| finixpnum 34871 | A finite Cartesian product... |
| fin2solem 34872 | Lemma for ~ fin2so . (Con... |
| fin2so 34873 | Any totally ordered Tarski... |
| ltflcei 34874 | Theorem to move the floor ... |
| leceifl 34875 | Theorem to move the floor ... |
| sin2h 34876 | Half-angle rule for sine. ... |
| cos2h 34877 | Half-angle rule for cosine... |
| tan2h 34878 | Half-angle rule for tangen... |
| lindsadd 34879 | In a vector space, the uni... |
| lindsdom 34880 | A linearly independent set... |
| lindsenlbs 34881 | A maximal linearly indepen... |
| matunitlindflem1 34882 | One direction of ~ matunit... |
| matunitlindflem2 34883 | One direction of ~ matunit... |
| matunitlindf 34884 | A matrix over a field is i... |
| ptrest 34885 | Expressing a restriction o... |
| ptrecube 34886 | Any point in an open set o... |
| poimirlem1 34887 | Lemma for ~ poimir - the v... |
| poimirlem2 34888 | Lemma for ~ poimir - conse... |
| poimirlem3 34889 | Lemma for ~ poimir to add ... |
| poimirlem4 34890 | Lemma for ~ poimir connect... |
| poimirlem5 34891 | Lemma for ~ poimir to esta... |
| poimirlem6 34892 | Lemma for ~ poimir establi... |
| poimirlem7 34893 | Lemma for ~ poimir , simil... |
| poimirlem8 34894 | Lemma for ~ poimir , estab... |
| poimirlem9 34895 | Lemma for ~ poimir , estab... |
| poimirlem10 34896 | Lemma for ~ poimir establi... |
| poimirlem11 34897 | Lemma for ~ poimir connect... |
| poimirlem12 34898 | Lemma for ~ poimir connect... |
| poimirlem13 34899 | Lemma for ~ poimir - for a... |
| poimirlem14 34900 | Lemma for ~ poimir - for a... |
| poimirlem15 34901 | Lemma for ~ poimir , that ... |
| poimirlem16 34902 | Lemma for ~ poimir establi... |
| poimirlem17 34903 | Lemma for ~ poimir establi... |
| poimirlem18 34904 | Lemma for ~ poimir stating... |
| poimirlem19 34905 | Lemma for ~ poimir establi... |
| poimirlem20 34906 | Lemma for ~ poimir establi... |
| poimirlem21 34907 | Lemma for ~ poimir stating... |
| poimirlem22 34908 | Lemma for ~ poimir , that ... |
| poimirlem23 34909 | Lemma for ~ poimir , two w... |
| poimirlem24 34910 | Lemma for ~ poimir , two w... |
| poimirlem25 34911 | Lemma for ~ poimir stating... |
| poimirlem26 34912 | Lemma for ~ poimir showing... |
| poimirlem27 34913 | Lemma for ~ poimir showing... |
| poimirlem28 34914 | Lemma for ~ poimir , a var... |
| poimirlem29 34915 | Lemma for ~ poimir connect... |
| poimirlem30 34916 | Lemma for ~ poimir combini... |
| poimirlem31 34917 | Lemma for ~ poimir , assig... |
| poimirlem32 34918 | Lemma for ~ poimir , combi... |
| poimir 34919 | Poincare-Miranda theorem. ... |
| broucube 34920 | Brouwer - or as Kulpa call... |
| heicant 34921 | Heine-Cantor theorem: a co... |
| opnmbllem0 34922 | Lemma for ~ ismblfin ; cou... |
| mblfinlem1 34923 | Lemma for ~ ismblfin , ord... |
| mblfinlem2 34924 | Lemma for ~ ismblfin , eff... |
| mblfinlem3 34925 | The difference between two... |
| mblfinlem4 34926 | Backward direction of ~ is... |
| ismblfin 34927 | Measurability in terms of ... |
| ovoliunnfl 34928 | ~ ovoliun is incompatible ... |
| ex-ovoliunnfl 34929 | Demonstration of ~ ovoliun... |
| voliunnfl 34930 | ~ voliun is incompatible w... |
| volsupnfl 34931 | ~ volsup is incompatible w... |
| mbfresfi 34932 | Measurability of a piecewi... |
| mbfposadd 34933 | If the sum of two measurab... |
| cnambfre 34934 | A real-valued, a.e. contin... |
| dvtanlem 34935 | Lemma for ~ dvtan - the do... |
| dvtan 34936 | Derivative of tangent. (C... |
| itg2addnclem 34937 | An alternate expression fo... |
| itg2addnclem2 34938 | Lemma for ~ itg2addnc . T... |
| itg2addnclem3 34939 | Lemma incomprehensible in ... |
| itg2addnc 34940 | Alternate proof of ~ itg2a... |
| itg2gt0cn 34941 | ~ itg2gt0 holds on functio... |
| ibladdnclem 34942 | Lemma for ~ ibladdnc ; cf ... |
| ibladdnc 34943 | Choice-free analogue of ~ ... |
| itgaddnclem1 34944 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnclem2 34945 | Lemma for ~ itgaddnc ; cf.... |
| itgaddnc 34946 | Choice-free analogue of ~ ... |
| iblsubnc 34947 | Choice-free analogue of ~ ... |
| itgsubnc 34948 | Choice-free analogue of ~ ... |
| iblabsnclem 34949 | Lemma for ~ iblabsnc ; cf.... |
| iblabsnc 34950 | Choice-free analogue of ~ ... |
| iblmulc2nc 34951 | Choice-free analogue of ~ ... |
| itgmulc2nclem1 34952 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nclem2 34953 | Lemma for ~ itgmulc2nc ; c... |
| itgmulc2nc 34954 | Choice-free analogue of ~ ... |
| itgabsnc 34955 | Choice-free analogue of ~ ... |
| bddiblnc 34956 | Choice-free proof of ~ bdd... |
| cnicciblnc 34957 | Choice-free proof of ~ cni... |
| itggt0cn 34958 | ~ itggt0 holds for continu... |
| ftc1cnnclem 34959 | Lemma for ~ ftc1cnnc ; cf.... |
| ftc1cnnc 34960 | Choice-free proof of ~ ftc... |
| ftc1anclem1 34961 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem2 34962 | Lemma for ~ ftc1anc - rest... |
| ftc1anclem3 34963 | Lemma for ~ ftc1anc - the ... |
| ftc1anclem4 34964 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem5 34965 | Lemma for ~ ftc1anc , the ... |
| ftc1anclem6 34966 | Lemma for ~ ftc1anc - cons... |
| ftc1anclem7 34967 | Lemma for ~ ftc1anc . (Co... |
| ftc1anclem8 34968 | Lemma for ~ ftc1anc . (Co... |
| ftc1anc 34969 | ~ ftc1a holds for function... |
| ftc2nc 34970 | Choice-free proof of ~ ftc... |
| asindmre 34971 | Real part of domain of dif... |
| dvasin 34972 | Derivative of arcsine. (C... |
| dvacos 34973 | Derivative of arccosine. ... |
| dvreasin 34974 | Real derivative of arcsine... |
| dvreacos 34975 | Real derivative of arccosi... |
| areacirclem1 34976 | Antiderivative of cross-se... |
| areacirclem2 34977 | Endpoint-inclusive continu... |
| areacirclem3 34978 | Integrability of cross-sec... |
| areacirclem4 34979 | Endpoint-inclusive continu... |
| areacirclem5 34980 | Finding the cross-section ... |
| areacirc 34981 | The area of a circle of ra... |
| unirep 34982 | Define a quantity whose de... |
| cover2 34983 | Two ways of expressing the... |
| cover2g 34984 | Two ways of expressing the... |
| brabg2 34985 | Relation by a binary relat... |
| opelopab3 34986 | Ordered pair membership in... |
| cocanfo 34987 | Cancellation of a surjecti... |
| brresi2 34988 | Restriction of a binary re... |
| fnopabeqd 34989 | Equality deduction for fun... |
| fvopabf4g 34990 | Function value of an opera... |
| eqfnun 34991 | Two functions on ` A u. B ... |
| fnopabco 34992 | Composition of a function ... |
| opropabco 34993 | Composition of an operator... |
| cocnv 34994 | Composition with a functio... |
| f1ocan1fv 34995 | Cancel a composition by a ... |
| f1ocan2fv 34996 | Cancel a composition by th... |
| inixp 34997 | Intersection of Cartesian ... |
| upixp 34998 | Universal property of the ... |
| abrexdom 34999 | An indexed set is dominate... |
| abrexdom2 35000 | An indexed set is dominate... |
| ac6gf 35001 | Axiom of Choice. (Contrib... |
| indexa 35002 | If for every element of an... |
| indexdom 35003 | If for every element of an... |
| frinfm 35004 | A subset of a well-founded... |
| welb 35005 | A nonempty subset of a wel... |
| supex2g 35006 | Existence of supremum. (C... |
| supclt 35007 | Closure of supremum. (Con... |
| supubt 35008 | Upper bound property of su... |
| filbcmb 35009 | Combine a finite set of lo... |
| fzmul 35010 | Membership of a product in... |
| sdclem2 35011 | Lemma for ~ sdc . (Contri... |
| sdclem1 35012 | Lemma for ~ sdc . (Contri... |
| sdc 35013 | Strong dependent choice. ... |
| fdc 35014 | Finite version of dependen... |
| fdc1 35015 | Variant of ~ fdc with no s... |
| seqpo 35016 | Two ways to say that a seq... |
| incsequz 35017 | An increasing sequence of ... |
| incsequz2 35018 | An increasing sequence of ... |
| nnubfi 35019 | A bounded above set of pos... |
| nninfnub 35020 | An infinite set of positiv... |
| subspopn 35021 | An open set is open in the... |
| neificl 35022 | Neighborhoods are closed u... |
| lpss2 35023 | Limit points of a subset a... |
| metf1o 35024 | Use a bijection with a met... |
| blssp 35025 | A ball in the subspace met... |
| mettrifi 35026 | Generalized triangle inequ... |
| lmclim2 35027 | A sequence in a metric spa... |
| geomcau 35028 | If the distance between co... |
| caures 35029 | The restriction of a Cauch... |
| caushft 35030 | A shifted Cauchy sequence ... |
| constcncf 35031 | A constant function is a c... |
| idcncf 35032 | The identity function is a... |
| sub1cncf 35033 | Subtracting a constant is ... |
| sub2cncf 35034 | Subtraction from a constan... |
| cnres2 35035 | The restriction of a conti... |
| cnresima 35036 | A continuous function is c... |
| cncfres 35037 | A continuous function on c... |
| istotbnd 35041 | The predicate "is a totall... |
| istotbnd2 35042 | The predicate "is a totall... |
| istotbnd3 35043 | A metric space is totally ... |
| totbndmet 35044 | The predicate "totally bou... |
| 0totbnd 35045 | The metric (there is only ... |
| sstotbnd2 35046 | Condition for a subset of ... |
| sstotbnd 35047 | Condition for a subset of ... |
| sstotbnd3 35048 | Use a net that is not nece... |
| totbndss 35049 | A subset of a totally boun... |
| equivtotbnd 35050 | If the metric ` M ` is "st... |
| isbnd 35052 | The predicate "is a bounde... |
| bndmet 35053 | A bounded metric space is ... |
| isbndx 35054 | A "bounded extended metric... |
| isbnd2 35055 | The predicate "is a bounde... |
| isbnd3 35056 | A metric space is bounded ... |
| isbnd3b 35057 | A metric space is bounded ... |
| bndss 35058 | A subset of a bounded metr... |
| blbnd 35059 | A ball is bounded. (Contr... |
| ssbnd 35060 | A subset of a metric space... |
| totbndbnd 35061 | A totally bounded metric s... |
| equivbnd 35062 | If the metric ` M ` is "st... |
| bnd2lem 35063 | Lemma for ~ equivbnd2 and ... |
| equivbnd2 35064 | If balls are totally bound... |
| prdsbnd 35065 | The product metric over fi... |
| prdstotbnd 35066 | The product metric over fi... |
| prdsbnd2 35067 | If balls are totally bound... |
| cntotbnd 35068 | A subset of the complex nu... |
| cnpwstotbnd 35069 | A subset of ` A ^ I ` , wh... |
| ismtyval 35072 | The set of isometries betw... |
| isismty 35073 | The condition "is an isome... |
| ismtycnv 35074 | The inverse of an isometry... |
| ismtyima 35075 | The image of a ball under ... |
| ismtyhmeolem 35076 | Lemma for ~ ismtyhmeo . (... |
| ismtyhmeo 35077 | An isometry is a homeomorp... |
| ismtybndlem 35078 | Lemma for ~ ismtybnd . (C... |
| ismtybnd 35079 | Isometries preserve bounde... |
| ismtyres 35080 | A restriction of an isomet... |
| heibor1lem 35081 | Lemma for ~ heibor1 . A c... |
| heibor1 35082 | One half of ~ heibor , tha... |
| heiborlem1 35083 | Lemma for ~ heibor . We w... |
| heiborlem2 35084 | Lemma for ~ heibor . Subs... |
| heiborlem3 35085 | Lemma for ~ heibor . Usin... |
| heiborlem4 35086 | Lemma for ~ heibor . Usin... |
| heiborlem5 35087 | Lemma for ~ heibor . The ... |
| heiborlem6 35088 | Lemma for ~ heibor . Sinc... |
| heiborlem7 35089 | Lemma for ~ heibor . Sinc... |
| heiborlem8 35090 | Lemma for ~ heibor . The ... |
| heiborlem9 35091 | Lemma for ~ heibor . Disc... |
| heiborlem10 35092 | Lemma for ~ heibor . The ... |
| heibor 35093 | Generalized Heine-Borel Th... |
| bfplem1 35094 | Lemma for ~ bfp . The seq... |
| bfplem2 35095 | Lemma for ~ bfp . Using t... |
| bfp 35096 | Banach fixed point theorem... |
| rrnval 35099 | The n-dimensional Euclidea... |
| rrnmval 35100 | The value of the Euclidean... |
| rrnmet 35101 | Euclidean space is a metri... |
| rrndstprj1 35102 | The distance between two p... |
| rrndstprj2 35103 | Bound on the distance betw... |
| rrncmslem 35104 | Lemma for ~ rrncms . (Con... |
| rrncms 35105 | Euclidean space is complet... |
| repwsmet 35106 | The supremum metric on ` R... |
| rrnequiv 35107 | The supremum metric on ` R... |
| rrntotbnd 35108 | A set in Euclidean space i... |
| rrnheibor 35109 | Heine-Borel theorem for Eu... |
| ismrer1 35110 | An isometry between ` RR `... |
| reheibor 35111 | Heine-Borel theorem for re... |
| iccbnd 35112 | A closed interval in ` RR ... |
| icccmpALT 35113 | A closed interval in ` RR ... |
| isass 35118 | The predicate "is an assoc... |
| isexid 35119 | The predicate ` G ` has a ... |
| ismgmOLD 35122 | Obsolete version of ~ ismg... |
| clmgmOLD 35123 | Obsolete version of ~ mgmc... |
| opidonOLD 35124 | Obsolete version of ~ mndp... |
| rngopidOLD 35125 | Obsolete version of ~ mndp... |
| opidon2OLD 35126 | Obsolete version of ~ mndp... |
| isexid2 35127 | If ` G e. ( Magma i^i ExId... |
| exidu1 35128 | Uniqueness of the left and... |
| idrval 35129 | The value of the identity ... |
| iorlid 35130 | A magma right and left ide... |
| cmpidelt 35131 | A magma right and left ide... |
| smgrpismgmOLD 35134 | Obsolete version of ~ sgrp... |
| issmgrpOLD 35135 | Obsolete version of ~ issg... |
| smgrpmgm 35136 | A semigroup is a magma. (... |
| smgrpassOLD 35137 | Obsolete version of ~ sgrp... |
| mndoissmgrpOLD 35140 | Obsolete version of ~ mnds... |
| mndoisexid 35141 | A monoid has an identity e... |
| mndoismgmOLD 35142 | Obsolete version of ~ mndm... |
| mndomgmid 35143 | A monoid is a magma with a... |
| ismndo 35144 | The predicate "is a monoid... |
| ismndo1 35145 | The predicate "is a monoid... |
| ismndo2 35146 | The predicate "is a monoid... |
| grpomndo 35147 | A group is a monoid. (Con... |
| exidcl 35148 | Closure of the binary oper... |
| exidreslem 35149 | Lemma for ~ exidres and ~ ... |
| exidres 35150 | The restriction of a binar... |
| exidresid 35151 | The restriction of a binar... |
| ablo4pnp 35152 | A commutative/associative ... |
| grpoeqdivid 35153 | Two group elements are equ... |
| grposnOLD 35154 | The group operation for th... |
| elghomlem1OLD 35157 | Obsolete as of 15-Mar-2020... |
| elghomlem2OLD 35158 | Obsolete as of 15-Mar-2020... |
| elghomOLD 35159 | Obsolete version of ~ isgh... |
| ghomlinOLD 35160 | Obsolete version of ~ ghml... |
| ghomidOLD 35161 | Obsolete version of ~ ghmi... |
| ghomf 35162 | Mapping property of a grou... |
| ghomco 35163 | The composition of two gro... |
| ghomdiv 35164 | Group homomorphisms preser... |
| grpokerinj 35165 | A group homomorphism is in... |
| relrngo 35168 | The class of all unital ri... |
| isrngo 35169 | The predicate "is a (unita... |
| isrngod 35170 | Conditions that determine ... |
| rngoi 35171 | The properties of a unital... |
| rngosm 35172 | Functionality of the multi... |
| rngocl 35173 | Closure of the multiplicat... |
| rngoid 35174 | The multiplication operati... |
| rngoideu 35175 | The unit element of a ring... |
| rngodi 35176 | Distributive law for the m... |
| rngodir 35177 | Distributive law for the m... |
| rngoass 35178 | Associative law for the mu... |
| rngo2 35179 | A ring element plus itself... |
| rngoablo 35180 | A ring's addition operatio... |
| rngoablo2 35181 | In a unital ring the addit... |
| rngogrpo 35182 | A ring's addition operatio... |
| rngone0 35183 | The base set of a ring is ... |
| rngogcl 35184 | Closure law for the additi... |
| rngocom 35185 | The addition operation of ... |
| rngoaass 35186 | The addition operation of ... |
| rngoa32 35187 | The addition operation of ... |
| rngoa4 35188 | Rearrangement of 4 terms i... |
| rngorcan 35189 | Right cancellation law for... |
| rngolcan 35190 | Left cancellation law for ... |
| rngo0cl 35191 | A ring has an additive ide... |
| rngo0rid 35192 | The additive identity of a... |
| rngo0lid 35193 | The additive identity of a... |
| rngolz 35194 | The zero of a unital ring ... |
| rngorz 35195 | The zero of a unital ring ... |
| rngosn3 35196 | Obsolete as of 25-Jan-2020... |
| rngosn4 35197 | Obsolete as of 25-Jan-2020... |
| rngosn6 35198 | Obsolete as of 25-Jan-2020... |
| rngonegcl 35199 | A ring is closed under neg... |
| rngoaddneg1 35200 | Adding the negative in a r... |
| rngoaddneg2 35201 | Adding the negative in a r... |
| rngosub 35202 | Subtraction in a ring, in ... |
| rngmgmbs4 35203 | The range of an internal o... |
| rngodm1dm2 35204 | In a unital ring the domai... |
| rngorn1 35205 | In a unital ring the range... |
| rngorn1eq 35206 | In a unital ring the range... |
| rngomndo 35207 | In a unital ring the multi... |
| rngoidmlem 35208 | The unit of a ring is an i... |
| rngolidm 35209 | The unit of a ring is an i... |
| rngoridm 35210 | The unit of a ring is an i... |
| rngo1cl 35211 | The unit of a ring belongs... |
| rngoueqz 35212 | Obsolete as of 23-Jan-2020... |
| rngonegmn1l 35213 | Negation in a ring is the ... |
| rngonegmn1r 35214 | Negation in a ring is the ... |
| rngoneglmul 35215 | Negation of a product in a... |
| rngonegrmul 35216 | Negation of a product in a... |
| rngosubdi 35217 | Ring multiplication distri... |
| rngosubdir 35218 | Ring multiplication distri... |
| zerdivemp1x 35219 | In a unitary ring a left i... |
| isdivrngo 35222 | The predicate "is a divisi... |
| drngoi 35223 | The properties of a divisi... |
| gidsn 35224 | Obsolete as of 23-Jan-2020... |
| zrdivrng 35225 | The zero ring is not a div... |
| dvrunz 35226 | In a division ring the uni... |
| isgrpda 35227 | Properties that determine ... |
| isdrngo1 35228 | The predicate "is a divisi... |
| divrngcl 35229 | The product of two nonzero... |
| isdrngo2 35230 | A division ring is a ring ... |
| isdrngo3 35231 | A division ring is a ring ... |
| rngohomval 35236 | The set of ring homomorphi... |
| isrngohom 35237 | The predicate "is a ring h... |
| rngohomf 35238 | A ring homomorphism is a f... |
| rngohomcl 35239 | Closure law for a ring hom... |
| rngohom1 35240 | A ring homomorphism preser... |
| rngohomadd 35241 | Ring homomorphisms preserv... |
| rngohommul 35242 | Ring homomorphisms preserv... |
| rngogrphom 35243 | A ring homomorphism is a g... |
| rngohom0 35244 | A ring homomorphism preser... |
| rngohomsub 35245 | Ring homomorphisms preserv... |
| rngohomco 35246 | The composition of two rin... |
| rngokerinj 35247 | A ring homomorphism is inj... |
| rngoisoval 35249 | The set of ring isomorphis... |
| isrngoiso 35250 | The predicate "is a ring i... |
| rngoiso1o 35251 | A ring isomorphism is a bi... |
| rngoisohom 35252 | A ring isomorphism is a ri... |
| rngoisocnv 35253 | The inverse of a ring isom... |
| rngoisoco 35254 | The composition of two rin... |
| isriscg 35256 | The ring isomorphism relat... |
| isrisc 35257 | The ring isomorphism relat... |
| risc 35258 | The ring isomorphism relat... |
| risci 35259 | Determine that two rings a... |
| riscer 35260 | Ring isomorphism is an equ... |
| iscom2 35267 | A device to add commutativ... |
| iscrngo 35268 | The predicate "is a commut... |
| iscrngo2 35269 | The predicate "is a commut... |
| iscringd 35270 | Conditions that determine ... |
| flddivrng 35271 | A field is a division ring... |
| crngorngo 35272 | A commutative ring is a ri... |
| crngocom 35273 | The multiplication operati... |
| crngm23 35274 | Commutative/associative la... |
| crngm4 35275 | Commutative/associative la... |
| fldcrng 35276 | A field is a commutative r... |
| isfld2 35277 | The predicate "is a field"... |
| crngohomfo 35278 | The image of a homomorphis... |
| idlval 35285 | The class of ideals of a r... |
| isidl 35286 | The predicate "is an ideal... |
| isidlc 35287 | The predicate "is an ideal... |
| idlss 35288 | An ideal of ` R ` is a sub... |
| idlcl 35289 | An element of an ideal is ... |
| idl0cl 35290 | An ideal contains ` 0 ` . ... |
| idladdcl 35291 | An ideal is closed under a... |
| idllmulcl 35292 | An ideal is closed under m... |
| idlrmulcl 35293 | An ideal is closed under m... |
| idlnegcl 35294 | An ideal is closed under n... |
| idlsubcl 35295 | An ideal is closed under s... |
| rngoidl 35296 | A ring ` R ` is an ` R ` i... |
| 0idl 35297 | The set containing only ` ... |
| 1idl 35298 | Two ways of expressing the... |
| 0rngo 35299 | In a ring, ` 0 = 1 ` iff t... |
| divrngidl 35300 | The only ideals in a divis... |
| intidl 35301 | The intersection of a none... |
| inidl 35302 | The intersection of two id... |
| unichnidl 35303 | The union of a nonempty ch... |
| keridl 35304 | The kernel of a ring homom... |
| pridlval 35305 | The class of prime ideals ... |
| ispridl 35306 | The predicate "is a prime ... |
| pridlidl 35307 | A prime ideal is an ideal.... |
| pridlnr 35308 | A prime ideal is a proper ... |
| pridl 35309 | The main property of a pri... |
| ispridl2 35310 | A condition that shows an ... |
| maxidlval 35311 | The set of maximal ideals ... |
| ismaxidl 35312 | The predicate "is a maxima... |
| maxidlidl 35313 | A maximal ideal is an idea... |
| maxidlnr 35314 | A maximal ideal is proper.... |
| maxidlmax 35315 | A maximal ideal is a maxim... |
| maxidln1 35316 | One is not contained in an... |
| maxidln0 35317 | A ring with a maximal idea... |
| isprrngo 35322 | The predicate "is a prime ... |
| prrngorngo 35323 | A prime ring is a ring. (... |
| smprngopr 35324 | A simple ring (one whose o... |
| divrngpr 35325 | A division ring is a prime... |
| isdmn 35326 | The predicate "is a domain... |
| isdmn2 35327 | The predicate "is a domain... |
| dmncrng 35328 | A domain is a commutative ... |
| dmnrngo 35329 | A domain is a ring. (Cont... |
| flddmn 35330 | A field is a domain. (Con... |
| igenval 35333 | The ideal generated by a s... |
| igenss 35334 | A set is a subset of the i... |
| igenidl 35335 | The ideal generated by a s... |
| igenmin 35336 | The ideal generated by a s... |
| igenidl2 35337 | The ideal generated by an ... |
| igenval2 35338 | The ideal generated by a s... |
| prnc 35339 | A principal ideal (an idea... |
| isfldidl 35340 | Determine if a ring is a f... |
| isfldidl2 35341 | Determine if a ring is a f... |
| ispridlc 35342 | The predicate "is a prime ... |
| pridlc 35343 | Property of a prime ideal ... |
| pridlc2 35344 | Property of a prime ideal ... |
| pridlc3 35345 | Property of a prime ideal ... |
| isdmn3 35346 | The predicate "is a domain... |
| dmnnzd 35347 | A domain has no zero-divis... |
| dmncan1 35348 | Cancellation law for domai... |
| dmncan2 35349 | Cancellation law for domai... |
| efald2 35350 | A proof by contradiction. ... |
| notbinot1 35351 | Simplification rule of neg... |
| bicontr 35352 | Biimplication of its own n... |
| impor 35353 | An equivalent formula for ... |
| orfa 35354 | The falsum ` F. ` can be r... |
| notbinot2 35355 | Commutation rule between n... |
| biimpor 35356 | A rewriting rule for biimp... |
| orfa1 35357 | Add a contradicting disjun... |
| orfa2 35358 | Remove a contradicting dis... |
| bifald 35359 | Infer the equivalence to a... |
| orsild 35360 | A lemma for not-or-not eli... |
| orsird 35361 | A lemma for not-or-not eli... |
| cnf1dd 35362 | A lemma for Conjunctive No... |
| cnf2dd 35363 | A lemma for Conjunctive No... |
| cnfn1dd 35364 | A lemma for Conjunctive No... |
| cnfn2dd 35365 | A lemma for Conjunctive No... |
| or32dd 35366 | A rearrangement of disjunc... |
| notornotel1 35367 | A lemma for not-or-not eli... |
| notornotel2 35368 | A lemma for not-or-not eli... |
| contrd 35369 | A proof by contradiction, ... |
| an12i 35370 | An inference from commutin... |
| exmid2 35371 | An excluded middle law. (... |
| selconj 35372 | An inference for selecting... |
| truconj 35373 | Add true as a conjunct. (... |
| orel 35374 | An inference for disjuncti... |
| negel 35375 | An inference for negation ... |
| botel 35376 | An inference for bottom el... |
| tradd 35377 | Add top ad a conjunct. (C... |
| gm-sbtru 35378 | Substitution does not chan... |
| sbfal 35379 | Substitution does not chan... |
| sbcani 35380 | Distribution of class subs... |
| sbcori 35381 | Distribution of class subs... |
| sbcimi 35382 | Distribution of class subs... |
| sbcni 35383 | Move class substitution in... |
| sbali 35384 | Discard class substitution... |
| sbexi 35385 | Discard class substitution... |
| sbcalf 35386 | Move universal quantifier ... |
| sbcexf 35387 | Move existential quantifie... |
| sbcalfi 35388 | Move universal quantifier ... |
| sbcexfi 35389 | Move existential quantifie... |
| spsbcdi 35390 | A lemma for eliminating a ... |
| alrimii 35391 | A lemma for introducing a ... |
| spesbcdi 35392 | A lemma for introducing an... |
| exlimddvf 35393 | A lemma for eliminating an... |
| exlimddvfi 35394 | A lemma for eliminating an... |
| sbceq1ddi 35395 | A lemma for eliminating in... |
| sbccom2lem 35396 | Lemma for ~ sbccom2 . (Co... |
| sbccom2 35397 | Commutative law for double... |
| sbccom2f 35398 | Commutative law for double... |
| sbccom2fi 35399 | Commutative law for double... |
| csbcom2fi 35400 | Commutative law for double... |
| fald 35401 | Refutation of falsity, in ... |
| tsim1 35402 | A Tseitin axiom for logica... |
| tsim2 35403 | A Tseitin axiom for logica... |
| tsim3 35404 | A Tseitin axiom for logica... |
| tsbi1 35405 | A Tseitin axiom for logica... |
| tsbi2 35406 | A Tseitin axiom for logica... |
| tsbi3 35407 | A Tseitin axiom for logica... |
| tsbi4 35408 | A Tseitin axiom for logica... |
| tsxo1 35409 | A Tseitin axiom for logica... |
| tsxo2 35410 | A Tseitin axiom for logica... |
| tsxo3 35411 | A Tseitin axiom for logica... |
| tsxo4 35412 | A Tseitin axiom for logica... |
| tsan1 35413 | A Tseitin axiom for logica... |
| tsan2 35414 | A Tseitin axiom for logica... |
| tsan3 35415 | A Tseitin axiom for logica... |
| tsna1 35416 | A Tseitin axiom for logica... |
| tsna2 35417 | A Tseitin axiom for logica... |
| tsna3 35418 | A Tseitin axiom for logica... |
| tsor1 35419 | A Tseitin axiom for logica... |
| tsor2 35420 | A Tseitin axiom for logica... |
| tsor3 35421 | A Tseitin axiom for logica... |
| ts3an1 35422 | A Tseitin axiom for triple... |
| ts3an2 35423 | A Tseitin axiom for triple... |
| ts3an3 35424 | A Tseitin axiom for triple... |
| ts3or1 35425 | A Tseitin axiom for triple... |
| ts3or2 35426 | A Tseitin axiom for triple... |
| ts3or3 35427 | A Tseitin axiom for triple... |
| iuneq2f 35428 | Equality deduction for ind... |
| rabeq12f 35429 | Equality deduction for res... |
| csbeq12 35430 | Equality deduction for sub... |
| sbeqi 35431 | Equality deduction for sub... |
| ralbi12f 35432 | Equality deduction for res... |
| oprabbi 35433 | Equality deduction for cla... |
| mpobi123f 35434 | Equality deduction for map... |
| iuneq12f 35435 | Equality deduction for ind... |
| iineq12f 35436 | Equality deduction for ind... |
| opabbi 35437 | Equality deduction for cla... |
| mptbi12f 35438 | Equality deduction for map... |
| orcomdd 35439 | Commutativity of logic dis... |
| scottexf 35440 | A version of ~ scottex wit... |
| scott0f 35441 | A version of ~ scott0 with... |
| scottn0f 35442 | A version of ~ scott0f wit... |
| ac6s3f 35443 | Generalization of the Axio... |
| ac6s6 35444 | Generalization of the Axio... |
| ac6s6f 35445 | Generalization of the Axio... |
| el2v1 35484 | New way ( ~ elv , and the ... |
| el3v 35485 | New way ( ~ elv , and the ... |
| el3v1 35486 | New way ( ~ elv , and the ... |
| el3v2 35487 | New way ( ~ elv , and the ... |
| el3v3 35488 | New way ( ~ elv , and the ... |
| el3v12 35489 | New way ( ~ elv , and the ... |
| el3v13 35490 | New way ( ~ elv , and the ... |
| el3v23 35491 | New way ( ~ elv , and the ... |
| an2anr 35492 | Double commutation in conj... |
| anan 35493 | Multiple commutations in c... |
| triantru3 35494 | A wff is equivalent to its... |
| eqeltr 35495 | Substitution of equal clas... |
| eqelb 35496 | Substitution of equal clas... |
| eqeqan2d 35497 | Implication of introducing... |
| ineqcom 35498 | Two ways of saying that tw... |
| ineqcomi 35499 | Disjointness inference (wh... |
| inres2 35500 | Two ways of expressing the... |
| coideq 35501 | Equality theorem for compo... |
| nexmo1 35502 | If there is no case where ... |
| 3albii 35503 | Inference adding three uni... |
| 3ralbii 35504 | Inference adding three res... |
| ssrabi 35505 | Inference of restricted ab... |
| rabbieq 35506 | Equivalent wff's correspon... |
| rabimbieq 35507 | Restricted equivalent wff'... |
| abeqin 35508 | Intersection with class ab... |
| abeqinbi 35509 | Intersection with class ab... |
| rabeqel 35510 | Class element of a restric... |
| eqrelf 35511 | The equality connective be... |
| releleccnv 35512 | Elementhood in a converse ... |
| releccnveq 35513 | Equality of converse ` R `... |
| opelvvdif 35514 | Negated elementhood of ord... |
| vvdifopab 35515 | Ordered-pair class abstrac... |
| brvdif 35516 | Binary relation with unive... |
| brvdif2 35517 | Binary relation with unive... |
| brvvdif 35518 | Binary relation with the c... |
| brvbrvvdif 35519 | Binary relation with the c... |
| brcnvep 35520 | The converse of the binary... |
| elecALTV 35521 | Elementhood in the ` R ` -... |
| brcnvepres 35522 | Restricted converse epsilo... |
| brres2 35523 | Binary relation on a restr... |
| eldmres 35524 | Elementhood in the domain ... |
| eldm4 35525 | Elementhood in a domain. ... |
| eldmres2 35526 | Elementhood in the domain ... |
| eceq1i 35527 | Equality theorem for ` C `... |
| elecres 35528 | Elementhood in the restric... |
| ecres 35529 | Restricted coset of ` B ` ... |
| ecres2 35530 | The restricted coset of ` ... |
| eccnvepres 35531 | Restricted converse epsilo... |
| eleccnvep 35532 | Elementhood in the convers... |
| eccnvep 35533 | The converse epsilon coset... |
| extep 35534 | Property of epsilon relati... |
| eccnvepres2 35535 | The restricted converse ep... |
| eccnvepres3 35536 | Condition for a restricted... |
| eldmqsres 35537 | Elementhood in a restricte... |
| eldmqsres2 35538 | Elementhood in a restricte... |
| qsss1 35539 | Subclass theorem for quoti... |
| qseq1i 35540 | Equality theorem for quoti... |
| qseq1d 35541 | Equality theorem for quoti... |
| brinxprnres 35542 | Binary relation on a restr... |
| inxprnres 35543 | Restriction of a class as ... |
| dfres4 35544 | Alternate definition of th... |
| exan3 35545 | Equivalent expressions wit... |
| exanres 35546 | Equivalent expressions wit... |
| exanres3 35547 | Equivalent expressions wit... |
| exanres2 35548 | Equivalent expressions wit... |
| cnvepres 35549 | Restricted converse epsilo... |
| ssrel3 35550 | Subclass relation in anoth... |
| eqrel2 35551 | Equality of relations. (C... |
| rncnv 35552 | Range of converse is the d... |
| dfdm6 35553 | Alternate definition of do... |
| dfrn6 35554 | Alternate definition of ra... |
| rncnvepres 35555 | The range of the restricte... |
| dmecd 35556 | Equality of the coset of `... |
| dmec2d 35557 | Equality of the coset of `... |
| brid 35558 | Property of the identity b... |
| ideq2 35559 | For sets, the identity bin... |
| idresssidinxp 35560 | Condition for the identity... |
| idreseqidinxp 35561 | Condition for the identity... |
| extid 35562 | Property of identity relat... |
| inxpss 35563 | Two ways to say that an in... |
| idinxpss 35564 | Two ways to say that an in... |
| inxpss3 35565 | Two ways to say that an in... |
| inxpss2 35566 | Two ways to say that inter... |
| inxpssidinxp 35567 | Two ways to say that inter... |
| idinxpssinxp 35568 | Two ways to say that inter... |
| idinxpssinxp2 35569 | Identity intersection with... |
| idinxpssinxp3 35570 | Identity intersection with... |
| idinxpssinxp4 35571 | Identity intersection with... |
| relcnveq3 35572 | Two ways of saying a relat... |
| relcnveq 35573 | Two ways of saying a relat... |
| relcnveq2 35574 | Two ways of saying a relat... |
| relcnveq4 35575 | Two ways of saying a relat... |
| qsresid 35576 | Simplification of a specia... |
| n0elqs 35577 | Two ways of expressing tha... |
| n0elqs2 35578 | Two ways of expressing tha... |
| ecex2 35579 | Condition for a coset to b... |
| uniqsALTV 35580 | The union of a quotient se... |
| imaexALTV 35581 | Existence of an image of a... |
| ecexALTV 35582 | Existence of a coset, like... |
| rnresequniqs 35583 | The range of a restriction... |
| n0el2 35584 | Two ways of expressing tha... |
| cnvepresex 35585 | Sethood condition for the ... |
| eccnvepex 35586 | The converse epsilon coset... |
| cnvepimaex 35587 | The image of converse epsi... |
| cnvepima 35588 | The image of converse epsi... |
| inex3 35589 | Sufficient condition for t... |
| inxpex 35590 | Sufficient condition for a... |
| eqres 35591 | Converting a class constan... |
| brrabga 35592 | The law of concretion for ... |
| brcnvrabga 35593 | The law of concretion for ... |
| opideq 35594 | Equality conditions for or... |
| iss2 35595 | A subclass of the identity... |
| eldmcnv 35596 | Elementhood in a domain of... |
| dfrel5 35597 | Alternate definition of th... |
| dfrel6 35598 | Alternate definition of th... |
| cnvresrn 35599 | Converse restricted to ran... |
| ecin0 35600 | Two ways of saying that th... |
| ecinn0 35601 | Two ways of saying that th... |
| ineleq 35602 | Equivalence of restricted ... |
| inecmo 35603 | Equivalence of a double re... |
| inecmo2 35604 | Equivalence of a double re... |
| ineccnvmo 35605 | Equivalence of a double re... |
| alrmomorn 35606 | Equivalence of an "at most... |
| alrmomodm 35607 | Equivalence of an "at most... |
| ineccnvmo2 35608 | Equivalence of a double un... |
| inecmo3 35609 | Equivalence of a double un... |
| moantr 35610 | Sufficient condition for t... |
| brabidgaw 35611 | The law of concretion for ... |
| brabidga 35612 | The law of concretion for ... |
| inxp2 35613 | Intersection with a Cartes... |
| opabf 35614 | A class abstraction of a c... |
| ec0 35615 | The empty-coset of a class... |
| 0qs 35616 | Quotient set with the empt... |
| xrnss3v 35618 | A range Cartesian product ... |
| xrnrel 35619 | A range Cartesian product ... |
| brxrn 35620 | Characterize a ternary rel... |
| brxrn2 35621 | A characterization of the ... |
| dfxrn2 35622 | Alternate definition of th... |
| xrneq1 35623 | Equality theorem for the r... |
| xrneq1i 35624 | Equality theorem for the r... |
| xrneq1d 35625 | Equality theorem for the r... |
| xrneq2 35626 | Equality theorem for the r... |
| xrneq2i 35627 | Equality theorem for the r... |
| xrneq2d 35628 | Equality theorem for the r... |
| xrneq12 35629 | Equality theorem for the r... |
| xrneq12i 35630 | Equality theorem for the r... |
| xrneq12d 35631 | Equality theorem for the r... |
| elecxrn 35632 | Elementhood in the ` ( R |... |
| ecxrn 35633 | The ` ( R |X. S ) ` -coset... |
| xrninxp 35634 | Intersection of a range Ca... |
| xrninxp2 35635 | Intersection of a range Ca... |
| xrninxpex 35636 | Sufficient condition for t... |
| inxpxrn 35637 | Two ways to express the in... |
| br1cnvxrn2 35638 | The converse of a binary r... |
| elec1cnvxrn2 35639 | Elementhood in the convers... |
| rnxrn 35640 | Range of the range Cartesi... |
| rnxrnres 35641 | Range of a range Cartesian... |
| rnxrncnvepres 35642 | Range of a range Cartesian... |
| rnxrnidres 35643 | Range of a range Cartesian... |
| xrnres 35644 | Two ways to express restri... |
| xrnres2 35645 | Two ways to express restri... |
| xrnres3 35646 | Two ways to express restri... |
| xrnres4 35647 | Two ways to express restri... |
| xrnresex 35648 | Sufficient condition for a... |
| xrnidresex 35649 | Sufficient condition for a... |
| xrncnvepresex 35650 | Sufficient condition for a... |
| brin2 35651 | Binary relation on an inte... |
| brin3 35652 | Binary relation on an inte... |
| dfcoss2 35655 | Alternate definition of th... |
| dfcoss3 35656 | Alternate definition of th... |
| dfcoss4 35657 | Alternate definition of th... |
| cossex 35658 | If ` A ` is a set then the... |
| cosscnvex 35659 | If ` A ` is a set then the... |
| 1cosscnvepresex 35660 | Sufficient condition for a... |
| 1cossxrncnvepresex 35661 | Sufficient condition for a... |
| relcoss 35662 | Cosets by ` R ` is a relat... |
| relcoels 35663 | Coelements on ` A ` is a r... |
| cossss 35664 | Subclass theorem for the c... |
| cosseq 35665 | Equality theorem for the c... |
| cosseqi 35666 | Equality theorem for the c... |
| cosseqd 35667 | Equality theorem for the c... |
| 1cossres 35668 | The class of cosets by a r... |
| dfcoels 35669 | Alternate definition of th... |
| brcoss 35670 | ` A ` and ` B ` are cosets... |
| brcoss2 35671 | Alternate form of the ` A ... |
| brcoss3 35672 | Alternate form of the ` A ... |
| brcosscnvcoss 35673 | For sets, the ` A ` and ` ... |
| brcoels 35674 | ` B ` and ` C ` are coelem... |
| cocossss 35675 | Two ways of saying that co... |
| cnvcosseq 35676 | The converse of cosets by ... |
| br2coss 35677 | Cosets by ` ,~ R ` binary ... |
| br1cossres 35678 | ` B ` and ` C ` are cosets... |
| br1cossres2 35679 | ` B ` and ` C ` are cosets... |
| relbrcoss 35680 | ` A ` and ` B ` are cosets... |
| br1cossinres 35681 | ` B ` and ` C ` are cosets... |
| br1cossxrnres 35682 | ` <. B , C >. ` and ` <. D... |
| br1cossinidres 35683 | ` B ` and ` C ` are cosets... |
| br1cossincnvepres 35684 | ` B ` and ` C ` are cosets... |
| br1cossxrnidres 35685 | ` <. B , C >. ` and ` <. D... |
| br1cossxrncnvepres 35686 | ` <. B , C >. ` and ` <. D... |
| dmcoss3 35687 | The domain of cosets is th... |
| dmcoss2 35688 | The domain of cosets is th... |
| rncossdmcoss 35689 | The range of cosets is the... |
| dm1cosscnvepres 35690 | The domain of cosets of th... |
| dmcoels 35691 | The domain of coelements i... |
| eldmcoss 35692 | Elementhood in the domain ... |
| eldmcoss2 35693 | Elementhood in the domain ... |
| eldm1cossres 35694 | Elementhood in the domain ... |
| eldm1cossres2 35695 | Elementhood in the domain ... |
| refrelcosslem 35696 | Lemma for the left side of... |
| refrelcoss3 35697 | The class of cosets by ` R... |
| refrelcoss2 35698 | The class of cosets by ` R... |
| symrelcoss3 35699 | The class of cosets by ` R... |
| symrelcoss2 35700 | The class of cosets by ` R... |
| cossssid 35701 | Equivalent expressions for... |
| cossssid2 35702 | Equivalent expressions for... |
| cossssid3 35703 | Equivalent expressions for... |
| cossssid4 35704 | Equivalent expressions for... |
| cossssid5 35705 | Equivalent expressions for... |
| brcosscnv 35706 | ` A ` and ` B ` are cosets... |
| brcosscnv2 35707 | ` A ` and ` B ` are cosets... |
| br1cosscnvxrn 35708 | ` A ` and ` B ` are cosets... |
| 1cosscnvxrn 35709 | Cosets by the converse ran... |
| cosscnvssid3 35710 | Equivalent expressions for... |
| cosscnvssid4 35711 | Equivalent expressions for... |
| cosscnvssid5 35712 | Equivalent expressions for... |
| coss0 35713 | Cosets by the empty set ar... |
| cossid 35714 | Cosets by the identity rel... |
| cosscnvid 35715 | Cosets by the converse ide... |
| trcoss 35716 | Sufficient condition for t... |
| eleccossin 35717 | Two ways of saying that th... |
| trcoss2 35718 | Equivalent expressions for... |
| elrels2 35720 | The element of the relatio... |
| elrelsrel 35721 | The element of the relatio... |
| elrelsrelim 35722 | The element of the relatio... |
| elrels5 35723 | Equivalent expressions for... |
| elrels6 35724 | Equivalent expressions for... |
| elrelscnveq3 35725 | Two ways of saying a relat... |
| elrelscnveq 35726 | Two ways of saying a relat... |
| elrelscnveq2 35727 | Two ways of saying a relat... |
| elrelscnveq4 35728 | Two ways of saying a relat... |
| cnvelrels 35729 | The converse of a set is a... |
| cosselrels 35730 | Cosets of sets are element... |
| cosscnvelrels 35731 | Cosets of converse sets ar... |
| dfssr2 35733 | Alternate definition of th... |
| relssr 35734 | The subset relation is a r... |
| brssr 35735 | The subset relation and su... |
| brssrid 35736 | Any set is a subset of its... |
| issetssr 35737 | Two ways of expressing set... |
| brssrres 35738 | Restricted subset binary r... |
| br1cnvssrres 35739 | Restricted converse subset... |
| brcnvssr 35740 | The converse of a subset r... |
| brcnvssrid 35741 | Any set is a converse subs... |
| br1cossxrncnvssrres 35742 | ` <. B , C >. ` and ` <. D... |
| extssr 35743 | Property of subset relatio... |
| dfrefrels2 35747 | Alternate definition of th... |
| dfrefrels3 35748 | Alternate definition of th... |
| dfrefrel2 35749 | Alternate definition of th... |
| dfrefrel3 35750 | Alternate definition of th... |
| elrefrels2 35751 | Element of the class of re... |
| elrefrels3 35752 | Element of the class of re... |
| elrefrelsrel 35753 | For sets, being an element... |
| refreleq 35754 | Equality theorem for refle... |
| refrelid 35755 | Identity relation is refle... |
| refrelcoss 35756 | The class of cosets by ` R... |
| dfcnvrefrels2 35760 | Alternate definition of th... |
| dfcnvrefrels3 35761 | Alternate definition of th... |
| dfcnvrefrel2 35762 | Alternate definition of th... |
| dfcnvrefrel3 35763 | Alternate definition of th... |
| elcnvrefrels2 35764 | Element of the class of co... |
| elcnvrefrels3 35765 | Element of the class of co... |
| elcnvrefrelsrel 35766 | For sets, being an element... |
| cnvrefrelcoss2 35767 | Necessary and sufficient c... |
| cosselcnvrefrels2 35768 | Necessary and sufficient c... |
| cosselcnvrefrels3 35769 | Necessary and sufficient c... |
| cosselcnvrefrels4 35770 | Necessary and sufficient c... |
| cosselcnvrefrels5 35771 | Necessary and sufficient c... |
| dfsymrels2 35775 | Alternate definition of th... |
| dfsymrels3 35776 | Alternate definition of th... |
| dfsymrels4 35777 | Alternate definition of th... |
| dfsymrels5 35778 | Alternate definition of th... |
| dfsymrel2 35779 | Alternate definition of th... |
| dfsymrel3 35780 | Alternate definition of th... |
| dfsymrel4 35781 | Alternate definition of th... |
| dfsymrel5 35782 | Alternate definition of th... |
| elsymrels2 35783 | Element of the class of sy... |
| elsymrels3 35784 | Element of the class of sy... |
| elsymrels4 35785 | Element of the class of sy... |
| elsymrels5 35786 | Element of the class of sy... |
| elsymrelsrel 35787 | For sets, being an element... |
| symreleq 35788 | Equality theorem for symme... |
| symrelim 35789 | Symmetric relation implies... |
| symrelcoss 35790 | The class of cosets by ` R... |
| idsymrel 35791 | The identity relation is s... |
| epnsymrel 35792 | The membership (epsilon) r... |
| symrefref2 35793 | Symmetry is a sufficient c... |
| symrefref3 35794 | Symmetry is a sufficient c... |
| refsymrels2 35795 | Elements of the class of r... |
| refsymrels3 35796 | Elements of the class of r... |
| refsymrel2 35797 | A relation which is reflex... |
| refsymrel3 35798 | A relation which is reflex... |
| elrefsymrels2 35799 | Elements of the class of r... |
| elrefsymrels3 35800 | Elements of the class of r... |
| elrefsymrelsrel 35801 | For sets, being an element... |
| dftrrels2 35805 | Alternate definition of th... |
| dftrrels3 35806 | Alternate definition of th... |
| dftrrel2 35807 | Alternate definition of th... |
| dftrrel3 35808 | Alternate definition of th... |
| eltrrels2 35809 | Element of the class of tr... |
| eltrrels3 35810 | Element of the class of tr... |
| eltrrelsrel 35811 | For sets, being an element... |
| trreleq 35812 | Equality theorem for the t... |
| dfeqvrels2 35817 | Alternate definition of th... |
| dfeqvrels3 35818 | Alternate definition of th... |
| dfeqvrel2 35819 | Alternate definition of th... |
| dfeqvrel3 35820 | Alternate definition of th... |
| eleqvrels2 35821 | Element of the class of eq... |
| eleqvrels3 35822 | Element of the class of eq... |
| eleqvrelsrel 35823 | For sets, being an element... |
| elcoeleqvrels 35824 | Elementhood in the coeleme... |
| elcoeleqvrelsrel 35825 | For sets, being an element... |
| eqvrelrel 35826 | An equivalence relation is... |
| eqvrelrefrel 35827 | An equivalence relation is... |
| eqvrelsymrel 35828 | An equivalence relation is... |
| eqvreltrrel 35829 | An equivalence relation is... |
| eqvrelim 35830 | Equivalence relation impli... |
| eqvreleq 35831 | Equality theorem for equiv... |
| eqvreleqi 35832 | Equality theorem for equiv... |
| eqvreleqd 35833 | Equality theorem for equiv... |
| eqvrelsym 35834 | An equivalence relation is... |
| eqvrelsymb 35835 | An equivalence relation is... |
| eqvreltr 35836 | An equivalence relation is... |
| eqvreltrd 35837 | A transitivity relation fo... |
| eqvreltr4d 35838 | A transitivity relation fo... |
| eqvrelref 35839 | An equivalence relation is... |
| eqvrelth 35840 | Basic property of equivale... |
| eqvrelcl 35841 | Elementhood in the field o... |
| eqvrelthi 35842 | Basic property of equivale... |
| eqvreldisj 35843 | Equivalence classes do not... |
| qsdisjALTV 35844 | Elements of a quotient set... |
| eqvrelqsel 35845 | If an element of a quotien... |
| eqvrelcoss 35846 | Two ways to express equiva... |
| eqvrelcoss3 35847 | Two ways to express equiva... |
| eqvrelcoss2 35848 | Two ways to express equiva... |
| eqvrelcoss4 35849 | Two ways to express equiva... |
| dfcoeleqvrels 35850 | Alternate definition of th... |
| dfcoeleqvrel 35851 | Alternate definition of th... |
| brredunds 35855 | Binary relation on the cla... |
| brredundsredund 35856 | For sets, binary relation ... |
| redundss3 35857 | Implication of redundancy ... |
| redundeq1 35858 | Equivalence of redundancy ... |
| redundpim3 35859 | Implication of redundancy ... |
| redundpbi1 35860 | Equivalence of redundancy ... |
| refrelsredund4 35861 | The naive version of the c... |
| refrelsredund2 35862 | The naive version of the c... |
| refrelsredund3 35863 | The naive version of the c... |
| refrelredund4 35864 | The naive version of the d... |
| refrelredund2 35865 | The naive version of the d... |
| refrelredund3 35866 | The naive version of the d... |
| dmqseq 35869 | Equality theorem for domai... |
| dmqseqi 35870 | Equality theorem for domai... |
| dmqseqd 35871 | Equality theorem for domai... |
| dmqseqeq1 35872 | Equality theorem for domai... |
| dmqseqeq1i 35873 | Equality theorem for domai... |
| dmqseqeq1d 35874 | Equality theorem for domai... |
| brdmqss 35875 | The domain quotient binary... |
| brdmqssqs 35876 | If ` A ` and ` R ` are set... |
| n0eldmqs 35877 | The empty set is not an el... |
| n0eldmqseq 35878 | The empty set is not an el... |
| n0el3 35879 | Two ways of expressing tha... |
| cnvepresdmqss 35880 | The domain quotient binary... |
| cnvepresdmqs 35881 | The domain quotient predic... |
| unidmqs 35882 | The range of a relation is... |
| unidmqseq 35883 | The union of the domain qu... |
| dmqseqim 35884 | If the domain quotient of ... |
| dmqseqim2 35885 | Lemma for ~ erim2 . (Cont... |
| releldmqs 35886 | Elementhood in the domain ... |
| eldmqs1cossres 35887 | Elementhood in the domain ... |
| releldmqscoss 35888 | Elementhood in the domain ... |
| dmqscoelseq 35889 | Two ways to express the eq... |
| dmqs1cosscnvepreseq 35890 | Two ways to express the eq... |
| brers 35895 | Binary equivalence relatio... |
| dferALTV2 35896 | Equivalence relation with ... |
| erALTVeq1 35897 | Equality theorem for equiv... |
| erALTVeq1i 35898 | Equality theorem for equiv... |
| erALTVeq1d 35899 | Equality theorem for equiv... |
| dfmember 35900 | Alternate definition of th... |
| dfmember2 35901 | Alternate definition of th... |
| dfmember3 35902 | Alternate definition of th... |
| eqvreldmqs 35903 | Two ways to express member... |
| brerser 35904 | Binary equivalence relatio... |
| erim2 35905 | Equivalence relation on it... |
| erim 35906 | Equivalence relation on it... |
| dffunsALTV 35910 | Alternate definition of th... |
| dffunsALTV2 35911 | Alternate definition of th... |
| dffunsALTV3 35912 | Alternate definition of th... |
| dffunsALTV4 35913 | Alternate definition of th... |
| dffunsALTV5 35914 | Alternate definition of th... |
| dffunALTV2 35915 | Alternate definition of th... |
| dffunALTV3 35916 | Alternate definition of th... |
| dffunALTV4 35917 | Alternate definition of th... |
| dffunALTV5 35918 | Alternate definition of th... |
| elfunsALTV 35919 | Elementhood in the class o... |
| elfunsALTV2 35920 | Elementhood in the class o... |
| elfunsALTV3 35921 | Elementhood in the class o... |
| elfunsALTV4 35922 | Elementhood in the class o... |
| elfunsALTV5 35923 | Elementhood in the class o... |
| elfunsALTVfunALTV 35924 | The element of the class o... |
| funALTVfun 35925 | Our definition of the func... |
| funALTVss 35926 | Subclass theorem for funct... |
| funALTVeq 35927 | Equality theorem for funct... |
| funALTVeqi 35928 | Equality inference for the... |
| funALTVeqd 35929 | Equality deduction for the... |
| dfdisjs 35935 | Alternate definition of th... |
| dfdisjs2 35936 | Alternate definition of th... |
| dfdisjs3 35937 | Alternate definition of th... |
| dfdisjs4 35938 | Alternate definition of th... |
| dfdisjs5 35939 | Alternate definition of th... |
| dfdisjALTV 35940 | Alternate definition of th... |
| dfdisjALTV2 35941 | Alternate definition of th... |
| dfdisjALTV3 35942 | Alternate definition of th... |
| dfdisjALTV4 35943 | Alternate definition of th... |
| dfdisjALTV5 35944 | Alternate definition of th... |
| dfeldisj2 35945 | Alternate definition of th... |
| dfeldisj3 35946 | Alternate definition of th... |
| dfeldisj4 35947 | Alternate definition of th... |
| dfeldisj5 35948 | Alternate definition of th... |
| eldisjs 35949 | Elementhood in the class o... |
| eldisjs2 35950 | Elementhood in the class o... |
| eldisjs3 35951 | Elementhood in the class o... |
| eldisjs4 35952 | Elementhood in the class o... |
| eldisjs5 35953 | Elementhood in the class o... |
| eldisjsdisj 35954 | The element of the class o... |
| eleldisjs 35955 | Elementhood in the disjoin... |
| eleldisjseldisj 35956 | The element of the disjoin... |
| disjrel 35957 | Disjoint relation is a rel... |
| disjss 35958 | Subclass theorem for disjo... |
| disjssi 35959 | Subclass theorem for disjo... |
| disjssd 35960 | Subclass theorem for disjo... |
| disjeq 35961 | Equality theorem for disjo... |
| disjeqi 35962 | Equality theorem for disjo... |
| disjeqd 35963 | Equality theorem for disjo... |
| disjdmqseqeq1 35964 | Lemma for the equality the... |
| eldisjss 35965 | Subclass theorem for disjo... |
| eldisjssi 35966 | Subclass theorem for disjo... |
| eldisjssd 35967 | Subclass theorem for disjo... |
| eldisjeq 35968 | Equality theorem for disjo... |
| eldisjeqi 35969 | Equality theorem for disjo... |
| eldisjeqd 35970 | Equality theorem for disjo... |
| disjxrn 35971 | Two ways of saying that a ... |
| disjorimxrn 35972 | Disjointness condition for... |
| disjimxrn 35973 | Disjointness condition for... |
| disjimres 35974 | Disjointness condition for... |
| disjimin 35975 | Disjointness condition for... |
| disjiminres 35976 | Disjointness condition for... |
| disjimxrnres 35977 | Disjointness condition for... |
| disjALTV0 35978 | The null class is disjoint... |
| disjALTVid 35979 | The class of identity rela... |
| disjALTVidres 35980 | The class of identity rela... |
| disjALTVinidres 35981 | The intersection with rest... |
| disjALTVxrnidres 35982 | The class of range Cartesi... |
| prtlem60 35983 | Lemma for ~ prter3 . (Con... |
| bicomdd 35984 | Commute two sides of a bic... |
| jca2r 35985 | Inference conjoining the c... |
| jca3 35986 | Inference conjoining the c... |
| prtlem70 35987 | Lemma for ~ prter3 : a rea... |
| ibdr 35988 | Reverse of ~ ibd . (Contr... |
| prtlem100 35989 | Lemma for ~ prter3 . (Con... |
| prtlem5 35990 | Lemma for ~ prter1 , ~ prt... |
| prtlem80 35991 | Lemma for ~ prter2 . (Con... |
| brabsb2 35992 | A closed form of ~ brabsb ... |
| eqbrrdv2 35993 | Other version of ~ eqbrrdi... |
| prtlem9 35994 | Lemma for ~ prter3 . (Con... |
| prtlem10 35995 | Lemma for ~ prter3 . (Con... |
| prtlem11 35996 | Lemma for ~ prter2 . (Con... |
| prtlem12 35997 | Lemma for ~ prtex and ~ pr... |
| prtlem13 35998 | Lemma for ~ prter1 , ~ prt... |
| prtlem16 35999 | Lemma for ~ prtex , ~ prte... |
| prtlem400 36000 | Lemma for ~ prter2 and als... |
| erprt 36003 | The quotient set of an equ... |
| prtlem14 36004 | Lemma for ~ prter1 , ~ prt... |
| prtlem15 36005 | Lemma for ~ prter1 and ~ p... |
| prtlem17 36006 | Lemma for ~ prter2 . (Con... |
| prtlem18 36007 | Lemma for ~ prter2 . (Con... |
| prtlem19 36008 | Lemma for ~ prter2 . (Con... |
| prter1 36009 | Every partition generates ... |
| prtex 36010 | The equivalence relation g... |
| prter2 36011 | The quotient set of the eq... |
| prter3 36012 | For every partition there ... |
| axc5 36023 | This theorem repeats ~ sp ... |
| ax4fromc4 36024 | Rederivation of axiom ~ ax... |
| ax10fromc7 36025 | Rederivation of axiom ~ ax... |
| ax6fromc10 36026 | Rederivation of axiom ~ ax... |
| hba1-o 36027 | The setvar ` x ` is not fr... |
| axc4i-o 36028 | Inference version of ~ ax-... |
| equid1 36029 | Proof of ~ equid from our ... |
| equcomi1 36030 | Proof of ~ equcomi from ~ ... |
| aecom-o 36031 | Commutation law for identi... |
| aecoms-o 36032 | A commutation rule for ide... |
| hbae-o 36033 | All variables are effectiv... |
| dral1-o 36034 | Formula-building lemma for... |
| ax12fromc15 36035 | Rederivation of axiom ~ ax... |
| ax13fromc9 36036 | Derive ~ ax-13 from ~ ax-c... |
| ax5ALT 36037 | Axiom to quantify a variab... |
| sps-o 36038 | Generalization of antecede... |
| hbequid 36039 | Bound-variable hypothesis ... |
| nfequid-o 36040 | Bound-variable hypothesis ... |
| axc5c7 36041 | Proof of a single axiom th... |
| axc5c7toc5 36042 | Rederivation of ~ ax-c5 fr... |
| axc5c7toc7 36043 | Rederivation of ~ ax-c7 fr... |
| axc711 36044 | Proof of a single axiom th... |
| nfa1-o 36045 | ` x ` is not free in ` A. ... |
| axc711toc7 36046 | Rederivation of ~ ax-c7 fr... |
| axc711to11 36047 | Rederivation of ~ ax-11 fr... |
| axc5c711 36048 | Proof of a single axiom th... |
| axc5c711toc5 36049 | Rederivation of ~ ax-c5 fr... |
| axc5c711toc7 36050 | Rederivation of ~ ax-c7 fr... |
| axc5c711to11 36051 | Rederivation of ~ ax-11 fr... |
| equidqe 36052 | ~ equid with existential q... |
| axc5sp1 36053 | A special case of ~ ax-c5 ... |
| equidq 36054 | ~ equid with universal qua... |
| equid1ALT 36055 | Alternate proof of ~ equid... |
| axc11nfromc11 36056 | Rederivation of ~ ax-c11n ... |
| naecoms-o 36057 | A commutation rule for dis... |
| hbnae-o 36058 | All variables are effectiv... |
| dvelimf-o 36059 | Proof of ~ dvelimh that us... |
| dral2-o 36060 | Formula-building lemma for... |
| aev-o 36061 | A "distinctor elimination"... |
| ax5eq 36062 | Theorem to add distinct qu... |
| dveeq2-o 36063 | Quantifier introduction wh... |
| axc16g-o 36064 | A generalization of axiom ... |
| dveeq1-o 36065 | Quantifier introduction wh... |
| dveeq1-o16 36066 | Version of ~ dveeq1 using ... |
| ax5el 36067 | Theorem to add distinct qu... |
| axc11n-16 36068 | This theorem shows that, g... |
| dveel2ALT 36069 | Alternate proof of ~ dveel... |
| ax12f 36070 | Basis step for constructin... |
| ax12eq 36071 | Basis step for constructin... |
| ax12el 36072 | Basis step for constructin... |
| ax12indn 36073 | Induction step for constru... |
| ax12indi 36074 | Induction step for constru... |
| ax12indalem 36075 | Lemma for ~ ax12inda2 and ... |
| ax12inda2ALT 36076 | Alternate proof of ~ ax12i... |
| ax12inda2 36077 | Induction step for constru... |
| ax12inda 36078 | Induction step for constru... |
| ax12v2-o 36079 | Rederivation of ~ ax-c15 f... |
| ax12a2-o 36080 | Derive ~ ax-c15 from a hyp... |
| axc11-o 36081 | Show that ~ ax-c11 can be ... |
| fsumshftd 36082 | Index shift of a finite su... |
| riotaclbgBAD 36084 | Closure of restricted iota... |
| riotaclbBAD 36085 | Closure of restricted iota... |
| riotasvd 36086 | Deduction version of ~ rio... |
| riotasv2d 36087 | Value of description binde... |
| riotasv2s 36088 | The value of description b... |
| riotasv 36089 | Value of description binde... |
| riotasv3d 36090 | A property ` ch ` holding ... |
| elimhyps 36091 | A version of ~ elimhyp usi... |
| dedths 36092 | A version of weak deductio... |
| renegclALT 36093 | Closure law for negative o... |
| elimhyps2 36094 | Generalization of ~ elimhy... |
| dedths2 36095 | Generalization of ~ dedths... |
| nfcxfrdf 36096 | A utility lemma to transfe... |
| nfded 36097 | A deduction theorem that c... |
| nfded2 36098 | A deduction theorem that c... |
| nfunidALT2 36099 | Deduction version of ~ nfu... |
| nfunidALT 36100 | Deduction version of ~ nfu... |
| nfopdALT 36101 | Deduction version of bound... |
| cnaddcom 36102 | Recover the commutative la... |
| toycom 36103 | Show the commutative law f... |
| lshpset 36108 | The set of all hyperplanes... |
| islshp 36109 | The predicate "is a hyperp... |
| islshpsm 36110 | Hyperplane properties expr... |
| lshplss 36111 | A hyperplane is a subspace... |
| lshpne 36112 | A hyperplane is not equal ... |
| lshpnel 36113 | A hyperplane's generating ... |
| lshpnelb 36114 | The subspace sum of a hype... |
| lshpnel2N 36115 | Condition that determines ... |
| lshpne0 36116 | The member of the span in ... |
| lshpdisj 36117 | A hyperplane and the span ... |
| lshpcmp 36118 | If two hyperplanes are com... |
| lshpinN 36119 | The intersection of two di... |
| lsatset 36120 | The set of all 1-dim subsp... |
| islsat 36121 | The predicate "is a 1-dim ... |
| lsatlspsn2 36122 | The span of a nonzero sing... |
| lsatlspsn 36123 | The span of a nonzero sing... |
| islsati 36124 | A 1-dim subspace (atom) (o... |
| lsateln0 36125 | A 1-dim subspace (atom) (o... |
| lsatlss 36126 | The set of 1-dim subspaces... |
| lsatlssel 36127 | An atom is a subspace. (C... |
| lsatssv 36128 | An atom is a set of vector... |
| lsatn0 36129 | A 1-dim subspace (atom) of... |
| lsatspn0 36130 | The span of a vector is an... |
| lsator0sp 36131 | The span of a vector is ei... |
| lsatssn0 36132 | A subspace (or any class) ... |
| lsatcmp 36133 | If two atoms are comparabl... |
| lsatcmp2 36134 | If an atom is included in ... |
| lsatel 36135 | A nonzero vector in an ato... |
| lsatelbN 36136 | A nonzero vector in an ato... |
| lsat2el 36137 | Two atoms sharing a nonzer... |
| lsmsat 36138 | Convert comparison of atom... |
| lsatfixedN 36139 | Show equality with the spa... |
| lsmsatcv 36140 | Subspace sum has the cover... |
| lssatomic 36141 | The lattice of subspaces i... |
| lssats 36142 | The lattice of subspaces i... |
| lpssat 36143 | Two subspaces in a proper ... |
| lrelat 36144 | Subspaces are relatively a... |
| lssatle 36145 | The ordering of two subspa... |
| lssat 36146 | Two subspaces in a proper ... |
| islshpat 36147 | Hyperplane properties expr... |
| lcvfbr 36150 | The covers relation for a ... |
| lcvbr 36151 | The covers relation for a ... |
| lcvbr2 36152 | The covers relation for a ... |
| lcvbr3 36153 | The covers relation for a ... |
| lcvpss 36154 | The covers relation implie... |
| lcvnbtwn 36155 | The covers relation implie... |
| lcvntr 36156 | The covers relation is not... |
| lcvnbtwn2 36157 | The covers relation implie... |
| lcvnbtwn3 36158 | The covers relation implie... |
| lsmcv2 36159 | Subspace sum has the cover... |
| lcvat 36160 | If a subspace covers anoth... |
| lsatcv0 36161 | An atom covers the zero su... |
| lsatcveq0 36162 | A subspace covered by an a... |
| lsat0cv 36163 | A subspace is an atom iff ... |
| lcvexchlem1 36164 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem2 36165 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem3 36166 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem4 36167 | Lemma for ~ lcvexch . (Co... |
| lcvexchlem5 36168 | Lemma for ~ lcvexch . (Co... |
| lcvexch 36169 | Subspaces satisfy the exch... |
| lcvp 36170 | Covering property of Defin... |
| lcv1 36171 | Covering property of a sub... |
| lcv2 36172 | Covering property of a sub... |
| lsatexch 36173 | The atom exchange property... |
| lsatnle 36174 | The meet of a subspace and... |
| lsatnem0 36175 | The meet of distinct atoms... |
| lsatexch1 36176 | The atom exch1ange propert... |
| lsatcv0eq 36177 | If the sum of two atoms co... |
| lsatcv1 36178 | Two atoms covering the zer... |
| lsatcvatlem 36179 | Lemma for ~ lsatcvat . (C... |
| lsatcvat 36180 | A nonzero subspace less th... |
| lsatcvat2 36181 | A subspace covered by the ... |
| lsatcvat3 36182 | A condition implying that ... |
| islshpcv 36183 | Hyperplane properties expr... |
| l1cvpat 36184 | A subspace covered by the ... |
| l1cvat 36185 | Create an atom under an el... |
| lshpat 36186 | Create an atom under a hyp... |
| lflset 36189 | The set of linear function... |
| islfl 36190 | The predicate "is a linear... |
| lfli 36191 | Property of a linear funct... |
| islfld 36192 | Properties that determine ... |
| lflf 36193 | A linear functional is a f... |
| lflcl 36194 | A linear functional value ... |
| lfl0 36195 | A linear functional is zer... |
| lfladd 36196 | Property of a linear funct... |
| lflsub 36197 | Property of a linear funct... |
| lflmul 36198 | Property of a linear funct... |
| lfl0f 36199 | The zero function is a fun... |
| lfl1 36200 | A nonzero functional has a... |
| lfladdcl 36201 | Closure of addition of two... |
| lfladdcom 36202 | Commutativity of functiona... |
| lfladdass 36203 | Associativity of functiona... |
| lfladd0l 36204 | Functional addition with t... |
| lflnegcl 36205 | Closure of the negative of... |
| lflnegl 36206 | A functional plus its nega... |
| lflvscl 36207 | Closure of a scalar produc... |
| lflvsdi1 36208 | Distributive law for (righ... |
| lflvsdi2 36209 | Reverse distributive law f... |
| lflvsdi2a 36210 | Reverse distributive law f... |
| lflvsass 36211 | Associative law for (right... |
| lfl0sc 36212 | The (right vector space) s... |
| lflsc0N 36213 | The scalar product with th... |
| lfl1sc 36214 | The (right vector space) s... |
| lkrfval 36217 | The kernel of a functional... |
| lkrval 36218 | Value of the kernel of a f... |
| ellkr 36219 | Membership in the kernel o... |
| lkrval2 36220 | Value of the kernel of a f... |
| ellkr2 36221 | Membership in the kernel o... |
| lkrcl 36222 | A member of the kernel of ... |
| lkrf0 36223 | The value of a functional ... |
| lkr0f 36224 | The kernel of the zero fun... |
| lkrlss 36225 | The kernel of a linear fun... |
| lkrssv 36226 | The kernel of a linear fun... |
| lkrsc 36227 | The kernel of a nonzero sc... |
| lkrscss 36228 | The kernel of a scalar pro... |
| eqlkr 36229 | Two functionals with the s... |
| eqlkr2 36230 | Two functionals with the s... |
| eqlkr3 36231 | Two functionals with the s... |
| lkrlsp 36232 | The subspace sum of a kern... |
| lkrlsp2 36233 | The subspace sum of a kern... |
| lkrlsp3 36234 | The subspace sum of a kern... |
| lkrshp 36235 | The kernel of a nonzero fu... |
| lkrshp3 36236 | The kernels of nonzero fun... |
| lkrshpor 36237 | The kernel of a functional... |
| lkrshp4 36238 | A kernel is a hyperplane i... |
| lshpsmreu 36239 | Lemma for ~ lshpkrex . Sh... |
| lshpkrlem1 36240 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem2 36241 | Lemma for ~ lshpkrex . Th... |
| lshpkrlem3 36242 | Lemma for ~ lshpkrex . De... |
| lshpkrlem4 36243 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem5 36244 | Lemma for ~ lshpkrex . Pa... |
| lshpkrlem6 36245 | Lemma for ~ lshpkrex . Sh... |
| lshpkrcl 36246 | The set ` G ` defined by h... |
| lshpkr 36247 | The kernel of functional `... |
| lshpkrex 36248 | There exists a functional ... |
| lshpset2N 36249 | The set of all hyperplanes... |
| islshpkrN 36250 | The predicate "is a hyperp... |
| lfl1dim 36251 | Equivalent expressions for... |
| lfl1dim2N 36252 | Equivalent expressions for... |
| ldualset 36255 | Define the (left) dual of ... |
| ldualvbase 36256 | The vectors of a dual spac... |
| ldualelvbase 36257 | Utility theorem for conver... |
| ldualfvadd 36258 | Vector addition in the dua... |
| ldualvadd 36259 | Vector addition in the dua... |
| ldualvaddcl 36260 | The value of vector additi... |
| ldualvaddval 36261 | The value of the value of ... |
| ldualsca 36262 | The ring of scalars of the... |
| ldualsbase 36263 | Base set of scalar ring fo... |
| ldualsaddN 36264 | Scalar addition for the du... |
| ldualsmul 36265 | Scalar multiplication for ... |
| ldualfvs 36266 | Scalar product operation f... |
| ldualvs 36267 | Scalar product operation v... |
| ldualvsval 36268 | Value of scalar product op... |
| ldualvscl 36269 | The scalar product operati... |
| ldualvaddcom 36270 | Commutative law for vector... |
| ldualvsass 36271 | Associative law for scalar... |
| ldualvsass2 36272 | Associative law for scalar... |
| ldualvsdi1 36273 | Distributive law for scala... |
| ldualvsdi2 36274 | Reverse distributive law f... |
| ldualgrplem 36275 | Lemma for ~ ldualgrp . (C... |
| ldualgrp 36276 | The dual of a vector space... |
| ldual0 36277 | The zero scalar of the dua... |
| ldual1 36278 | The unit scalar of the dua... |
| ldualneg 36279 | The negative of a scalar o... |
| ldual0v 36280 | The zero vector of the dua... |
| ldual0vcl 36281 | The dual zero vector is a ... |
| lduallmodlem 36282 | Lemma for ~ lduallmod . (... |
| lduallmod 36283 | The dual of a left module ... |
| lduallvec 36284 | The dual of a left vector ... |
| ldualvsub 36285 | The value of vector subtra... |
| ldualvsubcl 36286 | Closure of vector subtract... |
| ldualvsubval 36287 | The value of the value of ... |
| ldualssvscl 36288 | Closure of scalar product ... |
| ldualssvsubcl 36289 | Closure of vector subtract... |
| ldual0vs 36290 | Scalar zero times a functi... |
| lkr0f2 36291 | The kernel of the zero fun... |
| lduallkr3 36292 | The kernels of nonzero fun... |
| lkrpssN 36293 | Proper subset relation bet... |
| lkrin 36294 | Intersection of the kernel... |
| eqlkr4 36295 | Two functionals with the s... |
| ldual1dim 36296 | Equivalent expressions for... |
| ldualkrsc 36297 | The kernel of a nonzero sc... |
| lkrss 36298 | The kernel of a scalar pro... |
| lkrss2N 36299 | Two functionals with kerne... |
| lkreqN 36300 | Proportional functionals h... |
| lkrlspeqN 36301 | Condition for colinear fun... |
| isopos 36310 | The predicate "is an ortho... |
| opposet 36311 | Every orthoposet is a pose... |
| oposlem 36312 | Lemma for orthoposet prope... |
| op01dm 36313 | Conditions necessary for z... |
| op0cl 36314 | An orthoposet has a zero e... |
| op1cl 36315 | An orthoposet has a unit e... |
| op0le 36316 | Orthoposet zero is less th... |
| ople0 36317 | An element less than or eq... |
| opnlen0 36318 | An element not less than a... |
| lub0N 36319 | The least upper bound of t... |
| opltn0 36320 | A lattice element greater ... |
| ople1 36321 | Any element is less than t... |
| op1le 36322 | If the orthoposet unit is ... |
| glb0N 36323 | The greatest lower bound o... |
| opoccl 36324 | Closure of orthocomplement... |
| opococ 36325 | Double negative law for or... |
| opcon3b 36326 | Contraposition law for ort... |
| opcon2b 36327 | Orthocomplement contraposi... |
| opcon1b 36328 | Orthocomplement contraposi... |
| oplecon3 36329 | Contraposition law for ort... |
| oplecon3b 36330 | Contraposition law for ort... |
| oplecon1b 36331 | Contraposition law for str... |
| opoc1 36332 | Orthocomplement of orthopo... |
| opoc0 36333 | Orthocomplement of orthopo... |
| opltcon3b 36334 | Contraposition law for str... |
| opltcon1b 36335 | Contraposition law for str... |
| opltcon2b 36336 | Contraposition law for str... |
| opexmid 36337 | Law of excluded middle for... |
| opnoncon 36338 | Law of contradiction for o... |
| riotaocN 36339 | The orthocomplement of the... |
| cmtfvalN 36340 | Value of commutes relation... |
| cmtvalN 36341 | Equivalence for commutes r... |
| isolat 36342 | The predicate "is an ortho... |
| ollat 36343 | An ortholattice is a latti... |
| olop 36344 | An ortholattice is an orth... |
| olposN 36345 | An ortholattice is a poset... |
| isolatiN 36346 | Properties that determine ... |
| oldmm1 36347 | De Morgan's law for meet i... |
| oldmm2 36348 | De Morgan's law for meet i... |
| oldmm3N 36349 | De Morgan's law for meet i... |
| oldmm4 36350 | De Morgan's law for meet i... |
| oldmj1 36351 | De Morgan's law for join i... |
| oldmj2 36352 | De Morgan's law for join i... |
| oldmj3 36353 | De Morgan's law for join i... |
| oldmj4 36354 | De Morgan's law for join i... |
| olj01 36355 | An ortholattice element jo... |
| olj02 36356 | An ortholattice element jo... |
| olm11 36357 | The meet of an ortholattic... |
| olm12 36358 | The meet of an ortholattic... |
| latmassOLD 36359 | Ortholattice meet is assoc... |
| latm12 36360 | A rearrangement of lattice... |
| latm32 36361 | A rearrangement of lattice... |
| latmrot 36362 | Rotate lattice meet of 3 c... |
| latm4 36363 | Rearrangement of lattice m... |
| latmmdiN 36364 | Lattice meet distributes o... |
| latmmdir 36365 | Lattice meet distributes o... |
| olm01 36366 | Meet with lattice zero is ... |
| olm02 36367 | Meet with lattice zero is ... |
| isoml 36368 | The predicate "is an ortho... |
| isomliN 36369 | Properties that determine ... |
| omlol 36370 | An orthomodular lattice is... |
| omlop 36371 | An orthomodular lattice is... |
| omllat 36372 | An orthomodular lattice is... |
| omllaw 36373 | The orthomodular law. (Co... |
| omllaw2N 36374 | Variation of orthomodular ... |
| omllaw3 36375 | Orthomodular law equivalen... |
| omllaw4 36376 | Orthomodular law equivalen... |
| omllaw5N 36377 | The orthomodular law. Rem... |
| cmtcomlemN 36378 | Lemma for ~ cmtcomN . ( ~... |
| cmtcomN 36379 | Commutation is symmetric. ... |
| cmt2N 36380 | Commutation with orthocomp... |
| cmt3N 36381 | Commutation with orthocomp... |
| cmt4N 36382 | Commutation with orthocomp... |
| cmtbr2N 36383 | Alternate definition of th... |
| cmtbr3N 36384 | Alternate definition for t... |
| cmtbr4N 36385 | Alternate definition for t... |
| lecmtN 36386 | Ordered elements commute. ... |
| cmtidN 36387 | Any element commutes with ... |
| omlfh1N 36388 | Foulis-Holland Theorem, pa... |
| omlfh3N 36389 | Foulis-Holland Theorem, pa... |
| omlmod1i2N 36390 | Analogue of modular law ~ ... |
| omlspjN 36391 | Contraction of a Sasaki pr... |
| cvrfval 36398 | Value of covers relation "... |
| cvrval 36399 | Binary relation expressing... |
| cvrlt 36400 | The covers relation implie... |
| cvrnbtwn 36401 | There is no element betwee... |
| ncvr1 36402 | No element covers the latt... |
| cvrletrN 36403 | Property of an element abo... |
| cvrval2 36404 | Binary relation expressing... |
| cvrnbtwn2 36405 | The covers relation implie... |
| cvrnbtwn3 36406 | The covers relation implie... |
| cvrcon3b 36407 | Contraposition law for the... |
| cvrle 36408 | The covers relation implie... |
| cvrnbtwn4 36409 | The covers relation implie... |
| cvrnle 36410 | The covers relation implie... |
| cvrne 36411 | The covers relation implie... |
| cvrnrefN 36412 | The covers relation is not... |
| cvrcmp 36413 | If two lattice elements th... |
| cvrcmp2 36414 | If two lattice elements co... |
| pats 36415 | The set of atoms in a pose... |
| isat 36416 | The predicate "is an atom"... |
| isat2 36417 | The predicate "is an atom"... |
| atcvr0 36418 | An atom covers zero. ( ~ ... |
| atbase 36419 | An atom is a member of the... |
| atssbase 36420 | The set of atoms is a subs... |
| 0ltat 36421 | An atom is greater than ze... |
| leatb 36422 | A poset element less than ... |
| leat 36423 | A poset element less than ... |
| leat2 36424 | A nonzero poset element le... |
| leat3 36425 | A poset element less than ... |
| meetat 36426 | The meet of any element wi... |
| meetat2 36427 | The meet of any element wi... |
| isatl 36429 | The predicate "is an atomi... |
| atllat 36430 | An atomic lattice is a lat... |
| atlpos 36431 | An atomic lattice is a pos... |
| atl0dm 36432 | Condition necessary for ze... |
| atl0cl 36433 | An atomic lattice has a ze... |
| atl0le 36434 | Orthoposet zero is less th... |
| atlle0 36435 | An element less than or eq... |
| atlltn0 36436 | A lattice element greater ... |
| isat3 36437 | The predicate "is an atom"... |
| atn0 36438 | An atom is not zero. ( ~ ... |
| atnle0 36439 | An atom is not less than o... |
| atlen0 36440 | A lattice element is nonze... |
| atcmp 36441 | If two atoms are comparabl... |
| atncmp 36442 | Frequently-used variation ... |
| atnlt 36443 | Two atoms cannot satisfy t... |
| atcvreq0 36444 | An element covered by an a... |
| atncvrN 36445 | Two atoms cannot satisfy t... |
| atlex 36446 | Every nonzero element of a... |
| atnle 36447 | Two ways of expressing "an... |
| atnem0 36448 | The meet of distinct atoms... |
| atlatmstc 36449 | An atomic, complete, ortho... |
| atlatle 36450 | The ordering of two Hilber... |
| atlrelat1 36451 | An atomistic lattice with ... |
| iscvlat 36453 | The predicate "is an atomi... |
| iscvlat2N 36454 | The predicate "is an atomi... |
| cvlatl 36455 | An atomic lattice with the... |
| cvllat 36456 | An atomic lattice with the... |
| cvlposN 36457 | An atomic lattice with the... |
| cvlexch1 36458 | An atomic covering lattice... |
| cvlexch2 36459 | An atomic covering lattice... |
| cvlexchb1 36460 | An atomic covering lattice... |
| cvlexchb2 36461 | An atomic covering lattice... |
| cvlexch3 36462 | An atomic covering lattice... |
| cvlexch4N 36463 | An atomic covering lattice... |
| cvlatexchb1 36464 | A version of ~ cvlexchb1 f... |
| cvlatexchb2 36465 | A version of ~ cvlexchb2 f... |
| cvlatexch1 36466 | Atom exchange property. (... |
| cvlatexch2 36467 | Atom exchange property. (... |
| cvlatexch3 36468 | Atom exchange property. (... |
| cvlcvr1 36469 | The covering property. Pr... |
| cvlcvrp 36470 | A Hilbert lattice satisfie... |
| cvlatcvr1 36471 | An atom is covered by its ... |
| cvlatcvr2 36472 | An atom is covered by its ... |
| cvlsupr2 36473 | Two equivalent ways of exp... |
| cvlsupr3 36474 | Two equivalent ways of exp... |
| cvlsupr4 36475 | Consequence of superpositi... |
| cvlsupr5 36476 | Consequence of superpositi... |
| cvlsupr6 36477 | Consequence of superpositi... |
| cvlsupr7 36478 | Consequence of superpositi... |
| cvlsupr8 36479 | Consequence of superpositi... |
| ishlat1 36482 | The predicate "is a Hilber... |
| ishlat2 36483 | The predicate "is a Hilber... |
| ishlat3N 36484 | The predicate "is a Hilber... |
| ishlatiN 36485 | Properties that determine ... |
| hlomcmcv 36486 | A Hilbert lattice is ortho... |
| hloml 36487 | A Hilbert lattice is ortho... |
| hlclat 36488 | A Hilbert lattice is compl... |
| hlcvl 36489 | A Hilbert lattice is an at... |
| hlatl 36490 | A Hilbert lattice is atomi... |
| hlol 36491 | A Hilbert lattice is an or... |
| hlop 36492 | A Hilbert lattice is an or... |
| hllat 36493 | A Hilbert lattice is a lat... |
| hllatd 36494 | Deduction form of ~ hllat ... |
| hlomcmat 36495 | A Hilbert lattice is ortho... |
| hlpos 36496 | A Hilbert lattice is a pos... |
| hlatjcl 36497 | Closure of join operation.... |
| hlatjcom 36498 | Commutatitivity of join op... |
| hlatjidm 36499 | Idempotence of join operat... |
| hlatjass 36500 | Lattice join is associativ... |
| hlatj12 36501 | Swap 1st and 2nd members o... |
| hlatj32 36502 | Swap 2nd and 3rd members o... |
| hlatjrot 36503 | Rotate lattice join of 3 c... |
| hlatj4 36504 | Rearrangement of lattice j... |
| hlatlej1 36505 | A join's first argument is... |
| hlatlej2 36506 | A join's second argument i... |
| glbconN 36507 | De Morgan's law for GLB an... |
| glbconxN 36508 | De Morgan's law for GLB an... |
| atnlej1 36509 | If an atom is not less tha... |
| atnlej2 36510 | If an atom is not less tha... |
| hlsuprexch 36511 | A Hilbert lattice has the ... |
| hlexch1 36512 | A Hilbert lattice has the ... |
| hlexch2 36513 | A Hilbert lattice has the ... |
| hlexchb1 36514 | A Hilbert lattice has the ... |
| hlexchb2 36515 | A Hilbert lattice has the ... |
| hlsupr 36516 | A Hilbert lattice has the ... |
| hlsupr2 36517 | A Hilbert lattice has the ... |
| hlhgt4 36518 | A Hilbert lattice has a he... |
| hlhgt2 36519 | A Hilbert lattice has a he... |
| hl0lt1N 36520 | Lattice 0 is less than lat... |
| hlexch3 36521 | A Hilbert lattice has the ... |
| hlexch4N 36522 | A Hilbert lattice has the ... |
| hlatexchb1 36523 | A version of ~ hlexchb1 fo... |
| hlatexchb2 36524 | A version of ~ hlexchb2 fo... |
| hlatexch1 36525 | Atom exchange property. (... |
| hlatexch2 36526 | Atom exchange property. (... |
| hlatmstcOLDN 36527 | An atomic, complete, ortho... |
| hlatle 36528 | The ordering of two Hilber... |
| hlateq 36529 | The equality of two Hilber... |
| hlrelat1 36530 | An atomistic lattice with ... |
| hlrelat5N 36531 | An atomistic lattice with ... |
| hlrelat 36532 | A Hilbert lattice is relat... |
| hlrelat2 36533 | A consequence of relative ... |
| exatleN 36534 | A condition for an atom to... |
| hl2at 36535 | A Hilbert lattice has at l... |
| atex 36536 | At least one atom exists. ... |
| intnatN 36537 | If the intersection with a... |
| 2llnne2N 36538 | Condition implying that tw... |
| 2llnneN 36539 | Condition implying that tw... |
| cvr1 36540 | A Hilbert lattice has the ... |
| cvr2N 36541 | Less-than and covers equiv... |
| hlrelat3 36542 | The Hilbert lattice is rel... |
| cvrval3 36543 | Binary relation expressing... |
| cvrval4N 36544 | Binary relation expressing... |
| cvrval5 36545 | Binary relation expressing... |
| cvrp 36546 | A Hilbert lattice satisfie... |
| atcvr1 36547 | An atom is covered by its ... |
| atcvr2 36548 | An atom is covered by its ... |
| cvrexchlem 36549 | Lemma for ~ cvrexch . ( ~... |
| cvrexch 36550 | A Hilbert lattice satisfie... |
| cvratlem 36551 | Lemma for ~ cvrat . ( ~ a... |
| cvrat 36552 | A nonzero Hilbert lattice ... |
| ltltncvr 36553 | A chained strong ordering ... |
| ltcvrntr 36554 | Non-transitive condition f... |
| cvrntr 36555 | The covers relation is not... |
| atcvr0eq 36556 | The covers relation is not... |
| lnnat 36557 | A line (the join of two di... |
| atcvrj0 36558 | Two atoms covering the zer... |
| cvrat2 36559 | A Hilbert lattice element ... |
| atcvrneN 36560 | Inequality derived from at... |
| atcvrj1 36561 | Condition for an atom to b... |
| atcvrj2b 36562 | Condition for an atom to b... |
| atcvrj2 36563 | Condition for an atom to b... |
| atleneN 36564 | Inequality derived from at... |
| atltcvr 36565 | An equivalence of less-tha... |
| atle 36566 | Any nonzero element has an... |
| atlt 36567 | Two atoms are unequal iff ... |
| atlelt 36568 | Transfer less-than relatio... |
| 2atlt 36569 | Given an atom less than an... |
| atexchcvrN 36570 | Atom exchange property. V... |
| atexchltN 36571 | Atom exchange property. V... |
| cvrat3 36572 | A condition implying that ... |
| cvrat4 36573 | A condition implying exist... |
| cvrat42 36574 | Commuted version of ~ cvra... |
| 2atjm 36575 | The meet of a line (expres... |
| atbtwn 36576 | Property of a 3rd atom ` R... |
| atbtwnexOLDN 36577 | There exists a 3rd atom ` ... |
| atbtwnex 36578 | Given atoms ` P ` in ` X `... |
| 3noncolr2 36579 | Two ways to express 3 non-... |
| 3noncolr1N 36580 | Two ways to express 3 non-... |
| hlatcon3 36581 | Atom exchange combined wit... |
| hlatcon2 36582 | Atom exchange combined wit... |
| 4noncolr3 36583 | A way to express 4 non-col... |
| 4noncolr2 36584 | A way to express 4 non-col... |
| 4noncolr1 36585 | A way to express 4 non-col... |
| athgt 36586 | A Hilbert lattice, whose h... |
| 3dim0 36587 | There exists a 3-dimension... |
| 3dimlem1 36588 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem2 36589 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3a 36590 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem3 36591 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem3OLDN 36592 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4a 36593 | Lemma for ~ 3dim3 . (Cont... |
| 3dimlem4 36594 | Lemma for ~ 3dim1 . (Cont... |
| 3dimlem4OLDN 36595 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1lem5 36596 | Lemma for ~ 3dim1 . (Cont... |
| 3dim1 36597 | Construct a 3-dimensional ... |
| 3dim2 36598 | Construct 2 new layers on ... |
| 3dim3 36599 | Construct a new layer on t... |
| 2dim 36600 | Generate a height-3 elemen... |
| 1dimN 36601 | An atom is covered by a he... |
| 1cvrco 36602 | The orthocomplement of an ... |
| 1cvratex 36603 | There exists an atom less ... |
| 1cvratlt 36604 | An atom less than or equal... |
| 1cvrjat 36605 | An element covered by the ... |
| 1cvrat 36606 | Create an atom under an el... |
| ps-1 36607 | The join of two atoms ` R ... |
| ps-2 36608 | Lattice analogue for the p... |
| 2atjlej 36609 | Two atoms are different if... |
| hlatexch3N 36610 | Rearrange join of atoms in... |
| hlatexch4 36611 | Exchange 2 atoms. (Contri... |
| ps-2b 36612 | Variation of projective ge... |
| 3atlem1 36613 | Lemma for ~ 3at . (Contri... |
| 3atlem2 36614 | Lemma for ~ 3at . (Contri... |
| 3atlem3 36615 | Lemma for ~ 3at . (Contri... |
| 3atlem4 36616 | Lemma for ~ 3at . (Contri... |
| 3atlem5 36617 | Lemma for ~ 3at . (Contri... |
| 3atlem6 36618 | Lemma for ~ 3at . (Contri... |
| 3atlem7 36619 | Lemma for ~ 3at . (Contri... |
| 3at 36620 | Any three non-colinear ato... |
| llnset 36635 | The set of lattice lines i... |
| islln 36636 | The predicate "is a lattic... |
| islln4 36637 | The predicate "is a lattic... |
| llni 36638 | Condition implying a latti... |
| llnbase 36639 | A lattice line is a lattic... |
| islln3 36640 | The predicate "is a lattic... |
| islln2 36641 | The predicate "is a lattic... |
| llni2 36642 | The join of two different ... |
| llnnleat 36643 | An atom cannot majorize a ... |
| llnneat 36644 | A lattice line is not an a... |
| 2atneat 36645 | The join of two distinct a... |
| llnn0 36646 | A lattice line is nonzero.... |
| islln2a 36647 | The predicate "is a lattic... |
| llnle 36648 | Any element greater than 0... |
| atcvrlln2 36649 | An atom under a line is co... |
| atcvrlln 36650 | An element covering an ato... |
| llnexatN 36651 | Given an atom on a line, t... |
| llncmp 36652 | If two lattice lines are c... |
| llnnlt 36653 | Two lattice lines cannot s... |
| 2llnmat 36654 | Two intersecting lines int... |
| 2at0mat0 36655 | Special case of ~ 2atmat0 ... |
| 2atmat0 36656 | The meet of two unequal li... |
| 2atm 36657 | An atom majorized by two d... |
| ps-2c 36658 | Variation of projective ge... |
| lplnset 36659 | The set of lattice planes ... |
| islpln 36660 | The predicate "is a lattic... |
| islpln4 36661 | The predicate "is a lattic... |
| lplni 36662 | Condition implying a latti... |
| islpln3 36663 | The predicate "is a lattic... |
| lplnbase 36664 | A lattice plane is a latti... |
| islpln5 36665 | The predicate "is a lattic... |
| islpln2 36666 | The predicate "is a lattic... |
| lplni2 36667 | The join of 3 different at... |
| lvolex3N 36668 | There is an atom outside o... |
| llnmlplnN 36669 | The intersection of a line... |
| lplnle 36670 | Any element greater than 0... |
| lplnnle2at 36671 | A lattice line (or atom) c... |
| lplnnleat 36672 | A lattice plane cannot maj... |
| lplnnlelln 36673 | A lattice plane is not les... |
| 2atnelpln 36674 | The join of two atoms is n... |
| lplnneat 36675 | No lattice plane is an ato... |
| lplnnelln 36676 | No lattice plane is a latt... |
| lplnn0N 36677 | A lattice plane is nonzero... |
| islpln2a 36678 | The predicate "is a lattic... |
| islpln2ah 36679 | The predicate "is a lattic... |
| lplnriaN 36680 | Property of a lattice plan... |
| lplnribN 36681 | Property of a lattice plan... |
| lplnric 36682 | Property of a lattice plan... |
| lplnri1 36683 | Property of a lattice plan... |
| lplnri2N 36684 | Property of a lattice plan... |
| lplnri3N 36685 | Property of a lattice plan... |
| lplnllnneN 36686 | Two lattice lines defined ... |
| llncvrlpln2 36687 | A lattice line under a lat... |
| llncvrlpln 36688 | An element covering a latt... |
| 2lplnmN 36689 | If the join of two lattice... |
| 2llnmj 36690 | The meet of two lattice li... |
| 2atmat 36691 | The meet of two intersecti... |
| lplncmp 36692 | If two lattice planes are ... |
| lplnexatN 36693 | Given a lattice line on a ... |
| lplnexllnN 36694 | Given an atom on a lattice... |
| lplnnlt 36695 | Two lattice planes cannot ... |
| 2llnjaN 36696 | The join of two different ... |
| 2llnjN 36697 | The join of two different ... |
| 2llnm2N 36698 | The meet of two different ... |
| 2llnm3N 36699 | Two lattice lines in a lat... |
| 2llnm4 36700 | Two lattice lines that maj... |
| 2llnmeqat 36701 | An atom equals the interse... |
| lvolset 36702 | The set of 3-dim lattice v... |
| islvol 36703 | The predicate "is a 3-dim ... |
| islvol4 36704 | The predicate "is a 3-dim ... |
| lvoli 36705 | Condition implying a 3-dim... |
| islvol3 36706 | The predicate "is a 3-dim ... |
| lvoli3 36707 | Condition implying a 3-dim... |
| lvolbase 36708 | A 3-dim lattice volume is ... |
| islvol5 36709 | The predicate "is a 3-dim ... |
| islvol2 36710 | The predicate "is a 3-dim ... |
| lvoli2 36711 | The join of 4 different at... |
| lvolnle3at 36712 | A lattice plane (or lattic... |
| lvolnleat 36713 | An atom cannot majorize a ... |
| lvolnlelln 36714 | A lattice line cannot majo... |
| lvolnlelpln 36715 | A lattice plane cannot maj... |
| 3atnelvolN 36716 | The join of 3 atoms is not... |
| 2atnelvolN 36717 | The join of two atoms is n... |
| lvolneatN 36718 | No lattice volume is an at... |
| lvolnelln 36719 | No lattice volume is a lat... |
| lvolnelpln 36720 | No lattice volume is a lat... |
| lvoln0N 36721 | A lattice volume is nonzer... |
| islvol2aN 36722 | The predicate "is a lattic... |
| 4atlem0a 36723 | Lemma for ~ 4at . (Contri... |
| 4atlem0ae 36724 | Lemma for ~ 4at . (Contri... |
| 4atlem0be 36725 | Lemma for ~ 4at . (Contri... |
| 4atlem3 36726 | Lemma for ~ 4at . Break i... |
| 4atlem3a 36727 | Lemma for ~ 4at . Break i... |
| 4atlem3b 36728 | Lemma for ~ 4at . Break i... |
| 4atlem4a 36729 | Lemma for ~ 4at . Frequen... |
| 4atlem4b 36730 | Lemma for ~ 4at . Frequen... |
| 4atlem4c 36731 | Lemma for ~ 4at . Frequen... |
| 4atlem4d 36732 | Lemma for ~ 4at . Frequen... |
| 4atlem9 36733 | Lemma for ~ 4at . Substit... |
| 4atlem10a 36734 | Lemma for ~ 4at . Substit... |
| 4atlem10b 36735 | Lemma for ~ 4at . Substit... |
| 4atlem10 36736 | Lemma for ~ 4at . Combine... |
| 4atlem11a 36737 | Lemma for ~ 4at . Substit... |
| 4atlem11b 36738 | Lemma for ~ 4at . Substit... |
| 4atlem11 36739 | Lemma for ~ 4at . Combine... |
| 4atlem12a 36740 | Lemma for ~ 4at . Substit... |
| 4atlem12b 36741 | Lemma for ~ 4at . Substit... |
| 4atlem12 36742 | Lemma for ~ 4at . Combine... |
| 4at 36743 | Four atoms determine a lat... |
| 4at2 36744 | Four atoms determine a lat... |
| lplncvrlvol2 36745 | A lattice line under a lat... |
| lplncvrlvol 36746 | An element covering a latt... |
| lvolcmp 36747 | If two lattice planes are ... |
| lvolnltN 36748 | Two lattice volumes cannot... |
| 2lplnja 36749 | The join of two different ... |
| 2lplnj 36750 | The join of two different ... |
| 2lplnm2N 36751 | The meet of two different ... |
| 2lplnmj 36752 | The meet of two lattice pl... |
| dalemkehl 36753 | Lemma for ~ dath . Freque... |
| dalemkelat 36754 | Lemma for ~ dath . Freque... |
| dalemkeop 36755 | Lemma for ~ dath . Freque... |
| dalempea 36756 | Lemma for ~ dath . Freque... |
| dalemqea 36757 | Lemma for ~ dath . Freque... |
| dalemrea 36758 | Lemma for ~ dath . Freque... |
| dalemsea 36759 | Lemma for ~ dath . Freque... |
| dalemtea 36760 | Lemma for ~ dath . Freque... |
| dalemuea 36761 | Lemma for ~ dath . Freque... |
| dalemyeo 36762 | Lemma for ~ dath . Freque... |
| dalemzeo 36763 | Lemma for ~ dath . Freque... |
| dalemclpjs 36764 | Lemma for ~ dath . Freque... |
| dalemclqjt 36765 | Lemma for ~ dath . Freque... |
| dalemclrju 36766 | Lemma for ~ dath . Freque... |
| dalem-clpjq 36767 | Lemma for ~ dath . Freque... |
| dalemceb 36768 | Lemma for ~ dath . Freque... |
| dalempeb 36769 | Lemma for ~ dath . Freque... |
| dalemqeb 36770 | Lemma for ~ dath . Freque... |
| dalemreb 36771 | Lemma for ~ dath . Freque... |
| dalemseb 36772 | Lemma for ~ dath . Freque... |
| dalemteb 36773 | Lemma for ~ dath . Freque... |
| dalemueb 36774 | Lemma for ~ dath . Freque... |
| dalempjqeb 36775 | Lemma for ~ dath . Freque... |
| dalemsjteb 36776 | Lemma for ~ dath . Freque... |
| dalemtjueb 36777 | Lemma for ~ dath . Freque... |
| dalemqrprot 36778 | Lemma for ~ dath . Freque... |
| dalemyeb 36779 | Lemma for ~ dath . Freque... |
| dalemcnes 36780 | Lemma for ~ dath . Freque... |
| dalempnes 36781 | Lemma for ~ dath . Freque... |
| dalemqnet 36782 | Lemma for ~ dath . Freque... |
| dalempjsen 36783 | Lemma for ~ dath . Freque... |
| dalemply 36784 | Lemma for ~ dath . Freque... |
| dalemsly 36785 | Lemma for ~ dath . Freque... |
| dalemswapyz 36786 | Lemma for ~ dath . Swap t... |
| dalemrot 36787 | Lemma for ~ dath . Rotate... |
| dalemrotyz 36788 | Lemma for ~ dath . Rotate... |
| dalem1 36789 | Lemma for ~ dath . Show t... |
| dalemcea 36790 | Lemma for ~ dath . Freque... |
| dalem2 36791 | Lemma for ~ dath . Show t... |
| dalemdea 36792 | Lemma for ~ dath . Freque... |
| dalemeea 36793 | Lemma for ~ dath . Freque... |
| dalem3 36794 | Lemma for ~ dalemdnee . (... |
| dalem4 36795 | Lemma for ~ dalemdnee . (... |
| dalemdnee 36796 | Lemma for ~ dath . Axis o... |
| dalem5 36797 | Lemma for ~ dath . Atom `... |
| dalem6 36798 | Lemma for ~ dath . Analog... |
| dalem7 36799 | Lemma for ~ dath . Analog... |
| dalem8 36800 | Lemma for ~ dath . Plane ... |
| dalem-cly 36801 | Lemma for ~ dalem9 . Cent... |
| dalem9 36802 | Lemma for ~ dath . Since ... |
| dalem10 36803 | Lemma for ~ dath . Atom `... |
| dalem11 36804 | Lemma for ~ dath . Analog... |
| dalem12 36805 | Lemma for ~ dath . Analog... |
| dalem13 36806 | Lemma for ~ dalem14 . (Co... |
| dalem14 36807 | Lemma for ~ dath . Planes... |
| dalem15 36808 | Lemma for ~ dath . The ax... |
| dalem16 36809 | Lemma for ~ dath . The at... |
| dalem17 36810 | Lemma for ~ dath . When p... |
| dalem18 36811 | Lemma for ~ dath . Show t... |
| dalem19 36812 | Lemma for ~ dath . Show t... |
| dalemccea 36813 | Lemma for ~ dath . Freque... |
| dalemddea 36814 | Lemma for ~ dath . Freque... |
| dalem-ccly 36815 | Lemma for ~ dath . Freque... |
| dalem-ddly 36816 | Lemma for ~ dath . Freque... |
| dalemccnedd 36817 | Lemma for ~ dath . Freque... |
| dalemclccjdd 36818 | Lemma for ~ dath . Freque... |
| dalemcceb 36819 | Lemma for ~ dath . Freque... |
| dalemswapyzps 36820 | Lemma for ~ dath . Swap t... |
| dalemrotps 36821 | Lemma for ~ dath . Rotate... |
| dalemcjden 36822 | Lemma for ~ dath . Show t... |
| dalem20 36823 | Lemma for ~ dath . Show t... |
| dalem21 36824 | Lemma for ~ dath . Show t... |
| dalem22 36825 | Lemma for ~ dath . Show t... |
| dalem23 36826 | Lemma for ~ dath . Show t... |
| dalem24 36827 | Lemma for ~ dath . Show t... |
| dalem25 36828 | Lemma for ~ dath . Show t... |
| dalem27 36829 | Lemma for ~ dath . Show t... |
| dalem28 36830 | Lemma for ~ dath . Lemma ... |
| dalem29 36831 | Lemma for ~ dath . Analog... |
| dalem30 36832 | Lemma for ~ dath . Analog... |
| dalem31N 36833 | Lemma for ~ dath . Analog... |
| dalem32 36834 | Lemma for ~ dath . Analog... |
| dalem33 36835 | Lemma for ~ dath . Analog... |
| dalem34 36836 | Lemma for ~ dath . Analog... |
| dalem35 36837 | Lemma for ~ dath . Analog... |
| dalem36 36838 | Lemma for ~ dath . Analog... |
| dalem37 36839 | Lemma for ~ dath . Analog... |
| dalem38 36840 | Lemma for ~ dath . Plane ... |
| dalem39 36841 | Lemma for ~ dath . Auxili... |
| dalem40 36842 | Lemma for ~ dath . Analog... |
| dalem41 36843 | Lemma for ~ dath . (Contr... |
| dalem42 36844 | Lemma for ~ dath . Auxili... |
| dalem43 36845 | Lemma for ~ dath . Planes... |
| dalem44 36846 | Lemma for ~ dath . Dummy ... |
| dalem45 36847 | Lemma for ~ dath . Dummy ... |
| dalem46 36848 | Lemma for ~ dath . Analog... |
| dalem47 36849 | Lemma for ~ dath . Analog... |
| dalem48 36850 | Lemma for ~ dath . Analog... |
| dalem49 36851 | Lemma for ~ dath . Analog... |
| dalem50 36852 | Lemma for ~ dath . Analog... |
| dalem51 36853 | Lemma for ~ dath . Constr... |
| dalem52 36854 | Lemma for ~ dath . Lines ... |
| dalem53 36855 | Lemma for ~ dath . The au... |
| dalem54 36856 | Lemma for ~ dath . Line `... |
| dalem55 36857 | Lemma for ~ dath . Lines ... |
| dalem56 36858 | Lemma for ~ dath . Analog... |
| dalem57 36859 | Lemma for ~ dath . Axis o... |
| dalem58 36860 | Lemma for ~ dath . Analog... |
| dalem59 36861 | Lemma for ~ dath . Analog... |
| dalem60 36862 | Lemma for ~ dath . ` B ` i... |
| dalem61 36863 | Lemma for ~ dath . Show t... |
| dalem62 36864 | Lemma for ~ dath . Elimin... |
| dalem63 36865 | Lemma for ~ dath . Combin... |
| dath 36866 | Desargues's theorem of pro... |
| dath2 36867 | Version of Desargues's the... |
| lineset 36868 | The set of lines in a Hilb... |
| isline 36869 | The predicate "is a line".... |
| islinei 36870 | Condition implying "is a l... |
| pointsetN 36871 | The set of points in a Hil... |
| ispointN 36872 | The predicate "is a point"... |
| atpointN 36873 | The singleton of an atom i... |
| psubspset 36874 | The set of projective subs... |
| ispsubsp 36875 | The predicate "is a projec... |
| ispsubsp2 36876 | The predicate "is a projec... |
| psubspi 36877 | Property of a projective s... |
| psubspi2N 36878 | Property of a projective s... |
| 0psubN 36879 | The empty set is a project... |
| snatpsubN 36880 | The singleton of an atom i... |
| pointpsubN 36881 | A point (singleton of an a... |
| linepsubN 36882 | A line is a projective sub... |
| atpsubN 36883 | The set of all atoms is a ... |
| psubssat 36884 | A projective subspace cons... |
| psubatN 36885 | A member of a projective s... |
| pmapfval 36886 | The projective map of a Hi... |
| pmapval 36887 | Value of the projective ma... |
| elpmap 36888 | Member of a projective map... |
| pmapssat 36889 | The projective map of a Hi... |
| pmapssbaN 36890 | A weakening of ~ pmapssat ... |
| pmaple 36891 | The projective map of a Hi... |
| pmap11 36892 | The projective map of a Hi... |
| pmapat 36893 | The projective map of an a... |
| elpmapat 36894 | Member of the projective m... |
| pmap0 36895 | Value of the projective ma... |
| pmapeq0 36896 | A projective map value is ... |
| pmap1N 36897 | Value of the projective ma... |
| pmapsub 36898 | The projective map of a Hi... |
| pmapglbx 36899 | The projective map of the ... |
| pmapglb 36900 | The projective map of the ... |
| pmapglb2N 36901 | The projective map of the ... |
| pmapglb2xN 36902 | The projective map of the ... |
| pmapmeet 36903 | The projective map of a me... |
| isline2 36904 | Definition of line in term... |
| linepmap 36905 | A line described with a pr... |
| isline3 36906 | Definition of line in term... |
| isline4N 36907 | Definition of line in term... |
| lneq2at 36908 | A line equals the join of ... |
| lnatexN 36909 | There is an atom in a line... |
| lnjatN 36910 | Given an atom in a line, t... |
| lncvrelatN 36911 | A lattice element covered ... |
| lncvrat 36912 | A line covers the atoms it... |
| lncmp 36913 | If two lines are comparabl... |
| 2lnat 36914 | Two intersecting lines int... |
| 2atm2atN 36915 | Two joins with a common at... |
| 2llnma1b 36916 | Generalization of ~ 2llnma... |
| 2llnma1 36917 | Two different intersecting... |
| 2llnma3r 36918 | Two different intersecting... |
| 2llnma2 36919 | Two different intersecting... |
| 2llnma2rN 36920 | Two different intersecting... |
| cdlema1N 36921 | A condition for required f... |
| cdlema2N 36922 | A condition for required f... |
| cdlemblem 36923 | Lemma for ~ cdlemb . (Con... |
| cdlemb 36924 | Given two atoms not less t... |
| paddfval 36927 | Projective subspace sum op... |
| paddval 36928 | Projective subspace sum op... |
| elpadd 36929 | Member of a projective sub... |
| elpaddn0 36930 | Member of projective subsp... |
| paddvaln0N 36931 | Projective subspace sum op... |
| elpaddri 36932 | Condition implying members... |
| elpaddatriN 36933 | Condition implying members... |
| elpaddat 36934 | Membership in a projective... |
| elpaddatiN 36935 | Consequence of membership ... |
| elpadd2at 36936 | Membership in a projective... |
| elpadd2at2 36937 | Membership in a projective... |
| paddunssN 36938 | Projective subspace sum in... |
| elpadd0 36939 | Member of projective subsp... |
| paddval0 36940 | Projective subspace sum wi... |
| padd01 36941 | Projective subspace sum wi... |
| padd02 36942 | Projective subspace sum wi... |
| paddcom 36943 | Projective subspace sum co... |
| paddssat 36944 | A projective subspace sum ... |
| sspadd1 36945 | A projective subspace sum ... |
| sspadd2 36946 | A projective subspace sum ... |
| paddss1 36947 | Subset law for projective ... |
| paddss2 36948 | Subset law for projective ... |
| paddss12 36949 | Subset law for projective ... |
| paddasslem1 36950 | Lemma for ~ paddass . (Co... |
| paddasslem2 36951 | Lemma for ~ paddass . (Co... |
| paddasslem3 36952 | Lemma for ~ paddass . Res... |
| paddasslem4 36953 | Lemma for ~ paddass . Com... |
| paddasslem5 36954 | Lemma for ~ paddass . Sho... |
| paddasslem6 36955 | Lemma for ~ paddass . (Co... |
| paddasslem7 36956 | Lemma for ~ paddass . Com... |
| paddasslem8 36957 | Lemma for ~ paddass . (Co... |
| paddasslem9 36958 | Lemma for ~ paddass . Com... |
| paddasslem10 36959 | Lemma for ~ paddass . Use... |
| paddasslem11 36960 | Lemma for ~ paddass . The... |
| paddasslem12 36961 | Lemma for ~ paddass . The... |
| paddasslem13 36962 | Lemma for ~ paddass . The... |
| paddasslem14 36963 | Lemma for ~ paddass . Rem... |
| paddasslem15 36964 | Lemma for ~ paddass . Use... |
| paddasslem16 36965 | Lemma for ~ paddass . Use... |
| paddasslem17 36966 | Lemma for ~ paddass . The... |
| paddasslem18 36967 | Lemma for ~ paddass . Com... |
| paddass 36968 | Projective subspace sum is... |
| padd12N 36969 | Commutative/associative la... |
| padd4N 36970 | Rearrangement of 4 terms i... |
| paddidm 36971 | Projective subspace sum is... |
| paddclN 36972 | The projective sum of two ... |
| paddssw1 36973 | Subset law for projective ... |
| paddssw2 36974 | Subset law for projective ... |
| paddss 36975 | Subset law for projective ... |
| pmodlem1 36976 | Lemma for ~ pmod1i . (Con... |
| pmodlem2 36977 | Lemma for ~ pmod1i . (Con... |
| pmod1i 36978 | The modular law holds in a... |
| pmod2iN 36979 | Dual of the modular law. ... |
| pmodN 36980 | The modular law for projec... |
| pmodl42N 36981 | Lemma derived from modular... |
| pmapjoin 36982 | The projective map of the ... |
| pmapjat1 36983 | The projective map of the ... |
| pmapjat2 36984 | The projective map of the ... |
| pmapjlln1 36985 | The projective map of the ... |
| hlmod1i 36986 | A version of the modular l... |
| atmod1i1 36987 | Version of modular law ~ p... |
| atmod1i1m 36988 | Version of modular law ~ p... |
| atmod1i2 36989 | Version of modular law ~ p... |
| llnmod1i2 36990 | Version of modular law ~ p... |
| atmod2i1 36991 | Version of modular law ~ p... |
| atmod2i2 36992 | Version of modular law ~ p... |
| llnmod2i2 36993 | Version of modular law ~ p... |
| atmod3i1 36994 | Version of modular law tha... |
| atmod3i2 36995 | Version of modular law tha... |
| atmod4i1 36996 | Version of modular law tha... |
| atmod4i2 36997 | Version of modular law tha... |
| llnexchb2lem 36998 | Lemma for ~ llnexchb2 . (... |
| llnexchb2 36999 | Line exchange property (co... |
| llnexch2N 37000 | Line exchange property (co... |
| dalawlem1 37001 | Lemma for ~ dalaw . Speci... |
| dalawlem2 37002 | Lemma for ~ dalaw . Utili... |
| dalawlem3 37003 | Lemma for ~ dalaw . First... |
| dalawlem4 37004 | Lemma for ~ dalaw . Secon... |
| dalawlem5 37005 | Lemma for ~ dalaw . Speci... |
| dalawlem6 37006 | Lemma for ~ dalaw . First... |
| dalawlem7 37007 | Lemma for ~ dalaw . Secon... |
| dalawlem8 37008 | Lemma for ~ dalaw . Speci... |
| dalawlem9 37009 | Lemma for ~ dalaw . Speci... |
| dalawlem10 37010 | Lemma for ~ dalaw . Combi... |
| dalawlem11 37011 | Lemma for ~ dalaw . First... |
| dalawlem12 37012 | Lemma for ~ dalaw . Secon... |
| dalawlem13 37013 | Lemma for ~ dalaw . Speci... |
| dalawlem14 37014 | Lemma for ~ dalaw . Combi... |
| dalawlem15 37015 | Lemma for ~ dalaw . Swap ... |
| dalaw 37016 | Desargues's law, derived f... |
| pclfvalN 37019 | The projective subspace cl... |
| pclvalN 37020 | Value of the projective su... |
| pclclN 37021 | Closure of the projective ... |
| elpclN 37022 | Membership in the projecti... |
| elpcliN 37023 | Implication of membership ... |
| pclssN 37024 | Ordering is preserved by s... |
| pclssidN 37025 | A set of atoms is included... |
| pclidN 37026 | The projective subspace cl... |
| pclbtwnN 37027 | A projective subspace sand... |
| pclunN 37028 | The projective subspace cl... |
| pclun2N 37029 | The projective subspace cl... |
| pclfinN 37030 | The projective subspace cl... |
| pclcmpatN 37031 | The set of projective subs... |
| polfvalN 37034 | The projective subspace po... |
| polvalN 37035 | Value of the projective su... |
| polval2N 37036 | Alternate expression for v... |
| polsubN 37037 | The polarity of a set of a... |
| polssatN 37038 | The polarity of a set of a... |
| pol0N 37039 | The polarity of the empty ... |
| pol1N 37040 | The polarity of the whole ... |
| 2pol0N 37041 | The closed subspace closur... |
| polpmapN 37042 | The polarity of a projecti... |
| 2polpmapN 37043 | Double polarity of a proje... |
| 2polvalN 37044 | Value of double polarity. ... |
| 2polssN 37045 | A set of atoms is a subset... |
| 3polN 37046 | Triple polarity cancels to... |
| polcon3N 37047 | Contraposition law for pol... |
| 2polcon4bN 37048 | Contraposition law for pol... |
| polcon2N 37049 | Contraposition law for pol... |
| polcon2bN 37050 | Contraposition law for pol... |
| pclss2polN 37051 | The projective subspace cl... |
| pcl0N 37052 | The projective subspace cl... |
| pcl0bN 37053 | The projective subspace cl... |
| pmaplubN 37054 | The LUB of a projective ma... |
| sspmaplubN 37055 | A set of atoms is a subset... |
| 2pmaplubN 37056 | Double projective map of a... |
| paddunN 37057 | The closure of the project... |
| poldmj1N 37058 | De Morgan's law for polari... |
| pmapj2N 37059 | The projective map of the ... |
| pmapocjN 37060 | The projective map of the ... |
| polatN 37061 | The polarity of the single... |
| 2polatN 37062 | Double polarity of the sin... |
| pnonsingN 37063 | The intersection of a set ... |
| psubclsetN 37066 | The set of closed projecti... |
| ispsubclN 37067 | The predicate "is a closed... |
| psubcliN 37068 | Property of a closed proje... |
| psubcli2N 37069 | Property of a closed proje... |
| psubclsubN 37070 | A closed projective subspa... |
| psubclssatN 37071 | A closed projective subspa... |
| pmapidclN 37072 | Projective map of the LUB ... |
| 0psubclN 37073 | The empty set is a closed ... |
| 1psubclN 37074 | The set of all atoms is a ... |
| atpsubclN 37075 | A point (singleton of an a... |
| pmapsubclN 37076 | A projective map value is ... |
| ispsubcl2N 37077 | Alternate predicate for "i... |
| psubclinN 37078 | The intersection of two cl... |
| paddatclN 37079 | The projective sum of a cl... |
| pclfinclN 37080 | The projective subspace cl... |
| linepsubclN 37081 | A line is a closed project... |
| polsubclN 37082 | A polarity is a closed pro... |
| poml4N 37083 | Orthomodular law for proje... |
| poml5N 37084 | Orthomodular law for proje... |
| poml6N 37085 | Orthomodular law for proje... |
| osumcllem1N 37086 | Lemma for ~ osumclN . (Co... |
| osumcllem2N 37087 | Lemma for ~ osumclN . (Co... |
| osumcllem3N 37088 | Lemma for ~ osumclN . (Co... |
| osumcllem4N 37089 | Lemma for ~ osumclN . (Co... |
| osumcllem5N 37090 | Lemma for ~ osumclN . (Co... |
| osumcllem6N 37091 | Lemma for ~ osumclN . Use... |
| osumcllem7N 37092 | Lemma for ~ osumclN . (Co... |
| osumcllem8N 37093 | Lemma for ~ osumclN . (Co... |
| osumcllem9N 37094 | Lemma for ~ osumclN . (Co... |
| osumcllem10N 37095 | Lemma for ~ osumclN . Con... |
| osumcllem11N 37096 | Lemma for ~ osumclN . (Co... |
| osumclN 37097 | Closure of orthogonal sum.... |
| pmapojoinN 37098 | For orthogonal elements, p... |
| pexmidN 37099 | Excluded middle law for cl... |
| pexmidlem1N 37100 | Lemma for ~ pexmidN . Hol... |
| pexmidlem2N 37101 | Lemma for ~ pexmidN . (Co... |
| pexmidlem3N 37102 | Lemma for ~ pexmidN . Use... |
| pexmidlem4N 37103 | Lemma for ~ pexmidN . (Co... |
| pexmidlem5N 37104 | Lemma for ~ pexmidN . (Co... |
| pexmidlem6N 37105 | Lemma for ~ pexmidN . (Co... |
| pexmidlem7N 37106 | Lemma for ~ pexmidN . Con... |
| pexmidlem8N 37107 | Lemma for ~ pexmidN . The... |
| pexmidALTN 37108 | Excluded middle law for cl... |
| pl42lem1N 37109 | Lemma for ~ pl42N . (Cont... |
| pl42lem2N 37110 | Lemma for ~ pl42N . (Cont... |
| pl42lem3N 37111 | Lemma for ~ pl42N . (Cont... |
| pl42lem4N 37112 | Lemma for ~ pl42N . (Cont... |
| pl42N 37113 | Law holding in a Hilbert l... |
| watfvalN 37122 | The W atoms function. (Co... |
| watvalN 37123 | Value of the W atoms funct... |
| iswatN 37124 | The predicate "is a W atom... |
| lhpset 37125 | The set of co-atoms (latti... |
| islhp 37126 | The predicate "is a co-ato... |
| islhp2 37127 | The predicate "is a co-ato... |
| lhpbase 37128 | A co-atom is a member of t... |
| lhp1cvr 37129 | The lattice unit covers a ... |
| lhplt 37130 | An atom under a co-atom is... |
| lhp2lt 37131 | The join of two atoms unde... |
| lhpexlt 37132 | There exists an atom less ... |
| lhp0lt 37133 | A co-atom is greater than ... |
| lhpn0 37134 | A co-atom is nonzero. TOD... |
| lhpexle 37135 | There exists an atom under... |
| lhpexnle 37136 | There exists an atom not u... |
| lhpexle1lem 37137 | Lemma for ~ lhpexle1 and o... |
| lhpexle1 37138 | There exists an atom under... |
| lhpexle2lem 37139 | Lemma for ~ lhpexle2 . (C... |
| lhpexle2 37140 | There exists atom under a ... |
| lhpexle3lem 37141 | There exists atom under a ... |
| lhpexle3 37142 | There exists atom under a ... |
| lhpex2leN 37143 | There exist at least two d... |
| lhpoc 37144 | The orthocomplement of a c... |
| lhpoc2N 37145 | The orthocomplement of an ... |
| lhpocnle 37146 | The orthocomplement of a c... |
| lhpocat 37147 | The orthocomplement of a c... |
| lhpocnel 37148 | The orthocomplement of a c... |
| lhpocnel2 37149 | The orthocomplement of a c... |
| lhpjat1 37150 | The join of a co-atom (hyp... |
| lhpjat2 37151 | The join of a co-atom (hyp... |
| lhpj1 37152 | The join of a co-atom (hyp... |
| lhpmcvr 37153 | The meet of a lattice hype... |
| lhpmcvr2 37154 | Alternate way to express t... |
| lhpmcvr3 37155 | Specialization of ~ lhpmcv... |
| lhpmcvr4N 37156 | Specialization of ~ lhpmcv... |
| lhpmcvr5N 37157 | Specialization of ~ lhpmcv... |
| lhpmcvr6N 37158 | Specialization of ~ lhpmcv... |
| lhpm0atN 37159 | If the meet of a lattice h... |
| lhpmat 37160 | An element covered by the ... |
| lhpmatb 37161 | An element covered by the ... |
| lhp2at0 37162 | Join and meet with differe... |
| lhp2atnle 37163 | Inequality for 2 different... |
| lhp2atne 37164 | Inequality for joins with ... |
| lhp2at0nle 37165 | Inequality for 2 different... |
| lhp2at0ne 37166 | Inequality for joins with ... |
| lhpelim 37167 | Eliminate an atom not unde... |
| lhpmod2i2 37168 | Modular law for hyperplane... |
| lhpmod6i1 37169 | Modular law for hyperplane... |
| lhprelat3N 37170 | The Hilbert lattice is rel... |
| cdlemb2 37171 | Given two atoms not under ... |
| lhple 37172 | Property of a lattice elem... |
| lhpat 37173 | Create an atom under a co-... |
| lhpat4N 37174 | Property of an atom under ... |
| lhpat2 37175 | Create an atom under a co-... |
| lhpat3 37176 | There is only one atom und... |
| 4atexlemk 37177 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemw 37178 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempw 37179 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemp 37180 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemq 37181 | Lemma for ~ 4atexlem7 . (... |
| 4atexlems 37182 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemt 37183 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemutvt 37184 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempnq 37185 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnslpq 37186 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkl 37187 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemkc 37188 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemwb 37189 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempsb 37190 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemqtb 37191 | Lemma for ~ 4atexlem7 . (... |
| 4atexlempns 37192 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemswapqr 37193 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemu 37194 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemv 37195 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemunv 37196 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemtlw 37197 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemntlpq 37198 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemc 37199 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemnclw 37200 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex2 37201 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemcnd 37202 | Lemma for ~ 4atexlem7 . (... |
| 4atexlemex4 37203 | Lemma for ~ 4atexlem7 . S... |
| 4atexlemex6 37204 | Lemma for ~ 4atexlem7 . (... |
| 4atexlem7 37205 | Whenever there are at leas... |
| 4atex 37206 | Whenever there are at leas... |
| 4atex2 37207 | More general version of ~ ... |
| 4atex2-0aOLDN 37208 | Same as ~ 4atex2 except th... |
| 4atex2-0bOLDN 37209 | Same as ~ 4atex2 except th... |
| 4atex2-0cOLDN 37210 | Same as ~ 4atex2 except th... |
| 4atex3 37211 | More general version of ~ ... |
| lautset 37212 | The set of lattice automor... |
| islaut 37213 | The predictate "is a latti... |
| lautle 37214 | Less-than or equal propert... |
| laut1o 37215 | A lattice automorphism is ... |
| laut11 37216 | One-to-one property of a l... |
| lautcl 37217 | A lattice automorphism val... |
| lautcnvclN 37218 | Reverse closure of a latti... |
| lautcnvle 37219 | Less-than or equal propert... |
| lautcnv 37220 | The converse of a lattice ... |
| lautlt 37221 | Less-than property of a la... |
| lautcvr 37222 | Covering property of a lat... |
| lautj 37223 | Meet property of a lattice... |
| lautm 37224 | Meet property of a lattice... |
| lauteq 37225 | A lattice automorphism arg... |
| idlaut 37226 | The identity function is a... |
| lautco 37227 | The composition of two lat... |
| pautsetN 37228 | The set of projective auto... |
| ispautN 37229 | The predictate "is a proje... |
| ldilfset 37238 | The mapping from fiducial ... |
| ldilset 37239 | The set of lattice dilatio... |
| isldil 37240 | The predicate "is a lattic... |
| ldillaut 37241 | A lattice dilation is an a... |
| ldil1o 37242 | A lattice dilation is a on... |
| ldilval 37243 | Value of a lattice dilatio... |
| idldil 37244 | The identity function is a... |
| ldilcnv 37245 | The converse of a lattice ... |
| ldilco 37246 | The composition of two lat... |
| ltrnfset 37247 | The set of all lattice tra... |
| ltrnset 37248 | The set of lattice transla... |
| isltrn 37249 | The predicate "is a lattic... |
| isltrn2N 37250 | The predicate "is a lattic... |
| ltrnu 37251 | Uniqueness property of a l... |
| ltrnldil 37252 | A lattice translation is a... |
| ltrnlaut 37253 | A lattice translation is a... |
| ltrn1o 37254 | A lattice translation is a... |
| ltrncl 37255 | Closure of a lattice trans... |
| ltrn11 37256 | One-to-one property of a l... |
| ltrncnvnid 37257 | If a translation is differ... |
| ltrncoidN 37258 | Two translations are equal... |
| ltrnle 37259 | Less-than or equal propert... |
| ltrncnvleN 37260 | Less-than or equal propert... |
| ltrnm 37261 | Lattice translation of a m... |
| ltrnj 37262 | Lattice translation of a m... |
| ltrncvr 37263 | Covering property of a lat... |
| ltrnval1 37264 | Value of a lattice transla... |
| ltrnid 37265 | A lattice translation is t... |
| ltrnnid 37266 | If a lattice translation i... |
| ltrnatb 37267 | The lattice translation of... |
| ltrncnvatb 37268 | The converse of the lattic... |
| ltrnel 37269 | The lattice translation of... |
| ltrnat 37270 | The lattice translation of... |
| ltrncnvat 37271 | The converse of the lattic... |
| ltrncnvel 37272 | The converse of the lattic... |
| ltrncoelN 37273 | Composition of lattice tra... |
| ltrncoat 37274 | Composition of lattice tra... |
| ltrncoval 37275 | Two ways to express value ... |
| ltrncnv 37276 | The converse of a lattice ... |
| ltrn11at 37277 | Frequently used one-to-one... |
| ltrneq2 37278 | The equality of two transl... |
| ltrneq 37279 | The equality of two transl... |
| idltrn 37280 | The identity function is a... |
| ltrnmw 37281 | Property of lattice transl... |
| dilfsetN 37282 | The mapping from fiducial ... |
| dilsetN 37283 | The set of dilations for a... |
| isdilN 37284 | The predicate "is a dilati... |
| trnfsetN 37285 | The mapping from fiducial ... |
| trnsetN 37286 | The set of translations fo... |
| istrnN 37287 | The predicate "is a transl... |
| trlfset 37290 | The set of all traces of l... |
| trlset 37291 | The set of traces of latti... |
| trlval 37292 | The value of the trace of ... |
| trlval2 37293 | The value of the trace of ... |
| trlcl 37294 | Closure of the trace of a ... |
| trlcnv 37295 | The trace of the converse ... |
| trljat1 37296 | The value of a translation... |
| trljat2 37297 | The value of a translation... |
| trljat3 37298 | The value of a translation... |
| trlat 37299 | If an atom differs from it... |
| trl0 37300 | If an atom not under the f... |
| trlator0 37301 | The trace of a lattice tra... |
| trlatn0 37302 | The trace of a lattice tra... |
| trlnidat 37303 | The trace of a lattice tra... |
| ltrnnidn 37304 | If a lattice translation i... |
| ltrnideq 37305 | Property of the identity l... |
| trlid0 37306 | The trace of the identity ... |
| trlnidatb 37307 | A lattice translation is n... |
| trlid0b 37308 | A lattice translation is t... |
| trlnid 37309 | Different translations wit... |
| ltrn2ateq 37310 | Property of the equality o... |
| ltrnateq 37311 | If any atom (under ` W ` )... |
| ltrnatneq 37312 | If any atom (under ` W ` )... |
| ltrnatlw 37313 | If the value of an atom eq... |
| trlle 37314 | The trace of a lattice tra... |
| trlne 37315 | The trace of a lattice tra... |
| trlnle 37316 | The atom not under the fid... |
| trlval3 37317 | The value of the trace of ... |
| trlval4 37318 | The value of the trace of ... |
| trlval5 37319 | The value of the trace of ... |
| arglem1N 37320 | Lemma for Desargues's law.... |
| cdlemc1 37321 | Part of proof of Lemma C i... |
| cdlemc2 37322 | Part of proof of Lemma C i... |
| cdlemc3 37323 | Part of proof of Lemma C i... |
| cdlemc4 37324 | Part of proof of Lemma C i... |
| cdlemc5 37325 | Lemma for ~ cdlemc . (Con... |
| cdlemc6 37326 | Lemma for ~ cdlemc . (Con... |
| cdlemc 37327 | Lemma C in [Crawley] p. 11... |
| cdlemd1 37328 | Part of proof of Lemma D i... |
| cdlemd2 37329 | Part of proof of Lemma D i... |
| cdlemd3 37330 | Part of proof of Lemma D i... |
| cdlemd4 37331 | Part of proof of Lemma D i... |
| cdlemd5 37332 | Part of proof of Lemma D i... |
| cdlemd6 37333 | Part of proof of Lemma D i... |
| cdlemd7 37334 | Part of proof of Lemma D i... |
| cdlemd8 37335 | Part of proof of Lemma D i... |
| cdlemd9 37336 | Part of proof of Lemma D i... |
| cdlemd 37337 | If two translations agree ... |
| ltrneq3 37338 | Two translations agree at ... |
| cdleme00a 37339 | Part of proof of Lemma E i... |
| cdleme0aa 37340 | Part of proof of Lemma E i... |
| cdleme0a 37341 | Part of proof of Lemma E i... |
| cdleme0b 37342 | Part of proof of Lemma E i... |
| cdleme0c 37343 | Part of proof of Lemma E i... |
| cdleme0cp 37344 | Part of proof of Lemma E i... |
| cdleme0cq 37345 | Part of proof of Lemma E i... |
| cdleme0dN 37346 | Part of proof of Lemma E i... |
| cdleme0e 37347 | Part of proof of Lemma E i... |
| cdleme0fN 37348 | Part of proof of Lemma E i... |
| cdleme0gN 37349 | Part of proof of Lemma E i... |
| cdlemeulpq 37350 | Part of proof of Lemma E i... |
| cdleme01N 37351 | Part of proof of Lemma E i... |
| cdleme02N 37352 | Part of proof of Lemma E i... |
| cdleme0ex1N 37353 | Part of proof of Lemma E i... |
| cdleme0ex2N 37354 | Part of proof of Lemma E i... |
| cdleme0moN 37355 | Part of proof of Lemma E i... |
| cdleme1b 37356 | Part of proof of Lemma E i... |
| cdleme1 37357 | Part of proof of Lemma E i... |
| cdleme2 37358 | Part of proof of Lemma E i... |
| cdleme3b 37359 | Part of proof of Lemma E i... |
| cdleme3c 37360 | Part of proof of Lemma E i... |
| cdleme3d 37361 | Part of proof of Lemma E i... |
| cdleme3e 37362 | Part of proof of Lemma E i... |
| cdleme3fN 37363 | Part of proof of Lemma E i... |
| cdleme3g 37364 | Part of proof of Lemma E i... |
| cdleme3h 37365 | Part of proof of Lemma E i... |
| cdleme3fa 37366 | Part of proof of Lemma E i... |
| cdleme3 37367 | Part of proof of Lemma E i... |
| cdleme4 37368 | Part of proof of Lemma E i... |
| cdleme4a 37369 | Part of proof of Lemma E i... |
| cdleme5 37370 | Part of proof of Lemma E i... |
| cdleme6 37371 | Part of proof of Lemma E i... |
| cdleme7aa 37372 | Part of proof of Lemma E i... |
| cdleme7a 37373 | Part of proof of Lemma E i... |
| cdleme7b 37374 | Part of proof of Lemma E i... |
| cdleme7c 37375 | Part of proof of Lemma E i... |
| cdleme7d 37376 | Part of proof of Lemma E i... |
| cdleme7e 37377 | Part of proof of Lemma E i... |
| cdleme7ga 37378 | Part of proof of Lemma E i... |
| cdleme7 37379 | Part of proof of Lemma E i... |
| cdleme8 37380 | Part of proof of Lemma E i... |
| cdleme9a 37381 | Part of proof of Lemma E i... |
| cdleme9b 37382 | Utility lemma for Lemma E ... |
| cdleme9 37383 | Part of proof of Lemma E i... |
| cdleme10 37384 | Part of proof of Lemma E i... |
| cdleme8tN 37385 | Part of proof of Lemma E i... |
| cdleme9taN 37386 | Part of proof of Lemma E i... |
| cdleme9tN 37387 | Part of proof of Lemma E i... |
| cdleme10tN 37388 | Part of proof of Lemma E i... |
| cdleme16aN 37389 | Part of proof of Lemma E i... |
| cdleme11a 37390 | Part of proof of Lemma E i... |
| cdleme11c 37391 | Part of proof of Lemma E i... |
| cdleme11dN 37392 | Part of proof of Lemma E i... |
| cdleme11e 37393 | Part of proof of Lemma E i... |
| cdleme11fN 37394 | Part of proof of Lemma E i... |
| cdleme11g 37395 | Part of proof of Lemma E i... |
| cdleme11h 37396 | Part of proof of Lemma E i... |
| cdleme11j 37397 | Part of proof of Lemma E i... |
| cdleme11k 37398 | Part of proof of Lemma E i... |
| cdleme11l 37399 | Part of proof of Lemma E i... |
| cdleme11 37400 | Part of proof of Lemma E i... |
| cdleme12 37401 | Part of proof of Lemma E i... |
| cdleme13 37402 | Part of proof of Lemma E i... |
| cdleme14 37403 | Part of proof of Lemma E i... |
| cdleme15a 37404 | Part of proof of Lemma E i... |
| cdleme15b 37405 | Part of proof of Lemma E i... |
| cdleme15c 37406 | Part of proof of Lemma E i... |
| cdleme15d 37407 | Part of proof of Lemma E i... |
| cdleme15 37408 | Part of proof of Lemma E i... |
| cdleme16b 37409 | Part of proof of Lemma E i... |
| cdleme16c 37410 | Part of proof of Lemma E i... |
| cdleme16d 37411 | Part of proof of Lemma E i... |
| cdleme16e 37412 | Part of proof of Lemma E i... |
| cdleme16f 37413 | Part of proof of Lemma E i... |
| cdleme16g 37414 | Part of proof of Lemma E i... |
| cdleme16 37415 | Part of proof of Lemma E i... |
| cdleme17a 37416 | Part of proof of Lemma E i... |
| cdleme17b 37417 | Lemma leading to ~ cdleme1... |
| cdleme17c 37418 | Part of proof of Lemma E i... |
| cdleme17d1 37419 | Part of proof of Lemma E i... |
| cdleme0nex 37420 | Part of proof of Lemma E i... |
| cdleme18a 37421 | Part of proof of Lemma E i... |
| cdleme18b 37422 | Part of proof of Lemma E i... |
| cdleme18c 37423 | Part of proof of Lemma E i... |
| cdleme22gb 37424 | Utility lemma for Lemma E ... |
| cdleme18d 37425 | Part of proof of Lemma E i... |
| cdlemesner 37426 | Part of proof of Lemma E i... |
| cdlemedb 37427 | Part of proof of Lemma E i... |
| cdlemeda 37428 | Part of proof of Lemma E i... |
| cdlemednpq 37429 | Part of proof of Lemma E i... |
| cdlemednuN 37430 | Part of proof of Lemma E i... |
| cdleme20zN 37431 | Part of proof of Lemma E i... |
| cdleme20y 37432 | Part of proof of Lemma E i... |
| cdleme19a 37433 | Part of proof of Lemma E i... |
| cdleme19b 37434 | Part of proof of Lemma E i... |
| cdleme19c 37435 | Part of proof of Lemma E i... |
| cdleme19d 37436 | Part of proof of Lemma E i... |
| cdleme19e 37437 | Part of proof of Lemma E i... |
| cdleme19f 37438 | Part of proof of Lemma E i... |
| cdleme20aN 37439 | Part of proof of Lemma E i... |
| cdleme20bN 37440 | Part of proof of Lemma E i... |
| cdleme20c 37441 | Part of proof of Lemma E i... |
| cdleme20d 37442 | Part of proof of Lemma E i... |
| cdleme20e 37443 | Part of proof of Lemma E i... |
| cdleme20f 37444 | Part of proof of Lemma E i... |
| cdleme20g 37445 | Part of proof of Lemma E i... |
| cdleme20h 37446 | Part of proof of Lemma E i... |
| cdleme20i 37447 | Part of proof of Lemma E i... |
| cdleme20j 37448 | Part of proof of Lemma E i... |
| cdleme20k 37449 | Part of proof of Lemma E i... |
| cdleme20l1 37450 | Part of proof of Lemma E i... |
| cdleme20l2 37451 | Part of proof of Lemma E i... |
| cdleme20l 37452 | Part of proof of Lemma E i... |
| cdleme20m 37453 | Part of proof of Lemma E i... |
| cdleme20 37454 | Combine ~ cdleme19f and ~ ... |
| cdleme21a 37455 | Part of proof of Lemma E i... |
| cdleme21b 37456 | Part of proof of Lemma E i... |
| cdleme21c 37457 | Part of proof of Lemma E i... |
| cdleme21at 37458 | Part of proof of Lemma E i... |
| cdleme21ct 37459 | Part of proof of Lemma E i... |
| cdleme21d 37460 | Part of proof of Lemma E i... |
| cdleme21e 37461 | Part of proof of Lemma E i... |
| cdleme21f 37462 | Part of proof of Lemma E i... |
| cdleme21g 37463 | Part of proof of Lemma E i... |
| cdleme21h 37464 | Part of proof of Lemma E i... |
| cdleme21i 37465 | Part of proof of Lemma E i... |
| cdleme21j 37466 | Combine ~ cdleme20 and ~ c... |
| cdleme21 37467 | Part of proof of Lemma E i... |
| cdleme21k 37468 | Eliminate ` S =/= T ` cond... |
| cdleme22aa 37469 | Part of proof of Lemma E i... |
| cdleme22a 37470 | Part of proof of Lemma E i... |
| cdleme22b 37471 | Part of proof of Lemma E i... |
| cdleme22cN 37472 | Part of proof of Lemma E i... |
| cdleme22d 37473 | Part of proof of Lemma E i... |
| cdleme22e 37474 | Part of proof of Lemma E i... |
| cdleme22eALTN 37475 | Part of proof of Lemma E i... |
| cdleme22f 37476 | Part of proof of Lemma E i... |
| cdleme22f2 37477 | Part of proof of Lemma E i... |
| cdleme22g 37478 | Part of proof of Lemma E i... |
| cdleme23a 37479 | Part of proof of Lemma E i... |
| cdleme23b 37480 | Part of proof of Lemma E i... |
| cdleme23c 37481 | Part of proof of Lemma E i... |
| cdleme24 37482 | Quantified version of ~ cd... |
| cdleme25a 37483 | Lemma for ~ cdleme25b . (... |
| cdleme25b 37484 | Transform ~ cdleme24 . TO... |
| cdleme25c 37485 | Transform ~ cdleme25b . (... |
| cdleme25dN 37486 | Transform ~ cdleme25c . (... |
| cdleme25cl 37487 | Show closure of the unique... |
| cdleme25cv 37488 | Change bound variables in ... |
| cdleme26e 37489 | Part of proof of Lemma E i... |
| cdleme26ee 37490 | Part of proof of Lemma E i... |
| cdleme26eALTN 37491 | Part of proof of Lemma E i... |
| cdleme26fALTN 37492 | Part of proof of Lemma E i... |
| cdleme26f 37493 | Part of proof of Lemma E i... |
| cdleme26f2ALTN 37494 | Part of proof of Lemma E i... |
| cdleme26f2 37495 | Part of proof of Lemma E i... |
| cdleme27cl 37496 | Part of proof of Lemma E i... |
| cdleme27a 37497 | Part of proof of Lemma E i... |
| cdleme27b 37498 | Lemma for ~ cdleme27N . (... |
| cdleme27N 37499 | Part of proof of Lemma E i... |
| cdleme28a 37500 | Lemma for ~ cdleme25b . T... |
| cdleme28b 37501 | Lemma for ~ cdleme25b . T... |
| cdleme28c 37502 | Part of proof of Lemma E i... |
| cdleme28 37503 | Quantified version of ~ cd... |
| cdleme29ex 37504 | Lemma for ~ cdleme29b . (... |
| cdleme29b 37505 | Transform ~ cdleme28 . (C... |
| cdleme29c 37506 | Transform ~ cdleme28b . (... |
| cdleme29cl 37507 | Show closure of the unique... |
| cdleme30a 37508 | Part of proof of Lemma E i... |
| cdleme31so 37509 | Part of proof of Lemma E i... |
| cdleme31sn 37510 | Part of proof of Lemma E i... |
| cdleme31sn1 37511 | Part of proof of Lemma E i... |
| cdleme31se 37512 | Part of proof of Lemma D i... |
| cdleme31se2 37513 | Part of proof of Lemma D i... |
| cdleme31sc 37514 | Part of proof of Lemma E i... |
| cdleme31sde 37515 | Part of proof of Lemma D i... |
| cdleme31snd 37516 | Part of proof of Lemma D i... |
| cdleme31sdnN 37517 | Part of proof of Lemma E i... |
| cdleme31sn1c 37518 | Part of proof of Lemma E i... |
| cdleme31sn2 37519 | Part of proof of Lemma E i... |
| cdleme31fv 37520 | Part of proof of Lemma E i... |
| cdleme31fv1 37521 | Part of proof of Lemma E i... |
| cdleme31fv1s 37522 | Part of proof of Lemma E i... |
| cdleme31fv2 37523 | Part of proof of Lemma E i... |
| cdleme31id 37524 | Part of proof of Lemma E i... |
| cdlemefrs29pre00 37525 | ***START OF VALUE AT ATOM ... |
| cdlemefrs29bpre0 37526 | TODO fix comment. (Contri... |
| cdlemefrs29bpre1 37527 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29cpre1 37528 | TODO: FIX COMMENT. (Contr... |
| cdlemefrs29clN 37529 | TODO: NOT USED? Show clo... |
| cdlemefrs32fva 37530 | Part of proof of Lemma E i... |
| cdlemefrs32fva1 37531 | Part of proof of Lemma E i... |
| cdlemefr29exN 37532 | Lemma for ~ cdlemefs29bpre... |
| cdlemefr27cl 37533 | Part of proof of Lemma E i... |
| cdlemefr32sn2aw 37534 | Show that ` [_ R / s ]_ N ... |
| cdlemefr32snb 37535 | Show closure of ` [_ R / s... |
| cdlemefr29bpre0N 37536 | TODO fix comment. (Contri... |
| cdlemefr29clN 37537 | Show closure of the unique... |
| cdleme43frv1snN 37538 | Value of ` [_ R / s ]_ N `... |
| cdlemefr32fvaN 37539 | Part of proof of Lemma E i... |
| cdlemefr32fva1 37540 | Part of proof of Lemma E i... |
| cdlemefr31fv1 37541 | Value of ` ( F `` R ) ` wh... |
| cdlemefs29pre00N 37542 | FIX COMMENT. TODO: see if ... |
| cdlemefs27cl 37543 | Part of proof of Lemma E i... |
| cdlemefs32sn1aw 37544 | Show that ` [_ R / s ]_ N ... |
| cdlemefs32snb 37545 | Show closure of ` [_ R / s... |
| cdlemefs29bpre0N 37546 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29bpre1N 37547 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29cpre1N 37548 | TODO: FIX COMMENT. (Contr... |
| cdlemefs29clN 37549 | Show closure of the unique... |
| cdleme43fsv1snlem 37550 | Value of ` [_ R / s ]_ N `... |
| cdleme43fsv1sn 37551 | Value of ` [_ R / s ]_ N `... |
| cdlemefs32fvaN 37552 | Part of proof of Lemma E i... |
| cdlemefs32fva1 37553 | Part of proof of Lemma E i... |
| cdlemefs31fv1 37554 | Value of ` ( F `` R ) ` wh... |
| cdlemefr44 37555 | Value of f(r) when r is an... |
| cdlemefs44 37556 | Value of f_s(r) when r is ... |
| cdlemefr45 37557 | Value of f(r) when r is an... |
| cdlemefr45e 37558 | Explicit expansion of ~ cd... |
| cdlemefs45 37559 | Value of f_s(r) when r is ... |
| cdlemefs45ee 37560 | Explicit expansion of ~ cd... |
| cdlemefs45eN 37561 | Explicit expansion of ~ cd... |
| cdleme32sn1awN 37562 | Show that ` [_ R / s ]_ N ... |
| cdleme41sn3a 37563 | Show that ` [_ R / s ]_ N ... |
| cdleme32sn2awN 37564 | Show that ` [_ R / s ]_ N ... |
| cdleme32snaw 37565 | Show that ` [_ R / s ]_ N ... |
| cdleme32snb 37566 | Show closure of ` [_ R / s... |
| cdleme32fva 37567 | Part of proof of Lemma D i... |
| cdleme32fva1 37568 | Part of proof of Lemma D i... |
| cdleme32fvaw 37569 | Show that ` ( F `` R ) ` i... |
| cdleme32fvcl 37570 | Part of proof of Lemma D i... |
| cdleme32a 37571 | Part of proof of Lemma D i... |
| cdleme32b 37572 | Part of proof of Lemma D i... |
| cdleme32c 37573 | Part of proof of Lemma D i... |
| cdleme32d 37574 | Part of proof of Lemma D i... |
| cdleme32e 37575 | Part of proof of Lemma D i... |
| cdleme32f 37576 | Part of proof of Lemma D i... |
| cdleme32le 37577 | Part of proof of Lemma D i... |
| cdleme35a 37578 | Part of proof of Lemma E i... |
| cdleme35fnpq 37579 | Part of proof of Lemma E i... |
| cdleme35b 37580 | Part of proof of Lemma E i... |
| cdleme35c 37581 | Part of proof of Lemma E i... |
| cdleme35d 37582 | Part of proof of Lemma E i... |
| cdleme35e 37583 | Part of proof of Lemma E i... |
| cdleme35f 37584 | Part of proof of Lemma E i... |
| cdleme35g 37585 | Part of proof of Lemma E i... |
| cdleme35h 37586 | Part of proof of Lemma E i... |
| cdleme35h2 37587 | Part of proof of Lemma E i... |
| cdleme35sn2aw 37588 | Part of proof of Lemma E i... |
| cdleme35sn3a 37589 | Part of proof of Lemma E i... |
| cdleme36a 37590 | Part of proof of Lemma E i... |
| cdleme36m 37591 | Part of proof of Lemma E i... |
| cdleme37m 37592 | Part of proof of Lemma E i... |
| cdleme38m 37593 | Part of proof of Lemma E i... |
| cdleme38n 37594 | Part of proof of Lemma E i... |
| cdleme39a 37595 | Part of proof of Lemma E i... |
| cdleme39n 37596 | Part of proof of Lemma E i... |
| cdleme40m 37597 | Part of proof of Lemma E i... |
| cdleme40n 37598 | Part of proof of Lemma E i... |
| cdleme40v 37599 | Part of proof of Lemma E i... |
| cdleme40w 37600 | Part of proof of Lemma E i... |
| cdleme42a 37601 | Part of proof of Lemma E i... |
| cdleme42c 37602 | Part of proof of Lemma E i... |
| cdleme42d 37603 | Part of proof of Lemma E i... |
| cdleme41sn3aw 37604 | Part of proof of Lemma E i... |
| cdleme41sn4aw 37605 | Part of proof of Lemma E i... |
| cdleme41snaw 37606 | Part of proof of Lemma E i... |
| cdleme41fva11 37607 | Part of proof of Lemma E i... |
| cdleme42b 37608 | Part of proof of Lemma E i... |
| cdleme42e 37609 | Part of proof of Lemma E i... |
| cdleme42f 37610 | Part of proof of Lemma E i... |
| cdleme42g 37611 | Part of proof of Lemma E i... |
| cdleme42h 37612 | Part of proof of Lemma E i... |
| cdleme42i 37613 | Part of proof of Lemma E i... |
| cdleme42k 37614 | Part of proof of Lemma E i... |
| cdleme42ke 37615 | Part of proof of Lemma E i... |
| cdleme42keg 37616 | Part of proof of Lemma E i... |
| cdleme42mN 37617 | Part of proof of Lemma E i... |
| cdleme42mgN 37618 | Part of proof of Lemma E i... |
| cdleme43aN 37619 | Part of proof of Lemma E i... |
| cdleme43bN 37620 | Lemma for Lemma E in [Craw... |
| cdleme43cN 37621 | Part of proof of Lemma E i... |
| cdleme43dN 37622 | Part of proof of Lemma E i... |
| cdleme46f2g2 37623 | Conversion for ` G ` to re... |
| cdleme46f2g1 37624 | Conversion for ` G ` to re... |
| cdleme17d2 37625 | Part of proof of Lemma E i... |
| cdleme17d3 37626 | TODO: FIX COMMENT. (Contr... |
| cdleme17d4 37627 | TODO: FIX COMMENT. (Contr... |
| cdleme17d 37628 | Part of proof of Lemma E i... |
| cdleme48fv 37629 | Part of proof of Lemma D i... |
| cdleme48fvg 37630 | Remove ` P =/= Q ` conditi... |
| cdleme46fvaw 37631 | Show that ` ( F `` R ) ` i... |
| cdleme48bw 37632 | TODO: fix comment. TODO: ... |
| cdleme48b 37633 | TODO: fix comment. (Contr... |
| cdleme46frvlpq 37634 | Show that ` ( F `` S ) ` i... |
| cdleme46fsvlpq 37635 | Show that ` ( F `` R ) ` i... |
| cdlemeg46fvcl 37636 | TODO: fix comment. (Contr... |
| cdleme4gfv 37637 | Part of proof of Lemma D i... |
| cdlemeg47b 37638 | TODO: FIX COMMENT. (Contr... |
| cdlemeg47rv 37639 | Value of g_s(r) when r is ... |
| cdlemeg47rv2 37640 | Value of g_s(r) when r is ... |
| cdlemeg49le 37641 | Part of proof of Lemma D i... |
| cdlemeg46bOLDN 37642 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46c 37643 | TODO FIX COMMENT. (Contrib... |
| cdlemeg46rvOLDN 37644 | Value of g_s(r) when r is ... |
| cdlemeg46rv2OLDN 37645 | Value of g_s(r) when r is ... |
| cdlemeg46fvaw 37646 | Show that ` ( F `` R ) ` i... |
| cdlemeg46nlpq 37647 | Show that ` ( G `` S ) ` i... |
| cdlemeg46ngfr 37648 | TODO FIX COMMENT g(f(s))=s... |
| cdlemeg46nfgr 37649 | TODO FIX COMMENT f(g(s))=s... |
| cdlemeg46sfg 37650 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fjgN 37651 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46rjgN 37652 | NOT NEEDED? TODO FIX COMM... |
| cdlemeg46fjv 37653 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46fsfv 37654 | TODO FIX COMMENT f(r) ` \/... |
| cdlemeg46frv 37655 | TODO FIX COMMENT. (f(r) ` ... |
| cdlemeg46v1v2 37656 | TODO FIX COMMENT v_1 = v_2... |
| cdlemeg46vrg 37657 | TODO FIX COMMENT v_1 ` <_ ... |
| cdlemeg46rgv 37658 | TODO FIX COMMENT r ` <_ ` ... |
| cdlemeg46req 37659 | TODO FIX COMMENT r = (v_1 ... |
| cdlemeg46gfv 37660 | TODO FIX COMMENT p. 115 pe... |
| cdlemeg46gfr 37661 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gfre 37662 | TODO FIX COMMENT p. 116 pe... |
| cdlemeg46gf 37663 | TODO FIX COMMENT Eliminate... |
| cdlemeg46fgN 37664 | TODO FIX COMMENT p. 116 pe... |
| cdleme48d 37665 | TODO: fix comment. (Contr... |
| cdleme48gfv1 37666 | TODO: fix comment. (Contr... |
| cdleme48gfv 37667 | TODO: fix comment. (Contr... |
| cdleme48fgv 37668 | TODO: fix comment. (Contr... |
| cdlemeg49lebilem 37669 | Part of proof of Lemma D i... |
| cdleme50lebi 37670 | Part of proof of Lemma D i... |
| cdleme50eq 37671 | Part of proof of Lemma D i... |
| cdleme50f 37672 | Part of proof of Lemma D i... |
| cdleme50f1 37673 | Part of proof of Lemma D i... |
| cdleme50rnlem 37674 | Part of proof of Lemma D i... |
| cdleme50rn 37675 | Part of proof of Lemma D i... |
| cdleme50f1o 37676 | Part of proof of Lemma D i... |
| cdleme50laut 37677 | Part of proof of Lemma D i... |
| cdleme50ldil 37678 | Part of proof of Lemma D i... |
| cdleme50trn1 37679 | Part of proof that ` F ` i... |
| cdleme50trn2a 37680 | Part of proof that ` F ` i... |
| cdleme50trn2 37681 | Part of proof that ` F ` i... |
| cdleme50trn12 37682 | Part of proof that ` F ` i... |
| cdleme50trn3 37683 | Part of proof that ` F ` i... |
| cdleme50trn123 37684 | Part of proof that ` F ` i... |
| cdleme51finvfvN 37685 | Part of proof of Lemma E i... |
| cdleme51finvN 37686 | Part of proof of Lemma E i... |
| cdleme50ltrn 37687 | Part of proof of Lemma E i... |
| cdleme51finvtrN 37688 | Part of proof of Lemma E i... |
| cdleme50ex 37689 | Part of Lemma E in [Crawle... |
| cdleme 37690 | Lemma E in [Crawley] p. 11... |
| cdlemf1 37691 | Part of Lemma F in [Crawle... |
| cdlemf2 37692 | Part of Lemma F in [Crawle... |
| cdlemf 37693 | Lemma F in [Crawley] p. 11... |
| cdlemfnid 37694 | ~ cdlemf with additional c... |
| cdlemftr3 37695 | Special case of ~ cdlemf s... |
| cdlemftr2 37696 | Special case of ~ cdlemf s... |
| cdlemftr1 37697 | Part of proof of Lemma G o... |
| cdlemftr0 37698 | Special case of ~ cdlemf s... |
| trlord 37699 | The ordering of two Hilber... |
| cdlemg1a 37700 | Shorter expression for ` G... |
| cdlemg1b2 37701 | This theorem can be used t... |
| cdlemg1idlemN 37702 | Lemma for ~ cdlemg1idN . ... |
| cdlemg1fvawlemN 37703 | Lemma for ~ ltrniotafvawN ... |
| cdlemg1ltrnlem 37704 | Lemma for ~ ltrniotacl . ... |
| cdlemg1finvtrlemN 37705 | Lemma for ~ ltrniotacnvN .... |
| cdlemg1bOLDN 37706 | This theorem can be used t... |
| cdlemg1idN 37707 | Version of ~ cdleme31id wi... |
| ltrniotafvawN 37708 | Version of ~ cdleme46fvaw ... |
| ltrniotacl 37709 | Version of ~ cdleme50ltrn ... |
| ltrniotacnvN 37710 | Version of ~ cdleme51finvt... |
| ltrniotaval 37711 | Value of the unique transl... |
| ltrniotacnvval 37712 | Converse value of the uniq... |
| ltrniotaidvalN 37713 | Value of the unique transl... |
| ltrniotavalbN 37714 | Value of the unique transl... |
| cdlemeiota 37715 | A translation is uniquely ... |
| cdlemg1ci2 37716 | Any function of the form o... |
| cdlemg1cN 37717 | Any translation belongs to... |
| cdlemg1cex 37718 | Any translation is one of ... |
| cdlemg2cN 37719 | Any translation belongs to... |
| cdlemg2dN 37720 | This theorem can be used t... |
| cdlemg2cex 37721 | Any translation is one of ... |
| cdlemg2ce 37722 | Utility theorem to elimina... |
| cdlemg2jlemOLDN 37723 | Part of proof of Lemma E i... |
| cdlemg2fvlem 37724 | Lemma for ~ cdlemg2fv . (... |
| cdlemg2klem 37725 | ~ cdleme42keg with simpler... |
| cdlemg2idN 37726 | Version of ~ cdleme31id wi... |
| cdlemg3a 37727 | Part of proof of Lemma G i... |
| cdlemg2jOLDN 37728 | TODO: Replace this with ~... |
| cdlemg2fv 37729 | Value of a translation in ... |
| cdlemg2fv2 37730 | Value of a translation in ... |
| cdlemg2k 37731 | ~ cdleme42keg with simpler... |
| cdlemg2kq 37732 | ~ cdlemg2k with ` P ` and ... |
| cdlemg2l 37733 | TODO: FIX COMMENT. (Contr... |
| cdlemg2m 37734 | TODO: FIX COMMENT. (Contr... |
| cdlemg5 37735 | TODO: Is there a simpler ... |
| cdlemb3 37736 | Given two atoms not under ... |
| cdlemg7fvbwN 37737 | Properties of a translatio... |
| cdlemg4a 37738 | TODO: FIX COMMENT If fg(p... |
| cdlemg4b1 37739 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b2 37740 | TODO: FIX COMMENT. (Contr... |
| cdlemg4b12 37741 | TODO: FIX COMMENT. (Contr... |
| cdlemg4c 37742 | TODO: FIX COMMENT. (Contr... |
| cdlemg4d 37743 | TODO: FIX COMMENT. (Contr... |
| cdlemg4e 37744 | TODO: FIX COMMENT. (Contr... |
| cdlemg4f 37745 | TODO: FIX COMMENT. (Contr... |
| cdlemg4g 37746 | TODO: FIX COMMENT. (Contr... |
| cdlemg4 37747 | TODO: FIX COMMENT. (Contr... |
| cdlemg6a 37748 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6b 37749 | TODO: FIX COMMENT. TODO: ... |
| cdlemg6c 37750 | TODO: FIX COMMENT. (Contr... |
| cdlemg6d 37751 | TODO: FIX COMMENT. (Contr... |
| cdlemg6e 37752 | TODO: FIX COMMENT. (Contr... |
| cdlemg6 37753 | TODO: FIX COMMENT. (Contr... |
| cdlemg7fvN 37754 | Value of a translation com... |
| cdlemg7aN 37755 | TODO: FIX COMMENT. (Contr... |
| cdlemg7N 37756 | TODO: FIX COMMENT. (Contr... |
| cdlemg8a 37757 | TODO: FIX COMMENT. (Contr... |
| cdlemg8b 37758 | TODO: FIX COMMENT. (Contr... |
| cdlemg8c 37759 | TODO: FIX COMMENT. (Contr... |
| cdlemg8d 37760 | TODO: FIX COMMENT. (Contr... |
| cdlemg8 37761 | TODO: FIX COMMENT. (Contr... |
| cdlemg9a 37762 | TODO: FIX COMMENT. (Contr... |
| cdlemg9b 37763 | The triples ` <. P , ( F `... |
| cdlemg9 37764 | The triples ` <. P , ( F `... |
| cdlemg10b 37765 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10bALTN 37766 | TODO: FIX COMMENT. TODO: ... |
| cdlemg11a 37767 | TODO: FIX COMMENT. (Contr... |
| cdlemg11aq 37768 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10c 37769 | TODO: FIX COMMENT. TODO: ... |
| cdlemg10a 37770 | TODO: FIX COMMENT. (Contr... |
| cdlemg10 37771 | TODO: FIX COMMENT. (Contr... |
| cdlemg11b 37772 | TODO: FIX COMMENT. (Contr... |
| cdlemg12a 37773 | TODO: FIX COMMENT. (Contr... |
| cdlemg12b 37774 | The triples ` <. P , ( F `... |
| cdlemg12c 37775 | The triples ` <. P , ( F `... |
| cdlemg12d 37776 | TODO: FIX COMMENT. (Contr... |
| cdlemg12e 37777 | TODO: FIX COMMENT. (Contr... |
| cdlemg12f 37778 | TODO: FIX COMMENT. (Contr... |
| cdlemg12g 37779 | TODO: FIX COMMENT. TODO: ... |
| cdlemg12 37780 | TODO: FIX COMMENT. (Contr... |
| cdlemg13a 37781 | TODO: FIX COMMENT. (Contr... |
| cdlemg13 37782 | TODO: FIX COMMENT. (Contr... |
| cdlemg14f 37783 | TODO: FIX COMMENT. (Contr... |
| cdlemg14g 37784 | TODO: FIX COMMENT. (Contr... |
| cdlemg15a 37785 | Eliminate the ` ( F `` P )... |
| cdlemg15 37786 | Eliminate the ` ( (... |
| cdlemg16 37787 | Part of proof of Lemma G o... |
| cdlemg16ALTN 37788 | This version of ~ cdlemg16... |
| cdlemg16z 37789 | Eliminate ` ( ( F `... |
| cdlemg16zz 37790 | Eliminate ` P =/= Q ` from... |
| cdlemg17a 37791 | TODO: FIX COMMENT. (Contr... |
| cdlemg17b 37792 | Part of proof of Lemma G i... |
| cdlemg17dN 37793 | TODO: fix comment. (Contr... |
| cdlemg17dALTN 37794 | Same as ~ cdlemg17dN with ... |
| cdlemg17e 37795 | TODO: fix comment. (Contr... |
| cdlemg17f 37796 | TODO: fix comment. (Contr... |
| cdlemg17g 37797 | TODO: fix comment. (Contr... |
| cdlemg17h 37798 | TODO: fix comment. (Contr... |
| cdlemg17i 37799 | TODO: fix comment. (Contr... |
| cdlemg17ir 37800 | TODO: fix comment. (Contr... |
| cdlemg17j 37801 | TODO: fix comment. (Contr... |
| cdlemg17pq 37802 | Utility theorem for swappi... |
| cdlemg17bq 37803 | ~ cdlemg17b with ` P ` and... |
| cdlemg17iqN 37804 | ~ cdlemg17i with ` P ` and... |
| cdlemg17irq 37805 | ~ cdlemg17ir with ` P ` an... |
| cdlemg17jq 37806 | ~ cdlemg17j with ` P ` and... |
| cdlemg17 37807 | Part of Lemma G of [Crawle... |
| cdlemg18a 37808 | Show two lines are differe... |
| cdlemg18b 37809 | Lemma for ~ cdlemg18c . T... |
| cdlemg18c 37810 | Show two lines intersect a... |
| cdlemg18d 37811 | Show two lines intersect a... |
| cdlemg18 37812 | Show two lines intersect a... |
| cdlemg19a 37813 | Show two lines intersect a... |
| cdlemg19 37814 | Show two lines intersect a... |
| cdlemg20 37815 | Show two lines intersect a... |
| cdlemg21 37816 | Version of cdlemg19 with `... |
| cdlemg22 37817 | ~ cdlemg21 with ` ( F `` P... |
| cdlemg24 37818 | Combine ~ cdlemg16z and ~ ... |
| cdlemg37 37819 | Use ~ cdlemg8 to eliminate... |
| cdlemg25zz 37820 | ~ cdlemg16zz restated for ... |
| cdlemg26zz 37821 | ~ cdlemg16zz restated for ... |
| cdlemg27a 37822 | For use with case when ` (... |
| cdlemg28a 37823 | Part of proof of Lemma G o... |
| cdlemg31b0N 37824 | TODO: Fix comment. (Cont... |
| cdlemg31b0a 37825 | TODO: Fix comment. (Cont... |
| cdlemg27b 37826 | TODO: Fix comment. (Cont... |
| cdlemg31a 37827 | TODO: fix comment. (Contr... |
| cdlemg31b 37828 | TODO: fix comment. (Contr... |
| cdlemg31c 37829 | Show that when ` N ` is an... |
| cdlemg31d 37830 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33b0 37831 | TODO: Fix comment. (Cont... |
| cdlemg33c0 37832 | TODO: Fix comment. (Cont... |
| cdlemg28b 37833 | Part of proof of Lemma G o... |
| cdlemg28 37834 | Part of proof of Lemma G o... |
| cdlemg29 37835 | Eliminate ` ( F `` P ) =/=... |
| cdlemg33a 37836 | TODO: Fix comment. (Cont... |
| cdlemg33b 37837 | TODO: Fix comment. (Cont... |
| cdlemg33c 37838 | TODO: Fix comment. (Cont... |
| cdlemg33d 37839 | TODO: Fix comment. (Cont... |
| cdlemg33e 37840 | TODO: Fix comment. (Cont... |
| cdlemg33 37841 | Combine ~ cdlemg33b , ~ cd... |
| cdlemg34 37842 | Use cdlemg33 to eliminate ... |
| cdlemg35 37843 | TODO: Fix comment. TODO:... |
| cdlemg36 37844 | Use cdlemg35 to eliminate ... |
| cdlemg38 37845 | Use ~ cdlemg37 to eliminat... |
| cdlemg39 37846 | Eliminate ` =/= ` conditio... |
| cdlemg40 37847 | Eliminate ` P =/= Q ` cond... |
| cdlemg41 37848 | Convert ~ cdlemg40 to func... |
| ltrnco 37849 | The composition of two tra... |
| trlcocnv 37850 | Swap the arguments of the ... |
| trlcoabs 37851 | Absorption into a composit... |
| trlcoabs2N 37852 | Absorption of the trace of... |
| trlcoat 37853 | The trace of a composition... |
| trlcocnvat 37854 | Commonly used special case... |
| trlconid 37855 | The composition of two dif... |
| trlcolem 37856 | Lemma for ~ trlco . (Cont... |
| trlco 37857 | The trace of a composition... |
| trlcone 37858 | If two translations have d... |
| cdlemg42 37859 | Part of proof of Lemma G o... |
| cdlemg43 37860 | Part of proof of Lemma G o... |
| cdlemg44a 37861 | Part of proof of Lemma G o... |
| cdlemg44b 37862 | Eliminate ` ( F `` P ) =/=... |
| cdlemg44 37863 | Part of proof of Lemma G o... |
| cdlemg47a 37864 | TODO: fix comment. TODO: ... |
| cdlemg46 37865 | Part of proof of Lemma G o... |
| cdlemg47 37866 | Part of proof of Lemma G o... |
| cdlemg48 37867 | Eliminate ` h ` from ~ cdl... |
| ltrncom 37868 | Composition is commutative... |
| ltrnco4 37869 | Rearrange a composition of... |
| trljco 37870 | Trace joined with trace of... |
| trljco2 37871 | Trace joined with trace of... |
| tgrpfset 37874 | The translation group maps... |
| tgrpset 37875 | The translation group for ... |
| tgrpbase 37876 | The base set of the transl... |
| tgrpopr 37877 | The group operation of the... |
| tgrpov 37878 | The group operation value ... |
| tgrpgrplem 37879 | Lemma for ~ tgrpgrp . (Co... |
| tgrpgrp 37880 | The translation group is a... |
| tgrpabl 37881 | The translation group is a... |
| tendofset 37888 | The set of all trace-prese... |
| tendoset 37889 | The set of trace-preservin... |
| istendo 37890 | The predicate "is a trace-... |
| tendotp 37891 | Trace-preserving property ... |
| istendod 37892 | Deduce the predicate "is a... |
| tendof 37893 | Functionality of a trace-p... |
| tendoeq1 37894 | Condition determining equa... |
| tendovalco 37895 | Value of composition of tr... |
| tendocoval 37896 | Value of composition of en... |
| tendocl 37897 | Closure of a trace-preserv... |
| tendoco2 37898 | Distribution of compositio... |
| tendoidcl 37899 | The identity is a trace-pr... |
| tendo1mul 37900 | Multiplicative identity mu... |
| tendo1mulr 37901 | Multiplicative identity mu... |
| tendococl 37902 | The composition of two tra... |
| tendoid 37903 | The identity value of a tr... |
| tendoeq2 37904 | Condition determining equa... |
| tendoplcbv 37905 | Define sum operation for t... |
| tendopl 37906 | Value of endomorphism sum ... |
| tendopl2 37907 | Value of result of endomor... |
| tendoplcl2 37908 | Value of result of endomor... |
| tendoplco2 37909 | Value of result of endomor... |
| tendopltp 37910 | Trace-preserving property ... |
| tendoplcl 37911 | Endomorphism sum is a trac... |
| tendoplcom 37912 | The endomorphism sum opera... |
| tendoplass 37913 | The endomorphism sum opera... |
| tendodi1 37914 | Endomorphism composition d... |
| tendodi2 37915 | Endomorphism composition d... |
| tendo0cbv 37916 | Define additive identity f... |
| tendo02 37917 | Value of additive identity... |
| tendo0co2 37918 | The additive identity trac... |
| tendo0tp 37919 | Trace-preserving property ... |
| tendo0cl 37920 | The additive identity is a... |
| tendo0pl 37921 | Property of the additive i... |
| tendo0plr 37922 | Property of the additive i... |
| tendoicbv 37923 | Define inverse function fo... |
| tendoi 37924 | Value of inverse endomorph... |
| tendoi2 37925 | Value of additive inverse ... |
| tendoicl 37926 | Closure of the additive in... |
| tendoipl 37927 | Property of the additive i... |
| tendoipl2 37928 | Property of the additive i... |
| erngfset 37929 | The division rings on trac... |
| erngset 37930 | The division ring on trace... |
| erngbase 37931 | The base set of the divisi... |
| erngfplus 37932 | Ring addition operation. ... |
| erngplus 37933 | Ring addition operation. ... |
| erngplus2 37934 | Ring addition operation. ... |
| erngfmul 37935 | Ring multiplication operat... |
| erngmul 37936 | Ring addition operation. ... |
| erngfset-rN 37937 | The division rings on trac... |
| erngset-rN 37938 | The division ring on trace... |
| erngbase-rN 37939 | The base set of the divisi... |
| erngfplus-rN 37940 | Ring addition operation. ... |
| erngplus-rN 37941 | Ring addition operation. ... |
| erngplus2-rN 37942 | Ring addition operation. ... |
| erngfmul-rN 37943 | Ring multiplication operat... |
| erngmul-rN 37944 | Ring addition operation. ... |
| cdlemh1 37945 | Part of proof of Lemma H o... |
| cdlemh2 37946 | Part of proof of Lemma H o... |
| cdlemh 37947 | Lemma H of [Crawley] p. 11... |
| cdlemi1 37948 | Part of proof of Lemma I o... |
| cdlemi2 37949 | Part of proof of Lemma I o... |
| cdlemi 37950 | Lemma I of [Crawley] p. 11... |
| cdlemj1 37951 | Part of proof of Lemma J o... |
| cdlemj2 37952 | Part of proof of Lemma J o... |
| cdlemj3 37953 | Part of proof of Lemma J o... |
| tendocan 37954 | Cancellation law: if the v... |
| tendoid0 37955 | A trace-preserving endomor... |
| tendo0mul 37956 | Additive identity multipli... |
| tendo0mulr 37957 | Additive identity multipli... |
| tendo1ne0 37958 | The identity (unity) is no... |
| tendoconid 37959 | The composition (product) ... |
| tendotr 37960 | The trace of the value of ... |
| cdlemk1 37961 | Part of proof of Lemma K o... |
| cdlemk2 37962 | Part of proof of Lemma K o... |
| cdlemk3 37963 | Part of proof of Lemma K o... |
| cdlemk4 37964 | Part of proof of Lemma K o... |
| cdlemk5a 37965 | Part of proof of Lemma K o... |
| cdlemk5 37966 | Part of proof of Lemma K o... |
| cdlemk6 37967 | Part of proof of Lemma K o... |
| cdlemk8 37968 | Part of proof of Lemma K o... |
| cdlemk9 37969 | Part of proof of Lemma K o... |
| cdlemk9bN 37970 | Part of proof of Lemma K o... |
| cdlemki 37971 | Part of proof of Lemma K o... |
| cdlemkvcl 37972 | Part of proof of Lemma K o... |
| cdlemk10 37973 | Part of proof of Lemma K o... |
| cdlemksv 37974 | Part of proof of Lemma K o... |
| cdlemksel 37975 | Part of proof of Lemma K o... |
| cdlemksat 37976 | Part of proof of Lemma K o... |
| cdlemksv2 37977 | Part of proof of Lemma K o... |
| cdlemk7 37978 | Part of proof of Lemma K o... |
| cdlemk11 37979 | Part of proof of Lemma K o... |
| cdlemk12 37980 | Part of proof of Lemma K o... |
| cdlemkoatnle 37981 | Utility lemma. (Contribut... |
| cdlemk13 37982 | Part of proof of Lemma K o... |
| cdlemkole 37983 | Utility lemma. (Contribut... |
| cdlemk14 37984 | Part of proof of Lemma K o... |
| cdlemk15 37985 | Part of proof of Lemma K o... |
| cdlemk16a 37986 | Part of proof of Lemma K o... |
| cdlemk16 37987 | Part of proof of Lemma K o... |
| cdlemk17 37988 | Part of proof of Lemma K o... |
| cdlemk1u 37989 | Part of proof of Lemma K o... |
| cdlemk5auN 37990 | Part of proof of Lemma K o... |
| cdlemk5u 37991 | Part of proof of Lemma K o... |
| cdlemk6u 37992 | Part of proof of Lemma K o... |
| cdlemkj 37993 | Part of proof of Lemma K o... |
| cdlemkuvN 37994 | Part of proof of Lemma K o... |
| cdlemkuel 37995 | Part of proof of Lemma K o... |
| cdlemkuat 37996 | Part of proof of Lemma K o... |
| cdlemkuv2 37997 | Part of proof of Lemma K o... |
| cdlemk18 37998 | Part of proof of Lemma K o... |
| cdlemk19 37999 | Part of proof of Lemma K o... |
| cdlemk7u 38000 | Part of proof of Lemma K o... |
| cdlemk11u 38001 | Part of proof of Lemma K o... |
| cdlemk12u 38002 | Part of proof of Lemma K o... |
| cdlemk21N 38003 | Part of proof of Lemma K o... |
| cdlemk20 38004 | Part of proof of Lemma K o... |
| cdlemkoatnle-2N 38005 | Utility lemma. (Contribut... |
| cdlemk13-2N 38006 | Part of proof of Lemma K o... |
| cdlemkole-2N 38007 | Utility lemma. (Contribut... |
| cdlemk14-2N 38008 | Part of proof of Lemma K o... |
| cdlemk15-2N 38009 | Part of proof of Lemma K o... |
| cdlemk16-2N 38010 | Part of proof of Lemma K o... |
| cdlemk17-2N 38011 | Part of proof of Lemma K o... |
| cdlemkj-2N 38012 | Part of proof of Lemma K o... |
| cdlemkuv-2N 38013 | Part of proof of Lemma K o... |
| cdlemkuel-2N 38014 | Part of proof of Lemma K o... |
| cdlemkuv2-2 38015 | Part of proof of Lemma K o... |
| cdlemk18-2N 38016 | Part of proof of Lemma K o... |
| cdlemk19-2N 38017 | Part of proof of Lemma K o... |
| cdlemk7u-2N 38018 | Part of proof of Lemma K o... |
| cdlemk11u-2N 38019 | Part of proof of Lemma K o... |
| cdlemk12u-2N 38020 | Part of proof of Lemma K o... |
| cdlemk21-2N 38021 | Part of proof of Lemma K o... |
| cdlemk20-2N 38022 | Part of proof of Lemma K o... |
| cdlemk22 38023 | Part of proof of Lemma K o... |
| cdlemk30 38024 | Part of proof of Lemma K o... |
| cdlemkuu 38025 | Convert between function a... |
| cdlemk31 38026 | Part of proof of Lemma K o... |
| cdlemk32 38027 | Part of proof of Lemma K o... |
| cdlemkuel-3 38028 | Part of proof of Lemma K o... |
| cdlemkuv2-3N 38029 | Part of proof of Lemma K o... |
| cdlemk18-3N 38030 | Part of proof of Lemma K o... |
| cdlemk22-3 38031 | Part of proof of Lemma K o... |
| cdlemk23-3 38032 | Part of proof of Lemma K o... |
| cdlemk24-3 38033 | Part of proof of Lemma K o... |
| cdlemk25-3 38034 | Part of proof of Lemma K o... |
| cdlemk26b-3 38035 | Part of proof of Lemma K o... |
| cdlemk26-3 38036 | Part of proof of Lemma K o... |
| cdlemk27-3 38037 | Part of proof of Lemma K o... |
| cdlemk28-3 38038 | Part of proof of Lemma K o... |
| cdlemk33N 38039 | Part of proof of Lemma K o... |
| cdlemk34 38040 | Part of proof of Lemma K o... |
| cdlemk29-3 38041 | Part of proof of Lemma K o... |
| cdlemk35 38042 | Part of proof of Lemma K o... |
| cdlemk36 38043 | Part of proof of Lemma K o... |
| cdlemk37 38044 | Part of proof of Lemma K o... |
| cdlemk38 38045 | Part of proof of Lemma K o... |
| cdlemk39 38046 | Part of proof of Lemma K o... |
| cdlemk40 38047 | TODO: fix comment. (Contr... |
| cdlemk40t 38048 | TODO: fix comment. (Contr... |
| cdlemk40f 38049 | TODO: fix comment. (Contr... |
| cdlemk41 38050 | Part of proof of Lemma K o... |
| cdlemkfid1N 38051 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid1 38052 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid2N 38053 | Lemma for ~ cdlemkfid3N . ... |
| cdlemkid2 38054 | Lemma for ~ cdlemkid . (C... |
| cdlemkfid3N 38055 | TODO: is this useful or sh... |
| cdlemky 38056 | Part of proof of Lemma K o... |
| cdlemkyu 38057 | Convert between function a... |
| cdlemkyuu 38058 | ~ cdlemkyu with some hypot... |
| cdlemk11ta 38059 | Part of proof of Lemma K o... |
| cdlemk19ylem 38060 | Lemma for ~ cdlemk19y . (... |
| cdlemk11tb 38061 | Part of proof of Lemma K o... |
| cdlemk19y 38062 | ~ cdlemk19 with simpler hy... |
| cdlemkid3N 38063 | Lemma for ~ cdlemkid . (C... |
| cdlemkid4 38064 | Lemma for ~ cdlemkid . (C... |
| cdlemkid5 38065 | Lemma for ~ cdlemkid . (C... |
| cdlemkid 38066 | The value of the tau funct... |
| cdlemk35s 38067 | Substitution version of ~ ... |
| cdlemk35s-id 38068 | Substitution version of ~ ... |
| cdlemk39s 38069 | Substitution version of ~ ... |
| cdlemk39s-id 38070 | Substitution version of ~ ... |
| cdlemk42 38071 | Part of proof of Lemma K o... |
| cdlemk19xlem 38072 | Lemma for ~ cdlemk19x . (... |
| cdlemk19x 38073 | ~ cdlemk19 with simpler hy... |
| cdlemk42yN 38074 | Part of proof of Lemma K o... |
| cdlemk11tc 38075 | Part of proof of Lemma K o... |
| cdlemk11t 38076 | Part of proof of Lemma K o... |
| cdlemk45 38077 | Part of proof of Lemma K o... |
| cdlemk46 38078 | Part of proof of Lemma K o... |
| cdlemk47 38079 | Part of proof of Lemma K o... |
| cdlemk48 38080 | Part of proof of Lemma K o... |
| cdlemk49 38081 | Part of proof of Lemma K o... |
| cdlemk50 38082 | Part of proof of Lemma K o... |
| cdlemk51 38083 | Part of proof of Lemma K o... |
| cdlemk52 38084 | Part of proof of Lemma K o... |
| cdlemk53a 38085 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53b 38086 | Lemma for ~ cdlemk53 . (C... |
| cdlemk53 38087 | Part of proof of Lemma K o... |
| cdlemk54 38088 | Part of proof of Lemma K o... |
| cdlemk55a 38089 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55b 38090 | Lemma for ~ cdlemk55 . (C... |
| cdlemk55 38091 | Part of proof of Lemma K o... |
| cdlemkyyN 38092 | Part of proof of Lemma K o... |
| cdlemk43N 38093 | Part of proof of Lemma K o... |
| cdlemk35u 38094 | Substitution version of ~ ... |
| cdlemk55u1 38095 | Lemma for ~ cdlemk55u . (... |
| cdlemk55u 38096 | Part of proof of Lemma K o... |
| cdlemk39u1 38097 | Lemma for ~ cdlemk39u . (... |
| cdlemk39u 38098 | Part of proof of Lemma K o... |
| cdlemk19u1 38099 | ~ cdlemk19 with simpler hy... |
| cdlemk19u 38100 | Part of Lemma K of [Crawle... |
| cdlemk56 38101 | Part of Lemma K of [Crawle... |
| cdlemk19w 38102 | Use a fixed element to eli... |
| cdlemk56w 38103 | Use a fixed element to eli... |
| cdlemk 38104 | Lemma K of [Crawley] p. 11... |
| tendoex 38105 | Generalization of Lemma K ... |
| cdleml1N 38106 | Part of proof of Lemma L o... |
| cdleml2N 38107 | Part of proof of Lemma L o... |
| cdleml3N 38108 | Part of proof of Lemma L o... |
| cdleml4N 38109 | Part of proof of Lemma L o... |
| cdleml5N 38110 | Part of proof of Lemma L o... |
| cdleml6 38111 | Part of proof of Lemma L o... |
| cdleml7 38112 | Part of proof of Lemma L o... |
| cdleml8 38113 | Part of proof of Lemma L o... |
| cdleml9 38114 | Part of proof of Lemma L o... |
| dva1dim 38115 | Two expressions for the 1-... |
| dvhb1dimN 38116 | Two expressions for the 1-... |
| erng1lem 38117 | Value of the endomorphism ... |
| erngdvlem1 38118 | Lemma for ~ eringring . (... |
| erngdvlem2N 38119 | Lemma for ~ eringring . (... |
| erngdvlem3 38120 | Lemma for ~ eringring . (... |
| erngdvlem4 38121 | Lemma for ~ erngdv . (Con... |
| eringring 38122 | An endomorphism ring is a ... |
| erngdv 38123 | An endomorphism ring is a ... |
| erng0g 38124 | The division ring zero of ... |
| erng1r 38125 | The division ring unit of ... |
| erngdvlem1-rN 38126 | Lemma for ~ eringring . (... |
| erngdvlem2-rN 38127 | Lemma for ~ eringring . (... |
| erngdvlem3-rN 38128 | Lemma for ~ eringring . (... |
| erngdvlem4-rN 38129 | Lemma for ~ erngdv . (Con... |
| erngring-rN 38130 | An endomorphism ring is a ... |
| erngdv-rN 38131 | An endomorphism ring is a ... |
| dvafset 38134 | The constructed partial ve... |
| dvaset 38135 | The constructed partial ve... |
| dvasca 38136 | The ring base set of the c... |
| dvabase 38137 | The ring base set of the c... |
| dvafplusg 38138 | Ring addition operation fo... |
| dvaplusg 38139 | Ring addition operation fo... |
| dvaplusgv 38140 | Ring addition operation fo... |
| dvafmulr 38141 | Ring multiplication operat... |
| dvamulr 38142 | Ring multiplication operat... |
| dvavbase 38143 | The vectors (vector base s... |
| dvafvadd 38144 | The vector sum operation f... |
| dvavadd 38145 | Ring addition operation fo... |
| dvafvsca 38146 | Ring addition operation fo... |
| dvavsca 38147 | Ring addition operation fo... |
| tendospcl 38148 | Closure of endomorphism sc... |
| tendospass 38149 | Associative law for endomo... |
| tendospdi1 38150 | Forward distributive law f... |
| tendocnv 38151 | Converse of a trace-preser... |
| tendospdi2 38152 | Reverse distributive law f... |
| tendospcanN 38153 | Cancellation law for trace... |
| dvaabl 38154 | The constructed partial ve... |
| dvalveclem 38155 | Lemma for ~ dvalvec . (Co... |
| dvalvec 38156 | The constructed partial ve... |
| dva0g 38157 | The zero vector of partial... |
| diaffval 38160 | The partial isomorphism A ... |
| diafval 38161 | The partial isomorphism A ... |
| diaval 38162 | The partial isomorphism A ... |
| diaelval 38163 | Member of the partial isom... |
| diafn 38164 | Functionality and domain o... |
| diadm 38165 | Domain of the partial isom... |
| diaeldm 38166 | Member of domain of the pa... |
| diadmclN 38167 | A member of domain of the ... |
| diadmleN 38168 | A member of domain of the ... |
| dian0 38169 | The value of the partial i... |
| dia0eldmN 38170 | The lattice zero belongs t... |
| dia1eldmN 38171 | The fiducial hyperplane (t... |
| diass 38172 | The value of the partial i... |
| diael 38173 | A member of the value of t... |
| diatrl 38174 | Trace of a member of the p... |
| diaelrnN 38175 | Any value of the partial i... |
| dialss 38176 | The value of partial isomo... |
| diaord 38177 | The partial isomorphism A ... |
| dia11N 38178 | The partial isomorphism A ... |
| diaf11N 38179 | The partial isomorphism A ... |
| diaclN 38180 | Closure of partial isomorp... |
| diacnvclN 38181 | Closure of partial isomorp... |
| dia0 38182 | The value of the partial i... |
| dia1N 38183 | The value of the partial i... |
| dia1elN 38184 | The largest subspace in th... |
| diaglbN 38185 | Partial isomorphism A of a... |
| diameetN 38186 | Partial isomorphism A of a... |
| diainN 38187 | Inverse partial isomorphis... |
| diaintclN 38188 | The intersection of partia... |
| diasslssN 38189 | The partial isomorphism A ... |
| diassdvaN 38190 | The partial isomorphism A ... |
| dia1dim 38191 | Two expressions for the 1-... |
| dia1dim2 38192 | Two expressions for a 1-di... |
| dia1dimid 38193 | A vector (translation) bel... |
| dia2dimlem1 38194 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem2 38195 | Lemma for ~ dia2dim . Def... |
| dia2dimlem3 38196 | Lemma for ~ dia2dim . Def... |
| dia2dimlem4 38197 | Lemma for ~ dia2dim . Sho... |
| dia2dimlem5 38198 | Lemma for ~ dia2dim . The... |
| dia2dimlem6 38199 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem7 38200 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem8 38201 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem9 38202 | Lemma for ~ dia2dim . Eli... |
| dia2dimlem10 38203 | Lemma for ~ dia2dim . Con... |
| dia2dimlem11 38204 | Lemma for ~ dia2dim . Con... |
| dia2dimlem12 38205 | Lemma for ~ dia2dim . Obt... |
| dia2dimlem13 38206 | Lemma for ~ dia2dim . Eli... |
| dia2dim 38207 | A two-dimensional subspace... |
| dvhfset 38210 | The constructed full vecto... |
| dvhset 38211 | The constructed full vecto... |
| dvhsca 38212 | The ring of scalars of the... |
| dvhbase 38213 | The ring base set of the c... |
| dvhfplusr 38214 | Ring addition operation fo... |
| dvhfmulr 38215 | Ring multiplication operat... |
| dvhmulr 38216 | Ring multiplication operat... |
| dvhvbase 38217 | The vectors (vector base s... |
| dvhelvbasei 38218 | Vector membership in the c... |
| dvhvaddcbv 38219 | Change bound variables to ... |
| dvhvaddval 38220 | The vector sum operation f... |
| dvhfvadd 38221 | The vector sum operation f... |
| dvhvadd 38222 | The vector sum operation f... |
| dvhopvadd 38223 | The vector sum operation f... |
| dvhopvadd2 38224 | The vector sum operation f... |
| dvhvaddcl 38225 | Closure of the vector sum ... |
| dvhvaddcomN 38226 | Commutativity of vector su... |
| dvhvaddass 38227 | Associativity of vector su... |
| dvhvscacbv 38228 | Change bound variables to ... |
| dvhvscaval 38229 | The scalar product operati... |
| dvhfvsca 38230 | Scalar product operation f... |
| dvhvsca 38231 | Scalar product operation f... |
| dvhopvsca 38232 | Scalar product operation f... |
| dvhvscacl 38233 | Closure of the scalar prod... |
| tendoinvcl 38234 | Closure of multiplicative ... |
| tendolinv 38235 | Left multiplicative invers... |
| tendorinv 38236 | Right multiplicative inver... |
| dvhgrp 38237 | The full vector space ` U ... |
| dvhlveclem 38238 | Lemma for ~ dvhlvec . TOD... |
| dvhlvec 38239 | The full vector space ` U ... |
| dvhlmod 38240 | The full vector space ` U ... |
| dvh0g 38241 | The zero vector of vector ... |
| dvheveccl 38242 | Properties of a unit vecto... |
| dvhopclN 38243 | Closure of a ` DVecH ` vec... |
| dvhopaddN 38244 | Sum of ` DVecH ` vectors e... |
| dvhopspN 38245 | Scalar product of ` DVecH ... |
| dvhopN 38246 | Decompose a ` DVecH ` vect... |
| dvhopellsm 38247 | Ordered pair membership in... |
| cdlemm10N 38248 | The image of the map ` G `... |
| docaffvalN 38251 | Subspace orthocomplement f... |
| docafvalN 38252 | Subspace orthocomplement f... |
| docavalN 38253 | Subspace orthocomplement f... |
| docaclN 38254 | Closure of subspace orthoc... |
| diaocN 38255 | Value of partial isomorphi... |
| doca2N 38256 | Double orthocomplement of ... |
| doca3N 38257 | Double orthocomplement of ... |
| dvadiaN 38258 | Any closed subspace is a m... |
| diarnN 38259 | Partial isomorphism A maps... |
| diaf1oN 38260 | The partial isomorphism A ... |
| djaffvalN 38263 | Subspace join for ` DVecA ... |
| djafvalN 38264 | Subspace join for ` DVecA ... |
| djavalN 38265 | Subspace join for ` DVecA ... |
| djaclN 38266 | Closure of subspace join f... |
| djajN 38267 | Transfer lattice join to `... |
| dibffval 38270 | The partial isomorphism B ... |
| dibfval 38271 | The partial isomorphism B ... |
| dibval 38272 | The partial isomorphism B ... |
| dibopelvalN 38273 | Member of the partial isom... |
| dibval2 38274 | Value of the partial isomo... |
| dibopelval2 38275 | Member of the partial isom... |
| dibval3N 38276 | Value of the partial isomo... |
| dibelval3 38277 | Member of the partial isom... |
| dibopelval3 38278 | Member of the partial isom... |
| dibelval1st 38279 | Membership in value of the... |
| dibelval1st1 38280 | Membership in value of the... |
| dibelval1st2N 38281 | Membership in value of the... |
| dibelval2nd 38282 | Membership in value of the... |
| dibn0 38283 | The value of the partial i... |
| dibfna 38284 | Functionality and domain o... |
| dibdiadm 38285 | Domain of the partial isom... |
| dibfnN 38286 | Functionality and domain o... |
| dibdmN 38287 | Domain of the partial isom... |
| dibeldmN 38288 | Member of domain of the pa... |
| dibord 38289 | The isomorphism B for a la... |
| dib11N 38290 | The isomorphism B for a la... |
| dibf11N 38291 | The partial isomorphism A ... |
| dibclN 38292 | Closure of partial isomorp... |
| dibvalrel 38293 | The value of partial isomo... |
| dib0 38294 | The value of partial isomo... |
| dib1dim 38295 | Two expressions for the 1-... |
| dibglbN 38296 | Partial isomorphism B of a... |
| dibintclN 38297 | The intersection of partia... |
| dib1dim2 38298 | Two expressions for a 1-di... |
| dibss 38299 | The partial isomorphism B ... |
| diblss 38300 | The value of partial isomo... |
| diblsmopel 38301 | Membership in subspace sum... |
| dicffval 38304 | The partial isomorphism C ... |
| dicfval 38305 | The partial isomorphism C ... |
| dicval 38306 | The partial isomorphism C ... |
| dicopelval 38307 | Membership in value of the... |
| dicelvalN 38308 | Membership in value of the... |
| dicval2 38309 | The partial isomorphism C ... |
| dicelval3 38310 | Member of the partial isom... |
| dicopelval2 38311 | Membership in value of the... |
| dicelval2N 38312 | Membership in value of the... |
| dicfnN 38313 | Functionality and domain o... |
| dicdmN 38314 | Domain of the partial isom... |
| dicvalrelN 38315 | The value of partial isomo... |
| dicssdvh 38316 | The partial isomorphism C ... |
| dicelval1sta 38317 | Membership in value of the... |
| dicelval1stN 38318 | Membership in value of the... |
| dicelval2nd 38319 | Membership in value of the... |
| dicvaddcl 38320 | Membership in value of the... |
| dicvscacl 38321 | Membership in value of the... |
| dicn0 38322 | The value of the partial i... |
| diclss 38323 | The value of partial isomo... |
| diclspsn 38324 | The value of isomorphism C... |
| cdlemn2 38325 | Part of proof of Lemma N o... |
| cdlemn2a 38326 | Part of proof of Lemma N o... |
| cdlemn3 38327 | Part of proof of Lemma N o... |
| cdlemn4 38328 | Part of proof of Lemma N o... |
| cdlemn4a 38329 | Part of proof of Lemma N o... |
| cdlemn5pre 38330 | Part of proof of Lemma N o... |
| cdlemn5 38331 | Part of proof of Lemma N o... |
| cdlemn6 38332 | Part of proof of Lemma N o... |
| cdlemn7 38333 | Part of proof of Lemma N o... |
| cdlemn8 38334 | Part of proof of Lemma N o... |
| cdlemn9 38335 | Part of proof of Lemma N o... |
| cdlemn10 38336 | Part of proof of Lemma N o... |
| cdlemn11a 38337 | Part of proof of Lemma N o... |
| cdlemn11b 38338 | Part of proof of Lemma N o... |
| cdlemn11c 38339 | Part of proof of Lemma N o... |
| cdlemn11pre 38340 | Part of proof of Lemma N o... |
| cdlemn11 38341 | Part of proof of Lemma N o... |
| cdlemn 38342 | Lemma N of [Crawley] p. 12... |
| dihordlem6 38343 | Part of proof of Lemma N o... |
| dihordlem7 38344 | Part of proof of Lemma N o... |
| dihordlem7b 38345 | Part of proof of Lemma N o... |
| dihjustlem 38346 | Part of proof after Lemma ... |
| dihjust 38347 | Part of proof after Lemma ... |
| dihord1 38348 | Part of proof after Lemma ... |
| dihord2a 38349 | Part of proof after Lemma ... |
| dihord2b 38350 | Part of proof after Lemma ... |
| dihord2cN 38351 | Part of proof after Lemma ... |
| dihord11b 38352 | Part of proof after Lemma ... |
| dihord10 38353 | Part of proof after Lemma ... |
| dihord11c 38354 | Part of proof after Lemma ... |
| dihord2pre 38355 | Part of proof after Lemma ... |
| dihord2pre2 38356 | Part of proof after Lemma ... |
| dihord2 38357 | Part of proof after Lemma ... |
| dihffval 38360 | The isomorphism H for a la... |
| dihfval 38361 | Isomorphism H for a lattic... |
| dihval 38362 | Value of isomorphism H for... |
| dihvalc 38363 | Value of isomorphism H for... |
| dihlsscpre 38364 | Closure of isomorphism H f... |
| dihvalcqpre 38365 | Value of isomorphism H for... |
| dihvalcq 38366 | Value of isomorphism H for... |
| dihvalb 38367 | Value of isomorphism H for... |
| dihopelvalbN 38368 | Ordered pair member of the... |
| dihvalcqat 38369 | Value of isomorphism H for... |
| dih1dimb 38370 | Two expressions for a 1-di... |
| dih1dimb2 38371 | Isomorphism H at an atom u... |
| dih1dimc 38372 | Isomorphism H at an atom n... |
| dib2dim 38373 | Extend ~ dia2dim to partia... |
| dih2dimb 38374 | Extend ~ dib2dim to isomor... |
| dih2dimbALTN 38375 | Extend ~ dia2dim to isomor... |
| dihopelvalcqat 38376 | Ordered pair member of the... |
| dihvalcq2 38377 | Value of isomorphism H for... |
| dihopelvalcpre 38378 | Member of value of isomorp... |
| dihopelvalc 38379 | Member of value of isomorp... |
| dihlss 38380 | The value of isomorphism H... |
| dihss 38381 | The value of isomorphism H... |
| dihssxp 38382 | An isomorphism H value is ... |
| dihopcl 38383 | Closure of an ordered pair... |
| xihopellsmN 38384 | Ordered pair membership in... |
| dihopellsm 38385 | Ordered pair membership in... |
| dihord6apre 38386 | Part of proof that isomorp... |
| dihord3 38387 | The isomorphism H for a la... |
| dihord4 38388 | The isomorphism H for a la... |
| dihord5b 38389 | Part of proof that isomorp... |
| dihord6b 38390 | Part of proof that isomorp... |
| dihord6a 38391 | Part of proof that isomorp... |
| dihord5apre 38392 | Part of proof that isomorp... |
| dihord5a 38393 | Part of proof that isomorp... |
| dihord 38394 | The isomorphism H is order... |
| dih11 38395 | The isomorphism H is one-t... |
| dihf11lem 38396 | Functionality of the isomo... |
| dihf11 38397 | The isomorphism H for a la... |
| dihfn 38398 | Functionality and domain o... |
| dihdm 38399 | Domain of isomorphism H. (... |
| dihcl 38400 | Closure of isomorphism H. ... |
| dihcnvcl 38401 | Closure of isomorphism H c... |
| dihcnvid1 38402 | The converse isomorphism o... |
| dihcnvid2 38403 | The isomorphism of a conve... |
| dihcnvord 38404 | Ordering property for conv... |
| dihcnv11 38405 | The converse of isomorphis... |
| dihsslss 38406 | The isomorphism H maps to ... |
| dihrnlss 38407 | The isomorphism H maps to ... |
| dihrnss 38408 | The isomorphism H maps to ... |
| dihvalrel 38409 | The value of isomorphism H... |
| dih0 38410 | The value of isomorphism H... |
| dih0bN 38411 | A lattice element is zero ... |
| dih0vbN 38412 | A vector is zero iff its s... |
| dih0cnv 38413 | The isomorphism H converse... |
| dih0rn 38414 | The zero subspace belongs ... |
| dih0sb 38415 | A subspace is zero iff the... |
| dih1 38416 | The value of isomorphism H... |
| dih1rn 38417 | The full vector space belo... |
| dih1cnv 38418 | The isomorphism H converse... |
| dihwN 38419 | Value of isomorphism H at ... |
| dihmeetlem1N 38420 | Isomorphism H of a conjunc... |
| dihglblem5apreN 38421 | A conjunction property of ... |
| dihglblem5aN 38422 | A conjunction property of ... |
| dihglblem2aN 38423 | Lemma for isomorphism H of... |
| dihglblem2N 38424 | The GLB of a set of lattic... |
| dihglblem3N 38425 | Isomorphism H of a lattice... |
| dihglblem3aN 38426 | Isomorphism H of a lattice... |
| dihglblem4 38427 | Isomorphism H of a lattice... |
| dihglblem5 38428 | Isomorphism H of a lattice... |
| dihmeetlem2N 38429 | Isomorphism H of a conjunc... |
| dihglbcpreN 38430 | Isomorphism H of a lattice... |
| dihglbcN 38431 | Isomorphism H of a lattice... |
| dihmeetcN 38432 | Isomorphism H of a lattice... |
| dihmeetbN 38433 | Isomorphism H of a lattice... |
| dihmeetbclemN 38434 | Lemma for isomorphism H of... |
| dihmeetlem3N 38435 | Lemma for isomorphism H of... |
| dihmeetlem4preN 38436 | Lemma for isomorphism H of... |
| dihmeetlem4N 38437 | Lemma for isomorphism H of... |
| dihmeetlem5 38438 | Part of proof that isomorp... |
| dihmeetlem6 38439 | Lemma for isomorphism H of... |
| dihmeetlem7N 38440 | Lemma for isomorphism H of... |
| dihjatc1 38441 | Lemma for isomorphism H of... |
| dihjatc2N 38442 | Isomorphism H of join with... |
| dihjatc3 38443 | Isomorphism H of join with... |
| dihmeetlem8N 38444 | Lemma for isomorphism H of... |
| dihmeetlem9N 38445 | Lemma for isomorphism H of... |
| dihmeetlem10N 38446 | Lemma for isomorphism H of... |
| dihmeetlem11N 38447 | Lemma for isomorphism H of... |
| dihmeetlem12N 38448 | Lemma for isomorphism H of... |
| dihmeetlem13N 38449 | Lemma for isomorphism H of... |
| dihmeetlem14N 38450 | Lemma for isomorphism H of... |
| dihmeetlem15N 38451 | Lemma for isomorphism H of... |
| dihmeetlem16N 38452 | Lemma for isomorphism H of... |
| dihmeetlem17N 38453 | Lemma for isomorphism H of... |
| dihmeetlem18N 38454 | Lemma for isomorphism H of... |
| dihmeetlem19N 38455 | Lemma for isomorphism H of... |
| dihmeetlem20N 38456 | Lemma for isomorphism H of... |
| dihmeetALTN 38457 | Isomorphism H of a lattice... |
| dih1dimatlem0 38458 | Lemma for ~ dih1dimat . (... |
| dih1dimatlem 38459 | Lemma for ~ dih1dimat . (... |
| dih1dimat 38460 | Any 1-dimensional subspace... |
| dihlsprn 38461 | The span of a vector belon... |
| dihlspsnssN 38462 | A subspace included in a 1... |
| dihlspsnat 38463 | The inverse isomorphism H ... |
| dihatlat 38464 | The isomorphism H of an at... |
| dihat 38465 | There exists at least one ... |
| dihpN 38466 | The value of isomorphism H... |
| dihlatat 38467 | The reverse isomorphism H ... |
| dihatexv 38468 | There is a nonzero vector ... |
| dihatexv2 38469 | There is a nonzero vector ... |
| dihglblem6 38470 | Isomorphism H of a lattice... |
| dihglb 38471 | Isomorphism H of a lattice... |
| dihglb2 38472 | Isomorphism H of a lattice... |
| dihmeet 38473 | Isomorphism H of a lattice... |
| dihintcl 38474 | The intersection of closed... |
| dihmeetcl 38475 | Closure of closed subspace... |
| dihmeet2 38476 | Reverse isomorphism H of a... |
| dochffval 38479 | Subspace orthocomplement f... |
| dochfval 38480 | Subspace orthocomplement f... |
| dochval 38481 | Subspace orthocomplement f... |
| dochval2 38482 | Subspace orthocomplement f... |
| dochcl 38483 | Closure of subspace orthoc... |
| dochlss 38484 | A subspace orthocomplement... |
| dochssv 38485 | A subspace orthocomplement... |
| dochfN 38486 | Domain and codomain of the... |
| dochvalr 38487 | Orthocomplement of a close... |
| doch0 38488 | Orthocomplement of the zer... |
| doch1 38489 | Orthocomplement of the uni... |
| dochoc0 38490 | The zero subspace is close... |
| dochoc1 38491 | The unit subspace (all vec... |
| dochvalr2 38492 | Orthocomplement of a close... |
| dochvalr3 38493 | Orthocomplement of a close... |
| doch2val2 38494 | Double orthocomplement for... |
| dochss 38495 | Subset law for orthocomple... |
| dochocss 38496 | Double negative law for or... |
| dochoc 38497 | Double negative law for or... |
| dochsscl 38498 | If a set of vectors is inc... |
| dochoccl 38499 | A set of vectors is closed... |
| dochord 38500 | Ordering law for orthocomp... |
| dochord2N 38501 | Ordering law for orthocomp... |
| dochord3 38502 | Ordering law for orthocomp... |
| doch11 38503 | Orthocomplement is one-to-... |
| dochsordN 38504 | Strict ordering law for or... |
| dochn0nv 38505 | An orthocomplement is nonz... |
| dihoml4c 38506 | Version of ~ dihoml4 with ... |
| dihoml4 38507 | Orthomodular law for const... |
| dochspss 38508 | The span of a set of vecto... |
| dochocsp 38509 | The span of an orthocomple... |
| dochspocN 38510 | The span of an orthocomple... |
| dochocsn 38511 | The double orthocomplement... |
| dochsncom 38512 | Swap vectors in an orthoco... |
| dochsat 38513 | The double orthocomplement... |
| dochshpncl 38514 | If a hyperplane is not clo... |
| dochlkr 38515 | Equivalent conditions for ... |
| dochkrshp 38516 | The closure of a kernel is... |
| dochkrshp2 38517 | Properties of the closure ... |
| dochkrshp3 38518 | Properties of the closure ... |
| dochkrshp4 38519 | Properties of the closure ... |
| dochdmj1 38520 | De Morgan-like law for sub... |
| dochnoncon 38521 | Law of noncontradiction. ... |
| dochnel2 38522 | A nonzero member of a subs... |
| dochnel 38523 | A nonzero vector doesn't b... |
| djhffval 38526 | Subspace join for ` DVecH ... |
| djhfval 38527 | Subspace join for ` DVecH ... |
| djhval 38528 | Subspace join for ` DVecH ... |
| djhval2 38529 | Value of subspace join for... |
| djhcl 38530 | Closure of subspace join f... |
| djhlj 38531 | Transfer lattice join to `... |
| djhljjN 38532 | Lattice join in terms of `... |
| djhjlj 38533 | ` DVecH ` vector space clo... |
| djhj 38534 | ` DVecH ` vector space clo... |
| djhcom 38535 | Subspace join commutes. (... |
| djhspss 38536 | Subspace span of union is ... |
| djhsumss 38537 | Subspace sum is a subset o... |
| dihsumssj 38538 | The subspace sum of two is... |
| djhunssN 38539 | Subspace union is a subset... |
| dochdmm1 38540 | De Morgan-like law for clo... |
| djhexmid 38541 | Excluded middle property o... |
| djh01 38542 | Closed subspace join with ... |
| djh02 38543 | Closed subspace join with ... |
| djhlsmcl 38544 | A closed subspace sum equa... |
| djhcvat42 38545 | A covering property. ( ~ ... |
| dihjatb 38546 | Isomorphism H of lattice j... |
| dihjatc 38547 | Isomorphism H of lattice j... |
| dihjatcclem1 38548 | Lemma for isomorphism H of... |
| dihjatcclem2 38549 | Lemma for isomorphism H of... |
| dihjatcclem3 38550 | Lemma for ~ dihjatcc . (C... |
| dihjatcclem4 38551 | Lemma for isomorphism H of... |
| dihjatcc 38552 | Isomorphism H of lattice j... |
| dihjat 38553 | Isomorphism H of lattice j... |
| dihprrnlem1N 38554 | Lemma for ~ dihprrn , show... |
| dihprrnlem2 38555 | Lemma for ~ dihprrn . (Co... |
| dihprrn 38556 | The span of a vector pair ... |
| djhlsmat 38557 | The sum of two subspace at... |
| dihjat1lem 38558 | Subspace sum of a closed s... |
| dihjat1 38559 | Subspace sum of a closed s... |
| dihsmsprn 38560 | Subspace sum of a closed s... |
| dihjat2 38561 | The subspace sum of a clos... |
| dihjat3 38562 | Isomorphism H of lattice j... |
| dihjat4 38563 | Transfer the subspace sum ... |
| dihjat6 38564 | Transfer the subspace sum ... |
| dihsmsnrn 38565 | The subspace sum of two si... |
| dihsmatrn 38566 | The subspace sum of a clos... |
| dihjat5N 38567 | Transfer lattice join with... |
| dvh4dimat 38568 | There is an atom that is o... |
| dvh3dimatN 38569 | There is an atom that is o... |
| dvh2dimatN 38570 | Given an atom, there exist... |
| dvh1dimat 38571 | There exists an atom. (Co... |
| dvh1dim 38572 | There exists a nonzero vec... |
| dvh4dimlem 38573 | Lemma for ~ dvh4dimN . (C... |
| dvhdimlem 38574 | Lemma for ~ dvh2dim and ~ ... |
| dvh2dim 38575 | There is a vector that is ... |
| dvh3dim 38576 | There is a vector that is ... |
| dvh4dimN 38577 | There is a vector that is ... |
| dvh3dim2 38578 | There is a vector that is ... |
| dvh3dim3N 38579 | There is a vector that is ... |
| dochsnnz 38580 | The orthocomplement of a s... |
| dochsatshp 38581 | The orthocomplement of a s... |
| dochsatshpb 38582 | The orthocomplement of a s... |
| dochsnshp 38583 | The orthocomplement of a n... |
| dochshpsat 38584 | A hyperplane is closed iff... |
| dochkrsat 38585 | The orthocomplement of a k... |
| dochkrsat2 38586 | The orthocomplement of a k... |
| dochsat0 38587 | The orthocomplement of a k... |
| dochkrsm 38588 | The subspace sum of a clos... |
| dochexmidat 38589 | Special case of excluded m... |
| dochexmidlem1 38590 | Lemma for ~ dochexmid . H... |
| dochexmidlem2 38591 | Lemma for ~ dochexmid . (... |
| dochexmidlem3 38592 | Lemma for ~ dochexmid . U... |
| dochexmidlem4 38593 | Lemma for ~ dochexmid . (... |
| dochexmidlem5 38594 | Lemma for ~ dochexmid . (... |
| dochexmidlem6 38595 | Lemma for ~ dochexmid . (... |
| dochexmidlem7 38596 | Lemma for ~ dochexmid . C... |
| dochexmidlem8 38597 | Lemma for ~ dochexmid . T... |
| dochexmid 38598 | Excluded middle law for cl... |
| dochsnkrlem1 38599 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem2 38600 | Lemma for ~ dochsnkr . (C... |
| dochsnkrlem3 38601 | Lemma for ~ dochsnkr . (C... |
| dochsnkr 38602 | A (closed) kernel expresse... |
| dochsnkr2 38603 | Kernel of the explicit fun... |
| dochsnkr2cl 38604 | The ` X ` determining func... |
| dochflcl 38605 | Closure of the explicit fu... |
| dochfl1 38606 | The value of the explicit ... |
| dochfln0 38607 | The value of a functional ... |
| dochkr1 38608 | A nonzero functional has a... |
| dochkr1OLDN 38609 | A nonzero functional has a... |
| lpolsetN 38612 | The set of polarities of a... |
| islpolN 38613 | The predicate "is a polari... |
| islpoldN 38614 | Properties that determine ... |
| lpolfN 38615 | Functionality of a polarit... |
| lpolvN 38616 | The polarity of the whole ... |
| lpolconN 38617 | Contraposition property of... |
| lpolsatN 38618 | The polarity of an atomic ... |
| lpolpolsatN 38619 | Property of a polarity. (... |
| dochpolN 38620 | The subspace orthocompleme... |
| lcfl1lem 38621 | Property of a functional w... |
| lcfl1 38622 | Property of a functional w... |
| lcfl2 38623 | Property of a functional w... |
| lcfl3 38624 | Property of a functional w... |
| lcfl4N 38625 | Property of a functional w... |
| lcfl5 38626 | Property of a functional w... |
| lcfl5a 38627 | Property of a functional w... |
| lcfl6lem 38628 | Lemma for ~ lcfl6 . A fun... |
| lcfl7lem 38629 | Lemma for ~ lcfl7N . If t... |
| lcfl6 38630 | Property of a functional w... |
| lcfl7N 38631 | Property of a functional w... |
| lcfl8 38632 | Property of a functional w... |
| lcfl8a 38633 | Property of a functional w... |
| lcfl8b 38634 | Property of a nonzero func... |
| lcfl9a 38635 | Property implying that a f... |
| lclkrlem1 38636 | The set of functionals hav... |
| lclkrlem2a 38637 | Lemma for ~ lclkr . Use ~... |
| lclkrlem2b 38638 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2c 38639 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2d 38640 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2e 38641 | Lemma for ~ lclkr . The k... |
| lclkrlem2f 38642 | Lemma for ~ lclkr . Const... |
| lclkrlem2g 38643 | Lemma for ~ lclkr . Compa... |
| lclkrlem2h 38644 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2i 38645 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2j 38646 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2k 38647 | Lemma for ~ lclkr . Kerne... |
| lclkrlem2l 38648 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2m 38649 | Lemma for ~ lclkr . Const... |
| lclkrlem2n 38650 | Lemma for ~ lclkr . (Cont... |
| lclkrlem2o 38651 | Lemma for ~ lclkr . When ... |
| lclkrlem2p 38652 | Lemma for ~ lclkr . When ... |
| lclkrlem2q 38653 | Lemma for ~ lclkr . The s... |
| lclkrlem2r 38654 | Lemma for ~ lclkr . When ... |
| lclkrlem2s 38655 | Lemma for ~ lclkr . Thus,... |
| lclkrlem2t 38656 | Lemma for ~ lclkr . We el... |
| lclkrlem2u 38657 | Lemma for ~ lclkr . ~ lclk... |
| lclkrlem2v 38658 | Lemma for ~ lclkr . When ... |
| lclkrlem2w 38659 | Lemma for ~ lclkr . This ... |
| lclkrlem2x 38660 | Lemma for ~ lclkr . Elimi... |
| lclkrlem2y 38661 | Lemma for ~ lclkr . Resta... |
| lclkrlem2 38662 | The set of functionals hav... |
| lclkr 38663 | The set of functionals wit... |
| lcfls1lem 38664 | Property of a functional w... |
| lcfls1N 38665 | Property of a functional w... |
| lcfls1c 38666 | Property of a functional w... |
| lclkrslem1 38667 | The set of functionals hav... |
| lclkrslem2 38668 | The set of functionals hav... |
| lclkrs 38669 | The set of functionals hav... |
| lclkrs2 38670 | The set of functionals wit... |
| lcfrvalsnN 38671 | Reconstruction from the du... |
| lcfrlem1 38672 | Lemma for ~ lcfr . Note t... |
| lcfrlem2 38673 | Lemma for ~ lcfr . (Contr... |
| lcfrlem3 38674 | Lemma for ~ lcfr . (Contr... |
| lcfrlem4 38675 | Lemma for ~ lcfr . (Contr... |
| lcfrlem5 38676 | Lemma for ~ lcfr . The se... |
| lcfrlem6 38677 | Lemma for ~ lcfr . Closur... |
| lcfrlem7 38678 | Lemma for ~ lcfr . Closur... |
| lcfrlem8 38679 | Lemma for ~ lcf1o and ~ lc... |
| lcfrlem9 38680 | Lemma for ~ lcf1o . (This... |
| lcf1o 38681 | Define a function ` J ` th... |
| lcfrlem10 38682 | Lemma for ~ lcfr . (Contr... |
| lcfrlem11 38683 | Lemma for ~ lcfr . (Contr... |
| lcfrlem12N 38684 | Lemma for ~ lcfr . (Contr... |
| lcfrlem13 38685 | Lemma for ~ lcfr . (Contr... |
| lcfrlem14 38686 | Lemma for ~ lcfr . (Contr... |
| lcfrlem15 38687 | Lemma for ~ lcfr . (Contr... |
| lcfrlem16 38688 | Lemma for ~ lcfr . (Contr... |
| lcfrlem17 38689 | Lemma for ~ lcfr . Condit... |
| lcfrlem18 38690 | Lemma for ~ lcfr . (Contr... |
| lcfrlem19 38691 | Lemma for ~ lcfr . (Contr... |
| lcfrlem20 38692 | Lemma for ~ lcfr . (Contr... |
| lcfrlem21 38693 | Lemma for ~ lcfr . (Contr... |
| lcfrlem22 38694 | Lemma for ~ lcfr . (Contr... |
| lcfrlem23 38695 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem24 38696 | Lemma for ~ lcfr . (Contr... |
| lcfrlem25 38697 | Lemma for ~ lcfr . Specia... |
| lcfrlem26 38698 | Lemma for ~ lcfr . Specia... |
| lcfrlem27 38699 | Lemma for ~ lcfr . Specia... |
| lcfrlem28 38700 | Lemma for ~ lcfr . TODO: ... |
| lcfrlem29 38701 | Lemma for ~ lcfr . (Contr... |
| lcfrlem30 38702 | Lemma for ~ lcfr . (Contr... |
| lcfrlem31 38703 | Lemma for ~ lcfr . (Contr... |
| lcfrlem32 38704 | Lemma for ~ lcfr . (Contr... |
| lcfrlem33 38705 | Lemma for ~ lcfr . (Contr... |
| lcfrlem34 38706 | Lemma for ~ lcfr . (Contr... |
| lcfrlem35 38707 | Lemma for ~ lcfr . (Contr... |
| lcfrlem36 38708 | Lemma for ~ lcfr . (Contr... |
| lcfrlem37 38709 | Lemma for ~ lcfr . (Contr... |
| lcfrlem38 38710 | Lemma for ~ lcfr . Combin... |
| lcfrlem39 38711 | Lemma for ~ lcfr . Elimin... |
| lcfrlem40 38712 | Lemma for ~ lcfr . Elimin... |
| lcfrlem41 38713 | Lemma for ~ lcfr . Elimin... |
| lcfrlem42 38714 | Lemma for ~ lcfr . Elimin... |
| lcfr 38715 | Reconstruction of a subspa... |
| lcdfval 38718 | Dual vector space of funct... |
| lcdval 38719 | Dual vector space of funct... |
| lcdval2 38720 | Dual vector space of funct... |
| lcdlvec 38721 | The dual vector space of f... |
| lcdlmod 38722 | The dual vector space of f... |
| lcdvbase 38723 | Vector base set of a dual ... |
| lcdvbasess 38724 | The vector base set of the... |
| lcdvbaselfl 38725 | A vector in the base set o... |
| lcdvbasecl 38726 | Closure of the value of a ... |
| lcdvadd 38727 | Vector addition for the cl... |
| lcdvaddval 38728 | The value of the value of ... |
| lcdsca 38729 | The ring of scalars of the... |
| lcdsbase 38730 | Base set of scalar ring fo... |
| lcdsadd 38731 | Scalar addition for the cl... |
| lcdsmul 38732 | Scalar multiplication for ... |
| lcdvs 38733 | Scalar product for the clo... |
| lcdvsval 38734 | Value of scalar product op... |
| lcdvscl 38735 | The scalar product operati... |
| lcdlssvscl 38736 | Closure of scalar product ... |
| lcdvsass 38737 | Associative law for scalar... |
| lcd0 38738 | The zero scalar of the clo... |
| lcd1 38739 | The unit scalar of the clo... |
| lcdneg 38740 | The unit scalar of the clo... |
| lcd0v 38741 | The zero functional in the... |
| lcd0v2 38742 | The zero functional in the... |
| lcd0vvalN 38743 | Value of the zero function... |
| lcd0vcl 38744 | Closure of the zero functi... |
| lcd0vs 38745 | A scalar zero times a func... |
| lcdvs0N 38746 | A scalar times the zero fu... |
| lcdvsub 38747 | The value of vector subtra... |
| lcdvsubval 38748 | The value of the value of ... |
| lcdlss 38749 | Subspaces of a dual vector... |
| lcdlss2N 38750 | Subspaces of a dual vector... |
| lcdlsp 38751 | Span in the set of functio... |
| lcdlkreqN 38752 | Colinear functionals have ... |
| lcdlkreq2N 38753 | Colinear functionals have ... |
| mapdffval 38756 | Projectivity from vector s... |
| mapdfval 38757 | Projectivity from vector s... |
| mapdval 38758 | Value of projectivity from... |
| mapdvalc 38759 | Value of projectivity from... |
| mapdval2N 38760 | Value of projectivity from... |
| mapdval3N 38761 | Value of projectivity from... |
| mapdval4N 38762 | Value of projectivity from... |
| mapdval5N 38763 | Value of projectivity from... |
| mapdordlem1a 38764 | Lemma for ~ mapdord . (Co... |
| mapdordlem1bN 38765 | Lemma for ~ mapdord . (Co... |
| mapdordlem1 38766 | Lemma for ~ mapdord . (Co... |
| mapdordlem2 38767 | Lemma for ~ mapdord . Ord... |
| mapdord 38768 | Ordering property of the m... |
| mapd11 38769 | The map defined by ~ df-ma... |
| mapddlssN 38770 | The mapping of a subspace ... |
| mapdsn 38771 | Value of the map defined b... |
| mapdsn2 38772 | Value of the map defined b... |
| mapdsn3 38773 | Value of the map defined b... |
| mapd1dim2lem1N 38774 | Value of the map defined b... |
| mapdrvallem2 38775 | Lemma for ~ mapdrval . TO... |
| mapdrvallem3 38776 | Lemma for ~ mapdrval . (C... |
| mapdrval 38777 | Given a dual subspace ` R ... |
| mapd1o 38778 | The map defined by ~ df-ma... |
| mapdrn 38779 | Range of the map defined b... |
| mapdunirnN 38780 | Union of the range of the ... |
| mapdrn2 38781 | Range of the map defined b... |
| mapdcnvcl 38782 | Closure of the converse of... |
| mapdcl 38783 | Closure the value of the m... |
| mapdcnvid1N 38784 | Converse of the value of t... |
| mapdsord 38785 | Strong ordering property o... |
| mapdcl2 38786 | The mapping of a subspace ... |
| mapdcnvid2 38787 | Value of the converse of t... |
| mapdcnvordN 38788 | Ordering property of the c... |
| mapdcnv11N 38789 | The converse of the map de... |
| mapdcv 38790 | Covering property of the c... |
| mapdincl 38791 | Closure of dual subspace i... |
| mapdin 38792 | Subspace intersection is p... |
| mapdlsmcl 38793 | Closure of dual subspace s... |
| mapdlsm 38794 | Subspace sum is preserved ... |
| mapd0 38795 | Projectivity map of the ze... |
| mapdcnvatN 38796 | Atoms are preserved by the... |
| mapdat 38797 | Atoms are preserved by the... |
| mapdspex 38798 | The map of a span equals t... |
| mapdn0 38799 | Transfer nonzero property ... |
| mapdncol 38800 | Transfer non-colinearity f... |
| mapdindp 38801 | Transfer (part of) vector ... |
| mapdpglem1 38802 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2 38803 | Lemma for ~ mapdpg . Baer... |
| mapdpglem2a 38804 | Lemma for ~ mapdpg . (Con... |
| mapdpglem3 38805 | Lemma for ~ mapdpg . Baer... |
| mapdpglem4N 38806 | Lemma for ~ mapdpg . (Con... |
| mapdpglem5N 38807 | Lemma for ~ mapdpg . (Con... |
| mapdpglem6 38808 | Lemma for ~ mapdpg . Baer... |
| mapdpglem8 38809 | Lemma for ~ mapdpg . Baer... |
| mapdpglem9 38810 | Lemma for ~ mapdpg . Baer... |
| mapdpglem10 38811 | Lemma for ~ mapdpg . Baer... |
| mapdpglem11 38812 | Lemma for ~ mapdpg . (Con... |
| mapdpglem12 38813 | Lemma for ~ mapdpg . TODO... |
| mapdpglem13 38814 | Lemma for ~ mapdpg . (Con... |
| mapdpglem14 38815 | Lemma for ~ mapdpg . (Con... |
| mapdpglem15 38816 | Lemma for ~ mapdpg . (Con... |
| mapdpglem16 38817 | Lemma for ~ mapdpg . Baer... |
| mapdpglem17N 38818 | Lemma for ~ mapdpg . Baer... |
| mapdpglem18 38819 | Lemma for ~ mapdpg . Baer... |
| mapdpglem19 38820 | Lemma for ~ mapdpg . Baer... |
| mapdpglem20 38821 | Lemma for ~ mapdpg . Baer... |
| mapdpglem21 38822 | Lemma for ~ mapdpg . (Con... |
| mapdpglem22 38823 | Lemma for ~ mapdpg . Baer... |
| mapdpglem23 38824 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30a 38825 | Lemma for ~ mapdpg . (Con... |
| mapdpglem30b 38826 | Lemma for ~ mapdpg . (Con... |
| mapdpglem25 38827 | Lemma for ~ mapdpg . Baer... |
| mapdpglem26 38828 | Lemma for ~ mapdpg . Baer... |
| mapdpglem27 38829 | Lemma for ~ mapdpg . Baer... |
| mapdpglem29 38830 | Lemma for ~ mapdpg . Baer... |
| mapdpglem28 38831 | Lemma for ~ mapdpg . Baer... |
| mapdpglem30 38832 | Lemma for ~ mapdpg . Baer... |
| mapdpglem31 38833 | Lemma for ~ mapdpg . Baer... |
| mapdpglem24 38834 | Lemma for ~ mapdpg . Exis... |
| mapdpglem32 38835 | Lemma for ~ mapdpg . Uniq... |
| mapdpg 38836 | Part 1 of proof of the fir... |
| baerlem3lem1 38837 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem1 38838 | Lemma for ~ baerlem5a . (... |
| baerlem5blem1 38839 | Lemma for ~ baerlem5b . (... |
| baerlem3lem2 38840 | Lemma for ~ baerlem3 . (C... |
| baerlem5alem2 38841 | Lemma for ~ baerlem5a . (... |
| baerlem5blem2 38842 | Lemma for ~ baerlem5b . (... |
| baerlem3 38843 | An equality that holds whe... |
| baerlem5a 38844 | An equality that holds whe... |
| baerlem5b 38845 | An equality that holds whe... |
| baerlem5amN 38846 | An equality that holds whe... |
| baerlem5bmN 38847 | An equality that holds whe... |
| baerlem5abmN 38848 | An equality that holds whe... |
| mapdindp0 38849 | Vector independence lemma.... |
| mapdindp1 38850 | Vector independence lemma.... |
| mapdindp2 38851 | Vector independence lemma.... |
| mapdindp3 38852 | Vector independence lemma.... |
| mapdindp4 38853 | Vector independence lemma.... |
| mapdhval 38854 | Lemmma for ~~? mapdh . (C... |
| mapdhval0 38855 | Lemmma for ~~? mapdh . (C... |
| mapdhval2 38856 | Lemmma for ~~? mapdh . (C... |
| mapdhcl 38857 | Lemmma for ~~? mapdh . (C... |
| mapdheq 38858 | Lemmma for ~~? mapdh . Th... |
| mapdheq2 38859 | Lemmma for ~~? mapdh . On... |
| mapdheq2biN 38860 | Lemmma for ~~? mapdh . Pa... |
| mapdheq4lem 38861 | Lemma for ~ mapdheq4 . Pa... |
| mapdheq4 38862 | Lemma for ~~? mapdh . Par... |
| mapdh6lem1N 38863 | Lemma for ~ mapdh6N . Par... |
| mapdh6lem2N 38864 | Lemma for ~ mapdh6N . Par... |
| mapdh6aN 38865 | Lemma for ~ mapdh6N . Par... |
| mapdh6b0N 38866 | Lemmma for ~ mapdh6N . (C... |
| mapdh6bN 38867 | Lemmma for ~ mapdh6N . (C... |
| mapdh6cN 38868 | Lemmma for ~ mapdh6N . (C... |
| mapdh6dN 38869 | Lemmma for ~ mapdh6N . (C... |
| mapdh6eN 38870 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6fN 38871 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6gN 38872 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6hN 38873 | Lemmma for ~ mapdh6N . Pa... |
| mapdh6iN 38874 | Lemmma for ~ mapdh6N . El... |
| mapdh6jN 38875 | Lemmma for ~ mapdh6N . El... |
| mapdh6kN 38876 | Lemmma for ~ mapdh6N . El... |
| mapdh6N 38877 | Part (6) of [Baer] p. 47 l... |
| mapdh7eN 38878 | Part (7) of [Baer] p. 48 l... |
| mapdh7cN 38879 | Part (7) of [Baer] p. 48 l... |
| mapdh7dN 38880 | Part (7) of [Baer] p. 48 l... |
| mapdh7fN 38881 | Part (7) of [Baer] p. 48 l... |
| mapdh75e 38882 | Part (7) of [Baer] p. 48 l... |
| mapdh75cN 38883 | Part (7) of [Baer] p. 48 l... |
| mapdh75d 38884 | Part (7) of [Baer] p. 48 l... |
| mapdh75fN 38885 | Part (7) of [Baer] p. 48 l... |
| hvmapffval 38888 | Map from nonzero vectors t... |
| hvmapfval 38889 | Map from nonzero vectors t... |
| hvmapval 38890 | Value of map from nonzero ... |
| hvmapvalvalN 38891 | Value of value of map (i.e... |
| hvmapidN 38892 | The value of the vector to... |
| hvmap1o 38893 | The vector to functional m... |
| hvmapclN 38894 | Closure of the vector to f... |
| hvmap1o2 38895 | The vector to functional m... |
| hvmapcl2 38896 | Closure of the vector to f... |
| hvmaplfl 38897 | The vector to functional m... |
| hvmaplkr 38898 | Kernel of the vector to fu... |
| mapdhvmap 38899 | Relationship between ` map... |
| lspindp5 38900 | Obtain an independent vect... |
| hdmaplem1 38901 | Lemma to convert a frequen... |
| hdmaplem2N 38902 | Lemma to convert a frequen... |
| hdmaplem3 38903 | Lemma to convert a frequen... |
| hdmaplem4 38904 | Lemma to convert a frequen... |
| mapdh8a 38905 | Part of Part (8) in [Baer]... |
| mapdh8aa 38906 | Part of Part (8) in [Baer]... |
| mapdh8ab 38907 | Part of Part (8) in [Baer]... |
| mapdh8ac 38908 | Part of Part (8) in [Baer]... |
| mapdh8ad 38909 | Part of Part (8) in [Baer]... |
| mapdh8b 38910 | Part of Part (8) in [Baer]... |
| mapdh8c 38911 | Part of Part (8) in [Baer]... |
| mapdh8d0N 38912 | Part of Part (8) in [Baer]... |
| mapdh8d 38913 | Part of Part (8) in [Baer]... |
| mapdh8e 38914 | Part of Part (8) in [Baer]... |
| mapdh8g 38915 | Part of Part (8) in [Baer]... |
| mapdh8i 38916 | Part of Part (8) in [Baer]... |
| mapdh8j 38917 | Part of Part (8) in [Baer]... |
| mapdh8 38918 | Part (8) in [Baer] p. 48. ... |
| mapdh9a 38919 | Lemma for part (9) in [Bae... |
| mapdh9aOLDN 38920 | Lemma for part (9) in [Bae... |
| hdmap1ffval 38925 | Preliminary map from vecto... |
| hdmap1fval 38926 | Preliminary map from vecto... |
| hdmap1vallem 38927 | Value of preliminary map f... |
| hdmap1val 38928 | Value of preliminary map f... |
| hdmap1val0 38929 | Value of preliminary map f... |
| hdmap1val2 38930 | Value of preliminary map f... |
| hdmap1eq 38931 | The defining equation for ... |
| hdmap1cbv 38932 | Frequently used lemma to c... |
| hdmap1valc 38933 | Connect the value of the p... |
| hdmap1cl 38934 | Convert closure theorem ~ ... |
| hdmap1eq2 38935 | Convert ~ mapdheq2 to use ... |
| hdmap1eq4N 38936 | Convert ~ mapdheq4 to use ... |
| hdmap1l6lem1 38937 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6lem2 38938 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6a 38939 | Lemma for ~ hdmap1l6 . Pa... |
| hdmap1l6b0N 38940 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6b 38941 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6c 38942 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6d 38943 | Lemmma for ~ hdmap1l6 . (... |
| hdmap1l6e 38944 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6f 38945 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6g 38946 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6h 38947 | Lemmma for ~ hdmap1l6 . P... |
| hdmap1l6i 38948 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6j 38949 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6k 38950 | Lemmma for ~ hdmap1l6 . E... |
| hdmap1l6 38951 | Part (6) of [Baer] p. 47 l... |
| hdmap1eulem 38952 | Lemma for ~ hdmap1eu . TO... |
| hdmap1eulemOLDN 38953 | Lemma for ~ hdmap1euOLDN .... |
| hdmap1eu 38954 | Convert ~ mapdh9a to use t... |
| hdmap1euOLDN 38955 | Convert ~ mapdh9aOLDN to u... |
| hdmapffval 38956 | Map from vectors to functi... |
| hdmapfval 38957 | Map from vectors to functi... |
| hdmapval 38958 | Value of map from vectors ... |
| hdmapfnN 38959 | Functionality of map from ... |
| hdmapcl 38960 | Closure of map from vector... |
| hdmapval2lem 38961 | Lemma for ~ hdmapval2 . (... |
| hdmapval2 38962 | Value of map from vectors ... |
| hdmapval0 38963 | Value of map from vectors ... |
| hdmapeveclem 38964 | Lemma for ~ hdmapevec . T... |
| hdmapevec 38965 | Value of map from vectors ... |
| hdmapevec2 38966 | The inner product of the r... |
| hdmapval3lemN 38967 | Value of map from vectors ... |
| hdmapval3N 38968 | Value of map from vectors ... |
| hdmap10lem 38969 | Lemma for ~ hdmap10 . (Co... |
| hdmap10 38970 | Part 10 in [Baer] p. 48 li... |
| hdmap11lem1 38971 | Lemma for ~ hdmapadd . (C... |
| hdmap11lem2 38972 | Lemma for ~ hdmapadd . (C... |
| hdmapadd 38973 | Part 11 in [Baer] p. 48 li... |
| hdmapeq0 38974 | Part of proof of part 12 i... |
| hdmapnzcl 38975 | Nonzero vector closure of ... |
| hdmapneg 38976 | Part of proof of part 12 i... |
| hdmapsub 38977 | Part of proof of part 12 i... |
| hdmap11 38978 | Part of proof of part 12 i... |
| hdmaprnlem1N 38979 | Part of proof of part 12 i... |
| hdmaprnlem3N 38980 | Part of proof of part 12 i... |
| hdmaprnlem3uN 38981 | Part of proof of part 12 i... |
| hdmaprnlem4tN 38982 | Lemma for ~ hdmaprnN . TO... |
| hdmaprnlem4N 38983 | Part of proof of part 12 i... |
| hdmaprnlem6N 38984 | Part of proof of part 12 i... |
| hdmaprnlem7N 38985 | Part of proof of part 12 i... |
| hdmaprnlem8N 38986 | Part of proof of part 12 i... |
| hdmaprnlem9N 38987 | Part of proof of part 12 i... |
| hdmaprnlem3eN 38988 | Lemma for ~ hdmaprnN . (C... |
| hdmaprnlem10N 38989 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem11N 38990 | Lemma for ~ hdmaprnN . Sh... |
| hdmaprnlem15N 38991 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem16N 38992 | Lemma for ~ hdmaprnN . El... |
| hdmaprnlem17N 38993 | Lemma for ~ hdmaprnN . In... |
| hdmaprnN 38994 | Part of proof of part 12 i... |
| hdmapf1oN 38995 | Part 12 in [Baer] p. 49. ... |
| hdmap14lem1a 38996 | Prior to part 14 in [Baer]... |
| hdmap14lem2a 38997 | Prior to part 14 in [Baer]... |
| hdmap14lem1 38998 | Prior to part 14 in [Baer]... |
| hdmap14lem2N 38999 | Prior to part 14 in [Baer]... |
| hdmap14lem3 39000 | Prior to part 14 in [Baer]... |
| hdmap14lem4a 39001 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem4 39002 | Simplify ` ( A \ { Q } ) `... |
| hdmap14lem6 39003 | Case where ` F ` is zero. ... |
| hdmap14lem7 39004 | Combine cases of ` F ` . ... |
| hdmap14lem8 39005 | Part of proof of part 14 i... |
| hdmap14lem9 39006 | Part of proof of part 14 i... |
| hdmap14lem10 39007 | Part of proof of part 14 i... |
| hdmap14lem11 39008 | Part of proof of part 14 i... |
| hdmap14lem12 39009 | Lemma for proof of part 14... |
| hdmap14lem13 39010 | Lemma for proof of part 14... |
| hdmap14lem14 39011 | Part of proof of part 14 i... |
| hdmap14lem15 39012 | Part of proof of part 14 i... |
| hgmapffval 39015 | Map from the scalar divisi... |
| hgmapfval 39016 | Map from the scalar divisi... |
| hgmapval 39017 | Value of map from the scal... |
| hgmapfnN 39018 | Functionality of scalar si... |
| hgmapcl 39019 | Closure of scalar sigma ma... |
| hgmapdcl 39020 | Closure of the vector spac... |
| hgmapvs 39021 | Part 15 of [Baer] p. 50 li... |
| hgmapval0 39022 | Value of the scalar sigma ... |
| hgmapval1 39023 | Value of the scalar sigma ... |
| hgmapadd 39024 | Part 15 of [Baer] p. 50 li... |
| hgmapmul 39025 | Part 15 of [Baer] p. 50 li... |
| hgmaprnlem1N 39026 | Lemma for ~ hgmaprnN . (C... |
| hgmaprnlem2N 39027 | Lemma for ~ hgmaprnN . Pa... |
| hgmaprnlem3N 39028 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem4N 39029 | Lemma for ~ hgmaprnN . El... |
| hgmaprnlem5N 39030 | Lemma for ~ hgmaprnN . El... |
| hgmaprnN 39031 | Part of proof of part 16 i... |
| hgmap11 39032 | The scalar sigma map is on... |
| hgmapf1oN 39033 | The scalar sigma map is a ... |
| hgmapeq0 39034 | The scalar sigma map is ze... |
| hdmapipcl 39035 | The inner product (Hermiti... |
| hdmapln1 39036 | Linearity property that wi... |
| hdmaplna1 39037 | Additive property of first... |
| hdmaplns1 39038 | Subtraction property of fi... |
| hdmaplnm1 39039 | Multiplicative property of... |
| hdmaplna2 39040 | Additive property of secon... |
| hdmapglnm2 39041 | g-linear property of secon... |
| hdmapgln2 39042 | g-linear property that wil... |
| hdmaplkr 39043 | Kernel of the vector to du... |
| hdmapellkr 39044 | Membership in the kernel (... |
| hdmapip0 39045 | Zero property that will be... |
| hdmapip1 39046 | Construct a proportional v... |
| hdmapip0com 39047 | Commutation property of Ba... |
| hdmapinvlem1 39048 | Line 27 in [Baer] p. 110. ... |
| hdmapinvlem2 39049 | Line 28 in [Baer] p. 110, ... |
| hdmapinvlem3 39050 | Line 30 in [Baer] p. 110, ... |
| hdmapinvlem4 39051 | Part 1.1 of Proposition 1 ... |
| hdmapglem5 39052 | Part 1.2 in [Baer] p. 110 ... |
| hgmapvvlem1 39053 | Involution property of sca... |
| hgmapvvlem2 39054 | Lemma for ~ hgmapvv . Eli... |
| hgmapvvlem3 39055 | Lemma for ~ hgmapvv . Eli... |
| hgmapvv 39056 | Value of a double involuti... |
| hdmapglem7a 39057 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7b 39058 | Lemma for ~ hdmapg . (Con... |
| hdmapglem7 39059 | Lemma for ~ hdmapg . Line... |
| hdmapg 39060 | Apply the scalar sigma fun... |
| hdmapoc 39061 | Express our constructed or... |
| hlhilset 39064 | The final Hilbert space co... |
| hlhilsca 39065 | The scalar of the final co... |
| hlhilbase 39066 | The base set of the final ... |
| hlhilplus 39067 | The vector addition for th... |
| hlhilslem 39068 | Lemma for ~ hlhilsbase2 . ... |
| hlhilsbase 39069 | The scalar base set of the... |
| hlhilsplus 39070 | Scalar addition for the fi... |
| hlhilsmul 39071 | Scalar multiplication for ... |
| hlhilsbase2 39072 | The scalar base set of the... |
| hlhilsplus2 39073 | Scalar addition for the fi... |
| hlhilsmul2 39074 | Scalar multiplication for ... |
| hlhils0 39075 | The scalar ring zero for t... |
| hlhils1N 39076 | The scalar ring unity for ... |
| hlhilvsca 39077 | The scalar product for the... |
| hlhilip 39078 | Inner product operation fo... |
| hlhilipval 39079 | Value of inner product ope... |
| hlhilnvl 39080 | The involution operation o... |
| hlhillvec 39081 | The final constructed Hilb... |
| hlhildrng 39082 | The star division ring for... |
| hlhilsrnglem 39083 | Lemma for ~ hlhilsrng . (... |
| hlhilsrng 39084 | The star division ring for... |
| hlhil0 39085 | The zero vector for the fi... |
| hlhillsm 39086 | The vector sum operation f... |
| hlhilocv 39087 | The orthocomplement for th... |
| hlhillcs 39088 | The closed subspaces of th... |
| hlhilphllem 39089 | Lemma for ~ hlhil . (Cont... |
| hlhilhillem 39090 | Lemma for ~ hlhil . (Cont... |
| hlathil 39091 | Construction of a Hilbert ... |
| andiff 39092 | Adding biconditional when ... |
| ioin9i8 39093 | Miscellaneous inference cr... |
| jaodd 39094 | Double deduction form of ~... |
| nsb 39095 | Generalization rule for ne... |
| sbn1 39096 | One direction of ~ sbn , u... |
| sbor2 39097 | One direction of ~ sbor , ... |
| 3rspcedvd 39098 | Triple application of ~ rs... |
| rabeqcda 39099 | When ` ps ` is always true... |
| rabdif 39100 | Move difference in and out... |
| sn-axrep5v 39101 | A condensed form of ~ axre... |
| sn-axprlem3 39102 | ~ axprlem3 using only Tars... |
| sn-el 39103 | A version of ~ el with an ... |
| sn-dtru 39104 | ~ dtru without ~ ax-8 or ~... |
| pssexg 39105 | The proper subset of a set... |
| pssn0 39106 | A proper superset is nonem... |
| psspwb 39107 | Classes are proper subclas... |
| xppss12 39108 | Proper subset theorem for ... |
| elpwbi 39109 | Membership in a power set,... |
| opelxpii 39110 | Ordered pair membership in... |
| iunsn 39111 | Indexed union of a singlet... |
| imaopab 39112 | The image of a class of or... |
| fnsnbt 39113 | A function's domain is a s... |
| fnimasnd 39114 | The image of a function by... |
| dfqs2 39115 | Alternate definition of qu... |
| dfqs3 39116 | Alternate definition of qu... |
| qseq12d 39117 | Equality theorem for quoti... |
| qsalrel 39118 | The quotient set is equal ... |
| fzosumm1 39119 | Separate out the last term... |
| ccatcan2d 39120 | Cancellation law for conca... |
| nelsubginvcld 39121 | The inverse of a non-subgr... |
| nelsubgcld 39122 | A non-subgroup-member plus... |
| nelsubgsubcld 39123 | A non-subgroup-member minu... |
| rnasclg 39124 | The set of injected scalar... |
| selvval2lem1 39125 | ` T ` is an associative al... |
| selvval2lem2 39126 | ` D ` is a ring homomorphi... |
| selvval2lem3 39127 | The third argument passed ... |
| selvval2lemn 39128 | A lemma to illustrate the ... |
| selvval2lem4 39129 | The fourth argument passed... |
| selvval2lem5 39130 | The fifth argument passed ... |
| selvcl 39131 | Closure of the "variable s... |
| frlmfielbas 39132 | The vectors of a finite fr... |
| frlmfzwrd 39133 | A vector of a module with ... |
| frlmfzowrd 39134 | A vector of a module with ... |
| frlmfzolen 39135 | The dimension of a vector ... |
| frlmfzowrdb 39136 | The vectors of a module wi... |
| frlmfzoccat 39137 | The concatenation of two v... |
| frlmvscadiccat 39138 | Scalar multiplication dist... |
| lvecgrp 39139 | A left vector is a group. ... |
| lvecring 39140 | The scalar component of a ... |
| lmhmlvec 39141 | The property for modules t... |
| frlmsnic 39142 | Given a free module with a... |
| uvccl 39143 | A unit vector is a vector.... |
| uvcn0 39144 | A unit vector is nonzero. ... |
| c0exALT 39145 | Alternate proof of ~ c0ex ... |
| 0cnALT3 39146 | Alternate proof of ~ 0cn u... |
| elre0re 39147 | Specialized version of ~ 0... |
| 1t1e1ALT 39148 | Alternate proof of ~ 1t1e1... |
| remulcan2d 39149 | ~ mulcan2d for real number... |
| readdid1addid2d 39150 | Given some real number ` B... |
| sn-1ne2 39151 | A proof of ~ 1ne2 without ... |
| nnn1suc 39152 | A positive integer that is... |
| nnadd1com 39153 | Addition with 1 is commuta... |
| nnaddcom 39154 | Addition is commutative fo... |
| nnaddcomli 39155 | Version of ~ addcomli for ... |
| nnadddir 39156 | Right-distributivity for n... |
| nnmul1com 39157 | Multiplication with 1 is c... |
| nnmulcom 39158 | Multiplication is commutat... |
| addsubeq4com 39159 | Relation between sums and ... |
| sqsumi 39160 | A sum squared. (Contribut... |
| negn0nposznnd 39161 | Lemma for ~ dffltz . (Con... |
| sqmid3api 39162 | Value of the square of the... |
| decaddcom 39163 | Commute ones place in addi... |
| sqn5i 39164 | The square of a number end... |
| sqn5ii 39165 | The square of a number end... |
| decpmulnc 39166 | Partial products algorithm... |
| decpmul 39167 | Partial products algorithm... |
| sqdeccom12 39168 | The square of a number in ... |
| sq3deccom12 39169 | Variant of ~ sqdeccom12 wi... |
| 235t711 39170 | Calculate a product by lon... |
| ex-decpmul 39171 | Example usage of ~ decpmul... |
| oexpreposd 39172 | Lemma for ~ dffltz . (Con... |
| cxpgt0d 39173 | Exponentiation with a posi... |
| dvdsexpim 39174 | ~ dvdssqim generalized to ... |
| nn0rppwr 39175 | If ` A ` and ` B ` are rel... |
| expgcd 39176 | Exponentiation distributes... |
| nn0expgcd 39177 | Exponentiation distributes... |
| zexpgcd 39178 | Exponentiation distributes... |
| numdenexp 39179 | ~ numdensq extended to non... |
| numexp 39180 | ~ numsq extended to nonneg... |
| denexp 39181 | ~ densq extended to nonneg... |
| exp11d 39182 | ~ sq11d for positive real ... |
| ltexp1d 39183 | ~ ltmul1d for exponentiati... |
| ltexp1dd 39184 | Raising both sides of 'les... |
| zrtelqelz 39185 | ~ zsqrtelqelz generalized ... |
| zrtdvds 39186 | A positive integer root di... |
| rtprmirr 39187 | The root of a prime number... |
| resubval 39190 | Value of real subtraction,... |
| renegeulemv 39191 | Lemma for ~ renegeu and si... |
| renegeulem 39192 | Lemma for ~ renegeu and si... |
| renegeu 39193 | Existential uniqueness of ... |
| rernegcl 39194 | Closure law for negative r... |
| renegadd 39195 | Relationship between real ... |
| renegid 39196 | Addition of a real number ... |
| reneg0addid2 39197 | Negative zero is a left ad... |
| resubeulem1 39198 | Lemma for ~ resubeu . A v... |
| resubeulem2 39199 | Lemma for ~ resubeu . A v... |
| resubeu 39200 | Existential uniqueness of ... |
| rersubcl 39201 | Closure for real subtracti... |
| resubadd 39202 | Relation between real subt... |
| resubaddd 39203 | Relationship between subtr... |
| resubf 39204 | Real subtraction is an ope... |
| repncan2 39205 | Addition and subtraction o... |
| repncan3 39206 | Addition and subtraction o... |
| readdsub 39207 | Law for addition and subtr... |
| reladdrsub 39208 | Move LHS of a sum into RHS... |
| reltsub1 39209 | Subtraction from both side... |
| reltsubadd2 39210 | 'Less than' relationship b... |
| resubcan2 39211 | Cancellation law for real ... |
| resubsub4 39212 | Law for double subtraction... |
| rennncan2 39213 | Cancellation law for real ... |
| renpncan3 39214 | Cancellation law for real ... |
| repnpcan 39215 | Cancellation law for addit... |
| reppncan 39216 | Cancellation law for mixed... |
| resubidaddid1lem 39217 | Lemma for ~ resubidaddid1 ... |
| resubidaddid1 39218 | Any real number subtracted... |
| resubdi 39219 | Distribution of multiplica... |
| re1m1e0m0 39220 | Equality of two left-addit... |
| sn-00idlem1 39221 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem2 39222 | Lemma for ~ sn-00id . (Co... |
| sn-00idlem3 39223 | Lemma for ~ sn-00id . (Co... |
| sn-00id 39224 | ~ 00id proven without ~ ax... |
| re0m0e0 39225 | Real number version of ~ 0... |
| readdid2 39226 | Real number version of ~ a... |
| sn-addid2 39227 | ~ addid2 without ~ ax-mulc... |
| remul02 39228 | Real number version of ~ m... |
| sn-0ne2 39229 | ~ 0ne2 without ~ ax-mulcom... |
| remul01 39230 | Real number version of ~ m... |
| resubid 39231 | Subtraction of a real numb... |
| readdid1 39232 | Real number version of ~ a... |
| resubid1 39233 | Real number version of ~ s... |
| renegneg 39234 | A real number is equal to ... |
| readdcan2 39235 | Commuted version of ~ read... |
| sn-ltaddpos 39236 | ~ ltaddpos without ~ ax-mu... |
| relt0neg1 39237 | Comparison of a real and i... |
| relt0neg2 39238 | Comparison of a real and i... |
| sn-0lt1 39239 | ~ 0lt1 without ~ ax-mulcom... |
| sn-ltp1 39240 | ~ ltp1 without ~ ax-mulcom... |
| remulinvcom 39241 | A left multiplicative inve... |
| remulid2 39242 | Commuted version of ~ ax-1... |
| remulcand 39243 | Commuted version of ~ remu... |
| prjspval 39246 | Value of the projective sp... |
| prjsprel 39247 | Utility theorem regarding ... |
| prjspertr 39248 | The relation in ` PrjSp ` ... |
| prjsperref 39249 | The relation in ` PrjSp ` ... |
| prjspersym 39250 | The relation in ` PrjSp ` ... |
| prjsper 39251 | The relation in ` PrjSp ` ... |
| prjspreln0 39252 | Two nonzero vectors are eq... |
| prjspvs 39253 | A nonzero multiple of a ve... |
| prjsprellsp 39254 | Two vectors are equivalent... |
| prjspeclsp 39255 | The vectors equivalent to ... |
| prjspval2 39256 | Alternate definition of pr... |
| prjspnval 39259 | Value of the n-dimensional... |
| prjspnval2 39260 | Value of the n-dimensional... |
| 0prjspnlem 39261 | Lemma for ~ 0prjspn . The... |
| 0prjspnrel 39262 | In the zero-dimensional pr... |
| 0prjspn 39263 | A zero-dimensional project... |
| dffltz 39264 | Fermat's Last Theorem (FLT... |
| fltne 39265 | If a counterexample to FLT... |
| fltltc 39266 | ` ( C ^ N ) ` is the large... |
| fltnltalem 39267 | Lemma for ~ fltnlta . A l... |
| fltnlta 39268 | ` N ` is less than ` A ` .... |
| binom2d 39269 | Deduction form of binom2. ... |
| cu3addd 39270 | Cube of sum of three numbe... |
| sqnegd 39271 | The square of the negative... |
| negexpidd 39272 | The sum of a real number t... |
| rexlimdv3d 39273 | An extended version of ~ r... |
| 3cubeslem1 39274 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem2 39275 | Lemma for ~ 3cubes . Used... |
| 3cubeslem3l 39276 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3r 39277 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem3 39278 | Lemma for ~ 3cubes . (Con... |
| 3cubeslem4 39279 | Lemma for ~ 3cubes . This... |
| 3cubes 39280 | Every rational number is a... |
| rntrclfvOAI 39281 | The range of the transitiv... |
| moxfr 39282 | Transfer at-most-one betwe... |
| imaiinfv 39283 | Indexed intersection of an... |
| elrfi 39284 | Elementhood in a set of re... |
| elrfirn 39285 | Elementhood in a set of re... |
| elrfirn2 39286 | Elementhood in a set of re... |
| cmpfiiin 39287 | In a compact topology, a s... |
| ismrcd1 39288 | Any function from the subs... |
| ismrcd2 39289 | Second half of ~ ismrcd1 .... |
| istopclsd 39290 | A closure function which s... |
| ismrc 39291 | A function is a Moore clos... |
| isnacs 39294 | Expand definition of Noeth... |
| nacsfg 39295 | In a Noetherian-type closu... |
| isnacs2 39296 | Express Noetherian-type cl... |
| mrefg2 39297 | Slight variation on finite... |
| mrefg3 39298 | Slight variation on finite... |
| nacsacs 39299 | A closure system of Noethe... |
| isnacs3 39300 | A choice-free order equiva... |
| incssnn0 39301 | Transitivity induction of ... |
| nacsfix 39302 | An increasing sequence of ... |
| constmap 39303 | A constant (represented wi... |
| mapco2g 39304 | Renaming indices in a tupl... |
| mapco2 39305 | Post-composition (renaming... |
| mapfzcons 39306 | Extending a one-based mapp... |
| mapfzcons1 39307 | Recover prefix mapping fro... |
| mapfzcons1cl 39308 | A nonempty mapping has a p... |
| mapfzcons2 39309 | Recover added element from... |
| mptfcl 39310 | Interpret range of a maps-... |
| mzpclval 39315 | Substitution lemma for ` m... |
| elmzpcl 39316 | Double substitution lemma ... |
| mzpclall 39317 | The set of all functions w... |
| mzpcln0 39318 | Corrolary of ~ mzpclall : ... |
| mzpcl1 39319 | Defining property 1 of a p... |
| mzpcl2 39320 | Defining property 2 of a p... |
| mzpcl34 39321 | Defining properties 3 and ... |
| mzpval 39322 | Value of the ` mzPoly ` fu... |
| dmmzp 39323 | ` mzPoly ` is defined for ... |
| mzpincl 39324 | Polynomial closedness is a... |
| mzpconst 39325 | Constant functions are pol... |
| mzpf 39326 | A polynomial function is a... |
| mzpproj 39327 | A projection function is p... |
| mzpadd 39328 | The pointwise sum of two p... |
| mzpmul 39329 | The pointwise product of t... |
| mzpconstmpt 39330 | A constant function expres... |
| mzpaddmpt 39331 | Sum of polynomial function... |
| mzpmulmpt 39332 | Product of polynomial func... |
| mzpsubmpt 39333 | The difference of two poly... |
| mzpnegmpt 39334 | Negation of a polynomial f... |
| mzpexpmpt 39335 | Raise a polynomial functio... |
| mzpindd 39336 | "Structural" induction to ... |
| mzpmfp 39337 | Relationship between multi... |
| mzpsubst 39338 | Substituting polynomials f... |
| mzprename 39339 | Simplified version of ~ mz... |
| mzpresrename 39340 | A polynomial is a polynomi... |
| mzpcompact2lem 39341 | Lemma for ~ mzpcompact2 . ... |
| mzpcompact2 39342 | Polynomials are finitary o... |
| coeq0i 39343 | ~ coeq0 but without explic... |
| fzsplit1nn0 39344 | Split a finite 1-based set... |
| eldiophb 39347 | Initial expression of Diop... |
| eldioph 39348 | Condition for a set to be ... |
| diophrw 39349 | Renaming and adding unused... |
| eldioph2lem1 39350 | Lemma for ~ eldioph2 . Co... |
| eldioph2lem2 39351 | Lemma for ~ eldioph2 . Co... |
| eldioph2 39352 | Construct a Diophantine se... |
| eldioph2b 39353 | While Diophantine sets wer... |
| eldiophelnn0 39354 | Remove antecedent on ` B `... |
| eldioph3b 39355 | Define Diophantine sets in... |
| eldioph3 39356 | Inference version of ~ eld... |
| ellz1 39357 | Membership in a lower set ... |
| lzunuz 39358 | The union of a lower set o... |
| fz1eqin 39359 | Express a one-based finite... |
| lzenom 39360 | Lower integers are countab... |
| elmapresaunres2 39361 | ~ fresaunres2 transposed t... |
| diophin 39362 | If two sets are Diophantin... |
| diophun 39363 | If two sets are Diophantin... |
| eldiophss 39364 | Diophantine sets are sets ... |
| diophrex 39365 | Projecting a Diophantine s... |
| eq0rabdioph 39366 | This is the first of a num... |
| eqrabdioph 39367 | Diophantine set builder fo... |
| 0dioph 39368 | The null set is Diophantin... |
| vdioph 39369 | The "universal" set (as la... |
| anrabdioph 39370 | Diophantine set builder fo... |
| orrabdioph 39371 | Diophantine set builder fo... |
| 3anrabdioph 39372 | Diophantine set builder fo... |
| 3orrabdioph 39373 | Diophantine set builder fo... |
| 2sbcrex 39374 | Exchange an existential qu... |
| sbcrexgOLD 39375 | Interchange class substitu... |
| 2sbcrexOLD 39376 | Exchange an existential qu... |
| sbc2rex 39377 | Exchange a substitution wi... |
| sbc2rexgOLD 39378 | Exchange a substitution wi... |
| sbc4rex 39379 | Exchange a substitution wi... |
| sbc4rexgOLD 39380 | Exchange a substitution wi... |
| sbcrot3 39381 | Rotate a sequence of three... |
| sbcrot5 39382 | Rotate a sequence of five ... |
| sbccomieg 39383 | Commute two explicit subst... |
| rexrabdioph 39384 | Diophantine set builder fo... |
| rexfrabdioph 39385 | Diophantine set builder fo... |
| 2rexfrabdioph 39386 | Diophantine set builder fo... |
| 3rexfrabdioph 39387 | Diophantine set builder fo... |
| 4rexfrabdioph 39388 | Diophantine set builder fo... |
| 6rexfrabdioph 39389 | Diophantine set builder fo... |
| 7rexfrabdioph 39390 | Diophantine set builder fo... |
| rabdiophlem1 39391 | Lemma for arithmetic dioph... |
| rabdiophlem2 39392 | Lemma for arithmetic dioph... |
| elnn0rabdioph 39393 | Diophantine set builder fo... |
| rexzrexnn0 39394 | Rewrite a quantification o... |
| lerabdioph 39395 | Diophantine set builder fo... |
| eluzrabdioph 39396 | Diophantine set builder fo... |
| elnnrabdioph 39397 | Diophantine set builder fo... |
| ltrabdioph 39398 | Diophantine set builder fo... |
| nerabdioph 39399 | Diophantine set builder fo... |
| dvdsrabdioph 39400 | Divisibility is a Diophant... |
| eldioph4b 39401 | Membership in ` Dioph ` ex... |
| eldioph4i 39402 | Forward-only version of ~ ... |
| diophren 39403 | Change variables in a Diop... |
| rabrenfdioph 39404 | Change variable numbers in... |
| rabren3dioph 39405 | Change variable numbers in... |
| fphpd 39406 | Pigeonhole principle expre... |
| fphpdo 39407 | Pigeonhole principle for s... |
| ctbnfien 39408 | An infinite subset of a co... |
| fiphp3d 39409 | Infinite pigeonhole princi... |
| rencldnfilem 39410 | Lemma for ~ rencldnfi . (... |
| rencldnfi 39411 | A set of real numbers whic... |
| irrapxlem1 39412 | Lemma for ~ irrapx1 . Div... |
| irrapxlem2 39413 | Lemma for ~ irrapx1 . Two... |
| irrapxlem3 39414 | Lemma for ~ irrapx1 . By ... |
| irrapxlem4 39415 | Lemma for ~ irrapx1 . Eli... |
| irrapxlem5 39416 | Lemma for ~ irrapx1 . Swi... |
| irrapxlem6 39417 | Lemma for ~ irrapx1 . Exp... |
| irrapx1 39418 | Dirichlet's approximation ... |
| pellexlem1 39419 | Lemma for ~ pellex . Arit... |
| pellexlem2 39420 | Lemma for ~ pellex . Arit... |
| pellexlem3 39421 | Lemma for ~ pellex . To e... |
| pellexlem4 39422 | Lemma for ~ pellex . Invo... |
| pellexlem5 39423 | Lemma for ~ pellex . Invo... |
| pellexlem6 39424 | Lemma for ~ pellex . Doin... |
| pellex 39425 | Every Pell equation has a ... |
| pell1qrval 39436 | Value of the set of first-... |
| elpell1qr 39437 | Membership in a first-quad... |
| pell14qrval 39438 | Value of the set of positi... |
| elpell14qr 39439 | Membership in the set of p... |
| pell1234qrval 39440 | Value of the set of genera... |
| elpell1234qr 39441 | Membership in the set of g... |
| pell1234qrre 39442 | General Pell solutions are... |
| pell1234qrne0 39443 | No solution to a Pell equa... |
| pell1234qrreccl 39444 | General solutions of the P... |
| pell1234qrmulcl 39445 | General solutions of the P... |
| pell14qrss1234 39446 | A positive Pell solution i... |
| pell14qrre 39447 | A positive Pell solution i... |
| pell14qrne0 39448 | A positive Pell solution i... |
| pell14qrgt0 39449 | A positive Pell solution i... |
| pell14qrrp 39450 | A positive Pell solution i... |
| pell1234qrdich 39451 | A general Pell solution is... |
| elpell14qr2 39452 | A number is a positive Pel... |
| pell14qrmulcl 39453 | Positive Pell solutions ar... |
| pell14qrreccl 39454 | Positive Pell solutions ar... |
| pell14qrdivcl 39455 | Positive Pell solutions ar... |
| pell14qrexpclnn0 39456 | Lemma for ~ pell14qrexpcl ... |
| pell14qrexpcl 39457 | Positive Pell solutions ar... |
| pell1qrss14 39458 | First-quadrant Pell soluti... |
| pell14qrdich 39459 | A positive Pell solution i... |
| pell1qrge1 39460 | A Pell solution in the fir... |
| pell1qr1 39461 | 1 is a Pell solution and i... |
| elpell1qr2 39462 | The first quadrant solutio... |
| pell1qrgaplem 39463 | Lemma for ~ pell1qrgap . ... |
| pell1qrgap 39464 | First-quadrant Pell soluti... |
| pell14qrgap 39465 | Positive Pell solutions ar... |
| pell14qrgapw 39466 | Positive Pell solutions ar... |
| pellqrexplicit 39467 | Condition for a calculated... |
| infmrgelbi 39468 | Any lower bound of a nonem... |
| pellqrex 39469 | There is a nontrivial solu... |
| pellfundval 39470 | Value of the fundamental s... |
| pellfundre 39471 | The fundamental solution o... |
| pellfundge 39472 | Lower bound on the fundame... |
| pellfundgt1 39473 | Weak lower bound on the Pe... |
| pellfundlb 39474 | A nontrivial first quadran... |
| pellfundglb 39475 | If a real is larger than t... |
| pellfundex 39476 | The fundamental solution a... |
| pellfund14gap 39477 | There are no solutions bet... |
| pellfundrp 39478 | The fundamental Pell solut... |
| pellfundne1 39479 | The fundamental Pell solut... |
| reglogcl 39480 | General logarithm is a rea... |
| reglogltb 39481 | General logarithm preserve... |
| reglogleb 39482 | General logarithm preserve... |
| reglogmul 39483 | Multiplication law for gen... |
| reglogexp 39484 | Power law for general log.... |
| reglogbas 39485 | General log of the base is... |
| reglog1 39486 | General log of 1 is 0. (C... |
| reglogexpbas 39487 | General log of a power of ... |
| pellfund14 39488 | Every positive Pell soluti... |
| pellfund14b 39489 | The positive Pell solution... |
| rmxfval 39494 | Value of the X sequence. ... |
| rmyfval 39495 | Value of the Y sequence. ... |
| rmspecsqrtnq 39496 | The discriminant used to d... |
| rmspecnonsq 39497 | The discriminant used to d... |
| qirropth 39498 | This lemma implements the ... |
| rmspecfund 39499 | The base of exponent used ... |
| rmxyelqirr 39500 | The solutions used to cons... |
| rmxypairf1o 39501 | The function used to extra... |
| rmxyelxp 39502 | Lemma for ~ frmx and ~ frm... |
| frmx 39503 | The X sequence is a nonneg... |
| frmy 39504 | The Y sequence is an integ... |
| rmxyval 39505 | Main definition of the X a... |
| rmspecpos 39506 | The discriminant used to d... |
| rmxycomplete 39507 | The X and Y sequences take... |
| rmxynorm 39508 | The X and Y sequences defi... |
| rmbaserp 39509 | The base of exponentiation... |
| rmxyneg 39510 | Negation law for X and Y s... |
| rmxyadd 39511 | Addition formula for X and... |
| rmxy1 39512 | Value of the X and Y seque... |
| rmxy0 39513 | Value of the X and Y seque... |
| rmxneg 39514 | Negation law (even functio... |
| rmx0 39515 | Value of X sequence at 0. ... |
| rmx1 39516 | Value of X sequence at 1. ... |
| rmxadd 39517 | Addition formula for X seq... |
| rmyneg 39518 | Negation formula for Y seq... |
| rmy0 39519 | Value of Y sequence at 0. ... |
| rmy1 39520 | Value of Y sequence at 1. ... |
| rmyadd 39521 | Addition formula for Y seq... |
| rmxp1 39522 | Special addition-of-1 form... |
| rmyp1 39523 | Special addition of 1 form... |
| rmxm1 39524 | Subtraction of 1 formula f... |
| rmym1 39525 | Subtraction of 1 formula f... |
| rmxluc 39526 | The X sequence is a Lucas ... |
| rmyluc 39527 | The Y sequence is a Lucas ... |
| rmyluc2 39528 | Lucas sequence property of... |
| rmxdbl 39529 | "Double-angle formula" for... |
| rmydbl 39530 | "Double-angle formula" for... |
| monotuz 39531 | A function defined on an u... |
| monotoddzzfi 39532 | A function which is odd an... |
| monotoddzz 39533 | A function (given implicit... |
| oddcomabszz 39534 | An odd function which take... |
| 2nn0ind 39535 | Induction on nonnegative i... |
| zindbi 39536 | Inductively transfer a pro... |
| rmxypos 39537 | For all nonnegative indice... |
| ltrmynn0 39538 | The Y-sequence is strictly... |
| ltrmxnn0 39539 | The X-sequence is strictly... |
| lermxnn0 39540 | The X-sequence is monotoni... |
| rmxnn 39541 | The X-sequence is defined ... |
| ltrmy 39542 | The Y-sequence is strictly... |
| rmyeq0 39543 | Y is zero only at zero. (... |
| rmyeq 39544 | Y is one-to-one. (Contrib... |
| lermy 39545 | Y is monotonic (non-strict... |
| rmynn 39546 | ` rmY ` is positive for po... |
| rmynn0 39547 | ` rmY ` is nonnegative for... |
| rmyabs 39548 | ` rmY ` commutes with ` ab... |
| jm2.24nn 39549 | X(n) is strictly greater t... |
| jm2.17a 39550 | First half of lemma 2.17 o... |
| jm2.17b 39551 | Weak form of the second ha... |
| jm2.17c 39552 | Second half of lemma 2.17 ... |
| jm2.24 39553 | Lemma 2.24 of [JonesMatija... |
| rmygeid 39554 | Y(n) increases faster than... |
| congtr 39555 | A wff of the form ` A || (... |
| congadd 39556 | If two pairs of numbers ar... |
| congmul 39557 | If two pairs of numbers ar... |
| congsym 39558 | Congruence mod ` A ` is a ... |
| congneg 39559 | If two integers are congru... |
| congsub 39560 | If two pairs of numbers ar... |
| congid 39561 | Every integer is congruent... |
| mzpcong 39562 | Polynomials commute with c... |
| congrep 39563 | Every integer is congruent... |
| congabseq 39564 | If two integers are congru... |
| acongid 39565 | A wff like that in this th... |
| acongsym 39566 | Symmetry of alternating co... |
| acongneg2 39567 | Negate right side of alter... |
| acongtr 39568 | Transitivity of alternatin... |
| acongeq12d 39569 | Substitution deduction for... |
| acongrep 39570 | Every integer is alternati... |
| fzmaxdif 39571 | Bound on the difference be... |
| fzneg 39572 | Reflection of a finite ran... |
| acongeq 39573 | Two numbers in the fundame... |
| dvdsacongtr 39574 | Alternating congruence pas... |
| coprmdvdsb 39575 | Multiplication by a coprim... |
| modabsdifz 39576 | Divisibility in terms of m... |
| dvdsabsmod0 39577 | Divisibility in terms of m... |
| jm2.18 39578 | Theorem 2.18 of [JonesMati... |
| jm2.19lem1 39579 | Lemma for ~ jm2.19 . X an... |
| jm2.19lem2 39580 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem3 39581 | Lemma for ~ jm2.19 . (Con... |
| jm2.19lem4 39582 | Lemma for ~ jm2.19 . Exte... |
| jm2.19 39583 | Lemma 2.19 of [JonesMatija... |
| jm2.21 39584 | Lemma for ~ jm2.20nn . Ex... |
| jm2.22 39585 | Lemma for ~ jm2.20nn . Ap... |
| jm2.23 39586 | Lemma for ~ jm2.20nn . Tr... |
| jm2.20nn 39587 | Lemma 2.20 of [JonesMatija... |
| jm2.25lem1 39588 | Lemma for ~ jm2.26 . (Con... |
| jm2.25 39589 | Lemma for ~ jm2.26 . Rema... |
| jm2.26a 39590 | Lemma for ~ jm2.26 . Reve... |
| jm2.26lem3 39591 | Lemma for ~ jm2.26 . Use ... |
| jm2.26 39592 | Lemma 2.26 of [JonesMatija... |
| jm2.15nn0 39593 | Lemma 2.15 of [JonesMatija... |
| jm2.16nn0 39594 | Lemma 2.16 of [JonesMatija... |
| jm2.27a 39595 | Lemma for ~ jm2.27 . Reve... |
| jm2.27b 39596 | Lemma for ~ jm2.27 . Expa... |
| jm2.27c 39597 | Lemma for ~ jm2.27 . Forw... |
| jm2.27 39598 | Lemma 2.27 of [JonesMatija... |
| jm2.27dlem1 39599 | Lemma for ~ rmydioph . Su... |
| jm2.27dlem2 39600 | Lemma for ~ rmydioph . Th... |
| jm2.27dlem3 39601 | Lemma for ~ rmydioph . In... |
| jm2.27dlem4 39602 | Lemma for ~ rmydioph . In... |
| jm2.27dlem5 39603 | Lemma for ~ rmydioph . Us... |
| rmydioph 39604 | ~ jm2.27 restated in terms... |
| rmxdiophlem 39605 | X can be expressed in term... |
| rmxdioph 39606 | X is a Diophantine functio... |
| jm3.1lem1 39607 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem2 39608 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1lem3 39609 | Lemma for ~ jm3.1 . (Cont... |
| jm3.1 39610 | Diophantine expression for... |
| expdiophlem1 39611 | Lemma for ~ expdioph . Fu... |
| expdiophlem2 39612 | Lemma for ~ expdioph . Ex... |
| expdioph 39613 | The exponential function i... |
| setindtr 39614 | Set induction for sets con... |
| setindtrs 39615 | Set induction scheme witho... |
| dford3lem1 39616 | Lemma for ~ dford3 . (Con... |
| dford3lem2 39617 | Lemma for ~ dford3 . (Con... |
| dford3 39618 | Ordinals are precisely the... |
| dford4 39619 | ~ dford3 expressed in prim... |
| wopprc 39620 | Unrelated: Wiener pairs t... |
| rpnnen3lem 39621 | Lemma for ~ rpnnen3 . (Co... |
| rpnnen3 39622 | Dedekind cut injection of ... |
| axac10 39623 | Characterization of choice... |
| harinf 39624 | The Hartogs number of an i... |
| wdom2d2 39625 | Deduction for weak dominan... |
| ttac 39626 | Tarski's theorem about cho... |
| pw2f1ocnv 39627 | Define a bijection between... |
| pw2f1o2 39628 | Define a bijection between... |
| pw2f1o2val 39629 | Function value of the ~ pw... |
| pw2f1o2val2 39630 | Membership in a mapped set... |
| soeq12d 39631 | Equality deduction for tot... |
| freq12d 39632 | Equality deduction for fou... |
| weeq12d 39633 | Equality deduction for wel... |
| limsuc2 39634 | Limit ordinals in the sens... |
| wepwsolem 39635 | Transfer an ordering on ch... |
| wepwso 39636 | A well-ordering induces a ... |
| dnnumch1 39637 | Define an enumeration of a... |
| dnnumch2 39638 | Define an enumeration (wea... |
| dnnumch3lem 39639 | Value of the ordinal injec... |
| dnnumch3 39640 | Define an injection from a... |
| dnwech 39641 | Define a well-ordering fro... |
| fnwe2val 39642 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem1 39643 | Lemma for ~ fnwe2 . Subst... |
| fnwe2lem2 39644 | Lemma for ~ fnwe2 . An el... |
| fnwe2lem3 39645 | Lemma for ~ fnwe2 . Trich... |
| fnwe2 39646 | A well-ordering can be con... |
| aomclem1 39647 | Lemma for ~ dfac11 . This... |
| aomclem2 39648 | Lemma for ~ dfac11 . Succ... |
| aomclem3 39649 | Lemma for ~ dfac11 . Succ... |
| aomclem4 39650 | Lemma for ~ dfac11 . Limi... |
| aomclem5 39651 | Lemma for ~ dfac11 . Comb... |
| aomclem6 39652 | Lemma for ~ dfac11 . Tran... |
| aomclem7 39653 | Lemma for ~ dfac11 . ` ( R... |
| aomclem8 39654 | Lemma for ~ dfac11 . Perf... |
| dfac11 39655 | The right-hand side of thi... |
| kelac1 39656 | Kelley's choice, basic for... |
| kelac2lem 39657 | Lemma for ~ kelac2 and ~ d... |
| kelac2 39658 | Kelley's choice, most comm... |
| dfac21 39659 | Tychonoff's theorem is a c... |
| islmodfg 39662 | Property of a finitely gen... |
| islssfg 39663 | Property of a finitely gen... |
| islssfg2 39664 | Property of a finitely gen... |
| islssfgi 39665 | Finitely spanned subspaces... |
| fglmod 39666 | Finitely generated left mo... |
| lsmfgcl 39667 | The sum of two finitely ge... |
| islnm 39670 | Property of being a Noethe... |
| islnm2 39671 | Property of being a Noethe... |
| lnmlmod 39672 | A Noetherian left module i... |
| lnmlssfg 39673 | A submodule of Noetherian ... |
| lnmlsslnm 39674 | All submodules of a Noethe... |
| lnmfg 39675 | A Noetherian left module i... |
| kercvrlsm 39676 | The domain of a linear fun... |
| lmhmfgima 39677 | A homomorphism maps finite... |
| lnmepi 39678 | Epimorphic images of Noeth... |
| lmhmfgsplit 39679 | If the kernel and range of... |
| lmhmlnmsplit 39680 | If the kernel and range of... |
| lnmlmic 39681 | Noetherian is an invariant... |
| pwssplit4 39682 | Splitting for structure po... |
| filnm 39683 | Finite left modules are No... |
| pwslnmlem0 39684 | Zeroeth powers are Noether... |
| pwslnmlem1 39685 | First powers are Noetheria... |
| pwslnmlem2 39686 | A sum of powers is Noether... |
| pwslnm 39687 | Finite powers of Noetheria... |
| unxpwdom3 39688 | Weaker version of ~ unxpwd... |
| pwfi2f1o 39689 | The ~ pw2f1o bijection rel... |
| pwfi2en 39690 | Finitely supported indicat... |
| frlmpwfi 39691 | Formal linear combinations... |
| gicabl 39692 | Being Abelian is a group i... |
| imasgim 39693 | A relabeling of the elemen... |
| isnumbasgrplem1 39694 | A set which is equipollent... |
| harn0 39695 | The Hartogs number of a se... |
| numinfctb 39696 | A numerable infinite set c... |
| isnumbasgrplem2 39697 | If the (to be thought of a... |
| isnumbasgrplem3 39698 | Every nonempty numerable s... |
| isnumbasabl 39699 | A set is numerable iff it ... |
| isnumbasgrp 39700 | A set is numerable iff it ... |
| dfacbasgrp 39701 | A choice equivalent in abs... |
| islnr 39704 | Property of a left-Noether... |
| lnrring 39705 | Left-Noetherian rings are ... |
| lnrlnm 39706 | Left-Noetherian rings have... |
| islnr2 39707 | Property of being a left-N... |
| islnr3 39708 | Relate left-Noetherian rin... |
| lnr2i 39709 | Given an ideal in a left-N... |
| lpirlnr 39710 | Left principal ideal rings... |
| lnrfrlm 39711 | Finite-dimensional free mo... |
| lnrfg 39712 | Finitely-generated modules... |
| lnrfgtr 39713 | A submodule of a finitely ... |
| hbtlem1 39716 | Value of the leading coeff... |
| hbtlem2 39717 | Leading coefficient ideals... |
| hbtlem7 39718 | Functionality of leading c... |
| hbtlem4 39719 | The leading ideal function... |
| hbtlem3 39720 | The leading ideal function... |
| hbtlem5 39721 | The leading ideal function... |
| hbtlem6 39722 | There is a finite set of p... |
| hbt 39723 | The Hilbert Basis Theorem ... |
| dgrsub2 39728 | Subtracting two polynomial... |
| elmnc 39729 | Property of a monic polyno... |
| mncply 39730 | A monic polynomial is a po... |
| mnccoe 39731 | A monic polynomial has lea... |
| mncn0 39732 | A monic polynomial is not ... |
| dgraaval 39737 | Value of the degree functi... |
| dgraalem 39738 | Properties of the degree o... |
| dgraacl 39739 | Closure of the degree func... |
| dgraaf 39740 | Degree function on algebra... |
| dgraaub 39741 | Upper bound on degree of a... |
| dgraa0p 39742 | A rational polynomial of d... |
| mpaaeu 39743 | An algebraic number has ex... |
| mpaaval 39744 | Value of the minimal polyn... |
| mpaalem 39745 | Properties of the minimal ... |
| mpaacl 39746 | Minimal polynomial is a po... |
| mpaadgr 39747 | Minimal polynomial has deg... |
| mpaaroot 39748 | The minimal polynomial of ... |
| mpaamn 39749 | Minimal polynomial is moni... |
| itgoval 39754 | Value of the integral-over... |
| aaitgo 39755 | The standard algebraic num... |
| itgoss 39756 | An integral element is int... |
| itgocn 39757 | All integral elements are ... |
| cnsrexpcl 39758 | Exponentiation is closed i... |
| fsumcnsrcl 39759 | Finite sums are closed in ... |
| cnsrplycl 39760 | Polynomials are closed in ... |
| rgspnval 39761 | Value of the ring-span of ... |
| rgspncl 39762 | The ring-span of a set is ... |
| rgspnssid 39763 | The ring-span of a set con... |
| rgspnmin 39764 | The ring-span is contained... |
| rgspnid 39765 | The span of a subring is i... |
| rngunsnply 39766 | Adjoining one element to a... |
| flcidc 39767 | Finite linear combinations... |
| algstr 39770 | Lemma to shorten proofs of... |
| algbase 39771 | The base set of a construc... |
| algaddg 39772 | The additive operation of ... |
| algmulr 39773 | The multiplicative operati... |
| algsca 39774 | The set of scalars of a co... |
| algvsca 39775 | The scalar product operati... |
| mendval 39776 | Value of the module endomo... |
| mendbas 39777 | Base set of the module end... |
| mendplusgfval 39778 | Addition in the module end... |
| mendplusg 39779 | A specific addition in the... |
| mendmulrfval 39780 | Multiplication in the modu... |
| mendmulr 39781 | A specific multiplication ... |
| mendsca 39782 | The module endomorphism al... |
| mendvscafval 39783 | Scalar multiplication in t... |
| mendvsca 39784 | A specific scalar multipli... |
| mendring 39785 | The module endomorphism al... |
| mendlmod 39786 | The module endomorphism al... |
| mendassa 39787 | The module endomorphism al... |
| idomrootle 39788 | No element of an integral ... |
| idomodle 39789 | Limit on the number of ` N... |
| fiuneneq 39790 | Two finite sets of equal s... |
| idomsubgmo 39791 | The units of an integral d... |
| proot1mul 39792 | Any primitive ` N ` -th ro... |
| proot1hash 39793 | If an integral domain has ... |
| proot1ex 39794 | The complex field has prim... |
| isdomn3 39797 | Nonzero elements form a mu... |
| mon1pid 39798 | Monicity and degree of the... |
| mon1psubm 39799 | Monic polynomials are a mu... |
| deg1mhm 39800 | Homomorphic property of th... |
| cytpfn 39801 | Functionality of the cyclo... |
| cytpval 39802 | Substitutions for the Nth ... |
| fgraphopab 39803 | Express a function as a su... |
| fgraphxp 39804 | Express a function as a su... |
| hausgraph 39805 | The graph of a continuous ... |
| iocunico 39810 | Split an open interval int... |
| iocinico 39811 | The intersection of two se... |
| iocmbl 39812 | An open-below, closed-abov... |
| cnioobibld 39813 | A bounded, continuous func... |
| itgpowd 39814 | The integral of a monomial... |
| arearect 39815 | The area of a rectangle wh... |
| areaquad 39816 | The area of a quadrilatera... |
| ifpan123g 39817 | Conjunction of conditional... |
| ifpan23 39818 | Conjunction of conditional... |
| ifpdfor2 39819 | Define or in terms of cond... |
| ifporcor 39820 | Corollary of commutation o... |
| ifpdfan2 39821 | Define and with conditiona... |
| ifpancor 39822 | Corollary of commutation o... |
| ifpdfor 39823 | Define or in terms of cond... |
| ifpdfan 39824 | Define and with conditiona... |
| ifpbi2 39825 | Equivalence theorem for co... |
| ifpbi3 39826 | Equivalence theorem for co... |
| ifpim1 39827 | Restate implication as con... |
| ifpnot 39828 | Restate negated wff as con... |
| ifpid2 39829 | Restate wff as conditional... |
| ifpim2 39830 | Restate implication as con... |
| ifpbi23 39831 | Equivalence theorem for co... |
| ifpdfbi 39832 | Define biimplication as co... |
| ifpbiidcor 39833 | Restatement of ~ biid . (... |
| ifpbicor 39834 | Corollary of commutation o... |
| ifpxorcor 39835 | Corollary of commutation o... |
| ifpbi1 39836 | Equivalence theorem for co... |
| ifpnot23 39837 | Negation of conditional lo... |
| ifpnotnotb 39838 | Factor conditional logic o... |
| ifpnorcor 39839 | Corollary of commutation o... |
| ifpnancor 39840 | Corollary of commutation o... |
| ifpnot23b 39841 | Negation of conditional lo... |
| ifpbiidcor2 39842 | Restatement of ~ biid . (... |
| ifpnot23c 39843 | Negation of conditional lo... |
| ifpnot23d 39844 | Negation of conditional lo... |
| ifpdfnan 39845 | Define nand as conditional... |
| ifpdfxor 39846 | Define xor as conditional ... |
| ifpbi12 39847 | Equivalence theorem for co... |
| ifpbi13 39848 | Equivalence theorem for co... |
| ifpbi123 39849 | Equivalence theorem for co... |
| ifpidg 39850 | Restate wff as conditional... |
| ifpid3g 39851 | Restate wff as conditional... |
| ifpid2g 39852 | Restate wff as conditional... |
| ifpid1g 39853 | Restate wff as conditional... |
| ifpim23g 39854 | Restate implication as con... |
| ifpim3 39855 | Restate implication as con... |
| ifpnim1 39856 | Restate negated implicatio... |
| ifpim4 39857 | Restate implication as con... |
| ifpnim2 39858 | Restate negated implicatio... |
| ifpim123g 39859 | Implication of conditional... |
| ifpim1g 39860 | Implication of conditional... |
| ifp1bi 39861 | Substitute the first eleme... |
| ifpbi1b 39862 | When the first variable is... |
| ifpimimb 39863 | Factor conditional logic o... |
| ifpororb 39864 | Factor conditional logic o... |
| ifpananb 39865 | Factor conditional logic o... |
| ifpnannanb 39866 | Factor conditional logic o... |
| ifpor123g 39867 | Disjunction of conditional... |
| ifpimim 39868 | Consequnce of implication.... |
| ifpbibib 39869 | Factor conditional logic o... |
| ifpxorxorb 39870 | Factor conditional logic o... |
| rp-fakeimass 39871 | A special case where impli... |
| rp-fakeanorass 39872 | A special case where a mix... |
| rp-fakeoranass 39873 | A special case where a mix... |
| rp-fakeinunass 39874 | A special case where a mix... |
| rp-fakeuninass 39875 | A special case where a mix... |
| rp-isfinite5 39876 | A set is said to be finite... |
| rp-isfinite6 39877 | A set is said to be finite... |
| intabssd 39878 | When for each element ` y ... |
| eu0 39879 | There is only one empty se... |
| epelon2 39880 | Over the ordinal numbers, ... |
| ontric3g 39881 | For all ` x , y e. On ` , ... |
| dfsucon 39882 | ` A ` is called a successo... |
| snen1g 39883 | A singleton is equinumerou... |
| snen1el 39884 | A singleton is equinumerou... |
| sn1dom 39885 | A singleton is dominated b... |
| pr2dom 39886 | An unordered pair is domin... |
| tr3dom 39887 | An unordered triple is dom... |
| ensucne0 39888 | A class equinumerous to a ... |
| ensucne0OLD 39889 | A class equinumerous to a ... |
| nndomog 39890 | Cardinal ordering agrees w... |
| dfom6 39891 | Let ` _om ` be defined to ... |
| infordmin 39892 | ` _om ` is the smallest in... |
| iscard4 39893 | Two ways to express the pr... |
| iscard5 39894 | Two ways to express the pr... |
| elrncard 39895 | Let us define a cardinal n... |
| harsucnn 39896 | The next cardinal after a ... |
| harval3 39897 | ` ( har `` A ) ` is the le... |
| harval3on 39898 | For any ordinal number ` A... |
| en2pr 39899 | A class is equinumerous to... |
| pr2cv 39900 | If an unordered pair is eq... |
| pr2el1 39901 | If an unordered pair is eq... |
| pr2cv1 39902 | If an unordered pair is eq... |
| pr2el2 39903 | If an unordered pair is eq... |
| pr2cv2 39904 | If an unordered pair is eq... |
| pren2 39905 | An unordered pair is equin... |
| pr2eldif1 39906 | If an unordered pair is eq... |
| pr2eldif2 39907 | If an unordered pair is eq... |
| pren2d 39908 | A pair of two distinct set... |
| aleph1min 39909 | ` ( aleph `` 1o ) ` is the... |
| alephiso2 39910 | ` aleph ` is a strictly or... |
| alephiso3 39911 | ` aleph ` is a strictly or... |
| pwelg 39912 | The powerclass is an eleme... |
| pwinfig 39913 | The powerclass of an infin... |
| pwinfi2 39914 | The powerclass of an infin... |
| pwinfi3 39915 | The powerclass of an infin... |
| pwinfi 39916 | The powerclass of an infin... |
| fipjust 39917 | A definition of the finite... |
| cllem0 39918 | The class of all sets with... |
| superficl 39919 | The class of all supersets... |
| superuncl 39920 | The class of all supersets... |
| ssficl 39921 | The class of all subsets o... |
| ssuncl 39922 | The class of all subsets o... |
| ssdifcl 39923 | The class of all subsets o... |
| sssymdifcl 39924 | The class of all subsets o... |
| fiinfi 39925 | If two classes have the fi... |
| rababg 39926 | Condition when restricted ... |
| elintabg 39927 | Two ways of saying a set i... |
| elinintab 39928 | Two ways of saying a set i... |
| elmapintrab 39929 | Two ways to say a set is a... |
| elinintrab 39930 | Two ways of saying a set i... |
| inintabss 39931 | Upper bound on intersectio... |
| inintabd 39932 | Value of the intersection ... |
| xpinintabd 39933 | Value of the intersection ... |
| relintabex 39934 | If the intersection of a c... |
| elcnvcnvintab 39935 | Two ways of saying a set i... |
| relintab 39936 | Value of the intersection ... |
| nonrel 39937 | A non-relation is equal to... |
| elnonrel 39938 | Only an ordered pair where... |
| cnvssb 39939 | Subclass theorem for conve... |
| relnonrel 39940 | The non-relation part of a... |
| cnvnonrel 39941 | The converse of the non-re... |
| brnonrel 39942 | A non-relation cannot rela... |
| dmnonrel 39943 | The domain of the non-rela... |
| rnnonrel 39944 | The range of the non-relat... |
| resnonrel 39945 | A restriction of the non-r... |
| imanonrel 39946 | An image under the non-rel... |
| cononrel1 39947 | Composition with the non-r... |
| cononrel2 39948 | Composition with the non-r... |
| elmapintab 39949 | Two ways to say a set is a... |
| fvnonrel 39950 | The function value of any ... |
| elinlem 39951 | Two ways to say a set is a... |
| elcnvcnvlem 39952 | Two ways to say a set is a... |
| cnvcnvintabd 39953 | Value of the relationship ... |
| elcnvlem 39954 | Two ways to say a set is a... |
| elcnvintab 39955 | Two ways of saying a set i... |
| cnvintabd 39956 | Value of the converse of t... |
| undmrnresiss 39957 | Two ways of saying the ide... |
| reflexg 39958 | Two ways of saying a relat... |
| cnvssco 39959 | A condition weaker than re... |
| refimssco 39960 | Reflexive relations are su... |
| cleq2lem 39961 | Equality implies bijection... |
| cbvcllem 39962 | Change of bound variable i... |
| clublem 39963 | If a superset ` Y ` of ` X... |
| clss2lem 39964 | The closure of a property ... |
| dfid7 39965 | Definition of identity rel... |
| mptrcllem 39966 | Show two versions of a clo... |
| cotrintab 39967 | The intersection of a clas... |
| rclexi 39968 | The reflexive closure of a... |
| rtrclexlem 39969 | Existence of relation impl... |
| rtrclex 39970 | The reflexive-transitive c... |
| trclubgNEW 39971 | If a relation exists then ... |
| trclubNEW 39972 | If a relation exists then ... |
| trclexi 39973 | The transitive closure of ... |
| rtrclexi 39974 | The reflexive-transitive c... |
| clrellem 39975 | When the property ` ps ` h... |
| clcnvlem 39976 | When ` A ` , an upper boun... |
| cnvtrucl0 39977 | The converse of the trivia... |
| cnvrcl0 39978 | The converse of the reflex... |
| cnvtrcl0 39979 | The converse of the transi... |
| dmtrcl 39980 | The domain of the transiti... |
| rntrcl 39981 | The range of the transitiv... |
| dfrtrcl5 39982 | Definition of reflexive-tr... |
| trcleq2lemRP 39983 | Equality implies bijection... |
| al3im 39984 | Version of ~ ax-4 for a ne... |
| intima0 39985 | Two ways of expressing the... |
| elimaint 39986 | Element of image of inters... |
| csbcog 39987 | Distribute proper substitu... |
| cnviun 39988 | Converse of indexed union.... |
| imaiun1 39989 | The image of an indexed un... |
| coiun1 39990 | Composition with an indexe... |
| elintima 39991 | Element of intersection of... |
| intimass 39992 | The image under the inters... |
| intimass2 39993 | The image under the inters... |
| intimag 39994 | Requirement for the image ... |
| intimasn 39995 | Two ways to express the im... |
| intimasn2 39996 | Two ways to express the im... |
| ss2iundf 39997 | Subclass theorem for index... |
| ss2iundv 39998 | Subclass theorem for index... |
| cbviuneq12df 39999 | Rule used to change the bo... |
| cbviuneq12dv 40000 | Rule used to change the bo... |
| conrel1d 40001 | Deduction about compositio... |
| conrel2d 40002 | Deduction about compositio... |
| trrelind 40003 | The intersection of transi... |
| xpintrreld 40004 | The intersection of a tran... |
| restrreld 40005 | The restriction of a trans... |
| trrelsuperreldg 40006 | Concrete construction of a... |
| trficl 40007 | The class of all transitiv... |
| cnvtrrel 40008 | The converse of a transiti... |
| trrelsuperrel2dg 40009 | Concrete construction of a... |
| dfrcl2 40012 | Reflexive closure of a rel... |
| dfrcl3 40013 | Reflexive closure of a rel... |
| dfrcl4 40014 | Reflexive closure of a rel... |
| relexp2 40015 | A set operated on by the r... |
| relexpnul 40016 | If the domain and range of... |
| eliunov2 40017 | Membership in the indexed ... |
| eltrclrec 40018 | Membership in the indexed ... |
| elrtrclrec 40019 | Membership in the indexed ... |
| briunov2 40020 | Two classes related by the... |
| brmptiunrelexpd 40021 | If two elements are connec... |
| fvmptiunrelexplb0d 40022 | If the indexed union range... |
| fvmptiunrelexplb0da 40023 | If the indexed union range... |
| fvmptiunrelexplb1d 40024 | If the indexed union range... |
| brfvid 40025 | If two elements are connec... |
| brfvidRP 40026 | If two elements are connec... |
| fvilbd 40027 | A set is a subset of its i... |
| fvilbdRP 40028 | A set is a subset of its i... |
| brfvrcld 40029 | If two elements are connec... |
| brfvrcld2 40030 | If two elements are connec... |
| fvrcllb0d 40031 | A restriction of the ident... |
| fvrcllb0da 40032 | A restriction of the ident... |
| fvrcllb1d 40033 | A set is a subset of its i... |
| brtrclrec 40034 | Two classes related by the... |
| brrtrclrec 40035 | Two classes related by the... |
| briunov2uz 40036 | Two classes related by the... |
| eliunov2uz 40037 | Membership in the indexed ... |
| ov2ssiunov2 40038 | Any particular operator va... |
| relexp0eq 40039 | The zeroth power of relati... |
| iunrelexp0 40040 | Simplification of zeroth p... |
| relexpxpnnidm 40041 | Any positive power of a cr... |
| relexpiidm 40042 | Any power of any restricti... |
| relexpss1d 40043 | The relational power of a ... |
| comptiunov2i 40044 | The composition two indexe... |
| corclrcl 40045 | The reflexive closure is i... |
| iunrelexpmin1 40046 | The indexed union of relat... |
| relexpmulnn 40047 | With exponents limited to ... |
| relexpmulg 40048 | With ordered exponents, th... |
| trclrelexplem 40049 | The union of relational po... |
| iunrelexpmin2 40050 | The indexed union of relat... |
| relexp01min 40051 | With exponents limited to ... |
| relexp1idm 40052 | Repeated raising a relatio... |
| relexp0idm 40053 | Repeated raising a relatio... |
| relexp0a 40054 | Absorbtion law for zeroth ... |
| relexpxpmin 40055 | The composition of powers ... |
| relexpaddss 40056 | The composition of two pow... |
| iunrelexpuztr 40057 | The indexed union of relat... |
| dftrcl3 40058 | Transitive closure of a re... |
| brfvtrcld 40059 | If two elements are connec... |
| fvtrcllb1d 40060 | A set is a subset of its i... |
| trclfvcom 40061 | The transitive closure of ... |
| cnvtrclfv 40062 | The converse of the transi... |
| cotrcltrcl 40063 | The transitive closure is ... |
| trclimalb2 40064 | Lower bound for image unde... |
| brtrclfv2 40065 | Two ways to indicate two e... |
| trclfvdecomr 40066 | The transitive closure of ... |
| trclfvdecoml 40067 | The transitive closure of ... |
| dmtrclfvRP 40068 | The domain of the transiti... |
| rntrclfvRP 40069 | The range of the transitiv... |
| rntrclfv 40070 | The range of the transitiv... |
| dfrtrcl3 40071 | Reflexive-transitive closu... |
| brfvrtrcld 40072 | If two elements are connec... |
| fvrtrcllb0d 40073 | A restriction of the ident... |
| fvrtrcllb0da 40074 | A restriction of the ident... |
| fvrtrcllb1d 40075 | A set is a subset of its i... |
| dfrtrcl4 40076 | Reflexive-transitive closu... |
| corcltrcl 40077 | The composition of the ref... |
| cortrcltrcl 40078 | Composition with the refle... |
| corclrtrcl 40079 | Composition with the refle... |
| cotrclrcl 40080 | The composition of the ref... |
| cortrclrcl 40081 | Composition with the refle... |
| cotrclrtrcl 40082 | Composition with the refle... |
| cortrclrtrcl 40083 | The reflexive-transitive c... |
| frege77d 40084 | If the images of both ` { ... |
| frege81d 40085 | If the image of ` U ` is a... |
| frege83d 40086 | If the image of the union ... |
| frege96d 40087 | If ` C ` follows ` A ` in ... |
| frege87d 40088 | If the images of both ` { ... |
| frege91d 40089 | If ` B ` follows ` A ` in ... |
| frege97d 40090 | If ` A ` contains all elem... |
| frege98d 40091 | If ` C ` follows ` A ` and... |
| frege102d 40092 | If either ` A ` and ` C ` ... |
| frege106d 40093 | If ` B ` follows ` A ` in ... |
| frege108d 40094 | If either ` A ` and ` C ` ... |
| frege109d 40095 | If ` A ` contains all elem... |
| frege114d 40096 | If either ` R ` relates ` ... |
| frege111d 40097 | If either ` A ` and ` C ` ... |
| frege122d 40098 | If ` F ` is a function, ` ... |
| frege124d 40099 | If ` F ` is a function, ` ... |
| frege126d 40100 | If ` F ` is a function, ` ... |
| frege129d 40101 | If ` F ` is a function and... |
| frege131d 40102 | If ` F ` is a function and... |
| frege133d 40103 | If ` F ` is a function and... |
| dfxor4 40104 | Express exclusive-or in te... |
| dfxor5 40105 | Express exclusive-or in te... |
| df3or2 40106 | Express triple-or in terms... |
| df3an2 40107 | Express triple-and in term... |
| nev 40108 | Express that not every set... |
| 0pssin 40109 | Express that an intersecti... |
| rp-imass 40110 | If the ` R ` -image of a c... |
| dfhe2 40113 | The property of relation `... |
| dfhe3 40114 | The property of relation `... |
| heeq12 40115 | Equality law for relations... |
| heeq1 40116 | Equality law for relations... |
| heeq2 40117 | Equality law for relations... |
| sbcheg 40118 | Distribute proper substitu... |
| hess 40119 | Subclass law for relations... |
| xphe 40120 | Any Cartesian product is h... |
| 0he 40121 | The empty relation is here... |
| 0heALT 40122 | The empty relation is here... |
| he0 40123 | Any relation is hereditary... |
| unhe1 40124 | The union of two relations... |
| snhesn 40125 | Any singleton is hereditar... |
| idhe 40126 | The identity relation is h... |
| psshepw 40127 | The relation between sets ... |
| sshepw 40128 | The relation between sets ... |
| rp-simp2-frege 40131 | Simplification of triple c... |
| rp-simp2 40132 | Simplification of triple c... |
| rp-frege3g 40133 | Add antecedent to ~ ax-fre... |
| frege3 40134 | Add antecedent to ~ ax-fre... |
| rp-misc1-frege 40135 | Double-use of ~ ax-frege2 ... |
| rp-frege24 40136 | Introducing an embedded an... |
| rp-frege4g 40137 | Deduction related to distr... |
| frege4 40138 | Special case of closed for... |
| frege5 40139 | A closed form of ~ syl . ... |
| rp-7frege 40140 | Distribute antecedent and ... |
| rp-4frege 40141 | Elimination of a nested an... |
| rp-6frege 40142 | Elimination of a nested an... |
| rp-8frege 40143 | Eliminate antecedent when ... |
| rp-frege25 40144 | Closed form for ~ a1dd . ... |
| frege6 40145 | A closed form of ~ imim2d ... |
| axfrege8 40146 | Swap antecedents. Identic... |
| frege7 40147 | A closed form of ~ syl6 . ... |
| frege26 40149 | Identical to ~ idd . Prop... |
| frege27 40150 | We cannot (at the same tim... |
| frege9 40151 | Closed form of ~ syl with ... |
| frege12 40152 | A closed form of ~ com23 .... |
| frege11 40153 | Elimination of a nested an... |
| frege24 40154 | Closed form for ~ a1d . D... |
| frege16 40155 | A closed form of ~ com34 .... |
| frege25 40156 | Closed form for ~ a1dd . ... |
| frege18 40157 | Closed form of a syllogism... |
| frege22 40158 | A closed form of ~ com45 .... |
| frege10 40159 | Result commuting anteceden... |
| frege17 40160 | A closed form of ~ com3l .... |
| frege13 40161 | A closed form of ~ com3r .... |
| frege14 40162 | Closed form of a deduction... |
| frege19 40163 | A closed form of ~ syl6 . ... |
| frege23 40164 | Syllogism followed by rota... |
| frege15 40165 | A closed form of ~ com4r .... |
| frege21 40166 | Replace antecedent in ante... |
| frege20 40167 | A closed form of ~ syl8 . ... |
| axfrege28 40168 | Contraposition. Identical... |
| frege29 40170 | Closed form of ~ con3d . ... |
| frege30 40171 | Commuted, closed form of ~... |
| axfrege31 40172 | Identical to ~ notnotr . ... |
| frege32 40174 | Deduce ~ con1 from ~ con3 ... |
| frege33 40175 | If ` ph ` or ` ps ` takes ... |
| frege34 40176 | If as a conseqence of the ... |
| frege35 40177 | Commuted, closed form of ~... |
| frege36 40178 | The case in which ` ps ` i... |
| frege37 40179 | If ` ch ` is a necessary c... |
| frege38 40180 | Identical to ~ pm2.21 . P... |
| frege39 40181 | Syllogism between ~ pm2.18... |
| frege40 40182 | Anything implies ~ pm2.18 ... |
| axfrege41 40183 | Identical to ~ notnot . A... |
| frege42 40185 | Not not ~ id . Propositio... |
| frege43 40186 | If there is a choice only ... |
| frege44 40187 | Similar to a commuted ~ pm... |
| frege45 40188 | Deduce ~ pm2.6 from ~ con1... |
| frege46 40189 | If ` ps ` holds when ` ph ... |
| frege47 40190 | Deduce consequence follows... |
| frege48 40191 | Closed form of syllogism w... |
| frege49 40192 | Closed form of deduction w... |
| frege50 40193 | Closed form of ~ jaoi . P... |
| frege51 40194 | Compare with ~ jaod . Pro... |
| axfrege52a 40195 | Justification for ~ ax-fre... |
| frege52aid 40197 | The case when the content ... |
| frege53aid 40198 | Specialization of ~ frege5... |
| frege53a 40199 | Lemma for ~ frege55a . Pr... |
| axfrege54a 40200 | Justification for ~ ax-fre... |
| frege54cor0a 40202 | Synonym for logical equiva... |
| frege54cor1a 40203 | Reflexive equality. (Cont... |
| frege55aid 40204 | Lemma for ~ frege57aid . ... |
| frege55lem1a 40205 | Necessary deduction regard... |
| frege55lem2a 40206 | Core proof of Proposition ... |
| frege55a 40207 | Proposition 55 of [Frege18... |
| frege55cor1a 40208 | Proposition 55 of [Frege18... |
| frege56aid 40209 | Lemma for ~ frege57aid . ... |
| frege56a 40210 | Proposition 56 of [Frege18... |
| frege57aid 40211 | This is the all imporant f... |
| frege57a 40212 | Analogue of ~ frege57aid .... |
| axfrege58a 40213 | Identical to ~ anifp . Ju... |
| frege58acor 40215 | Lemma for ~ frege59a . (C... |
| frege59a 40216 | A kind of Aristotelian inf... |
| frege60a 40217 | Swap antecedents of ~ ax-f... |
| frege61a 40218 | Lemma for ~ frege65a . Pr... |
| frege62a 40219 | A kind of Aristotelian inf... |
| frege63a 40220 | Proposition 63 of [Frege18... |
| frege64a 40221 | Lemma for ~ frege65a . Pr... |
| frege65a 40222 | A kind of Aristotelian inf... |
| frege66a 40223 | Swap antecedents of ~ freg... |
| frege67a 40224 | Lemma for ~ frege68a . Pr... |
| frege68a 40225 | Combination of applying a ... |
| axfrege52c 40226 | Justification for ~ ax-fre... |
| frege52b 40228 | The case when the content ... |
| frege53b 40229 | Lemma for frege102 (via ~ ... |
| axfrege54c 40230 | Reflexive equality of clas... |
| frege54b 40232 | Reflexive equality of sets... |
| frege54cor1b 40233 | Reflexive equality. (Cont... |
| frege55lem1b 40234 | Necessary deduction regard... |
| frege55lem2b 40235 | Lemma for ~ frege55b . Co... |
| frege55b 40236 | Lemma for ~ frege57b . Pr... |
| frege56b 40237 | Lemma for ~ frege57b . Pr... |
| frege57b 40238 | Analogue of ~ frege57aid .... |
| axfrege58b 40239 | If ` A. x ph ` is affirmed... |
| frege58bid 40241 | If ` A. x ph ` is affirmed... |
| frege58bcor 40242 | Lemma for ~ frege59b . (C... |
| frege59b 40243 | A kind of Aristotelian inf... |
| frege60b 40244 | Swap antecedents of ~ ax-f... |
| frege61b 40245 | Lemma for ~ frege65b . Pr... |
| frege62b 40246 | A kind of Aristotelian inf... |
| frege63b 40247 | Lemma for ~ frege91 . Pro... |
| frege64b 40248 | Lemma for ~ frege65b . Pr... |
| frege65b 40249 | A kind of Aristotelian inf... |
| frege66b 40250 | Swap antecedents of ~ freg... |
| frege67b 40251 | Lemma for ~ frege68b . Pr... |
| frege68b 40252 | Combination of applying a ... |
| frege53c 40253 | Proposition 53 of [Frege18... |
| frege54cor1c 40254 | Reflexive equality. (Cont... |
| frege55lem1c 40255 | Necessary deduction regard... |
| frege55lem2c 40256 | Core proof of Proposition ... |
| frege55c 40257 | Proposition 55 of [Frege18... |
| frege56c 40258 | Lemma for ~ frege57c . Pr... |
| frege57c 40259 | Swap order of implication ... |
| frege58c 40260 | Principle related to ~ sp ... |
| frege59c 40261 | A kind of Aristotelian inf... |
| frege60c 40262 | Swap antecedents of ~ freg... |
| frege61c 40263 | Lemma for ~ frege65c . Pr... |
| frege62c 40264 | A kind of Aristotelian inf... |
| frege63c 40265 | Analogue of ~ frege63b . ... |
| frege64c 40266 | Lemma for ~ frege65c . Pr... |
| frege65c 40267 | A kind of Aristotelian inf... |
| frege66c 40268 | Swap antecedents of ~ freg... |
| frege67c 40269 | Lemma for ~ frege68c . Pr... |
| frege68c 40270 | Combination of applying a ... |
| dffrege69 40271 | If from the proposition th... |
| frege70 40272 | Lemma for ~ frege72 . Pro... |
| frege71 40273 | Lemma for ~ frege72 . Pro... |
| frege72 40274 | If property ` A ` is hered... |
| frege73 40275 | Lemma for ~ frege87 . Pro... |
| frege74 40276 | If ` X ` has a property ` ... |
| frege75 40277 | If from the proposition th... |
| dffrege76 40278 | If from the two propositio... |
| frege77 40279 | If ` Y ` follows ` X ` in ... |
| frege78 40280 | Commuted form of of ~ freg... |
| frege79 40281 | Distributed form of ~ freg... |
| frege80 40282 | Add additional condition t... |
| frege81 40283 | If ` X ` has a property ` ... |
| frege82 40284 | Closed-form deduction base... |
| frege83 40285 | Apply commuted form of ~ f... |
| frege84 40286 | Commuted form of ~ frege81... |
| frege85 40287 | Commuted form of ~ frege77... |
| frege86 40288 | Conclusion about element o... |
| frege87 40289 | If ` Z ` is a result of an... |
| frege88 40290 | Commuted form of ~ frege87... |
| frege89 40291 | One direction of ~ dffrege... |
| frege90 40292 | Add antecedent to ~ frege8... |
| frege91 40293 | Every result of an applica... |
| frege92 40294 | Inference from ~ frege91 .... |
| frege93 40295 | Necessary condition for tw... |
| frege94 40296 | Looking one past a pair re... |
| frege95 40297 | Looking one past a pair re... |
| frege96 40298 | Every result of an applica... |
| frege97 40299 | The property of following ... |
| frege98 40300 | If ` Y ` follows ` X ` and... |
| dffrege99 40301 | If ` Z ` is identical with... |
| frege100 40302 | One direction of ~ dffrege... |
| frege101 40303 | Lemma for ~ frege102 . Pr... |
| frege102 40304 | If ` Z ` belongs to the ` ... |
| frege103 40305 | Proposition 103 of [Frege1... |
| frege104 40306 | Proposition 104 of [Frege1... |
| frege105 40307 | Proposition 105 of [Frege1... |
| frege106 40308 | Whatever follows ` X ` in ... |
| frege107 40309 | Proposition 107 of [Frege1... |
| frege108 40310 | If ` Y ` belongs to the ` ... |
| frege109 40311 | The property of belonging ... |
| frege110 40312 | Proposition 110 of [Frege1... |
| frege111 40313 | If ` Y ` belongs to the ` ... |
| frege112 40314 | Identity implies belonging... |
| frege113 40315 | Proposition 113 of [Frege1... |
| frege114 40316 | If ` X ` belongs to the ` ... |
| dffrege115 40317 | If from the circumstance t... |
| frege116 40318 | One direction of ~ dffrege... |
| frege117 40319 | Lemma for ~ frege118 . Pr... |
| frege118 40320 | Simplified application of ... |
| frege119 40321 | Lemma for ~ frege120 . Pr... |
| frege120 40322 | Simplified application of ... |
| frege121 40323 | Lemma for ~ frege122 . Pr... |
| frege122 40324 | If ` X ` is a result of an... |
| frege123 40325 | Lemma for ~ frege124 . Pr... |
| frege124 40326 | If ` X ` is a result of an... |
| frege125 40327 | Lemma for ~ frege126 . Pr... |
| frege126 40328 | If ` M ` follows ` Y ` in ... |
| frege127 40329 | Communte antecedents of ~ ... |
| frege128 40330 | Lemma for ~ frege129 . Pr... |
| frege129 40331 | If the procedure ` R ` is ... |
| frege130 40332 | Lemma for ~ frege131 . Pr... |
| frege131 40333 | If the procedure ` R ` is ... |
| frege132 40334 | Lemma for ~ frege133 . Pr... |
| frege133 40335 | If the procedure ` R ` is ... |
| enrelmap 40336 | The set of all possible re... |
| enrelmapr 40337 | The set of all possible re... |
| enmappw 40338 | The set of all mappings fr... |
| enmappwid 40339 | The set of all mappings fr... |
| rfovd 40340 | Value of the operator, ` (... |
| rfovfvd 40341 | Value of the operator, ` (... |
| rfovfvfvd 40342 | Value of the operator, ` (... |
| rfovcnvf1od 40343 | Properties of the operator... |
| rfovcnvd 40344 | Value of the converse of t... |
| rfovf1od 40345 | The value of the operator,... |
| rfovcnvfvd 40346 | Value of the converse of t... |
| fsovd 40347 | Value of the operator, ` (... |
| fsovrfovd 40348 | The operator which gives a... |
| fsovfvd 40349 | Value of the operator, ` (... |
| fsovfvfvd 40350 | Value of the operator, ` (... |
| fsovfd 40351 | The operator, ` ( A O B ) ... |
| fsovcnvlem 40352 | The ` O ` operator, which ... |
| fsovcnvd 40353 | The value of the converse ... |
| fsovcnvfvd 40354 | The value of the converse ... |
| fsovf1od 40355 | The value of ` ( A O B ) `... |
| dssmapfvd 40356 | Value of the duality opera... |
| dssmapfv2d 40357 | Value of the duality opera... |
| dssmapfv3d 40358 | Value of the duality opera... |
| dssmapnvod 40359 | For any base set ` B ` the... |
| dssmapf1od 40360 | For any base set ` B ` the... |
| dssmap2d 40361 | For any base set ` B ` the... |
| sscon34b 40362 | Relative complementation r... |
| rcompleq 40363 | Two subclasses are equal i... |
| or3or 40364 | Decompose disjunction into... |
| andi3or 40365 | Distribute over triple dis... |
| uneqsn 40366 | If a union of classes is e... |
| df3o2 40367 | Ordinal 3 is the triplet c... |
| df3o3 40368 | Ordinal 3 , fully expanded... |
| brfvimex 40369 | If a binary relation holds... |
| brovmptimex 40370 | If a binary relation holds... |
| brovmptimex1 40371 | If a binary relation holds... |
| brovmptimex2 40372 | If a binary relation holds... |
| brcoffn 40373 | Conditions allowing the de... |
| brcofffn 40374 | Conditions allowing the de... |
| brco2f1o 40375 | Conditions allowing the de... |
| brco3f1o 40376 | Conditions allowing the de... |
| ntrclsbex 40377 | If (pseudo-)interior and (... |
| ntrclsrcomplex 40378 | The relative complement of... |
| neik0imk0p 40379 | Kuratowski's K0 axiom impl... |
| ntrk2imkb 40380 | If an interior function is... |
| ntrkbimka 40381 | If the interiors of disjoi... |
| ntrk0kbimka 40382 | If the interiors of disjoi... |
| clsk3nimkb 40383 | If the base set is not emp... |
| clsk1indlem0 40384 | The ansatz closure functio... |
| clsk1indlem2 40385 | The ansatz closure functio... |
| clsk1indlem3 40386 | The ansatz closure functio... |
| clsk1indlem4 40387 | The ansatz closure functio... |
| clsk1indlem1 40388 | The ansatz closure functio... |
| clsk1independent 40389 | For generalized closure fu... |
| neik0pk1imk0 40390 | Kuratowski's K0' and K1 ax... |
| isotone1 40391 | Two different ways to say ... |
| isotone2 40392 | Two different ways to say ... |
| ntrk1k3eqk13 40393 | An interior function is bo... |
| ntrclsf1o 40394 | If (pseudo-)interior and (... |
| ntrclsnvobr 40395 | If (pseudo-)interior and (... |
| ntrclsiex 40396 | If (pseudo-)interior and (... |
| ntrclskex 40397 | If (pseudo-)interior and (... |
| ntrclsfv1 40398 | If (pseudo-)interior and (... |
| ntrclsfv2 40399 | If (pseudo-)interior and (... |
| ntrclselnel1 40400 | If (pseudo-)interior and (... |
| ntrclselnel2 40401 | If (pseudo-)interior and (... |
| ntrclsfv 40402 | The value of the interior ... |
| ntrclsfveq1 40403 | If interior and closure fu... |
| ntrclsfveq2 40404 | If interior and closure fu... |
| ntrclsfveq 40405 | If interior and closure fu... |
| ntrclsss 40406 | If interior and closure fu... |
| ntrclsneine0lem 40407 | If (pseudo-)interior and (... |
| ntrclsneine0 40408 | If (pseudo-)interior and (... |
| ntrclscls00 40409 | If (pseudo-)interior and (... |
| ntrclsiso 40410 | If (pseudo-)interior and (... |
| ntrclsk2 40411 | An interior function is co... |
| ntrclskb 40412 | The interiors of disjoint ... |
| ntrclsk3 40413 | The intersection of interi... |
| ntrclsk13 40414 | The interior of the inters... |
| ntrclsk4 40415 | Idempotence of the interio... |
| ntrneibex 40416 | If (pseudo-)interior and (... |
| ntrneircomplex 40417 | The relative complement of... |
| ntrneif1o 40418 | If (pseudo-)interior and (... |
| ntrneiiex 40419 | If (pseudo-)interior and (... |
| ntrneinex 40420 | If (pseudo-)interior and (... |
| ntrneicnv 40421 | If (pseudo-)interior and (... |
| ntrneifv1 40422 | If (pseudo-)interior and (... |
| ntrneifv2 40423 | If (pseudo-)interior and (... |
| ntrneiel 40424 | If (pseudo-)interior and (... |
| ntrneifv3 40425 | The value of the neighbors... |
| ntrneineine0lem 40426 | If (pseudo-)interior and (... |
| ntrneineine1lem 40427 | If (pseudo-)interior and (... |
| ntrneifv4 40428 | The value of the interior ... |
| ntrneiel2 40429 | Membership in iterated int... |
| ntrneineine0 40430 | If (pseudo-)interior and (... |
| ntrneineine1 40431 | If (pseudo-)interior and (... |
| ntrneicls00 40432 | If (pseudo-)interior and (... |
| ntrneicls11 40433 | If (pseudo-)interior and (... |
| ntrneiiso 40434 | If (pseudo-)interior and (... |
| ntrneik2 40435 | An interior function is co... |
| ntrneix2 40436 | An interior (closure) func... |
| ntrneikb 40437 | The interiors of disjoint ... |
| ntrneixb 40438 | The interiors (closures) o... |
| ntrneik3 40439 | The intersection of interi... |
| ntrneix3 40440 | The closure of the union o... |
| ntrneik13 40441 | The interior of the inters... |
| ntrneix13 40442 | The closure of the union o... |
| ntrneik4w 40443 | Idempotence of the interio... |
| ntrneik4 40444 | Idempotence of the interio... |
| clsneibex 40445 | If (pseudo-)closure and (p... |
| clsneircomplex 40446 | The relative complement of... |
| clsneif1o 40447 | If a (pseudo-)closure func... |
| clsneicnv 40448 | If a (pseudo-)closure func... |
| clsneikex 40449 | If closure and neighborhoo... |
| clsneinex 40450 | If closure and neighborhoo... |
| clsneiel1 40451 | If a (pseudo-)closure func... |
| clsneiel2 40452 | If a (pseudo-)closure func... |
| clsneifv3 40453 | Value of the neighborhoods... |
| clsneifv4 40454 | Value of the closure (inte... |
| neicvgbex 40455 | If (pseudo-)neighborhood a... |
| neicvgrcomplex 40456 | The relative complement of... |
| neicvgf1o 40457 | If neighborhood and conver... |
| neicvgnvo 40458 | If neighborhood and conver... |
| neicvgnvor 40459 | If neighborhood and conver... |
| neicvgmex 40460 | If the neighborhoods and c... |
| neicvgnex 40461 | If the neighborhoods and c... |
| neicvgel1 40462 | A subset being an element ... |
| neicvgel2 40463 | The complement of a subset... |
| neicvgfv 40464 | The value of the neighborh... |
| ntrrn 40465 | The range of the interior ... |
| ntrf 40466 | The interior function of a... |
| ntrf2 40467 | The interior function is a... |
| ntrelmap 40468 | The interior function is a... |
| clsf2 40469 | The closure function is a ... |
| clselmap 40470 | The closure function is a ... |
| dssmapntrcls 40471 | The interior and closure o... |
| dssmapclsntr 40472 | The closure and interior o... |
| gneispa 40473 | Each point ` p ` of the ne... |
| gneispb 40474 | Given a neighborhood ` N `... |
| gneispace2 40475 | The predicate that ` F ` i... |
| gneispace3 40476 | The predicate that ` F ` i... |
| gneispace 40477 | The predicate that ` F ` i... |
| gneispacef 40478 | A generic neighborhood spa... |
| gneispacef2 40479 | A generic neighborhood spa... |
| gneispacefun 40480 | A generic neighborhood spa... |
| gneispacern 40481 | A generic neighborhood spa... |
| gneispacern2 40482 | A generic neighborhood spa... |
| gneispace0nelrn 40483 | A generic neighborhood spa... |
| gneispace0nelrn2 40484 | A generic neighborhood spa... |
| gneispace0nelrn3 40485 | A generic neighborhood spa... |
| gneispaceel 40486 | Every neighborhood of a po... |
| gneispaceel2 40487 | Every neighborhood of a po... |
| gneispacess 40488 | All supersets of a neighbo... |
| gneispacess2 40489 | All supersets of a neighbo... |
| k0004lem1 40490 | Application of ~ ssin to r... |
| k0004lem2 40491 | A mapping with a particula... |
| k0004lem3 40492 | When the value of a mappin... |
| k0004val 40493 | The topological simplex of... |
| k0004ss1 40494 | The topological simplex of... |
| k0004ss2 40495 | The topological simplex of... |
| k0004ss3 40496 | The topological simplex of... |
| k0004val0 40497 | The topological simplex of... |
| inductionexd 40498 | Simple induction example. ... |
| wwlemuld 40499 | Natural deduction form of ... |
| leeq1d 40500 | Specialization of ~ breq1d... |
| leeq2d 40501 | Specialization of ~ breq2d... |
| absmulrposd 40502 | Specialization of absmuld ... |
| imadisjld 40503 | Natural dduction form of o... |
| imadisjlnd 40504 | Natural deduction form of ... |
| wnefimgd 40505 | The image of a mapping fro... |
| fco2d 40506 | Natural deduction form of ... |
| wfximgfd 40507 | The value of a function on... |
| extoimad 40508 | If |f(x)| <= C for all x t... |
| imo72b2lem0 40509 | Lemma for ~ imo72b2 . (Co... |
| suprleubrd 40510 | Natural deduction form of ... |
| imo72b2lem2 40511 | Lemma for ~ imo72b2 . (Co... |
| syldbl2 40512 | Stacked hypotheseis implie... |
| suprlubrd 40513 | Natural deduction form of ... |
| imo72b2lem1 40514 | Lemma for ~ imo72b2 . (Co... |
| lemuldiv3d 40515 | 'Less than or equal to' re... |
| lemuldiv4d 40516 | 'Less than or equal to' re... |
| rspcdvinvd 40517 | If something is true for a... |
| imo72b2 40518 | IMO 1972 B2. (14th Intern... |
| int-addcomd 40519 | AdditionCommutativity gene... |
| int-addassocd 40520 | AdditionAssociativity gene... |
| int-addsimpd 40521 | AdditionSimplification gen... |
| int-mulcomd 40522 | MultiplicationCommutativit... |
| int-mulassocd 40523 | MultiplicationAssociativit... |
| int-mulsimpd 40524 | MultiplicationSimplificati... |
| int-leftdistd 40525 | AdditionMultiplicationLeft... |
| int-rightdistd 40526 | AdditionMultiplicationRigh... |
| int-sqdefd 40527 | SquareDefinition generator... |
| int-mul11d 40528 | First MultiplicationOne ge... |
| int-mul12d 40529 | Second MultiplicationOne g... |
| int-add01d 40530 | First AdditionZero generat... |
| int-add02d 40531 | Second AdditionZero genera... |
| int-sqgeq0d 40532 | SquareGEQZero generator ru... |
| int-eqprincd 40533 | PrincipleOfEquality genera... |
| int-eqtransd 40534 | EqualityTransitivity gener... |
| int-eqmvtd 40535 | EquMoveTerm generator rule... |
| int-eqineqd 40536 | EquivalenceImpliesDoubleIn... |
| int-ineqmvtd 40537 | IneqMoveTerm generator rul... |
| int-ineq1stprincd 40538 | FirstPrincipleOfInequality... |
| int-ineq2ndprincd 40539 | SecondPrincipleOfInequalit... |
| int-ineqtransd 40540 | InequalityTransitivity gen... |
| unitadd 40541 | Theorem used in conjunctio... |
| gsumws3 40542 | Valuation of a length 3 wo... |
| gsumws4 40543 | Valuation of a length 4 wo... |
| amgm2d 40544 | Arithmetic-geometric mean ... |
| amgm3d 40545 | Arithmetic-geometric mean ... |
| amgm4d 40546 | Arithmetic-geometric mean ... |
| spALT 40547 | ~ sp can be proven from th... |
| elnelneqd 40548 | Two classes are not equal ... |
| elnelneq2d 40549 | Two classes are not equal ... |
| rr-spce 40550 | Prove an existential. (Co... |
| rexlimdvaacbv 40551 | Unpack a restricted existe... |
| rexlimddvcbvw 40552 | Unpack a restricted existe... |
| rexlimddvcbv 40553 | Unpack a restricted existe... |
| rr-elrnmpt3d 40554 | Elementhood in an image se... |
| rr-phpd 40555 | Equivalent of ~ php withou... |
| suceqd 40556 | Deduction associated with ... |
| tfindsd 40557 | Deduction associated with ... |
| gru0eld 40558 | A nonempty Grothendieck un... |
| grusucd 40559 | Grothendieck universes are... |
| r1rankcld 40560 | Any rank of the cumulative... |
| grur1cld 40561 | Grothendieck universes are... |
| grurankcld 40562 | Grothendieck universes are... |
| grurankrcld 40563 | If a Grothendieck universe... |
| scotteqd 40566 | Equality theorem for the S... |
| scotteq 40567 | Closed form of ~ scotteqd ... |
| nfscott 40568 | Bound-variable hypothesis ... |
| scottabf 40569 | Value of the Scott operati... |
| scottab 40570 | Value of the Scott operati... |
| scottabes 40571 | Value of the Scott operati... |
| scottss 40572 | Scott's trick produces a s... |
| elscottab 40573 | An element of the output o... |
| scottex2 40574 | ~ scottex expressed using ... |
| scotteld 40575 | The Scott operation sends ... |
| scottelrankd 40576 | Property of a Scott's tric... |
| scottrankd 40577 | Rank of a nonempty Scott's... |
| gruscottcld 40578 | If a Grothendieck universe... |
| dfcoll2 40581 | Alternate definition of th... |
| colleq12d 40582 | Equality theorem for the c... |
| colleq1 40583 | Equality theorem for the c... |
| colleq2 40584 | Equality theorem for the c... |
| nfcoll 40585 | Bound-variable hypothesis ... |
| collexd 40586 | The output of the collecti... |
| cpcolld 40587 | Property of the collection... |
| cpcoll2d 40588 | ~ cpcolld with an extra ex... |
| grucollcld 40589 | A Grothendieck universe co... |
| ismnu 40590 | The hypothesis of this the... |
| mnuop123d 40591 | Operations of a minimal un... |
| mnussd 40592 | Minimal universes are clos... |
| mnuss2d 40593 | ~ mnussd with arguments pr... |
| mnu0eld 40594 | A nonempty minimal univers... |
| mnuop23d 40595 | Second and third operation... |
| mnupwd 40596 | Minimal universes are clos... |
| mnusnd 40597 | Minimal universes are clos... |
| mnuprssd 40598 | A minimal universe contain... |
| mnuprss2d 40599 | Special case of ~ mnuprssd... |
| mnuop3d 40600 | Third operation of a minim... |
| mnuprdlem1 40601 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem2 40602 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem3 40603 | Lemma for ~ mnuprd . (Con... |
| mnuprdlem4 40604 | Lemma for ~ mnuprd . Gene... |
| mnuprd 40605 | Minimal universes are clos... |
| mnuunid 40606 | Minimal universes are clos... |
| mnuund 40607 | Minimal universes are clos... |
| mnutrcld 40608 | Minimal universes contain ... |
| mnutrd 40609 | Minimal universes are tran... |
| mnurndlem1 40610 | Lemma for ~ mnurnd . (Con... |
| mnurndlem2 40611 | Lemma for ~ mnurnd . Dedu... |
| mnurnd 40612 | Minimal universes contain ... |
| mnugrud 40613 | Minimal universes are Grot... |
| grumnudlem 40614 | Lemma for ~ grumnud . (Co... |
| grumnud 40615 | Grothendieck universes are... |
| grumnueq 40616 | The class of Grothendieck ... |
| expandan 40617 | Expand conjunction to prim... |
| expandexn 40618 | Expand an existential quan... |
| expandral 40619 | Expand a restricted univer... |
| expandrexn 40620 | Expand a restricted existe... |
| expandrex 40621 | Expand a restricted existe... |
| expanduniss 40622 | Expand ` U. A C_ B ` to pr... |
| ismnuprim 40623 | Express the predicate on `... |
| rr-grothprimbi 40624 | Express "every set is cont... |
| inagrud 40625 | Inaccessible levels of the... |
| inaex 40626 | Assuming the Tarski-Grothe... |
| gruex 40627 | Assuming the Tarski-Grothe... |
| rr-groth 40628 | An equivalent of ~ ax-grot... |
| rr-grothprim 40629 | An equivalent of ~ ax-grot... |
| nanorxor 40630 | 'nand' is equivalent to th... |
| undisjrab 40631 | Union of two disjoint rest... |
| iso0 40632 | The empty set is an ` R , ... |
| ssrecnpr 40633 | ` RR ` is a subset of both... |
| seff 40634 | Let set ` S ` be the real ... |
| sblpnf 40635 | The infinity ball in the a... |
| prmunb2 40636 | The primes are unbounded. ... |
| dvgrat 40637 | Ratio test for divergence ... |
| cvgdvgrat 40638 | Ratio test for convergence... |
| radcnvrat 40639 | Let ` L ` be the limit, if... |
| reldvds 40640 | The divides relation is in... |
| nznngen 40641 | All positive integers in t... |
| nzss 40642 | The set of multiples of _m... |
| nzin 40643 | The intersection of the se... |
| nzprmdif 40644 | Subtract one prime's multi... |
| hashnzfz 40645 | Special case of ~ hashdvds... |
| hashnzfz2 40646 | Special case of ~ hashnzfz... |
| hashnzfzclim 40647 | As the upper bound ` K ` o... |
| caofcan 40648 | Transfer a cancellation la... |
| ofsubid 40649 | Function analogue of ~ sub... |
| ofmul12 40650 | Function analogue of ~ mul... |
| ofdivrec 40651 | Function analogue of ~ div... |
| ofdivcan4 40652 | Function analogue of ~ div... |
| ofdivdiv2 40653 | Function analogue of ~ div... |
| lhe4.4ex1a 40654 | Example of the Fundamental... |
| dvsconst 40655 | Derivative of a constant f... |
| dvsid 40656 | Derivative of the identity... |
| dvsef 40657 | Derivative of the exponent... |
| expgrowthi 40658 | Exponential growth and dec... |
| dvconstbi 40659 | The derivative of a functi... |
| expgrowth 40660 | Exponential growth and dec... |
| bccval 40663 | Value of the generalized b... |
| bcccl 40664 | Closure of the generalized... |
| bcc0 40665 | The generalized binomial c... |
| bccp1k 40666 | Generalized binomial coeff... |
| bccm1k 40667 | Generalized binomial coeff... |
| bccn0 40668 | Generalized binomial coeff... |
| bccn1 40669 | Generalized binomial coeff... |
| bccbc 40670 | The binomial coefficient a... |
| uzmptshftfval 40671 | When ` F ` is a maps-to fu... |
| dvradcnv2 40672 | The radius of convergence ... |
| binomcxplemwb 40673 | Lemma for ~ binomcxp . Th... |
| binomcxplemnn0 40674 | Lemma for ~ binomcxp . Wh... |
| binomcxplemrat 40675 | Lemma for ~ binomcxp . As... |
| binomcxplemfrat 40676 | Lemma for ~ binomcxp . ~ b... |
| binomcxplemradcnv 40677 | Lemma for ~ binomcxp . By... |
| binomcxplemdvbinom 40678 | Lemma for ~ binomcxp . By... |
| binomcxplemcvg 40679 | Lemma for ~ binomcxp . Th... |
| binomcxplemdvsum 40680 | Lemma for ~ binomcxp . Th... |
| binomcxplemnotnn0 40681 | Lemma for ~ binomcxp . Wh... |
| binomcxp 40682 | Generalize the binomial th... |
| pm10.12 40683 | Theorem *10.12 in [Whitehe... |
| pm10.14 40684 | Theorem *10.14 in [Whitehe... |
| pm10.251 40685 | Theorem *10.251 in [Whiteh... |
| pm10.252 40686 | Theorem *10.252 in [Whiteh... |
| pm10.253 40687 | Theorem *10.253 in [Whiteh... |
| albitr 40688 | Theorem *10.301 in [Whiteh... |
| pm10.42 40689 | Theorem *10.42 in [Whitehe... |
| pm10.52 40690 | Theorem *10.52 in [Whitehe... |
| pm10.53 40691 | Theorem *10.53 in [Whitehe... |
| pm10.541 40692 | Theorem *10.541 in [Whiteh... |
| pm10.542 40693 | Theorem *10.542 in [Whiteh... |
| pm10.55 40694 | Theorem *10.55 in [Whitehe... |
| pm10.56 40695 | Theorem *10.56 in [Whitehe... |
| pm10.57 40696 | Theorem *10.57 in [Whitehe... |
| 2alanimi 40697 | Removes two universal quan... |
| 2al2imi 40698 | Removes two universal quan... |
| pm11.11 40699 | Theorem *11.11 in [Whitehe... |
| pm11.12 40700 | Theorem *11.12 in [Whitehe... |
| 19.21vv 40701 | Compare Theorem *11.3 in [... |
| 2alim 40702 | Theorem *11.32 in [Whitehe... |
| 2albi 40703 | Theorem *11.33 in [Whitehe... |
| 2exim 40704 | Theorem *11.34 in [Whitehe... |
| 2exbi 40705 | Theorem *11.341 in [Whiteh... |
| spsbce-2 40706 | Theorem *11.36 in [Whitehe... |
| 19.33-2 40707 | Theorem *11.421 in [Whiteh... |
| 19.36vv 40708 | Theorem *11.43 in [Whitehe... |
| 19.31vv 40709 | Theorem *11.44 in [Whitehe... |
| 19.37vv 40710 | Theorem *11.46 in [Whitehe... |
| 19.28vv 40711 | Theorem *11.47 in [Whitehe... |
| pm11.52 40712 | Theorem *11.52 in [Whitehe... |
| aaanv 40713 | Theorem *11.56 in [Whitehe... |
| pm11.57 40714 | Theorem *11.57 in [Whitehe... |
| pm11.58 40715 | Theorem *11.58 in [Whitehe... |
| pm11.59 40716 | Theorem *11.59 in [Whitehe... |
| pm11.6 40717 | Theorem *11.6 in [Whitehea... |
| pm11.61 40718 | Theorem *11.61 in [Whitehe... |
| pm11.62 40719 | Theorem *11.62 in [Whitehe... |
| pm11.63 40720 | Theorem *11.63 in [Whitehe... |
| pm11.7 40721 | Theorem *11.7 in [Whitehea... |
| pm11.71 40722 | Theorem *11.71 in [Whitehe... |
| sbeqal1 40723 | If ` x = y ` always implie... |
| sbeqal1i 40724 | Suppose you know ` x = y `... |
| sbeqal2i 40725 | If ` x = y ` implies ` x =... |
| axc5c4c711 40726 | Proof of a theorem that ca... |
| axc5c4c711toc5 40727 | Rederivation of ~ sp from ... |
| axc5c4c711toc4 40728 | Rederivation of ~ axc4 fro... |
| axc5c4c711toc7 40729 | Rederivation of ~ axc7 fro... |
| axc5c4c711to11 40730 | Rederivation of ~ ax-11 fr... |
| axc11next 40731 | This theorem shows that, g... |
| pm13.13a 40732 | One result of theorem *13.... |
| pm13.13b 40733 | Theorem *13.13 in [Whitehe... |
| pm13.14 40734 | Theorem *13.14 in [Whitehe... |
| pm13.192 40735 | Theorem *13.192 in [Whiteh... |
| pm13.193 40736 | Theorem *13.193 in [Whiteh... |
| pm13.194 40737 | Theorem *13.194 in [Whiteh... |
| pm13.195 40738 | Theorem *13.195 in [Whiteh... |
| pm13.196a 40739 | Theorem *13.196 in [Whiteh... |
| 2sbc6g 40740 | Theorem *13.21 in [Whitehe... |
| 2sbc5g 40741 | Theorem *13.22 in [Whitehe... |
| iotain 40742 | Equivalence between two di... |
| iotaexeu 40743 | The iota class exists. Th... |
| iotasbc 40744 | Definition *14.01 in [Whit... |
| iotasbc2 40745 | Theorem *14.111 in [Whiteh... |
| pm14.12 40746 | Theorem *14.12 in [Whitehe... |
| pm14.122a 40747 | Theorem *14.122 in [Whiteh... |
| pm14.122b 40748 | Theorem *14.122 in [Whiteh... |
| pm14.122c 40749 | Theorem *14.122 in [Whiteh... |
| pm14.123a 40750 | Theorem *14.123 in [Whiteh... |
| pm14.123b 40751 | Theorem *14.123 in [Whiteh... |
| pm14.123c 40752 | Theorem *14.123 in [Whiteh... |
| pm14.18 40753 | Theorem *14.18 in [Whitehe... |
| iotaequ 40754 | Theorem *14.2 in [Whitehea... |
| iotavalb 40755 | Theorem *14.202 in [Whiteh... |
| iotasbc5 40756 | Theorem *14.205 in [Whiteh... |
| pm14.24 40757 | Theorem *14.24 in [Whitehe... |
| iotavalsb 40758 | Theorem *14.242 in [Whiteh... |
| sbiota1 40759 | Theorem *14.25 in [Whitehe... |
| sbaniota 40760 | Theorem *14.26 in [Whitehe... |
| eubiOLD 40761 | Obsolete proof of ~ eubi a... |
| iotasbcq 40762 | Theorem *14.272 in [Whiteh... |
| elnev 40763 | Any set that contains one ... |
| rusbcALT 40764 | A version of Russell's par... |
| compeq 40765 | Equality between two ways ... |
| compne 40766 | The complement of ` A ` is... |
| compab 40767 | Two ways of saying "the co... |
| conss2 40768 | Contrapositive law for sub... |
| conss1 40769 | Contrapositive law for sub... |
| ralbidar 40770 | More general form of ~ ral... |
| rexbidar 40771 | More general form of ~ rex... |
| dropab1 40772 | Theorem to aid use of the ... |
| dropab2 40773 | Theorem to aid use of the ... |
| ipo0 40774 | If the identity relation p... |
| ifr0 40775 | A class that is founded by... |
| ordpss 40776 | ~ ordelpss with an anteced... |
| fvsb 40777 | Explicit substitution of a... |
| fveqsb 40778 | Implicit substitution of a... |
| xpexb 40779 | A Cartesian product exists... |
| trelpss 40780 | An element of a transitive... |
| addcomgi 40781 | Generalization of commutat... |
| addrval 40791 | Value of the operation of ... |
| subrval 40792 | Value of the operation of ... |
| mulvval 40793 | Value of the operation of ... |
| addrfv 40794 | Vector addition at a value... |
| subrfv 40795 | Vector subtraction at a va... |
| mulvfv 40796 | Scalar multiplication at a... |
| addrfn 40797 | Vector addition produces a... |
| subrfn 40798 | Vector subtraction produce... |
| mulvfn 40799 | Scalar multiplication prod... |
| addrcom 40800 | Vector addition is commuta... |
| idiALT 40804 | Placeholder for ~ idi . T... |
| exbir 40805 | Exportation implication al... |
| 3impexpbicom 40806 | Version of ~ 3impexp where... |
| 3impexpbicomi 40807 | Inference associated with ... |
| bi1imp 40808 | Importation inference simi... |
| bi2imp 40809 | Importation inference simi... |
| bi3impb 40810 | Similar to ~ 3impb with im... |
| bi3impa 40811 | Similar to ~ 3impa with im... |
| bi23impib 40812 | ~ 3impib with the inner im... |
| bi13impib 40813 | ~ 3impib with the outer im... |
| bi123impib 40814 | ~ 3impib with the implicat... |
| bi13impia 40815 | ~ 3impia with the outer im... |
| bi123impia 40816 | ~ 3impia with the implicat... |
| bi33imp12 40817 | ~ 3imp with innermost impl... |
| bi23imp13 40818 | ~ 3imp with middle implica... |
| bi13imp23 40819 | ~ 3imp with outermost impl... |
| bi13imp2 40820 | Similar to ~ 3imp except t... |
| bi12imp3 40821 | Similar to ~ 3imp except a... |
| bi23imp1 40822 | Similar to ~ 3imp except a... |
| bi123imp0 40823 | Similar to ~ 3imp except a... |
| 4animp1 40824 | A single hypothesis unific... |
| 4an31 40825 | A rearrangement of conjunc... |
| 4an4132 40826 | A rearrangement of conjunc... |
| expcomdg 40827 | Biconditional form of ~ ex... |
| iidn3 40828 | ~ idn3 without virtual ded... |
| ee222 40829 | ~ e222 without virtual ded... |
| ee3bir 40830 | Right-biconditional form o... |
| ee13 40831 | ~ e13 without virtual dedu... |
| ee121 40832 | ~ e121 without virtual ded... |
| ee122 40833 | ~ e122 without virtual ded... |
| ee333 40834 | ~ e333 without virtual ded... |
| ee323 40835 | ~ e323 without virtual ded... |
| 3ornot23 40836 | If the second and third di... |
| orbi1r 40837 | ~ orbi1 with order of disj... |
| 3orbi123 40838 | ~ pm4.39 with a 3-conjunct... |
| syl5imp 40839 | Closed form of ~ syl5 . D... |
| impexpd 40840 | The following User's Proof... |
| com3rgbi 40841 | The following User's Proof... |
| impexpdcom 40842 | The following User's Proof... |
| ee1111 40843 | Non-virtual deduction form... |
| pm2.43bgbi 40844 | Logical equivalence of a 2... |
| pm2.43cbi 40845 | Logical equivalence of a 3... |
| ee233 40846 | Non-virtual deduction form... |
| imbi13 40847 | Join three logical equival... |
| ee33 40848 | Non-virtual deduction form... |
| con5 40849 | Biconditional contrapositi... |
| con5i 40850 | Inference form of ~ con5 .... |
| exlimexi 40851 | Inference similar to Theor... |
| sb5ALT 40852 | Equivalence for substituti... |
| eexinst01 40853 | ~ exinst01 without virtual... |
| eexinst11 40854 | ~ exinst11 without virtual... |
| vk15.4j 40855 | Excercise 4j of Unit 15 of... |
| notnotrALT 40856 | Converse of double negatio... |
| con3ALT2 40857 | Contraposition. Alternate... |
| ssralv2 40858 | Quantification restricted ... |
| sbc3or 40859 | ~ sbcor with a 3-disjuncts... |
| alrim3con13v 40860 | Closed form of ~ alrimi wi... |
| rspsbc2 40861 | ~ rspsbc with two quantify... |
| sbcoreleleq 40862 | Substitution of a setvar v... |
| tratrb 40863 | If a class is transitive a... |
| ordelordALT 40864 | An element of an ordinal c... |
| sbcim2g 40865 | Distribution of class subs... |
| sbcbi 40866 | Implication form of ~ sbcb... |
| trsbc 40867 | Formula-building inference... |
| truniALT 40868 | The union of a class of tr... |
| onfrALTlem5 40869 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem4 40870 | Lemma for ~ onfrALT . (Co... |
| onfrALTlem3 40871 | Lemma for ~ onfrALT . (Co... |
| ggen31 40872 | ~ gen31 without virtual de... |
| onfrALTlem2 40873 | Lemma for ~ onfrALT . (Co... |
| cbvexsv 40874 | A theorem pertaining to th... |
| onfrALTlem1 40875 | Lemma for ~ onfrALT . (Co... |
| onfrALT 40876 | The membership relation is... |
| 19.41rg 40877 | Closed form of right-to-le... |
| opelopab4 40878 | Ordered pair membership in... |
| 2pm13.193 40879 | ~ pm13.193 for two variabl... |
| hbntal 40880 | A closed form of ~ hbn . ~... |
| hbimpg 40881 | A closed form of ~ hbim . ... |
| hbalg 40882 | Closed form of ~ hbal . D... |
| hbexg 40883 | Closed form of ~ nfex . D... |
| ax6e2eq 40884 | Alternate form of ~ ax6e f... |
| ax6e2nd 40885 | If at least two sets exist... |
| ax6e2ndeq 40886 | "At least two sets exist" ... |
| 2sb5nd 40887 | Equivalence for double sub... |
| 2uasbanh 40888 | Distribute the unabbreviat... |
| 2uasban 40889 | Distribute the unabbreviat... |
| e2ebind 40890 | Absorption of an existenti... |
| elpwgded 40891 | ~ elpwgdedVD in convention... |
| trelded 40892 | Deduction form of ~ trel .... |
| jaoded 40893 | Deduction form of ~ jao . ... |
| sbtT 40894 | A substitution into a theo... |
| not12an2impnot1 40895 | If a double conjunction is... |
| in1 40898 | Inference form of ~ df-vd1... |
| iin1 40899 | ~ in1 without virtual dedu... |
| dfvd1ir 40900 | Inference form of ~ df-vd1... |
| idn1 40901 | Virtual deduction identity... |
| dfvd1imp 40902 | Left-to-right part of defi... |
| dfvd1impr 40903 | Right-to-left part of defi... |
| dfvd2 40906 | Definition of a 2-hypothes... |
| dfvd2an 40909 | Definition of a 2-hypothes... |
| dfvd2ani 40910 | Inference form of ~ dfvd2a... |
| dfvd2anir 40911 | Right-to-left inference fo... |
| dfvd2i 40912 | Inference form of ~ dfvd2 ... |
| dfvd2ir 40913 | Right-to-left inference fo... |
| dfvd3 40918 | Definition of a 3-hypothes... |
| dfvd3i 40919 | Inference form of ~ dfvd3 ... |
| dfvd3ir 40920 | Right-to-left inference fo... |
| dfvd3an 40921 | Definition of a 3-hypothes... |
| dfvd3ani 40922 | Inference form of ~ dfvd3a... |
| dfvd3anir 40923 | Right-to-left inference fo... |
| vd01 40924 | A virtual hypothesis virtu... |
| vd02 40925 | Two virtual hypotheses vir... |
| vd03 40926 | A theorem is virtually inf... |
| vd12 40927 | A virtual deduction with 1... |
| vd13 40928 | A virtual deduction with 1... |
| vd23 40929 | A virtual deduction with 2... |
| dfvd2imp 40930 | The virtual deduction form... |
| dfvd2impr 40931 | A 2-antecedent nested impl... |
| in2 40932 | The virtual deduction intr... |
| int2 40933 | The virtual deduction intr... |
| iin2 40934 | ~ in2 without virtual dedu... |
| in2an 40935 | The virtual deduction intr... |
| in3 40936 | The virtual deduction intr... |
| iin3 40937 | ~ in3 without virtual dedu... |
| in3an 40938 | The virtual deduction intr... |
| int3 40939 | The virtual deduction intr... |
| idn2 40940 | Virtual deduction identity... |
| iden2 40941 | Virtual deduction identity... |
| idn3 40942 | Virtual deduction identity... |
| gen11 40943 | Virtual deduction generali... |
| gen11nv 40944 | Virtual deduction generali... |
| gen12 40945 | Virtual deduction generali... |
| gen21 40946 | Virtual deduction generali... |
| gen21nv 40947 | Virtual deduction form of ... |
| gen31 40948 | Virtual deduction generali... |
| gen22 40949 | Virtual deduction generali... |
| ggen22 40950 | ~ gen22 without virtual de... |
| exinst 40951 | Existential Instantiation.... |
| exinst01 40952 | Existential Instantiation.... |
| exinst11 40953 | Existential Instantiation.... |
| e1a 40954 | A Virtual deduction elimin... |
| el1 40955 | A Virtual deduction elimin... |
| e1bi 40956 | Biconditional form of ~ e1... |
| e1bir 40957 | Right biconditional form o... |
| e2 40958 | A virtual deduction elimin... |
| e2bi 40959 | Biconditional form of ~ e2... |
| e2bir 40960 | Right biconditional form o... |
| ee223 40961 | ~ e223 without virtual ded... |
| e223 40962 | A virtual deduction elimin... |
| e222 40963 | A virtual deduction elimin... |
| e220 40964 | A virtual deduction elimin... |
| ee220 40965 | ~ e220 without virtual ded... |
| e202 40966 | A virtual deduction elimin... |
| ee202 40967 | ~ e202 without virtual ded... |
| e022 40968 | A virtual deduction elimin... |
| ee022 40969 | ~ e022 without virtual ded... |
| e002 40970 | A virtual deduction elimin... |
| ee002 40971 | ~ e002 without virtual ded... |
| e020 40972 | A virtual deduction elimin... |
| ee020 40973 | ~ e020 without virtual ded... |
| e200 40974 | A virtual deduction elimin... |
| ee200 40975 | ~ e200 without virtual ded... |
| e221 40976 | A virtual deduction elimin... |
| ee221 40977 | ~ e221 without virtual ded... |
| e212 40978 | A virtual deduction elimin... |
| ee212 40979 | ~ e212 without virtual ded... |
| e122 40980 | A virtual deduction elimin... |
| e112 40981 | A virtual deduction elimin... |
| ee112 40982 | ~ e112 without virtual ded... |
| e121 40983 | A virtual deduction elimin... |
| e211 40984 | A virtual deduction elimin... |
| ee211 40985 | ~ e211 without virtual ded... |
| e210 40986 | A virtual deduction elimin... |
| ee210 40987 | ~ e210 without virtual ded... |
| e201 40988 | A virtual deduction elimin... |
| ee201 40989 | ~ e201 without virtual ded... |
| e120 40990 | A virtual deduction elimin... |
| ee120 40991 | Virtual deduction rule ~ e... |
| e021 40992 | A virtual deduction elimin... |
| ee021 40993 | ~ e021 without virtual ded... |
| e012 40994 | A virtual deduction elimin... |
| ee012 40995 | ~ e012 without virtual ded... |
| e102 40996 | A virtual deduction elimin... |
| ee102 40997 | ~ e102 without virtual ded... |
| e22 40998 | A virtual deduction elimin... |
| e22an 40999 | Conjunction form of ~ e22 ... |
| ee22an 41000 | ~ e22an without virtual de... |
| e111 41001 | A virtual deduction elimin... |
| e1111 41002 | A virtual deduction elimin... |
| e110 41003 | A virtual deduction elimin... |
| ee110 41004 | ~ e110 without virtual ded... |
| e101 41005 | A virtual deduction elimin... |
| ee101 41006 | ~ e101 without virtual ded... |
| e011 41007 | A virtual deduction elimin... |
| ee011 41008 | ~ e011 without virtual ded... |
| e100 41009 | A virtual deduction elimin... |
| ee100 41010 | ~ e100 without virtual ded... |
| e010 41011 | A virtual deduction elimin... |
| ee010 41012 | ~ e010 without virtual ded... |
| e001 41013 | A virtual deduction elimin... |
| ee001 41014 | ~ e001 without virtual ded... |
| e11 41015 | A virtual deduction elimin... |
| e11an 41016 | Conjunction form of ~ e11 ... |
| ee11an 41017 | ~ e11an without virtual de... |
| e01 41018 | A virtual deduction elimin... |
| e01an 41019 | Conjunction form of ~ e01 ... |
| ee01an 41020 | ~ e01an without virtual de... |
| e10 41021 | A virtual deduction elimin... |
| e10an 41022 | Conjunction form of ~ e10 ... |
| ee10an 41023 | ~ e10an without virtual de... |
| e02 41024 | A virtual deduction elimin... |
| e02an 41025 | Conjunction form of ~ e02 ... |
| ee02an 41026 | ~ e02an without virtual de... |
| eel021old 41027 | ~ el021old without virtual... |
| el021old 41028 | A virtual deduction elimin... |
| eel132 41029 | ~ syl2an with antecedents ... |
| eel000cT 41030 | An elimination deduction. ... |
| eel0TT 41031 | An elimination deduction. ... |
| eelT00 41032 | An elimination deduction. ... |
| eelTTT 41033 | An elimination deduction. ... |
| eelT11 41034 | An elimination deduction. ... |
| eelT1 41035 | Syllogism inference combin... |
| eelT12 41036 | An elimination deduction. ... |
| eelTT1 41037 | An elimination deduction. ... |
| eelT01 41038 | An elimination deduction. ... |
| eel0T1 41039 | An elimination deduction. ... |
| eel12131 41040 | An elimination deduction. ... |
| eel2131 41041 | ~ syl2an with antecedents ... |
| eel3132 41042 | ~ syl2an with antecedents ... |
| eel0321old 41043 | ~ el0321old without virtua... |
| el0321old 41044 | A virtual deduction elimin... |
| eel2122old 41045 | ~ el2122old without virtua... |
| el2122old 41046 | A virtual deduction elimin... |
| eel0000 41047 | Elimination rule similar t... |
| eel00001 41048 | An elimination deduction. ... |
| eel00000 41049 | Elimination rule similar ~... |
| eel11111 41050 | Five-hypothesis eliminatio... |
| e12 41051 | A virtual deduction elimin... |
| e12an 41052 | Conjunction form of ~ e12 ... |
| el12 41053 | Virtual deduction form of ... |
| e20 41054 | A virtual deduction elimin... |
| e20an 41055 | Conjunction form of ~ e20 ... |
| ee20an 41056 | ~ e20an without virtual de... |
| e21 41057 | A virtual deduction elimin... |
| e21an 41058 | Conjunction form of ~ e21 ... |
| ee21an 41059 | ~ e21an without virtual de... |
| e333 41060 | A virtual deduction elimin... |
| e33 41061 | A virtual deduction elimin... |
| e33an 41062 | Conjunction form of ~ e33 ... |
| ee33an 41063 | ~ e33an without virtual de... |
| e3 41064 | Meta-connective form of ~ ... |
| e3bi 41065 | Biconditional form of ~ e3... |
| e3bir 41066 | Right biconditional form o... |
| e03 41067 | A virtual deduction elimin... |
| ee03 41068 | ~ e03 without virtual dedu... |
| e03an 41069 | Conjunction form of ~ e03 ... |
| ee03an 41070 | Conjunction form of ~ ee03... |
| e30 41071 | A virtual deduction elimin... |
| ee30 41072 | ~ e30 without virtual dedu... |
| e30an 41073 | A virtual deduction elimin... |
| ee30an 41074 | Conjunction form of ~ ee30... |
| e13 41075 | A virtual deduction elimin... |
| e13an 41076 | A virtual deduction elimin... |
| ee13an 41077 | ~ e13an without virtual de... |
| e31 41078 | A virtual deduction elimin... |
| ee31 41079 | ~ e31 without virtual dedu... |
| e31an 41080 | A virtual deduction elimin... |
| ee31an 41081 | ~ e31an without virtual de... |
| e23 41082 | A virtual deduction elimin... |
| e23an 41083 | A virtual deduction elimin... |
| ee23an 41084 | ~ e23an without virtual de... |
| e32 41085 | A virtual deduction elimin... |
| ee32 41086 | ~ e32 without virtual dedu... |
| e32an 41087 | A virtual deduction elimin... |
| ee32an 41088 | ~ e33an without virtual de... |
| e123 41089 | A virtual deduction elimin... |
| ee123 41090 | ~ e123 without virtual ded... |
| el123 41091 | A virtual deduction elimin... |
| e233 41092 | A virtual deduction elimin... |
| e323 41093 | A virtual deduction elimin... |
| e000 41094 | A virtual deduction elimin... |
| e00 41095 | Elimination rule identical... |
| e00an 41096 | Elimination rule identical... |
| eel00cT 41097 | An elimination deduction. ... |
| eelTT 41098 | An elimination deduction. ... |
| e0a 41099 | Elimination rule identical... |
| eelT 41100 | An elimination deduction. ... |
| eel0cT 41101 | An elimination deduction. ... |
| eelT0 41102 | An elimination deduction. ... |
| e0bi 41103 | Elimination rule identical... |
| e0bir 41104 | Elimination rule identical... |
| uun0.1 41105 | Convention notation form o... |
| un0.1 41106 | ` T. ` is the constant tru... |
| uunT1 41107 | A deduction unionizing a n... |
| uunT1p1 41108 | A deduction unionizing a n... |
| uunT21 41109 | A deduction unionizing a n... |
| uun121 41110 | A deduction unionizing a n... |
| uun121p1 41111 | A deduction unionizing a n... |
| uun132 41112 | A deduction unionizing a n... |
| uun132p1 41113 | A deduction unionizing a n... |
| anabss7p1 41114 | A deduction unionizing a n... |
| un10 41115 | A unionizing deduction. (... |
| un01 41116 | A unionizing deduction. (... |
| un2122 41117 | A deduction unionizing a n... |
| uun2131 41118 | A deduction unionizing a n... |
| uun2131p1 41119 | A deduction unionizing a n... |
| uunTT1 41120 | A deduction unionizing a n... |
| uunTT1p1 41121 | A deduction unionizing a n... |
| uunTT1p2 41122 | A deduction unionizing a n... |
| uunT11 41123 | A deduction unionizing a n... |
| uunT11p1 41124 | A deduction unionizing a n... |
| uunT11p2 41125 | A deduction unionizing a n... |
| uunT12 41126 | A deduction unionizing a n... |
| uunT12p1 41127 | A deduction unionizing a n... |
| uunT12p2 41128 | A deduction unionizing a n... |
| uunT12p3 41129 | A deduction unionizing a n... |
| uunT12p4 41130 | A deduction unionizing a n... |
| uunT12p5 41131 | A deduction unionizing a n... |
| uun111 41132 | A deduction unionizing a n... |
| 3anidm12p1 41133 | A deduction unionizing a n... |
| 3anidm12p2 41134 | A deduction unionizing a n... |
| uun123 41135 | A deduction unionizing a n... |
| uun123p1 41136 | A deduction unionizing a n... |
| uun123p2 41137 | A deduction unionizing a n... |
| uun123p3 41138 | A deduction unionizing a n... |
| uun123p4 41139 | A deduction unionizing a n... |
| uun2221 41140 | A deduction unionizing a n... |
| uun2221p1 41141 | A deduction unionizing a n... |
| uun2221p2 41142 | A deduction unionizing a n... |
| 3impdirp1 41143 | A deduction unionizing a n... |
| 3impcombi 41144 | A 1-hypothesis proposition... |
| trsspwALT 41145 | Virtual deduction proof of... |
| trsspwALT2 41146 | Virtual deduction proof of... |
| trsspwALT3 41147 | Short predicate calculus p... |
| sspwtr 41148 | Virtual deduction proof of... |
| sspwtrALT 41149 | Virtual deduction proof of... |
| sspwtrALT2 41150 | Short predicate calculus p... |
| pwtrVD 41151 | Virtual deduction proof of... |
| pwtrrVD 41152 | Virtual deduction proof of... |
| suctrALT 41153 | The successor of a transit... |
| snssiALTVD 41154 | Virtual deduction proof of... |
| snssiALT 41155 | If a class is an element o... |
| snsslVD 41156 | Virtual deduction proof of... |
| snssl 41157 | If a singleton is a subcla... |
| snelpwrVD 41158 | Virtual deduction proof of... |
| unipwrVD 41159 | Virtual deduction proof of... |
| unipwr 41160 | A class is a subclass of t... |
| sstrALT2VD 41161 | Virtual deduction proof of... |
| sstrALT2 41162 | Virtual deduction proof of... |
| suctrALT2VD 41163 | Virtual deduction proof of... |
| suctrALT2 41164 | Virtual deduction proof of... |
| elex2VD 41165 | Virtual deduction proof of... |
| elex22VD 41166 | Virtual deduction proof of... |
| eqsbc3rVD 41167 | Virtual deduction proof of... |
| zfregs2VD 41168 | Virtual deduction proof of... |
| tpid3gVD 41169 | Virtual deduction proof of... |
| en3lplem1VD 41170 | Virtual deduction proof of... |
| en3lplem2VD 41171 | Virtual deduction proof of... |
| en3lpVD 41172 | Virtual deduction proof of... |
| simplbi2VD 41173 | Virtual deduction proof of... |
| 3ornot23VD 41174 | Virtual deduction proof of... |
| orbi1rVD 41175 | Virtual deduction proof of... |
| bitr3VD 41176 | Virtual deduction proof of... |
| 3orbi123VD 41177 | Virtual deduction proof of... |
| sbc3orgVD 41178 | Virtual deduction proof of... |
| 19.21a3con13vVD 41179 | Virtual deduction proof of... |
| exbirVD 41180 | Virtual deduction proof of... |
| exbiriVD 41181 | Virtual deduction proof of... |
| rspsbc2VD 41182 | Virtual deduction proof of... |
| 3impexpVD 41183 | Virtual deduction proof of... |
| 3impexpbicomVD 41184 | Virtual deduction proof of... |
| 3impexpbicomiVD 41185 | Virtual deduction proof of... |
| sbcoreleleqVD 41186 | Virtual deduction proof of... |
| hbra2VD 41187 | Virtual deduction proof of... |
| tratrbVD 41188 | Virtual deduction proof of... |
| al2imVD 41189 | Virtual deduction proof of... |
| syl5impVD 41190 | Virtual deduction proof of... |
| idiVD 41191 | Virtual deduction proof of... |
| ancomstVD 41192 | Closed form of ~ ancoms . ... |
| ssralv2VD 41193 | Quantification restricted ... |
| ordelordALTVD 41194 | An element of an ordinal c... |
| equncomVD 41195 | If a class equals the unio... |
| equncomiVD 41196 | Inference form of ~ equnco... |
| sucidALTVD 41197 | A set belongs to its succe... |
| sucidALT 41198 | A set belongs to its succe... |
| sucidVD 41199 | A set belongs to its succe... |
| imbi12VD 41200 | Implication form of ~ imbi... |
| imbi13VD 41201 | Join three logical equival... |
| sbcim2gVD 41202 | Distribution of class subs... |
| sbcbiVD 41203 | Implication form of ~ sbcb... |
| trsbcVD 41204 | Formula-building inference... |
| truniALTVD 41205 | The union of a class of tr... |
| ee33VD 41206 | Non-virtual deduction form... |
| trintALTVD 41207 | The intersection of a clas... |
| trintALT 41208 | The intersection of a clas... |
| undif3VD 41209 | The first equality of Exer... |
| sbcssgVD 41210 | Virtual deduction proof of... |
| csbingVD 41211 | Virtual deduction proof of... |
| onfrALTlem5VD 41212 | Virtual deduction proof of... |
| onfrALTlem4VD 41213 | Virtual deduction proof of... |
| onfrALTlem3VD 41214 | Virtual deduction proof of... |
| simplbi2comtVD 41215 | Virtual deduction proof of... |
| onfrALTlem2VD 41216 | Virtual deduction proof of... |
| onfrALTlem1VD 41217 | Virtual deduction proof of... |
| onfrALTVD 41218 | Virtual deduction proof of... |
| csbeq2gVD 41219 | Virtual deduction proof of... |
| csbsngVD 41220 | Virtual deduction proof of... |
| csbxpgVD 41221 | Virtual deduction proof of... |
| csbresgVD 41222 | Virtual deduction proof of... |
| csbrngVD 41223 | Virtual deduction proof of... |
| csbima12gALTVD 41224 | Virtual deduction proof of... |
| csbunigVD 41225 | Virtual deduction proof of... |
| csbfv12gALTVD 41226 | Virtual deduction proof of... |
| con5VD 41227 | Virtual deduction proof of... |
| relopabVD 41228 | Virtual deduction proof of... |
| 19.41rgVD 41229 | Virtual deduction proof of... |
| 2pm13.193VD 41230 | Virtual deduction proof of... |
| hbimpgVD 41231 | Virtual deduction proof of... |
| hbalgVD 41232 | Virtual deduction proof of... |
| hbexgVD 41233 | Virtual deduction proof of... |
| ax6e2eqVD 41234 | The following User's Proof... |
| ax6e2ndVD 41235 | The following User's Proof... |
| ax6e2ndeqVD 41236 | The following User's Proof... |
| 2sb5ndVD 41237 | The following User's Proof... |
| 2uasbanhVD 41238 | The following User's Proof... |
| e2ebindVD 41239 | The following User's Proof... |
| sb5ALTVD 41240 | The following User's Proof... |
| vk15.4jVD 41241 | The following User's Proof... |
| notnotrALTVD 41242 | The following User's Proof... |
| con3ALTVD 41243 | The following User's Proof... |
| elpwgdedVD 41244 | Membership in a power clas... |
| sspwimp 41245 | If a class is a subclass o... |
| sspwimpVD 41246 | The following User's Proof... |
| sspwimpcf 41247 | If a class is a subclass o... |
| sspwimpcfVD 41248 | The following User's Proof... |
| suctrALTcf 41249 | The sucessor of a transiti... |
| suctrALTcfVD 41250 | The following User's Proof... |
| suctrALT3 41251 | The successor of a transit... |
| sspwimpALT 41252 | If a class is a subclass o... |
| unisnALT 41253 | A set equals the union of ... |
| notnotrALT2 41254 | Converse of double negatio... |
| sspwimpALT2 41255 | If a class is a subclass o... |
| e2ebindALT 41256 | Absorption of an existenti... |
| ax6e2ndALT 41257 | If at least two sets exist... |
| ax6e2ndeqALT 41258 | "At least two sets exist" ... |
| 2sb5ndALT 41259 | Equivalence for double sub... |
| chordthmALT 41260 | The intersecting chords th... |
| isosctrlem1ALT 41261 | Lemma for ~ isosctr . Thi... |
| iunconnlem2 41262 | The indexed union of conne... |
| iunconnALT 41263 | The indexed union of conne... |
| sineq0ALT 41264 | A complex number whose sin... |
| evth2f 41265 | A version of ~ evth2 using... |
| elunif 41266 | A version of ~ eluni using... |
| rzalf 41267 | A version of ~ rzal using ... |
| fvelrnbf 41268 | A version of ~ fvelrnb usi... |
| rfcnpre1 41269 | If F is a continuous funct... |
| ubelsupr 41270 | If U belongs to A and U is... |
| fsumcnf 41271 | A finite sum of functions ... |
| mulltgt0 41272 | The product of a negative ... |
| rspcegf 41273 | A version of ~ rspcev usin... |
| rabexgf 41274 | A version of ~ rabexg usin... |
| fcnre 41275 | A function continuous with... |
| sumsnd 41276 | A sum of a singleton is th... |
| evthf 41277 | A version of ~ evth using ... |
| cnfex 41278 | The class of continuous fu... |
| fnchoice 41279 | For a finite set, a choice... |
| refsumcn 41280 | A finite sum of continuous... |
| rfcnpre2 41281 | If ` F ` is a continuous f... |
| cncmpmax 41282 | When the hypothesis for th... |
| rfcnpre3 41283 | If F is a continuous funct... |
| rfcnpre4 41284 | If F is a continuous funct... |
| sumpair 41285 | Sum of two distinct comple... |
| rfcnnnub 41286 | Given a real continuous fu... |
| refsum2cnlem1 41287 | This is the core Lemma for... |
| refsum2cn 41288 | The sum of two continuus r... |
| elunnel2 41289 | A member of a union that i... |
| adantlllr 41290 | Deduction adding a conjunc... |
| 3adantlr3 41291 | Deduction adding a conjunc... |
| nnxrd 41292 | A natural number is an ext... |
| 3adantll2 41293 | Deduction adding a conjunc... |
| 3adantll3 41294 | Deduction adding a conjunc... |
| ssnel 41295 | If not element of a set, t... |
| elabrexg 41296 | Elementhood in an image se... |
| sncldre 41297 | A singleton is closed w.r.... |
| n0p 41298 | A polynomial with a nonzer... |
| pm2.65ni 41299 | Inference rule for proof b... |
| pwssfi 41300 | Every element of the power... |
| iuneq2df 41301 | Equality deduction for ind... |
| nnfoctb 41302 | There exists a mapping fro... |
| ssinss1d 41303 | Intersection preserves sub... |
| elpwinss 41304 | An element of the powerset... |
| unidmex 41305 | If ` F ` is a set, then ` ... |
| ndisj2 41306 | A non-disjointness conditi... |
| zenom 41307 | The set of integer numbers... |
| uzwo4 41308 | Well-ordering principle: a... |
| unisn0 41309 | The union of the singleton... |
| ssin0 41310 | If two classes are disjoin... |
| inabs3 41311 | Absorption law for interse... |
| pwpwuni 41312 | Relationship between power... |
| disjiun2 41313 | In a disjoint collection, ... |
| 0pwfi 41314 | The empty set is in any po... |
| ssinss2d 41315 | Intersection preserves sub... |
| zct 41316 | The set of integer numbers... |
| pwfin0 41317 | A finite set always belong... |
| uzct 41318 | An upper integer set is co... |
| iunxsnf 41319 | A singleton index picks ou... |
| fiiuncl 41320 | If a set is closed under t... |
| iunp1 41321 | The addition of the next s... |
| fiunicl 41322 | If a set is closed under t... |
| ixpeq2d 41323 | Equality theorem for infin... |
| disjxp1 41324 | The sets of a cartesian pr... |
| disjsnxp 41325 | The sets in the cartesian ... |
| eliind 41326 | Membership in indexed inte... |
| rspcef 41327 | Restricted existential spe... |
| inn0f 41328 | A nonempty intersection. ... |
| ixpssmapc 41329 | An infinite Cartesian prod... |
| inn0 41330 | A nonempty intersection. ... |
| elintd 41331 | Membership in class inters... |
| ssdf 41332 | A sufficient condition for... |
| brneqtrd 41333 | Substitution of equal clas... |
| ssnct 41334 | A set containing an uncoun... |
| ssuniint 41335 | Sufficient condition for b... |
| elintdv 41336 | Membership in class inters... |
| ssd 41337 | A sufficient condition for... |
| ralimralim 41338 | Introducing any antecedent... |
| snelmap 41339 | Membership of the element ... |
| xrnmnfpnf 41340 | An extended real that is n... |
| nelrnmpt 41341 | Non-membership in the rang... |
| snn0d 41342 | The singleton of a set is ... |
| iuneq1i 41343 | Equality theorem for index... |
| nssrex 41344 | Negation of subclass relat... |
| iunssf 41345 | Subset theorem for an inde... |
| ssinc 41346 | Inclusion relation for a m... |
| ssdec 41347 | Inclusion relation for a m... |
| elixpconstg 41348 | Membership in an infinite ... |
| iineq1d 41349 | Equality theorem for index... |
| metpsmet 41350 | A metric is a pseudometric... |
| ixpssixp 41351 | Subclass theorem for infin... |
| ballss3 41352 | A sufficient condition for... |
| iunincfi 41353 | Given a sequence of increa... |
| nsstr 41354 | If it's not a subclass, it... |
| rexanuz3 41355 | Combine two different uppe... |
| cbvmpo2 41356 | Rule to change the second ... |
| cbvmpo1 41357 | Rule to change the first b... |
| eliuniin 41358 | Indexed union of indexed i... |
| ssabf 41359 | Subclass of a class abstra... |
| pssnssi 41360 | A proper subclass does not... |
| rabidim2 41361 | Membership in a restricted... |
| eluni2f 41362 | Membership in class union.... |
| eliin2f 41363 | Membership in indexed inte... |
| nssd 41364 | Negation of subclass relat... |
| iineq12dv 41365 | Equality deduction for ind... |
| supxrcld 41366 | The supremum of an arbitra... |
| elrestd 41367 | A sufficient condition for... |
| eliuniincex 41368 | Counterexample to show tha... |
| eliincex 41369 | Counterexample to show tha... |
| eliinid 41370 | Membership in an indexed i... |
| abssf 41371 | Class abstraction in a sub... |
| fexd 41372 | If the domain of a mapping... |
| supxrubd 41373 | A member of a set of exten... |
| ssrabf 41374 | Subclass of a restricted c... |
| eliin2 41375 | Membership in indexed inte... |
| ssrab2f 41376 | Subclass relation for a re... |
| restuni3 41377 | The underlying set of a su... |
| rabssf 41378 | Restricted class abstracti... |
| eliuniin2 41379 | Indexed union of indexed i... |
| restuni4 41380 | The underlying set of a su... |
| restuni6 41381 | The underlying set of a su... |
| restuni5 41382 | The underlying set of a su... |
| unirestss 41383 | The union of an elementwis... |
| iniin1 41384 | Indexed intersection of in... |
| iniin2 41385 | Indexed intersection of in... |
| cbvrabv2 41386 | A more general version of ... |
| cbvrabv2w 41387 | A more general version of ... |
| iinssiin 41388 | Subset implication for an ... |
| eliind2 41389 | Membership in indexed inte... |
| iinssd 41390 | Subset implication for an ... |
| ralrimia 41391 | Inference from Theorem 19.... |
| rabbida2 41392 | Equivalent wff's yield equ... |
| iinexd 41393 | The existence of an indexe... |
| rabexf 41394 | Separation Scheme in terms... |
| rabbida3 41395 | Equivalent wff's yield equ... |
| resexd 41396 | The restriction of a set i... |
| r19.36vf 41397 | Restricted quantifier vers... |
| raleqd 41398 | Equality deduction for res... |
| ralimda 41399 | Deduction quantifying both... |
| iinssf 41400 | Subset implication for an ... |
| iinssdf 41401 | Subset implication for an ... |
| resabs2i 41402 | Absorption law for restric... |
| ssdf2 41403 | A sufficient condition for... |
| rabssd 41404 | Restricted class abstracti... |
| rexnegd 41405 | Minus a real number. (Con... |
| rexlimd3 41406 | * Inference from Theorem 1... |
| resabs1i 41407 | Absorption law for restric... |
| nel1nelin 41408 | Membership in an intersect... |
| nel2nelin 41409 | Membership in an intersect... |
| nel1nelini 41410 | Membership in an intersect... |
| nel2nelini 41411 | Membership in an intersect... |
| eliunid 41412 | Membership in indexed unio... |
| reximddv3 41413 | Deduction from Theorem 19.... |
| reximdd 41414 | Deduction from Theorem 19.... |
| unfid 41415 | The union of two finite se... |
| feq1dd 41416 | Equality deduction for fun... |
| fnresdmss 41417 | A function does not change... |
| fmptsnxp 41418 | Maps-to notation and cross... |
| fvmpt2bd 41419 | Value of a function given ... |
| rnmptfi 41420 | The range of a function wi... |
| fresin2 41421 | Restriction of a function ... |
| ffi 41422 | A function with finite dom... |
| suprnmpt 41423 | An explicit bound for the ... |
| rnffi 41424 | The range of a function wi... |
| mptelpm 41425 | A function in maps-to nota... |
| rnmptpr 41426 | Range of a function define... |
| resmpti 41427 | Restriction of the mapping... |
| founiiun 41428 | Union expressed as an inde... |
| rnresun 41429 | Distribution law for range... |
| f1oeq1d 41430 | Equality deduction for one... |
| dffo3f 41431 | An onto mapping expressed ... |
| rnresss 41432 | The range of a restriction... |
| elrnmptd 41433 | The range of a function in... |
| elrnmptf 41434 | The range of a function in... |
| rnmptssrn 41435 | Inclusion relation for two... |
| disjf1 41436 | A 1 to 1 mapping built fro... |
| rnsnf 41437 | The range of a function wh... |
| wessf1ornlem 41438 | Given a function ` F ` on ... |
| wessf1orn 41439 | Given a function ` F ` on ... |
| foelrnf 41440 | Property of a surjective f... |
| nelrnres 41441 | If ` A ` is not in the ran... |
| disjrnmpt2 41442 | Disjointness of the range ... |
| elrnmpt1sf 41443 | Elementhood in an image se... |
| founiiun0 41444 | Union expressed as an inde... |
| disjf1o 41445 | A bijection built from dis... |
| fompt 41446 | Express being onto for a m... |
| disjinfi 41447 | Only a finite number of di... |
| fvovco 41448 | Value of the composition o... |
| ssnnf1octb 41449 | There exists a bijection b... |
| nnf1oxpnn 41450 | There is a bijection betwe... |
| rnmptssd 41451 | The range of an operation ... |
| projf1o 41452 | A biijection from a set to... |
| fvmap 41453 | Function value for a membe... |
| fvixp2 41454 | Projection of a factor of ... |
| fidmfisupp 41455 | A function with a finite d... |
| choicefi 41456 | For a finite set, a choice... |
| mpct 41457 | The exponentiation of a co... |
| cnmetcoval 41458 | Value of the distance func... |
| fcomptss 41459 | Express composition of two... |
| elmapsnd 41460 | Membership in a set expone... |
| mapss2 41461 | Subset inheritance for set... |
| fsneq 41462 | Equality condition for two... |
| difmap 41463 | Difference of two sets exp... |
| unirnmap 41464 | Given a subset of a set ex... |
| inmap 41465 | Intersection of two sets e... |
| fcoss 41466 | Composition of two mapping... |
| fsneqrn 41467 | Equality condition for two... |
| difmapsn 41468 | Difference of two sets exp... |
| mapssbi 41469 | Subset inheritance for set... |
| unirnmapsn 41470 | Equality theorem for a sub... |
| iunmapss 41471 | The indexed union of set e... |
| ssmapsn 41472 | A subset ` C ` of a set ex... |
| iunmapsn 41473 | The indexed union of set e... |
| absfico 41474 | Mapping domain and codomai... |
| icof 41475 | The set of left-closed rig... |
| rnmpt0 41476 | The range of a function in... |
| rnmptn0 41477 | The range of a function in... |
| elpmrn 41478 | The range of a partial fun... |
| imaexi 41479 | The image of a set is a se... |
| axccdom 41480 | Relax the constraint on ax... |
| dmmptdf 41481 | The domain of the mapping ... |
| elpmi2 41482 | The domain of a partial fu... |
| dmrelrnrel 41483 | A relation preserving func... |
| fco3 41484 | Functionality of a composi... |
| fvcod 41485 | Value of a function compos... |
| freld 41486 | A mapping is a relation. ... |
| elrnmpoid 41487 | Membership in the range of... |
| axccd 41488 | An alternative version of ... |
| axccd2 41489 | An alternative version of ... |
| funimassd 41490 | Sufficient condition for t... |
| fimassd 41491 | The image of a class is a ... |
| feqresmptf 41492 | Express a restricted funct... |
| elrnmpt1d 41493 | Elementhood in an image se... |
| dmresss 41494 | The domain of a restrictio... |
| dmmptssf 41495 | The domain of a mapping is... |
| dmmptdf2 41496 | The domain of the mapping ... |
| dmuz 41497 | Domain of the upper intege... |
| fmptd2f 41498 | Domain and codomain of the... |
| mpteq1df 41499 | An equality theorem for th... |
| mptexf 41500 | If the domain of a functio... |
| fvmpt4 41501 | Value of a function given ... |
| fmptf 41502 | Functionality of the mappi... |
| resimass 41503 | The image of a restriction... |
| mptssid 41504 | The mapping operation expr... |
| mptfnd 41505 | The maps-to notation defin... |
| mpteq12da 41506 | An equality inference for ... |
| rnmptlb 41507 | Boundness below of the ran... |
| rnmptbddlem 41508 | Boundness of the range of ... |
| rnmptbdd 41509 | Boundness of the range of ... |
| mptima2 41510 | Image of a function in map... |
| funimaeq 41511 | Membership relation for th... |
| rnmptssf 41512 | The range of an operation ... |
| rnmptbd2lem 41513 | Boundness below of the ran... |
| rnmptbd2 41514 | Boundness below of the ran... |
| infnsuprnmpt 41515 | The indexed infimum of rea... |
| suprclrnmpt 41516 | Closure of the indexed sup... |
| suprubrnmpt2 41517 | A member of a nonempty ind... |
| suprubrnmpt 41518 | A member of a nonempty ind... |
| rnmptssdf 41519 | The range of an operation ... |
| rnmptbdlem 41520 | Boundness above of the ran... |
| rnmptbd 41521 | Boundness above of the ran... |
| rnmptss2 41522 | The range of an operation ... |
| elmptima 41523 | The image of a function in... |
| ralrnmpt3 41524 | A restricted quantifier ov... |
| fvelima2 41525 | Function value in an image... |
| funresd 41526 | A restriction of a functio... |
| rnmptssbi 41527 | The range of an operation ... |
| fnfvelrnd 41528 | A function's value belongs... |
| imass2d 41529 | Subset theorem for image. ... |
| imassmpt 41530 | Membership relation for th... |
| fpmd 41531 | A total function is a part... |
| fconst7 41532 | An alternative way to expr... |
| sub2times 41533 | Subtracting from a number,... |
| abssubrp 41534 | The distance of two distin... |
| elfzfzo 41535 | Relationship between membe... |
| oddfl 41536 | Odd number representation ... |
| abscosbd 41537 | Bound for the absolute val... |
| mul13d 41538 | Commutative/associative la... |
| negpilt0 41539 | Negative ` _pi ` is negati... |
| dstregt0 41540 | A complex number ` A ` tha... |
| subadd4b 41541 | Rearrangement of 4 terms i... |
| xrlttri5d 41542 | Not equal and not larger i... |
| neglt 41543 | The negative of a positive... |
| zltlesub 41544 | If an integer ` N ` is les... |
| divlt0gt0d 41545 | The ratio of a negative nu... |
| subsub23d 41546 | Swap subtrahend and result... |
| 2timesgt 41547 | Double of a positive real ... |
| reopn 41548 | The reals are open with re... |
| elfzop1le2 41549 | A member in a half-open in... |
| sub31 41550 | Swap the first and third t... |
| nnne1ge2 41551 | A positive integer which i... |
| lefldiveq 41552 | A closed enough, smaller r... |
| negsubdi3d 41553 | Distribution of negative o... |
| ltdiv2dd 41554 | Division of a positive num... |
| abssinbd 41555 | Bound for the absolute val... |
| halffl 41556 | Floor of ` ( 1 / 2 ) ` . ... |
| monoords 41557 | Ordering relation for a st... |
| hashssle 41558 | The size of a subset of a ... |
| lttri5d 41559 | Not equal and not larger i... |
| fzisoeu 41560 | A finite ordered set has a... |
| lt3addmuld 41561 | If three real numbers are ... |
| absnpncan2d 41562 | Triangular inequality, com... |
| fperiodmullem 41563 | A function with period T i... |
| fperiodmul 41564 | A function with period T i... |
| upbdrech 41565 | Choice of an upper bound f... |
| lt4addmuld 41566 | If four real numbers are l... |
| absnpncan3d 41567 | Triangular inequality, com... |
| upbdrech2 41568 | Choice of an upper bound f... |
| ssfiunibd 41569 | A finite union of bounded ... |
| fzdifsuc2 41570 | Remove a successor from th... |
| fzsscn 41571 | A finite sequence of integ... |
| divcan8d 41572 | A cancellation law for div... |
| dmmcand 41573 | Cancellation law for divis... |
| fzssre 41574 | A finite sequence of integ... |
| elfzelzd 41575 | A member of a finite set o... |
| bccld 41576 | A binomial coefficient, in... |
| leadd12dd 41577 | Addition to both sides of ... |
| fzssnn0 41578 | A finite set of sequential... |
| xreqle 41579 | Equality implies 'less tha... |
| xaddid2d 41580 | ` 0 ` is a left identity f... |
| xadd0ge 41581 | A number is less than or e... |
| elfzolem1 41582 | A member in a half-open in... |
| xrgtned 41583 | 'Greater than' implies not... |
| xrleneltd 41584 | 'Less than or equal to' an... |
| xaddcomd 41585 | The extended real addition... |
| supxrre3 41586 | The supremum of a nonempty... |
| uzfissfz 41587 | For any finite subset of t... |
| xleadd2d 41588 | Addition of extended reals... |
| suprltrp 41589 | The supremum of a nonempty... |
| xleadd1d 41590 | Addition of extended reals... |
| xreqled 41591 | Equality implies 'less tha... |
| xrgepnfd 41592 | An extended real greater t... |
| xrge0nemnfd 41593 | A nonnegative extended rea... |
| supxrgere 41594 | If a real number can be ap... |
| iuneqfzuzlem 41595 | Lemma for ~ iuneqfzuz : he... |
| iuneqfzuz 41596 | If two unions indexed by u... |
| xle2addd 41597 | Adding both side of two in... |
| supxrgelem 41598 | If an extended real number... |
| supxrge 41599 | If an extended real number... |
| suplesup 41600 | If any element of ` A ` ca... |
| infxrglb 41601 | The infimum of a set of ex... |
| xadd0ge2 41602 | A number is less than or e... |
| nepnfltpnf 41603 | An extended real that is n... |
| ltadd12dd 41604 | Addition to both sides of ... |
| nemnftgtmnft 41605 | An extended real that is n... |
| xrgtso 41606 | 'Greater than' is a strict... |
| rpex 41607 | The positive reals form a ... |
| xrge0ge0 41608 | A nonnegative extended rea... |
| xrssre 41609 | A subset of extended reals... |
| ssuzfz 41610 | A finite subset of the upp... |
| absfun 41611 | The absolute value is a fu... |
| infrpge 41612 | The infimum of a nonempty,... |
| xrlexaddrp 41613 | If an extended real number... |
| supsubc 41614 | The supremum function dist... |
| xralrple2 41615 | Show that ` A ` is less th... |
| nnuzdisj 41616 | The first ` N ` elements o... |
| ltdivgt1 41617 | Divsion by a number greate... |
| xrltned 41618 | 'Less than' implies not eq... |
| nnsplit 41619 | Express the set of positiv... |
| divdiv3d 41620 | Division into a fraction. ... |
| abslt2sqd 41621 | Comparison of the square o... |
| qenom 41622 | The set of rational number... |
| qct 41623 | The set of rational number... |
| xrltnled 41624 | 'Less than' in terms of 'l... |
| lenlteq 41625 | 'less than or equal to' bu... |
| xrred 41626 | An extended real that is n... |
| rr2sscn2 41627 | The cartesian square of ` ... |
| infxr 41628 | The infimum of a set of ex... |
| infxrunb2 41629 | The infimum of an unbounde... |
| infxrbnd2 41630 | The infimum of a bounded-b... |
| infleinflem1 41631 | Lemma for ~ infleinf , cas... |
| infleinflem2 41632 | Lemma for ~ infleinf , whe... |
| infleinf 41633 | If any element of ` B ` ca... |
| xralrple4 41634 | Show that ` A ` is less th... |
| xralrple3 41635 | Show that ` A ` is less th... |
| eluzelzd 41636 | A member of an upper set o... |
| suplesup2 41637 | If any element of ` A ` is... |
| recnnltrp 41638 | ` N ` is a natural number ... |
| fiminre2 41639 | A nonempty finite set of r... |
| nnn0 41640 | The set of positive intege... |
| fzct 41641 | A finite set of sequential... |
| rpgtrecnn 41642 | Any positive real number i... |
| fzossuz 41643 | A half-open integer interv... |
| infrefilb 41644 | The infimum of a finite se... |
| infxrrefi 41645 | The real and extended real... |
| xrralrecnnle 41646 | Show that ` A ` is less th... |
| fzoct 41647 | A finite set of sequential... |
| frexr 41648 | A function taking real val... |
| nnrecrp 41649 | The reciprocal of a positi... |
| qred 41650 | A rational number is a rea... |
| reclt0d 41651 | The reciprocal of a negati... |
| lt0neg1dd 41652 | If a number is negative, i... |
| mnfled 41653 | Minus infinity is less tha... |
| infxrcld 41654 | The infimum of an arbitrar... |
| xrralrecnnge 41655 | Show that ` A ` is less th... |
| reclt0 41656 | The reciprocal of a negati... |
| ltmulneg 41657 | Multiplying by a negative ... |
| allbutfi 41658 | For all but finitely many.... |
| ltdiv23neg 41659 | Swap denominator with othe... |
| xreqnltd 41660 | A consequence of trichotom... |
| mnfnre2 41661 | Minus infinity is not a re... |
| uzssre 41662 | An upper set of integers i... |
| zssxr 41663 | The integers are a subset ... |
| fisupclrnmpt 41664 | A nonempty finite indexed ... |
| supxrunb3 41665 | The supremum of an unbound... |
| elfzod 41666 | Membership in a half-open ... |
| fimaxre4 41667 | A nonempty finite set of r... |
| ren0 41668 | The set of reals is nonemp... |
| eluzelz2 41669 | A member of an upper set o... |
| resabs2d 41670 | Absorption law for restric... |
| uzid2 41671 | Membership of the least me... |
| supxrleubrnmpt 41672 | The supremum of a nonempty... |
| uzssre2 41673 | An upper set of integers i... |
| uzssd 41674 | Subset relationship for tw... |
| eluzd 41675 | Membership in an upper set... |
| elfzd 41676 | Membership in a finite set... |
| infxrlbrnmpt2 41677 | A member of a nonempty ind... |
| xrre4 41678 | An extended real is real i... |
| uz0 41679 | The upper integers functio... |
| eluzelz2d 41680 | A member of an upper set o... |
| infleinf2 41681 | If any element in ` B ` is... |
| unb2ltle 41682 | "Unbounded below" expresse... |
| uzidd2 41683 | Membership of the least me... |
| uzssd2 41684 | Subset relationship for tw... |
| rexabslelem 41685 | An indexed set of absolute... |
| rexabsle 41686 | An indexed set of absolute... |
| allbutfiinf 41687 | Given a "for all but finit... |
| supxrrernmpt 41688 | The real and extended real... |
| suprleubrnmpt 41689 | The supremum of a nonempty... |
| infrnmptle 41690 | An indexed infimum of exte... |
| infxrunb3 41691 | The infimum of an unbounde... |
| uzn0d 41692 | The upper integers are all... |
| uzssd3 41693 | Subset relationship for tw... |
| rexabsle2 41694 | An indexed set of absolute... |
| infxrunb3rnmpt 41695 | The infimum of an unbounde... |
| supxrre3rnmpt 41696 | The indexed supremum of a ... |
| uzublem 41697 | A set of reals, indexed by... |
| uzub 41698 | A set of reals, indexed by... |
| ssrexr 41699 | A subset of the reals is a... |
| supxrmnf2 41700 | Removing minus infinity fr... |
| supxrcli 41701 | The supremum of an arbitra... |
| uzid3 41702 | Membership of the least me... |
| infxrlesupxr 41703 | The supremum of a nonempty... |
| xnegeqd 41704 | Equality of two extended n... |
| xnegrecl 41705 | The extended real negative... |
| xnegnegi 41706 | Extended real version of ~... |
| xnegeqi 41707 | Equality of two extended n... |
| nfxnegd 41708 | Deduction version of ~ nfx... |
| xnegnegd 41709 | Extended real version of ~... |
| uzred 41710 | An upper integer is a real... |
| xnegcli 41711 | Closure of extended real n... |
| supminfrnmpt 41712 | The indexed supremum of a ... |
| ceilged 41713 | The ceiling of a real numb... |
| infxrpnf 41714 | Adding plus infinity to a ... |
| infxrrnmptcl 41715 | The infimum of an arbitrar... |
| leneg2d 41716 | Negative of one side of 'l... |
| supxrltinfxr 41717 | The supremum of the empty ... |
| max1d 41718 | A number is less than or e... |
| ceilcld 41719 | Closure of the ceiling fun... |
| supxrleubrnmptf 41720 | The supremum of a nonempty... |
| nleltd 41721 | 'Not less than or equal to... |
| zxrd 41722 | An integer is an extended ... |
| infxrgelbrnmpt 41723 | The infimum of an indexed ... |
| rphalfltd 41724 | Half of a positive real is... |
| uzssz2 41725 | An upper set of integers i... |
| leneg3d 41726 | Negative of one side of 'l... |
| max2d 41727 | A number is less than or e... |
| uzn0bi 41728 | The upper integers functio... |
| xnegrecl2 41729 | If the extended real negat... |
| nfxneg 41730 | Bound-variable hypothesis ... |
| uzxrd 41731 | An upper integer is an ext... |
| infxrpnf2 41732 | Removing plus infinity fro... |
| supminfxr 41733 | The extended real suprema ... |
| infrpgernmpt 41734 | The infimum of a nonempty,... |
| xnegre 41735 | An extended real is real i... |
| xnegrecl2d 41736 | If the extended real negat... |
| uzxr 41737 | An upper integer is an ext... |
| supminfxr2 41738 | The extended real suprema ... |
| xnegred 41739 | An extended real is real i... |
| supminfxrrnmpt 41740 | The indexed supremum of a ... |
| min1d 41741 | The minimum of two numbers... |
| min2d 41742 | The minimum of two numbers... |
| pnfged 41743 | Plus infinity is an upper ... |
| xrnpnfmnf 41744 | An extended real that is n... |
| uzsscn 41745 | An upper set of integers i... |
| absimnre 41746 | The absolute value of the ... |
| uzsscn2 41747 | An upper set of integers i... |
| xrtgcntopre 41748 | The standard topologies on... |
| absimlere 41749 | The absolute value of the ... |
| rpssxr 41750 | The positive reals are a s... |
| monoordxrv 41751 | Ordering relation for a mo... |
| monoordxr 41752 | Ordering relation for a mo... |
| monoord2xrv 41753 | Ordering relation for a mo... |
| monoord2xr 41754 | Ordering relation for a mo... |
| xrpnf 41755 | An extended real is plus i... |
| xlenegcon1 41756 | Extended real version of ~... |
| xlenegcon2 41757 | Extended real version of ~... |
| gtnelioc 41758 | A real number larger than ... |
| ioossioc 41759 | An open interval is a subs... |
| ioondisj2 41760 | A condition for two open i... |
| ioondisj1 41761 | A condition for two open i... |
| ioosscn 41762 | An open interval is a set ... |
| ioogtlb 41763 | An element of a closed int... |
| evthiccabs 41764 | Extreme Value Theorem on y... |
| ltnelicc 41765 | A real number smaller than... |
| eliood 41766 | Membership in an open real... |
| iooabslt 41767 | An upper bound for the dis... |
| gtnelicc 41768 | A real number greater than... |
| iooinlbub 41769 | An open interval has empty... |
| iocgtlb 41770 | An element of a left-open ... |
| iocleub 41771 | An element of a left-open ... |
| eliccd 41772 | Membership in a closed rea... |
| iccssred 41773 | A closed real interval is ... |
| eliccre 41774 | A member of a closed inter... |
| eliooshift 41775 | Element of an open interva... |
| eliocd 41776 | Membership in a left-open ... |
| icoltub 41777 | An element of a left-close... |
| eliocre 41778 | A member of a left-open ri... |
| iooltub 41779 | An element of an open inte... |
| ioontr 41780 | The interior of an interva... |
| snunioo1 41781 | The closure of one end of ... |
| lbioc 41782 | A left-open right-closed i... |
| ioomidp 41783 | The midpoint is an element... |
| iccdifioo 41784 | If the open inverval is re... |
| iccdifprioo 41785 | An open interval is the cl... |
| ioossioobi 41786 | Biconditional form of ~ io... |
| iccshift 41787 | A closed interval shifted ... |
| iccsuble 41788 | An upper bound to the dist... |
| iocopn 41789 | A left-open right-closed i... |
| eliccelioc 41790 | Membership in a closed int... |
| iooshift 41791 | An open interval shifted b... |
| iccintsng 41792 | Intersection of two adiace... |
| icoiccdif 41793 | Left-closed right-open int... |
| icoopn 41794 | A left-closed right-open i... |
| icoub 41795 | A left-closed, right-open ... |
| eliccxrd 41796 | Membership in a closed rea... |
| pnfel0pnf 41797 | ` +oo ` is a nonnegative e... |
| eliccnelico 41798 | An element of a closed int... |
| eliccelicod 41799 | A member of a closed inter... |
| ge0xrre 41800 | A nonnegative extended rea... |
| ge0lere 41801 | A nonnegative extended Rea... |
| elicores 41802 | Membership in a left-close... |
| inficc 41803 | The infimum of a nonempty ... |
| qinioo 41804 | The rational numbers are d... |
| lenelioc 41805 | A real number smaller than... |
| ioonct 41806 | A nonempty open interval i... |
| xrgtnelicc 41807 | A real number greater than... |
| iccdificc 41808 | The difference of two clos... |
| iocnct 41809 | A nonempty left-open, righ... |
| iccnct 41810 | A closed interval, with mo... |
| iooiinicc 41811 | A closed interval expresse... |
| iccgelbd 41812 | An element of a closed int... |
| iooltubd 41813 | An element of an open inte... |
| icoltubd 41814 | An element of a left-close... |
| qelioo 41815 | The rational numbers are d... |
| tgqioo2 41816 | Every open set of reals is... |
| iccleubd 41817 | An element of a closed int... |
| elioored 41818 | A member of an open interv... |
| ioogtlbd 41819 | An element of a closed int... |
| ioofun 41820 | ` (,) ` is a function. (C... |
| icomnfinre 41821 | A left-closed, right-open,... |
| sqrlearg 41822 | The square compared with i... |
| ressiocsup 41823 | If the supremum belongs to... |
| ressioosup 41824 | If the supremum does not b... |
| iooiinioc 41825 | A left-open, right-closed ... |
| ressiooinf 41826 | If the infimum does not be... |
| icogelbd 41827 | An element of a left-close... |
| iocleubd 41828 | An element of a left-open ... |
| uzinico 41829 | An upper interval of integ... |
| preimaiocmnf 41830 | Preimage of a right-closed... |
| uzinico2 41831 | An upper interval of integ... |
| uzinico3 41832 | An upper interval of integ... |
| icossico2 41833 | Condition for a closed-bel... |
| dmico 41834 | The domain of the closed-b... |
| ndmico 41835 | The closed-below, open-abo... |
| uzubioo 41836 | The upper integers are unb... |
| uzubico 41837 | The upper integers are unb... |
| uzubioo2 41838 | The upper integers are unb... |
| uzubico2 41839 | The upper integers are unb... |
| iocgtlbd 41840 | An element of a left-open ... |
| xrtgioo2 41841 | The topology on the extend... |
| tgioo4 41842 | The standard topology on t... |
| fsumclf 41843 | Closure of a finite sum of... |
| fsummulc1f 41844 | Closure of a finite sum of... |
| fsumnncl 41845 | Closure of a nonempty, fin... |
| fsumsplit1 41846 | Separate out a term in a f... |
| fsumge0cl 41847 | The finite sum of nonnegat... |
| fsumf1of 41848 | Re-index a finite sum usin... |
| fsumiunss 41849 | Sum over a disjoint indexe... |
| fsumreclf 41850 | Closure of a finite sum of... |
| fsumlessf 41851 | A shorter sum of nonnegati... |
| fsumsupp0 41852 | Finite sum of function val... |
| fsumsermpt 41853 | A finite sum expressed in ... |
| fmul01 41854 | Multiplying a finite numbe... |
| fmulcl 41855 | If ' Y ' is closed under t... |
| fmuldfeqlem1 41856 | induction step for the pro... |
| fmuldfeq 41857 | X and Z are two equivalent... |
| fmul01lt1lem1 41858 | Given a finite multiplicat... |
| fmul01lt1lem2 41859 | Given a finite multiplicat... |
| fmul01lt1 41860 | Given a finite multiplicat... |
| cncfmptss 41861 | A continuous complex funct... |
| rrpsscn 41862 | The positive reals are a s... |
| mulc1cncfg 41863 | A version of ~ mulc1cncf u... |
| infrglb 41864 | The infimum of a nonempty ... |
| expcnfg 41865 | If ` F ` is a complex cont... |
| prodeq2ad 41866 | Equality deduction for pro... |
| fprodsplit1 41867 | Separate out a term in a f... |
| fprodexp 41868 | Positive integer exponenti... |
| fprodabs2 41869 | The absolute value of a fi... |
| fprod0 41870 | A finite product with a ze... |
| mccllem 41871 | * Induction step for ~ mcc... |
| mccl 41872 | A multinomial coefficient,... |
| fprodcnlem 41873 | A finite product of functi... |
| fprodcn 41874 | A finite product of functi... |
| clim1fr1 41875 | A class of sequences of fr... |
| isumneg 41876 | Negation of a converging s... |
| climrec 41877 | Limit of the reciprocal of... |
| climmulf 41878 | A version of ~ climmul usi... |
| climexp 41879 | The limit of natural power... |
| climinf 41880 | A bounded monotonic noninc... |
| climsuselem1 41881 | The subsequence index ` I ... |
| climsuse 41882 | A subsequence ` G ` of a c... |
| climrecf 41883 | A version of ~ climrec usi... |
| climneg 41884 | Complex limit of the negat... |
| climinff 41885 | A version of ~ climinf usi... |
| climdivf 41886 | Limit of the ratio of two ... |
| climreeq 41887 | If ` F ` is a real functio... |
| ellimciota 41888 | An explicit value for the ... |
| climaddf 41889 | A version of ~ climadd usi... |
| mullimc 41890 | Limit of the product of tw... |
| ellimcabssub0 41891 | An equivalent condition fo... |
| limcdm0 41892 | If a function has empty do... |
| islptre 41893 | An equivalence condition f... |
| limccog 41894 | Limit of the composition o... |
| limciccioolb 41895 | The limit of a function at... |
| climf 41896 | Express the predicate: Th... |
| mullimcf 41897 | Limit of the multiplicatio... |
| constlimc 41898 | Limit of constant function... |
| rexlim2d 41899 | Inference removing two res... |
| idlimc 41900 | Limit of the identity func... |
| divcnvg 41901 | The sequence of reciprocal... |
| limcperiod 41902 | If ` F ` is a periodic fun... |
| limcrecl 41903 | If ` F ` is a real-valued ... |
| sumnnodd 41904 | A series indexed by ` NN `... |
| lptioo2 41905 | The upper bound of an open... |
| lptioo1 41906 | The lower bound of an open... |
| elprn1 41907 | A member of an unordered p... |
| elprn2 41908 | A member of an unordered p... |
| limcmptdm 41909 | The domain of a maps-to fu... |
| clim2f 41910 | Express the predicate: Th... |
| limcicciooub 41911 | The limit of a function at... |
| ltmod 41912 | A sufficient condition for... |
| islpcn 41913 | A characterization for a l... |
| lptre2pt 41914 | If a set in the real line ... |
| limsupre 41915 | If a sequence is bounded, ... |
| limcresiooub 41916 | The left limit doesn't cha... |
| limcresioolb 41917 | The right limit doesn't ch... |
| limcleqr 41918 | If the left and the right ... |
| lptioo2cn 41919 | The upper bound of an open... |
| lptioo1cn 41920 | The lower bound of an open... |
| neglimc 41921 | Limit of the negative func... |
| addlimc 41922 | Sum of two limits. (Contr... |
| 0ellimcdiv 41923 | If the numerator converges... |
| clim2cf 41924 | Express the predicate ` F ... |
| limclner 41925 | For a limit point, both fr... |
| sublimc 41926 | Subtraction of two limits.... |
| reclimc 41927 | Limit of the reciprocal of... |
| clim0cf 41928 | Express the predicate ` F ... |
| limclr 41929 | For a limit point, both fr... |
| divlimc 41930 | Limit of the quotient of t... |
| expfac 41931 | Factorial grows faster tha... |
| climconstmpt 41932 | A constant sequence conver... |
| climresmpt 41933 | A function restricted to u... |
| climsubmpt 41934 | Limit of the difference of... |
| climsubc2mpt 41935 | Limit of the difference of... |
| climsubc1mpt 41936 | Limit of the difference of... |
| fnlimfv 41937 | The value of the limit fun... |
| climreclf 41938 | The limit of a convergent ... |
| climeldmeq 41939 | Two functions that are eve... |
| climf2 41940 | Express the predicate: Th... |
| fnlimcnv 41941 | The sequence of function v... |
| climeldmeqmpt 41942 | Two functions that are eve... |
| climfveq 41943 | Two functions that are eve... |
| clim2f2 41944 | Express the predicate: Th... |
| climfveqmpt 41945 | Two functions that are eve... |
| climd 41946 | Express the predicate: Th... |
| clim2d 41947 | The limit of complex numbe... |
| fnlimfvre 41948 | The limit function of real... |
| allbutfifvre 41949 | Given a sequence of real-v... |
| climleltrp 41950 | The limit of complex numbe... |
| fnlimfvre2 41951 | The limit function of real... |
| fnlimf 41952 | The limit function of real... |
| fnlimabslt 41953 | A sequence of function val... |
| climfveqf 41954 | Two functions that are eve... |
| climmptf 41955 | Exhibit a function ` G ` w... |
| climfveqmpt3 41956 | Two functions that are eve... |
| climeldmeqf 41957 | Two functions that are eve... |
| climreclmpt 41958 | The limit of B convergent ... |
| limsupref 41959 | If a sequence is bounded, ... |
| limsupbnd1f 41960 | If a sequence is eventuall... |
| climbddf 41961 | A converging sequence of c... |
| climeqf 41962 | Two functions that are eve... |
| climeldmeqmpt3 41963 | Two functions that are eve... |
| limsupcld 41964 | Closure of the superior li... |
| climfv 41965 | The limit of a convergent ... |
| limsupval3 41966 | The superior limit of an i... |
| climfveqmpt2 41967 | Two functions that are eve... |
| limsup0 41968 | The superior limit of the ... |
| climeldmeqmpt2 41969 | Two functions that are eve... |
| limsupresre 41970 | The supremum limit of a fu... |
| climeqmpt 41971 | Two functions that are eve... |
| climfvd 41972 | The limit of a convergent ... |
| limsuplesup 41973 | An upper bound for the sup... |
| limsupresico 41974 | The superior limit doesn't... |
| limsuppnfdlem 41975 | If the restriction of a fu... |
| limsuppnfd 41976 | If the restriction of a fu... |
| limsupresuz 41977 | If the real part of the do... |
| limsupub 41978 | If the limsup is not ` +oo... |
| limsupres 41979 | The superior limit of a re... |
| climinf2lem 41980 | A convergent, nonincreasin... |
| climinf2 41981 | A convergent, nonincreasin... |
| limsupvaluz 41982 | The superior limit, when t... |
| limsupresuz2 41983 | If the domain of a functio... |
| limsuppnflem 41984 | If the restriction of a fu... |
| limsuppnf 41985 | If the restriction of a fu... |
| limsupubuzlem 41986 | If the limsup is not ` +oo... |
| limsupubuz 41987 | For a real-valued function... |
| climinf2mpt 41988 | A bounded below, monotonic... |
| climinfmpt 41989 | A bounded below, monotonic... |
| climinf3 41990 | A convergent, nonincreasin... |
| limsupvaluzmpt 41991 | The superior limit, when t... |
| limsupequzmpt2 41992 | Two functions that are eve... |
| limsupubuzmpt 41993 | If the limsup is not ` +oo... |
| limsupmnflem 41994 | The superior limit of a fu... |
| limsupmnf 41995 | The superior limit of a fu... |
| limsupequzlem 41996 | Two functions that are eve... |
| limsupequz 41997 | Two functions that are eve... |
| limsupre2lem 41998 | Given a function on the ex... |
| limsupre2 41999 | Given a function on the ex... |
| limsupmnfuzlem 42000 | The superior limit of a fu... |
| limsupmnfuz 42001 | The superior limit of a fu... |
| limsupequzmptlem 42002 | Two functions that are eve... |
| limsupequzmpt 42003 | Two functions that are eve... |
| limsupre2mpt 42004 | Given a function on the ex... |
| limsupequzmptf 42005 | Two functions that are eve... |
| limsupre3lem 42006 | Given a function on the ex... |
| limsupre3 42007 | Given a function on the ex... |
| limsupre3mpt 42008 | Given a function on the ex... |
| limsupre3uzlem 42009 | Given a function on the ex... |
| limsupre3uz 42010 | Given a function on the ex... |
| limsupreuz 42011 | Given a function on the re... |
| limsupvaluz2 42012 | The superior limit, when t... |
| limsupreuzmpt 42013 | Given a function on the re... |
| supcnvlimsup 42014 | If a function on a set of ... |
| supcnvlimsupmpt 42015 | If a function on a set of ... |
| 0cnv 42016 | If (/) is a complex number... |
| climuzlem 42017 | Express the predicate: Th... |
| climuz 42018 | Express the predicate: Th... |
| lmbr3v 42019 | Express the binary relatio... |
| climisp 42020 | If a sequence converges to... |
| lmbr3 42021 | Express the binary relatio... |
| climrescn 42022 | A sequence converging w.r.... |
| climxrrelem 42023 | If a seqence ranging over ... |
| climxrre 42024 | If a sequence ranging over... |
| limsuplt2 42027 | The defining property of t... |
| liminfgord 42028 | Ordering property of the i... |
| limsupvald 42029 | The superior limit of a se... |
| limsupresicompt 42030 | The superior limit doesn't... |
| limsupcli 42031 | Closure of the superior li... |
| liminfgf 42032 | Closure of the inferior li... |
| liminfval 42033 | The inferior limit of a se... |
| climlimsup 42034 | A sequence of real numbers... |
| limsupge 42035 | The defining property of t... |
| liminfgval 42036 | Value of the inferior limi... |
| liminfcl 42037 | Closure of the inferior li... |
| liminfvald 42038 | The inferior limit of a se... |
| liminfval5 42039 | The inferior limit of an i... |
| limsupresxr 42040 | The superior limit of a fu... |
| liminfresxr 42041 | The inferior limit of a fu... |
| liminfval2 42042 | The superior limit, relati... |
| climlimsupcex 42043 | Counterexample for ~ climl... |
| liminfcld 42044 | Closure of the inferior li... |
| liminfresico 42045 | The inferior limit doesn't... |
| limsup10exlem 42046 | The range of the given fun... |
| limsup10ex 42047 | The superior limit of a fu... |
| liminf10ex 42048 | The inferior limit of a fu... |
| liminflelimsuplem 42049 | The superior limit is grea... |
| liminflelimsup 42050 | The superior limit is grea... |
| limsupgtlem 42051 | For any positive real, the... |
| limsupgt 42052 | Given a sequence of real n... |
| liminfresre 42053 | The inferior limit of a fu... |
| liminfresicompt 42054 | The inferior limit doesn't... |
| liminfltlimsupex 42055 | An example where the ` lim... |
| liminfgelimsup 42056 | The inferior limit is grea... |
| liminfvalxr 42057 | Alternate definition of ` ... |
| liminfresuz 42058 | If the real part of the do... |
| liminflelimsupuz 42059 | The superior limit is grea... |
| liminfvalxrmpt 42060 | Alternate definition of ` ... |
| liminfresuz2 42061 | If the domain of a functio... |
| liminfgelimsupuz 42062 | The inferior limit is grea... |
| liminfval4 42063 | Alternate definition of ` ... |
| liminfval3 42064 | Alternate definition of ` ... |
| liminfequzmpt2 42065 | Two functions that are eve... |
| liminfvaluz 42066 | Alternate definition of ` ... |
| liminf0 42067 | The inferior limit of the ... |
| limsupval4 42068 | Alternate definition of ` ... |
| liminfvaluz2 42069 | Alternate definition of ` ... |
| liminfvaluz3 42070 | Alternate definition of ` ... |
| liminflelimsupcex 42071 | A counterexample for ~ lim... |
| limsupvaluz3 42072 | Alternate definition of ` ... |
| liminfvaluz4 42073 | Alternate definition of ` ... |
| limsupvaluz4 42074 | Alternate definition of ` ... |
| climliminflimsupd 42075 | If a sequence of real numb... |
| liminfreuzlem 42076 | Given a function on the re... |
| liminfreuz 42077 | Given a function on the re... |
| liminfltlem 42078 | Given a sequence of real n... |
| liminflt 42079 | Given a sequence of real n... |
| climliminf 42080 | A sequence of real numbers... |
| liminflimsupclim 42081 | A sequence of real numbers... |
| climliminflimsup 42082 | A sequence of real numbers... |
| climliminflimsup2 42083 | A sequence of real numbers... |
| climliminflimsup3 42084 | A sequence of real numbers... |
| climliminflimsup4 42085 | A sequence of real numbers... |
| limsupub2 42086 | A extended real valued fun... |
| limsupubuz2 42087 | A sequence with values in ... |
| xlimpnfxnegmnf 42088 | A sequence converges to ` ... |
| liminflbuz2 42089 | A sequence with values in ... |
| liminfpnfuz 42090 | The inferior limit of a fu... |
| liminflimsupxrre 42091 | A sequence with values in ... |
| xlimrel 42094 | The limit on extended real... |
| xlimres 42095 | A function converges iff i... |
| xlimcl 42096 | The limit of a sequence of... |
| rexlimddv2 42097 | Restricted existential eli... |
| xlimclim 42098 | Given a sequence of reals,... |
| xlimconst 42099 | A constant sequence conver... |
| climxlim 42100 | A converging sequence in t... |
| xlimbr 42101 | Express the binary relatio... |
| fuzxrpmcn 42102 | A function mapping from an... |
| cnrefiisplem 42103 | Lemma for ~ cnrefiisp (som... |
| cnrefiisp 42104 | A non-real, complex number... |
| xlimxrre 42105 | If a sequence ranging over... |
| xlimmnfvlem1 42106 | Lemma for ~ xlimmnfv : the... |
| xlimmnfvlem2 42107 | Lemma for ~ xlimmnf : the ... |
| xlimmnfv 42108 | A function converges to mi... |
| xlimconst2 42109 | A sequence that eventually... |
| xlimpnfvlem1 42110 | Lemma for ~ xlimpnfv : the... |
| xlimpnfvlem2 42111 | Lemma for ~ xlimpnfv : the... |
| xlimpnfv 42112 | A function converges to pl... |
| xlimclim2lem 42113 | Lemma for ~ xlimclim2 . H... |
| xlimclim2 42114 | Given a sequence of extend... |
| xlimmnf 42115 | A function converges to mi... |
| xlimpnf 42116 | A function converges to pl... |
| xlimmnfmpt 42117 | A function converges to pl... |
| xlimpnfmpt 42118 | A function converges to pl... |
| climxlim2lem 42119 | In this lemma for ~ climxl... |
| climxlim2 42120 | A sequence of extended rea... |
| dfxlim2v 42121 | An alternative definition ... |
| dfxlim2 42122 | An alternative definition ... |
| climresd 42123 | A function restricted to u... |
| climresdm 42124 | A real function converges ... |
| dmclimxlim 42125 | A real valued sequence tha... |
| xlimmnflimsup2 42126 | A sequence of extended rea... |
| xlimuni 42127 | An infinite sequence conve... |
| xlimclimdm 42128 | A sequence of extended rea... |
| xlimfun 42129 | The convergence relation o... |
| xlimmnflimsup 42130 | If a sequence of extended ... |
| xlimdm 42131 | Two ways to express that a... |
| xlimpnfxnegmnf2 42132 | A sequence converges to ` ... |
| xlimresdm 42133 | A function converges in th... |
| xlimpnfliminf 42134 | If a sequence of extended ... |
| xlimpnfliminf2 42135 | A sequence of extended rea... |
| xlimliminflimsup 42136 | A sequence of extended rea... |
| xlimlimsupleliminf 42137 | A sequence of extended rea... |
| coseq0 42138 | A complex number whose cos... |
| sinmulcos 42139 | Multiplication formula for... |
| coskpi2 42140 | The cosine of an integer m... |
| cosnegpi 42141 | The cosine of negative ` _... |
| sinaover2ne0 42142 | If ` A ` in ` ( 0 , 2 _pi ... |
| cosknegpi 42143 | The cosine of an integer m... |
| mulcncff 42144 | The multiplication of two ... |
| subcncf 42145 | The addition of two contin... |
| cncfmptssg 42146 | A continuous complex funct... |
| constcncfg 42147 | A constant function is a c... |
| idcncfg 42148 | The identity function is a... |
| addcncf 42149 | The addition of two contin... |
| cncfshift 42150 | A periodic continuous func... |
| resincncf 42151 | ` sin ` restricted to real... |
| addccncf2 42152 | Adding a constant is a con... |
| 0cnf 42153 | The empty set is a continu... |
| fsumcncf 42154 | The finite sum of continuo... |
| cncfperiod 42155 | A periodic continuous func... |
| subcncff 42156 | The subtraction of two con... |
| negcncfg 42157 | The opposite of a continuo... |
| cnfdmsn 42158 | A function with a singleto... |
| cncfcompt 42159 | Composition of continuous ... |
| addcncff 42160 | The sum of two continuous ... |
| ioccncflimc 42161 | Limit at the upper bound o... |
| cncfuni 42162 | A complex function on a su... |
| icccncfext 42163 | A continuous function on a... |
| cncficcgt0 42164 | A the absolute value of a ... |
| icocncflimc 42165 | Limit at the lower bound, ... |
| cncfdmsn 42166 | A complex function with a ... |
| divcncff 42167 | The quotient of two contin... |
| cncfshiftioo 42168 | A periodic continuous func... |
| cncfiooicclem1 42169 | A continuous function ` F ... |
| cncfiooicc 42170 | A continuous function ` F ... |
| cncfiooiccre 42171 | A continuous function ` F ... |
| cncfioobdlem 42172 | ` G ` actually extends ` F... |
| cncfioobd 42173 | A continuous function ` F ... |
| jumpncnp 42174 | Jump discontinuity or disc... |
| cncfcompt2 42175 | Composition of continuous ... |
| cxpcncf2 42176 | The complex power function... |
| fprodcncf 42177 | The finite product of cont... |
| add1cncf 42178 | Addition to a constant is ... |
| add2cncf 42179 | Addition to a constant is ... |
| sub1cncfd 42180 | Subtracting a constant is ... |
| sub2cncfd 42181 | Subtraction from a constan... |
| fprodsub2cncf 42182 | ` F ` is continuous. (Con... |
| fprodadd2cncf 42183 | ` F ` is continuous. (Con... |
| fprodsubrecnncnvlem 42184 | The sequence ` S ` of fini... |
| fprodsubrecnncnv 42185 | The sequence ` S ` of fini... |
| fprodaddrecnncnvlem 42186 | The sequence ` S ` of fini... |
| fprodaddrecnncnv 42187 | The sequence ` S ` of fini... |
| dvsinexp 42188 | The derivative of sin^N . ... |
| dvcosre 42189 | The real derivative of the... |
| dvsinax 42190 | Derivative exercise: the d... |
| dvsubf 42191 | The subtraction rule for e... |
| dvmptconst 42192 | Function-builder for deriv... |
| dvcnre 42193 | From compex differentiatio... |
| dvmptidg 42194 | Function-builder for deriv... |
| dvresntr 42195 | Function-builder for deriv... |
| fperdvper 42196 | The derivative of a period... |
| dvmptresicc 42197 | Derivative of a function r... |
| dvasinbx 42198 | Derivative exercise: the d... |
| dvresioo 42199 | Restriction of a derivativ... |
| dvdivf 42200 | The quotient rule for ever... |
| dvdivbd 42201 | A sufficient condition for... |
| dvsubcncf 42202 | A sufficient condition for... |
| dvmulcncf 42203 | A sufficient condition for... |
| dvcosax 42204 | Derivative exercise: the d... |
| dvdivcncf 42205 | A sufficient condition for... |
| dvbdfbdioolem1 42206 | Given a function with boun... |
| dvbdfbdioolem2 42207 | A function on an open inte... |
| dvbdfbdioo 42208 | A function on an open inte... |
| ioodvbdlimc1lem1 42209 | If ` F ` has bounded deriv... |
| ioodvbdlimc1lem2 42210 | Limit at the lower bound o... |
| ioodvbdlimc1 42211 | A real function with bound... |
| ioodvbdlimc2lem 42212 | Limit at the upper bound o... |
| ioodvbdlimc2 42213 | A real function with bound... |
| dvdmsscn 42214 | ` X ` is a subset of ` CC ... |
| dvmptmulf 42215 | Function-builder for deriv... |
| dvnmptdivc 42216 | Function-builder for itera... |
| dvdsn1add 42217 | If ` K ` divides ` N ` but... |
| dvxpaek 42218 | Derivative of the polynomi... |
| dvnmptconst 42219 | The ` N ` -th derivative o... |
| dvnxpaek 42220 | The ` n ` -th derivative o... |
| dvnmul 42221 | Function-builder for the `... |
| dvmptfprodlem 42222 | Induction step for ~ dvmpt... |
| dvmptfprod 42223 | Function-builder for deriv... |
| dvnprodlem1 42224 | ` D ` is bijective. (Cont... |
| dvnprodlem2 42225 | Induction step for ~ dvnpr... |
| dvnprodlem3 42226 | The multinomial formula fo... |
| dvnprod 42227 | The multinomial formula fo... |
| itgsin0pilem1 42228 | Calculation of the integra... |
| ibliccsinexp 42229 | sin^n on a closed interval... |
| itgsin0pi 42230 | Calculation of the integra... |
| iblioosinexp 42231 | sin^n on an open integral ... |
| itgsinexplem1 42232 | Integration by parts is ap... |
| itgsinexp 42233 | A recursive formula for th... |
| iblconstmpt 42234 | A constant function is int... |
| itgeq1d 42235 | Equality theorem for an in... |
| mbfres2cn 42236 | Measurability of a piecewi... |
| vol0 42237 | The measure of the empty s... |
| ditgeqiooicc 42238 | A function ` F ` on an ope... |
| volge0 42239 | The volume of a set is alw... |
| cnbdibl 42240 | A continuous bounded funct... |
| snmbl 42241 | A singleton is measurable.... |
| ditgeq3d 42242 | Equality theorem for the d... |
| iblempty 42243 | The empty function is inte... |
| iblsplit 42244 | The union of two integrabl... |
| volsn 42245 | A singleton has 0 Lebesgue... |
| itgvol0 42246 | If the domani is negligibl... |
| itgcoscmulx 42247 | Exercise: the integral of ... |
| iblsplitf 42248 | A version of ~ iblsplit us... |
| ibliooicc 42249 | If a function is integrabl... |
| volioc 42250 | The measure of a left-open... |
| iblspltprt 42251 | If a function is integrabl... |
| itgsincmulx 42252 | Exercise: the integral of ... |
| itgsubsticclem 42253 | lemma for ~ itgsubsticc . ... |
| itgsubsticc 42254 | Integration by u-substitut... |
| itgioocnicc 42255 | The integral of a piecewis... |
| iblcncfioo 42256 | A continuous function ` F ... |
| itgspltprt 42257 | The ` S. ` integral splits... |
| itgiccshift 42258 | The integral of a function... |
| itgperiod 42259 | The integral of a periodic... |
| itgsbtaddcnst 42260 | Integral substitution, add... |
| itgeq2d 42261 | Equality theorem for an in... |
| volico 42262 | The measure of left-closed... |
| sublevolico 42263 | The Lebesgue measure of a ... |
| dmvolss 42264 | Lebesgue measurable sets a... |
| ismbl3 42265 | The predicate " ` A ` is L... |
| volioof 42266 | The function that assigns ... |
| ovolsplit 42267 | The Lebesgue outer measure... |
| fvvolioof 42268 | The function value of the ... |
| volioore 42269 | The measure of an open int... |
| fvvolicof 42270 | The function value of the ... |
| voliooico 42271 | An open interval and a lef... |
| ismbl4 42272 | The predicate " ` A ` is L... |
| volioofmpt 42273 | ` ( ( vol o. (,) ) o. F ) ... |
| volicoff 42274 | ` ( ( vol o. [,) ) o. F ) ... |
| voliooicof 42275 | The Lebesgue measure of op... |
| volicofmpt 42276 | ` ( ( vol o. [,) ) o. F ) ... |
| volicc 42277 | The Lebesgue measure of a ... |
| voliccico 42278 | A closed interval and a le... |
| mbfdmssre 42279 | The domain of a measurable... |
| stoweidlem1 42280 | Lemma for ~ stoweid . Thi... |
| stoweidlem2 42281 | lemma for ~ stoweid : here... |
| stoweidlem3 42282 | Lemma for ~ stoweid : if `... |
| stoweidlem4 42283 | Lemma for ~ stoweid : a cl... |
| stoweidlem5 42284 | There exists a δ as ... |
| stoweidlem6 42285 | Lemma for ~ stoweid : two ... |
| stoweidlem7 42286 | This lemma is used to prov... |
| stoweidlem8 42287 | Lemma for ~ stoweid : two ... |
| stoweidlem9 42288 | Lemma for ~ stoweid : here... |
| stoweidlem10 42289 | Lemma for ~ stoweid . Thi... |
| stoweidlem11 42290 | This lemma is used to prov... |
| stoweidlem12 42291 | Lemma for ~ stoweid . Thi... |
| stoweidlem13 42292 | Lemma for ~ stoweid . Thi... |
| stoweidlem14 42293 | There exists a ` k ` as in... |
| stoweidlem15 42294 | This lemma is used to prov... |
| stoweidlem16 42295 | Lemma for ~ stoweid . The... |
| stoweidlem17 42296 | This lemma proves that the... |
| stoweidlem18 42297 | This theorem proves Lemma ... |
| stoweidlem19 42298 | If a set of real functions... |
| stoweidlem20 42299 | If a set A of real functio... |
| stoweidlem21 42300 | Once the Stone Weierstrass... |
| stoweidlem22 42301 | If a set of real functions... |
| stoweidlem23 42302 | This lemma is used to prov... |
| stoweidlem24 42303 | This lemma proves that for... |
| stoweidlem25 42304 | This lemma proves that for... |
| stoweidlem26 42305 | This lemma is used to prov... |
| stoweidlem27 42306 | This lemma is used to prov... |
| stoweidlem28 42307 | There exists a δ as ... |
| stoweidlem29 42308 | When the hypothesis for th... |
| stoweidlem30 42309 | This lemma is used to prov... |
| stoweidlem31 42310 | This lemma is used to prov... |
| stoweidlem32 42311 | If a set A of real functio... |
| stoweidlem33 42312 | If a set of real functions... |
| stoweidlem34 42313 | This lemma proves that for... |
| stoweidlem35 42314 | This lemma is used to prov... |
| stoweidlem36 42315 | This lemma is used to prov... |
| stoweidlem37 42316 | This lemma is used to prov... |
| stoweidlem38 42317 | This lemma is used to prov... |
| stoweidlem39 42318 | This lemma is used to prov... |
| stoweidlem40 42319 | This lemma proves that q_n... |
| stoweidlem41 42320 | This lemma is used to prov... |
| stoweidlem42 42321 | This lemma is used to prov... |
| stoweidlem43 42322 | This lemma is used to prov... |
| stoweidlem44 42323 | This lemma is used to prov... |
| stoweidlem45 42324 | This lemma proves that, gi... |
| stoweidlem46 42325 | This lemma proves that set... |
| stoweidlem47 42326 | Subtracting a constant fro... |
| stoweidlem48 42327 | This lemma is used to prov... |
| stoweidlem49 42328 | There exists a function q_... |
| stoweidlem50 42329 | This lemma proves that set... |
| stoweidlem51 42330 | There exists a function x ... |
| stoweidlem52 42331 | There exists a neighborood... |
| stoweidlem53 42332 | This lemma is used to prov... |
| stoweidlem54 42333 | There exists a function ` ... |
| stoweidlem55 42334 | This lemma proves the exis... |
| stoweidlem56 42335 | This theorem proves Lemma ... |
| stoweidlem57 42336 | There exists a function x ... |
| stoweidlem58 42337 | This theorem proves Lemma ... |
| stoweidlem59 42338 | This lemma proves that the... |
| stoweidlem60 42339 | This lemma proves that the... |
| stoweidlem61 42340 | This lemma proves that the... |
| stoweidlem62 42341 | This theorem proves the St... |
| stoweid 42342 | This theorem proves the St... |
| stowei 42343 | This theorem proves the St... |
| wallispilem1 42344 | ` I ` is monotone: increas... |
| wallispilem2 42345 | A first set of properties ... |
| wallispilem3 42346 | I maps to real values. (C... |
| wallispilem4 42347 | ` F ` maps to explicit exp... |
| wallispilem5 42348 | The sequence ` H ` converg... |
| wallispi 42349 | Wallis' formula for π :... |
| wallispi2lem1 42350 | An intermediate step betwe... |
| wallispi2lem2 42351 | Two expressions are proven... |
| wallispi2 42352 | An alternative version of ... |
| stirlinglem1 42353 | A simple limit of fraction... |
| stirlinglem2 42354 | ` A ` maps to positive rea... |
| stirlinglem3 42355 | Long but simple algebraic ... |
| stirlinglem4 42356 | Algebraic manipulation of ... |
| stirlinglem5 42357 | If ` T ` is between ` 0 ` ... |
| stirlinglem6 42358 | A series that converges to... |
| stirlinglem7 42359 | Algebraic manipulation of ... |
| stirlinglem8 42360 | If ` A ` converges to ` C ... |
| stirlinglem9 42361 | ` ( ( B `` N ) - ( B `` ( ... |
| stirlinglem10 42362 | A bound for any B(N)-B(N +... |
| stirlinglem11 42363 | ` B ` is decreasing. (Con... |
| stirlinglem12 42364 | The sequence ` B ` is boun... |
| stirlinglem13 42365 | ` B ` is decreasing and ha... |
| stirlinglem14 42366 | The sequence ` A ` converg... |
| stirlinglem15 42367 | The Stirling's formula is ... |
| stirling 42368 | Stirling's approximation f... |
| stirlingr 42369 | Stirling's approximation f... |
| dirkerval 42370 | The N_th Dirichlet Kernel.... |
| dirker2re 42371 | The Dirchlet Kernel value ... |
| dirkerdenne0 42372 | The Dirchlet Kernel denomi... |
| dirkerval2 42373 | The N_th Dirichlet Kernel ... |
| dirkerre 42374 | The Dirichlet Kernel at an... |
| dirkerper 42375 | the Dirichlet Kernel has p... |
| dirkerf 42376 | For any natural number ` N... |
| dirkertrigeqlem1 42377 | Sum of an even number of a... |
| dirkertrigeqlem2 42378 | Trigonomic equality lemma ... |
| dirkertrigeqlem3 42379 | Trigonometric equality lem... |
| dirkertrigeq 42380 | Trigonometric equality for... |
| dirkeritg 42381 | The definite integral of t... |
| dirkercncflem1 42382 | If ` Y ` is a multiple of ... |
| dirkercncflem2 42383 | Lemma used to prove that t... |
| dirkercncflem3 42384 | The Dirichlet Kernel is co... |
| dirkercncflem4 42385 | The Dirichlet Kernel is co... |
| dirkercncf 42386 | For any natural number ` N... |
| fourierdlem1 42387 | A partition interval is a ... |
| fourierdlem2 42388 | Membership in a partition.... |
| fourierdlem3 42389 | Membership in a partition.... |
| fourierdlem4 42390 | ` E ` is a function that m... |
| fourierdlem5 42391 | ` S ` is a function. (Con... |
| fourierdlem6 42392 | ` X ` is in the periodic p... |
| fourierdlem7 42393 | The difference between the... |
| fourierdlem8 42394 | A partition interval is a ... |
| fourierdlem9 42395 | ` H ` is a complex functio... |
| fourierdlem10 42396 | Condition on the bounds of... |
| fourierdlem11 42397 | If there is a partition, t... |
| fourierdlem12 42398 | A point of a partition is ... |
| fourierdlem13 42399 | Value of ` V ` in terms of... |
| fourierdlem14 42400 | Given the partition ` V ` ... |
| fourierdlem15 42401 | The range of the partition... |
| fourierdlem16 42402 | The coefficients of the fo... |
| fourierdlem17 42403 | The defined ` L ` is actua... |
| fourierdlem18 42404 | The function ` S ` is cont... |
| fourierdlem19 42405 | If two elements of ` D ` h... |
| fourierdlem20 42406 | Every interval in the part... |
| fourierdlem21 42407 | The coefficients of the fo... |
| fourierdlem22 42408 | The coefficients of the fo... |
| fourierdlem23 42409 | If ` F ` is continuous and... |
| fourierdlem24 42410 | A sufficient condition for... |
| fourierdlem25 42411 | If ` C ` is not in the ran... |
| fourierdlem26 42412 | Periodic image of a point ... |
| fourierdlem27 42413 | A partition open interval ... |
| fourierdlem28 42414 | Derivative of ` ( F `` ( X... |
| fourierdlem29 42415 | Explicit function value fo... |
| fourierdlem30 42416 | Sum of three small pieces ... |
| fourierdlem31 42417 | If ` A ` is finite and for... |
| fourierdlem32 42418 | Limit of a continuous func... |
| fourierdlem33 42419 | Limit of a continuous func... |
| fourierdlem34 42420 | A partition is one to one.... |
| fourierdlem35 42421 | There is a single point in... |
| fourierdlem36 42422 | ` F ` is an isomorphism. ... |
| fourierdlem37 42423 | ` I ` is a function that m... |
| fourierdlem38 42424 | The function ` F ` is cont... |
| fourierdlem39 42425 | Integration by parts of ... |
| fourierdlem40 42426 | ` H ` is a continuous func... |
| fourierdlem41 42427 | Lemma used to prove that e... |
| fourierdlem42 42428 | The set of points in a mov... |
| fourierdlem43 42429 | ` K ` is a real function. ... |
| fourierdlem44 42430 | A condition for having ` (... |
| fourierdlem46 42431 | The function ` F ` has a l... |
| fourierdlem47 42432 | For ` r ` large enough, th... |
| fourierdlem48 42433 | The given periodic functio... |
| fourierdlem49 42434 | The given periodic functio... |
| fourierdlem50 42435 | Continuity of ` O ` and it... |
| fourierdlem51 42436 | ` X ` is in the periodic p... |
| fourierdlem52 42437 | d16:d17,d18:jca |- ( ph ->... |
| fourierdlem53 42438 | The limit of ` F ( s ) ` a... |
| fourierdlem54 42439 | Given a partition ` Q ` an... |
| fourierdlem55 42440 | ` U ` is a real function. ... |
| fourierdlem56 42441 | Derivative of the ` K ` fu... |
| fourierdlem57 42442 | The derivative of ` O ` . ... |
| fourierdlem58 42443 | The derivative of ` K ` is... |
| fourierdlem59 42444 | The derivative of ` H ` is... |
| fourierdlem60 42445 | Given a differentiable fun... |
| fourierdlem61 42446 | Given a differentiable fun... |
| fourierdlem62 42447 | The function ` K ` is cont... |
| fourierdlem63 42448 | The upper bound of interva... |
| fourierdlem64 42449 | The partition ` V ` is fin... |
| fourierdlem65 42450 | The distance of two adjace... |
| fourierdlem66 42451 | Value of the ` G ` functio... |
| fourierdlem67 42452 | ` G ` is a function. (Con... |
| fourierdlem68 42453 | The derivative of ` O ` is... |
| fourierdlem69 42454 | A piecewise continuous fun... |
| fourierdlem70 42455 | A piecewise continuous fun... |
| fourierdlem71 42456 | A periodic piecewise conti... |
| fourierdlem72 42457 | The derivative of ` O ` is... |
| fourierdlem73 42458 | A version of the Riemann L... |
| fourierdlem74 42459 | Given a piecewise smooth f... |
| fourierdlem75 42460 | Given a piecewise smooth f... |
| fourierdlem76 42461 | Continuity of ` O ` and it... |
| fourierdlem77 42462 | If ` H ` is bounded, then ... |
| fourierdlem78 42463 | ` G ` is continuous when r... |
| fourierdlem79 42464 | ` E ` projects every inter... |
| fourierdlem80 42465 | The derivative of ` O ` is... |
| fourierdlem81 42466 | The integral of a piecewis... |
| fourierdlem82 42467 | Integral by substitution, ... |
| fourierdlem83 42468 | The fourier partial sum fo... |
| fourierdlem84 42469 | If ` F ` is piecewise coni... |
| fourierdlem85 42470 | Limit of the function ` G ... |
| fourierdlem86 42471 | Continuity of ` O ` and it... |
| fourierdlem87 42472 | The integral of ` G ` goes... |
| fourierdlem88 42473 | Given a piecewise continuo... |
| fourierdlem89 42474 | Given a piecewise continuo... |
| fourierdlem90 42475 | Given a piecewise continuo... |
| fourierdlem91 42476 | Given a piecewise continuo... |
| fourierdlem92 42477 | The integral of a piecewis... |
| fourierdlem93 42478 | Integral by substitution (... |
| fourierdlem94 42479 | For a piecewise smooth fun... |
| fourierdlem95 42480 | Algebraic manipulation of ... |
| fourierdlem96 42481 | limit for ` F ` at the low... |
| fourierdlem97 42482 | ` F ` is continuous on the... |
| fourierdlem98 42483 | ` F ` is continuous on the... |
| fourierdlem99 42484 | limit for ` F ` at the upp... |
| fourierdlem100 42485 | A piecewise continuous fun... |
| fourierdlem101 42486 | Integral by substitution f... |
| fourierdlem102 42487 | For a piecewise smooth fun... |
| fourierdlem103 42488 | The half lower part of the... |
| fourierdlem104 42489 | The half upper part of the... |
| fourierdlem105 42490 | A piecewise continuous fun... |
| fourierdlem106 42491 | For a piecewise smooth fun... |
| fourierdlem107 42492 | The integral of a piecewis... |
| fourierdlem108 42493 | The integral of a piecewis... |
| fourierdlem109 42494 | The integral of a piecewis... |
| fourierdlem110 42495 | The integral of a piecewis... |
| fourierdlem111 42496 | The fourier partial sum fo... |
| fourierdlem112 42497 | Here abbreviations (local ... |
| fourierdlem113 42498 | Fourier series convergence... |
| fourierdlem114 42499 | Fourier series convergence... |
| fourierdlem115 42500 | Fourier serier convergence... |
| fourierd 42501 | Fourier series convergence... |
| fourierclimd 42502 | Fourier series convergence... |
| fourierclim 42503 | Fourier series convergence... |
| fourier 42504 | Fourier series convergence... |
| fouriercnp 42505 | If ` F ` is continuous at ... |
| fourier2 42506 | Fourier series convergence... |
| sqwvfoura 42507 | Fourier coefficients for t... |
| sqwvfourb 42508 | Fourier series ` B ` coeff... |
| fourierswlem 42509 | The Fourier series for the... |
| fouriersw 42510 | Fourier series convergence... |
| fouriercn 42511 | If the derivative of ` F `... |
| elaa2lem 42512 | Elementhood in the set of ... |
| elaa2 42513 | Elementhood in the set of ... |
| etransclem1 42514 | ` H ` is a function. (Con... |
| etransclem2 42515 | Derivative of ` G ` . (Co... |
| etransclem3 42516 | The given ` if ` term is a... |
| etransclem4 42517 | ` F ` expressed as a finit... |
| etransclem5 42518 | A change of bound variable... |
| etransclem6 42519 | A change of bound variable... |
| etransclem7 42520 | The given product is an in... |
| etransclem8 42521 | ` F ` is a function. (Con... |
| etransclem9 42522 | If ` K ` divides ` N ` but... |
| etransclem10 42523 | The given ` if ` term is a... |
| etransclem11 42524 | A change of bound variable... |
| etransclem12 42525 | ` C ` applied to ` N ` . ... |
| etransclem13 42526 | ` F ` applied to ` Y ` . ... |
| etransclem14 42527 | Value of the term ` T ` , ... |
| etransclem15 42528 | Value of the term ` T ` , ... |
| etransclem16 42529 | Every element in the range... |
| etransclem17 42530 | The ` N ` -th derivative o... |
| etransclem18 42531 | The given function is inte... |
| etransclem19 42532 | The ` N ` -th derivative o... |
| etransclem20 42533 | ` H ` is smooth. (Contrib... |
| etransclem21 42534 | The ` N ` -th derivative o... |
| etransclem22 42535 | The ` N ` -th derivative o... |
| etransclem23 42536 | This is the claim proof in... |
| etransclem24 42537 | ` P ` divides the I -th de... |
| etransclem25 42538 | ` P ` factorial divides th... |
| etransclem26 42539 | Every term in the sum of t... |
| etransclem27 42540 | The ` N ` -th derivative o... |
| etransclem28 42541 | ` ( P - 1 ) ` factorial di... |
| etransclem29 42542 | The ` N ` -th derivative o... |
| etransclem30 42543 | The ` N ` -th derivative o... |
| etransclem31 42544 | The ` N ` -th derivative o... |
| etransclem32 42545 | This is the proof for the ... |
| etransclem33 42546 | ` F ` is smooth. (Contrib... |
| etransclem34 42547 | The ` N ` -th derivative o... |
| etransclem35 42548 | ` P ` does not divide the ... |
| etransclem36 42549 | The ` N ` -th derivative o... |
| etransclem37 42550 | ` ( P - 1 ) ` factorial di... |
| etransclem38 42551 | ` P ` divides the I -th de... |
| etransclem39 42552 | ` G ` is a function. (Con... |
| etransclem40 42553 | The ` N ` -th derivative o... |
| etransclem41 42554 | ` P ` does not divide the ... |
| etransclem42 42555 | The ` N ` -th derivative o... |
| etransclem43 42556 | ` G ` is a continuous func... |
| etransclem44 42557 | The given finite sum is no... |
| etransclem45 42558 | ` K ` is an integer. (Con... |
| etransclem46 42559 | This is the proof for equa... |
| etransclem47 42560 | ` _e ` is transcendental. ... |
| etransclem48 42561 | ` _e ` is transcendental. ... |
| etransc 42562 | ` _e ` is transcendental. ... |
| rrxtopn 42563 | The topology of the genera... |
| rrxngp 42564 | Generalized Euclidean real... |
| rrxtps 42565 | Generalized Euclidean real... |
| rrxtopnfi 42566 | The topology of the n-dime... |
| rrxtopon 42567 | The topology on generalize... |
| rrxtop 42568 | The topology on generalize... |
| rrndistlt 42569 | Given two points in the sp... |
| rrxtoponfi 42570 | The topology on n-dimensio... |
| rrxunitopnfi 42571 | The base set of the standa... |
| rrxtopn0 42572 | The topology of the zero-d... |
| qndenserrnbllem 42573 | n-dimensional rational num... |
| qndenserrnbl 42574 | n-dimensional rational num... |
| rrxtopn0b 42575 | The topology of the zero-d... |
| qndenserrnopnlem 42576 | n-dimensional rational num... |
| qndenserrnopn 42577 | n-dimensional rational num... |
| qndenserrn 42578 | n-dimensional rational num... |
| rrxsnicc 42579 | A multidimensional singlet... |
| rrnprjdstle 42580 | The distance between two p... |
| rrndsmet 42581 | ` D ` is a metric for the ... |
| rrndsxmet 42582 | ` D ` is an extended metri... |
| ioorrnopnlem 42583 | The a point in an indexed ... |
| ioorrnopn 42584 | The indexed product of ope... |
| ioorrnopnxrlem 42585 | Given a point ` F ` that b... |
| ioorrnopnxr 42586 | The indexed product of ope... |
| issal 42593 | Express the predicate " ` ... |
| pwsal 42594 | The power set of a given s... |
| salunicl 42595 | SAlg sigma-algebra is clos... |
| saluncl 42596 | The union of two sets in a... |
| prsal 42597 | The pair of the empty set ... |
| saldifcl 42598 | The complement of an eleme... |
| 0sal 42599 | The empty set belongs to e... |
| salgenval 42600 | The sigma-algebra generate... |
| saliuncl 42601 | SAlg sigma-algebra is clos... |
| salincl 42602 | The intersection of two se... |
| saluni 42603 | A set is an element of any... |
| saliincl 42604 | SAlg sigma-algebra is clos... |
| saldifcl2 42605 | The difference of two elem... |
| intsaluni 42606 | The union of an arbitrary ... |
| intsal 42607 | The arbitrary intersection... |
| salgenn0 42608 | The set used in the defini... |
| salgencl 42609 | ` SalGen ` actually genera... |
| issald 42610 | Sufficient condition to pr... |
| salexct 42611 | An example of nontrivial s... |
| sssalgen 42612 | A set is a subset of the s... |
| salgenss 42613 | The sigma-algebra generate... |
| salgenuni 42614 | The base set of the sigma-... |
| issalgend 42615 | One side of ~ dfsalgen2 . ... |
| salexct2 42616 | An example of a subset tha... |
| unisalgen 42617 | The union of a set belongs... |
| dfsalgen2 42618 | Alternate characterization... |
| salexct3 42619 | An example of a sigma-alge... |
| salgencntex 42620 | This counterexample shows ... |
| salgensscntex 42621 | This counterexample shows ... |
| issalnnd 42622 | Sufficient condition to pr... |
| dmvolsal 42623 | Lebesgue measurable sets f... |
| saldifcld 42624 | The complement of an eleme... |
| saluncld 42625 | The union of two sets in a... |
| salgencld 42626 | ` SalGen ` actually genera... |
| 0sald 42627 | The empty set belongs to e... |
| iooborel 42628 | An open interval is a Bore... |
| salincld 42629 | The intersection of two se... |
| salunid 42630 | A set is an element of any... |
| unisalgen2 42631 | The union of a set belongs... |
| bor1sal 42632 | The Borel sigma-algebra on... |
| iocborel 42633 | A left-open, right-closed ... |
| subsaliuncllem 42634 | A subspace sigma-algebra i... |
| subsaliuncl 42635 | A subspace sigma-algebra i... |
| subsalsal 42636 | A subspace sigma-algebra i... |
| subsaluni 42637 | A set belongs to the subsp... |
| sge0rnre 42640 | When ` sum^ ` is applied t... |
| fge0icoicc 42641 | If ` F ` maps to nonnegati... |
| sge0val 42642 | The value of the sum of no... |
| fge0npnf 42643 | If ` F ` maps to nonnegati... |
| sge0rnn0 42644 | The range used in the defi... |
| sge0vald 42645 | The value of the sum of no... |
| fge0iccico 42646 | A range of nonnegative ext... |
| gsumge0cl 42647 | Closure of group sum, for ... |
| sge0reval 42648 | Value of the sum of nonneg... |
| sge0pnfval 42649 | If a term in the sum of no... |
| fge0iccre 42650 | A range of nonnegative ext... |
| sge0z 42651 | Any nonnegative extended s... |
| sge00 42652 | The sum of nonnegative ext... |
| fsumlesge0 42653 | Every finite subsum of non... |
| sge0revalmpt 42654 | Value of the sum of nonneg... |
| sge0sn 42655 | A sum of a nonnegative ext... |
| sge0tsms 42656 | ` sum^ ` applied to a nonn... |
| sge0cl 42657 | The arbitrary sum of nonne... |
| sge0f1o 42658 | Re-index a nonnegative ext... |
| sge0snmpt 42659 | A sum of a nonnegative ext... |
| sge0ge0 42660 | The sum of nonnegative ext... |
| sge0xrcl 42661 | The arbitrary sum of nonne... |
| sge0repnf 42662 | The of nonnegative extende... |
| sge0fsum 42663 | The arbitrary sum of a fin... |
| sge0rern 42664 | If the sum of nonnegative ... |
| sge0supre 42665 | If the arbitrary sum of no... |
| sge0fsummpt 42666 | The arbitrary sum of a fin... |
| sge0sup 42667 | The arbitrary sum of nonne... |
| sge0less 42668 | A shorter sum of nonnegati... |
| sge0rnbnd 42669 | The range used in the defi... |
| sge0pr 42670 | Sum of a pair of nonnegati... |
| sge0gerp 42671 | The arbitrary sum of nonne... |
| sge0pnffigt 42672 | If the sum of nonnegative ... |
| sge0ssre 42673 | If a sum of nonnegative ex... |
| sge0lefi 42674 | A sum of nonnegative exten... |
| sge0lessmpt 42675 | A shorter sum of nonnegati... |
| sge0ltfirp 42676 | If the sum of nonnegative ... |
| sge0prle 42677 | The sum of a pair of nonne... |
| sge0gerpmpt 42678 | The arbitrary sum of nonne... |
| sge0resrnlem 42679 | The sum of nonnegative ext... |
| sge0resrn 42680 | The sum of nonnegative ext... |
| sge0ssrempt 42681 | If a sum of nonnegative ex... |
| sge0resplit 42682 | ` sum^ ` splits into two p... |
| sge0le 42683 | If all of the terms of sum... |
| sge0ltfirpmpt 42684 | If the extended sum of non... |
| sge0split 42685 | Split a sum of nonnegative... |
| sge0lempt 42686 | If all of the terms of sum... |
| sge0splitmpt 42687 | Split a sum of nonnegative... |
| sge0ss 42688 | Change the index set to a ... |
| sge0iunmptlemfi 42689 | Sum of nonnegative extende... |
| sge0p1 42690 | The addition of the next t... |
| sge0iunmptlemre 42691 | Sum of nonnegative extende... |
| sge0fodjrnlem 42692 | Re-index a nonnegative ext... |
| sge0fodjrn 42693 | Re-index a nonnegative ext... |
| sge0iunmpt 42694 | Sum of nonnegative extende... |
| sge0iun 42695 | Sum of nonnegative extende... |
| sge0nemnf 42696 | The generalized sum of non... |
| sge0rpcpnf 42697 | The sum of an infinite num... |
| sge0rernmpt 42698 | If the sum of nonnegative ... |
| sge0lefimpt 42699 | A sum of nonnegative exten... |
| nn0ssge0 42700 | Nonnegative integers are n... |
| sge0clmpt 42701 | The generalized sum of non... |
| sge0ltfirpmpt2 42702 | If the extended sum of non... |
| sge0isum 42703 | If a series of nonnegative... |
| sge0xrclmpt 42704 | The generalized sum of non... |
| sge0xp 42705 | Combine two generalized su... |
| sge0isummpt 42706 | If a series of nonnegative... |
| sge0ad2en 42707 | The value of the infinite ... |
| sge0isummpt2 42708 | If a series of nonnegative... |
| sge0xaddlem1 42709 | The extended addition of t... |
| sge0xaddlem2 42710 | The extended addition of t... |
| sge0xadd 42711 | The extended addition of t... |
| sge0fsummptf 42712 | The generalized sum of a f... |
| sge0snmptf 42713 | A sum of a nonnegative ext... |
| sge0ge0mpt 42714 | The sum of nonnegative ext... |
| sge0repnfmpt 42715 | The of nonnegative extende... |
| sge0pnffigtmpt 42716 | If the generalized sum of ... |
| sge0splitsn 42717 | Separate out a term in a g... |
| sge0pnffsumgt 42718 | If the sum of nonnegative ... |
| sge0gtfsumgt 42719 | If the generalized sum of ... |
| sge0uzfsumgt 42720 | If a real number is smalle... |
| sge0pnfmpt 42721 | If a term in the sum of no... |
| sge0seq 42722 | A series of nonnegative re... |
| sge0reuz 42723 | Value of the generalized s... |
| sge0reuzb 42724 | Value of the generalized s... |
| ismea 42727 | Express the predicate " ` ... |
| dmmeasal 42728 | The domain of a measure is... |
| meaf 42729 | A measure is a function th... |
| mea0 42730 | The measure of the empty s... |
| nnfoctbdjlem 42731 | There exists a mapping fro... |
| nnfoctbdj 42732 | There exists a mapping fro... |
| meadjuni 42733 | The measure of the disjoin... |
| meacl 42734 | The measure of a set is a ... |
| iundjiunlem 42735 | The sets in the sequence `... |
| iundjiun 42736 | Given a sequence ` E ` of ... |
| meaxrcl 42737 | The measure of a set is an... |
| meadjun 42738 | The measure of the union o... |
| meassle 42739 | The measure of a set is gr... |
| meaunle 42740 | The measure of the union o... |
| meadjiunlem 42741 | The sum of nonnegative ext... |
| meadjiun 42742 | The measure of the disjoin... |
| ismeannd 42743 | Sufficient condition to pr... |
| meaiunlelem 42744 | The measure of the union o... |
| meaiunle 42745 | The measure of the union o... |
| psmeasurelem 42746 | ` M ` applied to a disjoin... |
| psmeasure 42747 | Point supported measure, R... |
| voliunsge0lem 42748 | The Lebesgue measure funct... |
| voliunsge0 42749 | The Lebesgue measure funct... |
| volmea 42750 | The Lebeasgue measure on t... |
| meage0 42751 | If the measure of a measur... |
| meadjunre 42752 | The measure of the union o... |
| meassre 42753 | If the measure of a measur... |
| meale0eq0 42754 | A measure that is less tha... |
| meadif 42755 | The measure of the differe... |
| meaiuninclem 42756 | Measures are continuous fr... |
| meaiuninc 42757 | Measures are continuous fr... |
| meaiuninc2 42758 | Measures are continuous fr... |
| meaiunincf 42759 | Measures are continuous fr... |
| meaiuninc3v 42760 | Measures are continuous fr... |
| meaiuninc3 42761 | Measures are continuous fr... |
| meaiininclem 42762 | Measures are continuous fr... |
| meaiininc 42763 | Measures are continuous fr... |
| meaiininc2 42764 | Measures are continuous fr... |
| caragenval 42769 | The sigma-algebra generate... |
| isome 42770 | Express the predicate " ` ... |
| caragenel 42771 | Membership in the Caratheo... |
| omef 42772 | An outer measure is a func... |
| ome0 42773 | The outer measure of the e... |
| omessle 42774 | The outer measure of a set... |
| omedm 42775 | The domain of an outer mea... |
| caragensplit 42776 | If ` E ` is in the set gen... |
| caragenelss 42777 | An element of the Caratheo... |
| carageneld 42778 | Membership in the Caratheo... |
| omecl 42779 | The outer measure of a set... |
| caragenss 42780 | The sigma-algebra generate... |
| omeunile 42781 | The outer measure of the u... |
| caragen0 42782 | The empty set belongs to a... |
| omexrcl 42783 | The outer measure of a set... |
| caragenunidm 42784 | The base set of an outer m... |
| caragensspw 42785 | The sigma-algebra generate... |
| omessre 42786 | If the outer measure of a ... |
| caragenuni 42787 | The base set of the sigma-... |
| caragenuncllem 42788 | The Caratheodory's constru... |
| caragenuncl 42789 | The Caratheodory's constru... |
| caragendifcl 42790 | The Caratheodory's constru... |
| caragenfiiuncl 42791 | The Caratheodory's constru... |
| omeunle 42792 | The outer measure of the u... |
| omeiunle 42793 | The outer measure of the i... |
| omelesplit 42794 | The outer measure of a set... |
| omeiunltfirp 42795 | If the outer measure of a ... |
| omeiunlempt 42796 | The outer measure of the i... |
| carageniuncllem1 42797 | The outer measure of ` A i... |
| carageniuncllem2 42798 | The Caratheodory's constru... |
| carageniuncl 42799 | The Caratheodory's constru... |
| caragenunicl 42800 | The Caratheodory's constru... |
| caragensal 42801 | Caratheodory's method gene... |
| caratheodorylem1 42802 | Lemma used to prove that C... |
| caratheodorylem2 42803 | Caratheodory's constructio... |
| caratheodory 42804 | Caratheodory's constructio... |
| 0ome 42805 | The map that assigns 0 to ... |
| isomenndlem 42806 | ` O ` is sub-additive w.r.... |
| isomennd 42807 | Sufficient condition to pr... |
| caragenel2d 42808 | Membership in the Caratheo... |
| omege0 42809 | If the outer measure of a ... |
| omess0 42810 | If the outer measure of a ... |
| caragencmpl 42811 | A measure built with the C... |
| vonval 42816 | Value of the Lebesgue meas... |
| ovnval 42817 | Value of the Lebesgue oute... |
| elhoi 42818 | Membership in a multidimen... |
| icoresmbl 42819 | A closed-below, open-above... |
| hoissre 42820 | The projection of a half-o... |
| ovnval2 42821 | Value of the Lebesgue oute... |
| volicorecl 42822 | The Lebesgue measure of a ... |
| hoiprodcl 42823 | The pre-measure of half-op... |
| hoicvr 42824 | ` I ` is a countable set o... |
| hoissrrn 42825 | A half-open interval is a ... |
| ovn0val 42826 | The Lebesgue outer measure... |
| ovnn0val 42827 | The value of a (multidimen... |
| ovnval2b 42828 | Value of the Lebesgue oute... |
| volicorescl 42829 | The Lebesgue measure of a ... |
| ovnprodcl 42830 | The product used in the de... |
| hoiprodcl2 42831 | The pre-measure of half-op... |
| hoicvrrex 42832 | Any subset of the multidim... |
| ovnsupge0 42833 | The set used in the defini... |
| ovnlecvr 42834 | Given a subset of multidim... |
| ovnpnfelsup 42835 | ` +oo ` is an element of t... |
| ovnsslelem 42836 | The (multidimensional, non... |
| ovnssle 42837 | The (multidimensional) Leb... |
| ovnlerp 42838 | The Lebesgue outer measure... |
| ovnf 42839 | The Lebesgue outer measure... |
| ovncvrrp 42840 | The Lebesgue outer measure... |
| ovn0lem 42841 | For any finite dimension, ... |
| ovn0 42842 | For any finite dimension, ... |
| ovncl 42843 | The Lebesgue outer measure... |
| ovn02 42844 | For the zero-dimensional s... |
| ovnxrcl 42845 | The Lebesgue outer measure... |
| ovnsubaddlem1 42846 | The Lebesgue outer measure... |
| ovnsubaddlem2 42847 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd 42848 | ` ( voln* `` X ) ` is suba... |
| ovnome 42849 | ` ( voln* `` X ) ` is an o... |
| vonmea 42850 | ` ( voln `` X ) ` is a mea... |
| volicon0 42851 | The measure of a nonempty ... |
| hsphoif 42852 | ` H ` is a function (that ... |
| hoidmvval 42853 | The dimensional volume of ... |
| hoissrrn2 42854 | A half-open interval is a ... |
| hsphoival 42855 | ` H ` is a function (that ... |
| hoiprodcl3 42856 | The pre-measure of half-op... |
| volicore 42857 | The Lebesgue measure of a ... |
| hoidmvcl 42858 | The dimensional volume of ... |
| hoidmv0val 42859 | The dimensional volume of ... |
| hoidmvn0val 42860 | The dimensional volume of ... |
| hsphoidmvle2 42861 | The dimensional volume of ... |
| hsphoidmvle 42862 | The dimensional volume of ... |
| hoidmvval0 42863 | The dimensional volume of ... |
| hoiprodp1 42864 | The dimensional volume of ... |
| sge0hsphoire 42865 | If the generalized sum of ... |
| hoidmvval0b 42866 | The dimensional volume of ... |
| hoidmv1lelem1 42867 | The supremum of ` U ` belo... |
| hoidmv1lelem2 42868 | This is the contradiction ... |
| hoidmv1lelem3 42869 | The dimensional volume of ... |
| hoidmv1le 42870 | The dimensional volume of ... |
| hoidmvlelem1 42871 | The supremum of ` U ` belo... |
| hoidmvlelem2 42872 | This is the contradiction ... |
| hoidmvlelem3 42873 | This is the contradiction ... |
| hoidmvlelem4 42874 | The dimensional volume of ... |
| hoidmvlelem5 42875 | The dimensional volume of ... |
| hoidmvle 42876 | The dimensional volume of ... |
| ovnhoilem1 42877 | The Lebesgue outer measure... |
| ovnhoilem2 42878 | The Lebesgue outer measure... |
| ovnhoi 42879 | The Lebesgue outer measure... |
| dmovn 42880 | The domain of the Lebesgue... |
| hoicoto2 42881 | The half-open interval exp... |
| dmvon 42882 | Lebesgue measurable n-dime... |
| hoi2toco 42883 | The half-open interval exp... |
| hoidifhspval 42884 | ` D ` is a function that r... |
| hspval 42885 | The value of the half-spac... |
| ovnlecvr2 42886 | Given a subset of multidim... |
| ovncvr2 42887 | ` B ` and ` T ` are the le... |
| dmovnsal 42888 | The domain of the Lebesgue... |
| unidmovn 42889 | Base set of the n-dimensio... |
| rrnmbl 42890 | The set of n-dimensional R... |
| hoidifhspval2 42891 | ` D ` is a function that r... |
| hspdifhsp 42892 | A n-dimensional half-open ... |
| unidmvon 42893 | Base set of the n-dimensio... |
| hoidifhspf 42894 | ` D ` is a function that r... |
| hoidifhspval3 42895 | ` D ` is a function that r... |
| hoidifhspdmvle 42896 | The dimensional volume of ... |
| voncmpl 42897 | The Lebesgue measure is co... |
| hoiqssbllem1 42898 | The center of the n-dimens... |
| hoiqssbllem2 42899 | The center of the n-dimens... |
| hoiqssbllem3 42900 | A n-dimensional ball conta... |
| hoiqssbl 42901 | A n-dimensional ball conta... |
| hspmbllem1 42902 | Any half-space of the n-di... |
| hspmbllem2 42903 | Any half-space of the n-di... |
| hspmbllem3 42904 | Any half-space of the n-di... |
| hspmbl 42905 | Any half-space of the n-di... |
| hoimbllem 42906 | Any n-dimensional half-ope... |
| hoimbl 42907 | Any n-dimensional half-ope... |
| opnvonmbllem1 42908 | The half-open interval exp... |
| opnvonmbllem2 42909 | An open subset of the n-di... |
| opnvonmbl 42910 | An open subset of the n-di... |
| opnssborel 42911 | Open sets of a generalized... |
| borelmbl 42912 | All Borel subsets of the n... |
| volicorege0 42913 | The Lebesgue measure of a ... |
| isvonmbl 42914 | The predicate " ` A ` is m... |
| mblvon 42915 | The n-dimensional Lebesgue... |
| vonmblss 42916 | n-dimensional Lebesgue mea... |
| volico2 42917 | The measure of left-closed... |
| vonmblss2 42918 | n-dimensional Lebesgue mea... |
| ovolval2lem 42919 | The value of the Lebesgue ... |
| ovolval2 42920 | The value of the Lebesgue ... |
| ovnsubadd2lem 42921 | ` ( voln* `` X ) ` is suba... |
| ovnsubadd2 42922 | ` ( voln* `` X ) ` is suba... |
| ovolval3 42923 | The value of the Lebesgue ... |
| ovnsplit 42924 | The n-dimensional Lebesgue... |
| ovolval4lem1 42925 | |- ( ( ph /\ n e. A ) -> ... |
| ovolval4lem2 42926 | The value of the Lebesgue ... |
| ovolval4 42927 | The value of the Lebesgue ... |
| ovolval5lem1 42928 | ` |- ( ph -> ( sum^ `` ( n... |
| ovolval5lem2 42929 | |- ( ( ph /\ n e. NN ) ->... |
| ovolval5lem3 42930 | The value of the Lebesgue ... |
| ovolval5 42931 | The value of the Lebesgue ... |
| ovnovollem1 42932 | if ` F ` is a cover of ` B... |
| ovnovollem2 42933 | if ` I ` is a cover of ` (... |
| ovnovollem3 42934 | The 1-dimensional Lebesgue... |
| ovnovol 42935 | The 1-dimensional Lebesgue... |
| vonvolmbllem 42936 | If a subset ` B ` of real ... |
| vonvolmbl 42937 | A subset of Real numbers i... |
| vonvol 42938 | The 1-dimensional Lebesgue... |
| vonvolmbl2 42939 | A subset ` X ` of the spac... |
| vonvol2 42940 | The 1-dimensional Lebesgue... |
| hoimbl2 42941 | Any n-dimensional half-ope... |
| voncl 42942 | The Lebesgue measure of a ... |
| vonhoi 42943 | The Lebesgue outer measure... |
| vonxrcl 42944 | The Lebesgue measure of a ... |
| ioosshoi 42945 | A n-dimensional open inter... |
| vonn0hoi 42946 | The Lebesgue outer measure... |
| von0val 42947 | The Lebesgue measure (for ... |
| vonhoire 42948 | The Lebesgue measure of a ... |
| iinhoiicclem 42949 | A n-dimensional closed int... |
| iinhoiicc 42950 | A n-dimensional closed int... |
| iunhoiioolem 42951 | A n-dimensional open inter... |
| iunhoiioo 42952 | A n-dimensional open inter... |
| ioovonmbl 42953 | Any n-dimensional open int... |
| iccvonmbllem 42954 | Any n-dimensional closed i... |
| iccvonmbl 42955 | Any n-dimensional closed i... |
| vonioolem1 42956 | The sequence of the measur... |
| vonioolem2 42957 | The n-dimensional Lebesgue... |
| vonioo 42958 | The n-dimensional Lebesgue... |
| vonicclem1 42959 | The sequence of the measur... |
| vonicclem2 42960 | The n-dimensional Lebesgue... |
| vonicc 42961 | The n-dimensional Lebesgue... |
| snvonmbl 42962 | A n-dimensional singleton ... |
| vonn0ioo 42963 | The n-dimensional Lebesgue... |
| vonn0icc 42964 | The n-dimensional Lebesgue... |
| ctvonmbl 42965 | Any n-dimensional countabl... |
| vonn0ioo2 42966 | The n-dimensional Lebesgue... |
| vonsn 42967 | The n-dimensional Lebesgue... |
| vonn0icc2 42968 | The n-dimensional Lebesgue... |
| vonct 42969 | The n-dimensional Lebesgue... |
| vitali2 42970 | There are non-measurable s... |
| pimltmnf2 42973 | Given a real-valued functi... |
| preimagelt 42974 | The preimage of a right-op... |
| preimalegt 42975 | The preimage of a left-ope... |
| pimconstlt0 42976 | Given a constant function,... |
| pimconstlt1 42977 | Given a constant function,... |
| pimltpnf 42978 | Given a real-valued functi... |
| pimgtpnf2 42979 | Given a real-valued functi... |
| salpreimagelt 42980 | If all the preimages of le... |
| pimrecltpos 42981 | The preimage of an unbound... |
| salpreimalegt 42982 | If all the preimages of ri... |
| pimiooltgt 42983 | The preimage of an open in... |
| preimaicomnf 42984 | Preimage of an open interv... |
| pimltpnf2 42985 | Given a real-valued functi... |
| pimgtmnf2 42986 | Given a real-valued functi... |
| pimdecfgtioc 42987 | Given a nonincreasing func... |
| pimincfltioc 42988 | Given a nondecreasing func... |
| pimdecfgtioo 42989 | Given a nondecreasing func... |
| pimincfltioo 42990 | Given a nondecreasing func... |
| preimaioomnf 42991 | Preimage of an open interv... |
| preimageiingt 42992 | A preimage of a left-close... |
| preimaleiinlt 42993 | A preimage of a left-open,... |
| pimgtmnf 42994 | Given a real-valued functi... |
| pimrecltneg 42995 | The preimage of an unbound... |
| salpreimagtge 42996 | If all the preimages of le... |
| salpreimaltle 42997 | If all the preimages of ri... |
| issmflem 42998 | The predicate " ` F ` is a... |
| issmf 42999 | The predicate " ` F ` is a... |
| salpreimalelt 43000 | If all the preimages of ri... |
| salpreimagtlt 43001 | If all the preimages of le... |
| smfpreimalt 43002 | Given a function measurabl... |
| smff 43003 | A function measurable w.r.... |
| smfdmss 43004 | The domain of a function m... |
| issmff 43005 | The predicate " ` F ` is a... |
| issmfd 43006 | A sufficient condition for... |
| smfpreimaltf 43007 | Given a function measurabl... |
| issmfdf 43008 | A sufficient condition for... |
| sssmf 43009 | The restriction of a sigma... |
| mbfresmf 43010 | A real-valued measurable f... |
| cnfsmf 43011 | A continuous function is m... |
| incsmflem 43012 | A nondecreasing function i... |
| incsmf 43013 | A real-valued, nondecreasi... |
| smfsssmf 43014 | If a function is measurabl... |
| issmflelem 43015 | The predicate " ` F ` is a... |
| issmfle 43016 | The predicate " ` F ` is a... |
| smfpimltmpt 43017 | Given a function measurabl... |
| smfpimltxr 43018 | Given a function measurabl... |
| issmfdmpt 43019 | A sufficient condition for... |
| smfconst 43020 | Given a sigma-algebra over... |
| sssmfmpt 43021 | The restriction of a sigma... |
| cnfrrnsmf 43022 | A function, continuous fro... |
| smfid 43023 | The identity function is B... |
| bormflebmf 43024 | A Borel measurable functio... |
| smfpreimale 43025 | Given a function measurabl... |
| issmfgtlem 43026 | The predicate " ` F ` is a... |
| issmfgt 43027 | The predicate " ` F ` is a... |
| issmfled 43028 | A sufficient condition for... |
| smfpimltxrmpt 43029 | Given a function measurabl... |
| smfmbfcex 43030 | A constant function, with ... |
| issmfgtd 43031 | A sufficient condition for... |
| smfpreimagt 43032 | Given a function measurabl... |
| smfaddlem1 43033 | Given the sum of two funct... |
| smfaddlem2 43034 | The sum of two sigma-measu... |
| smfadd 43035 | The sum of two sigma-measu... |
| decsmflem 43036 | A nonincreasing function i... |
| decsmf 43037 | A real-valued, nonincreasi... |
| smfpreimagtf 43038 | Given a function measurabl... |
| issmfgelem 43039 | The predicate " ` F ` is a... |
| issmfge 43040 | The predicate " ` F ` is a... |
| smflimlem1 43041 | Lemma for the proof that t... |
| smflimlem2 43042 | Lemma for the proof that t... |
| smflimlem3 43043 | The limit of sigma-measura... |
| smflimlem4 43044 | Lemma for the proof that t... |
| smflimlem5 43045 | Lemma for the proof that t... |
| smflimlem6 43046 | Lemma for the proof that t... |
| smflim 43047 | The limit of sigma-measura... |
| nsssmfmbflem 43048 | The sigma-measurable funct... |
| nsssmfmbf 43049 | The sigma-measurable funct... |
| smfpimgtxr 43050 | Given a function measurabl... |
| smfpimgtmpt 43051 | Given a function measurabl... |
| smfpreimage 43052 | Given a function measurabl... |
| mbfpsssmf 43053 | Real-valued measurable fun... |
| smfpimgtxrmpt 43054 | Given a function measurabl... |
| smfpimioompt 43055 | Given a function measurabl... |
| smfpimioo 43056 | Given a function measurabl... |
| smfresal 43057 | Given a sigma-measurable f... |
| smfrec 43058 | The reciprocal of a sigma-... |
| smfres 43059 | The restriction of sigma-m... |
| smfmullem1 43060 | The multiplication of two ... |
| smfmullem2 43061 | The multiplication of two ... |
| smfmullem3 43062 | The multiplication of two ... |
| smfmullem4 43063 | The multiplication of two ... |
| smfmul 43064 | The multiplication of two ... |
| smfmulc1 43065 | A sigma-measurable functio... |
| smfdiv 43066 | The fraction of two sigma-... |
| smfpimbor1lem1 43067 | Every open set belongs to ... |
| smfpimbor1lem2 43068 | Given a sigma-measurable f... |
| smfpimbor1 43069 | Given a sigma-measurable f... |
| smf2id 43070 | Twice the identity functio... |
| smfco 43071 | The composition of a Borel... |
| smfneg 43072 | The negative of a sigma-me... |
| smffmpt 43073 | A function measurable w.r.... |
| smflim2 43074 | The limit of a sequence of... |
| smfpimcclem 43075 | Lemma for ~ smfpimcc given... |
| smfpimcc 43076 | Given a countable set of s... |
| issmfle2d 43077 | A sufficient condition for... |
| smflimmpt 43078 | The limit of a sequence of... |
| smfsuplem1 43079 | The supremum of a countabl... |
| smfsuplem2 43080 | The supremum of a countabl... |
| smfsuplem3 43081 | The supremum of a countabl... |
| smfsup 43082 | The supremum of a countabl... |
| smfsupmpt 43083 | The supremum of a countabl... |
| smfsupxr 43084 | The supremum of a countabl... |
| smfinflem 43085 | The infimum of a countable... |
| smfinf 43086 | The infimum of a countable... |
| smfinfmpt 43087 | The infimum of a countable... |
| smflimsuplem1 43088 | If ` H ` converges, the ` ... |
| smflimsuplem2 43089 | The superior limit of a se... |
| smflimsuplem3 43090 | The limit of the ` ( H `` ... |
| smflimsuplem4 43091 | If ` H ` converges, the ` ... |
| smflimsuplem5 43092 | ` H ` converges to the sup... |
| smflimsuplem6 43093 | The superior limit of a se... |
| smflimsuplem7 43094 | The superior limit of a se... |
| smflimsuplem8 43095 | The superior limit of a se... |
| smflimsup 43096 | The superior limit of a se... |
| smflimsupmpt 43097 | The superior limit of a se... |
| smfliminflem 43098 | The inferior limit of a co... |
| smfliminf 43099 | The inferior limit of a co... |
| smfliminfmpt 43100 | The inferior limit of a co... |
| sigarval 43101 | Define the signed area by ... |
| sigarim 43102 | Signed area takes value in... |
| sigarac 43103 | Signed area is anticommuta... |
| sigaraf 43104 | Signed area is additive by... |
| sigarmf 43105 | Signed area is additive (w... |
| sigaras 43106 | Signed area is additive by... |
| sigarms 43107 | Signed area is additive (w... |
| sigarls 43108 | Signed area is linear by t... |
| sigarid 43109 | Signed area of a flat para... |
| sigarexp 43110 | Expand the signed area for... |
| sigarperm 43111 | Signed area ` ( A - C ) G ... |
| sigardiv 43112 | If signed area between vec... |
| sigarimcd 43113 | Signed area takes value in... |
| sigariz 43114 | If signed area is zero, th... |
| sigarcol 43115 | Given three points ` A ` ,... |
| sharhght 43116 | Let ` A B C ` be a triangl... |
| sigaradd 43117 | Subtracting (double) area ... |
| cevathlem1 43118 | Ceva's theorem first lemma... |
| cevathlem2 43119 | Ceva's theorem second lemm... |
| cevath 43120 | Ceva's theorem. Let ` A B... |
| simpcntrab 43121 | The center of a simple gro... |
| hirstL-ax3 43122 | The third axiom of a syste... |
| ax3h 43123 | Recover ~ ax-3 from ~ hirs... |
| aibandbiaiffaiffb 43124 | A closed form showing (a i... |
| aibandbiaiaiffb 43125 | A closed form showing (a i... |
| notatnand 43126 | Do not use. Use intnanr i... |
| aistia 43127 | Given a is equivalent to `... |
| aisfina 43128 | Given a is equivalent to `... |
| bothtbothsame 43129 | Given both a, b are equiva... |
| bothfbothsame 43130 | Given both a, b are equiva... |
| aiffbbtat 43131 | Given a is equivalent to b... |
| aisbbisfaisf 43132 | Given a is equivalent to b... |
| axorbtnotaiffb 43133 | Given a is exclusive to b,... |
| aiffnbandciffatnotciffb 43134 | Given a is equivalent to (... |
| axorbciffatcxorb 43135 | Given a is equivalent to (... |
| aibnbna 43136 | Given a implies b, (not b)... |
| aibnbaif 43137 | Given a implies b, not b, ... |
| aiffbtbat 43138 | Given a is equivalent to b... |
| astbstanbst 43139 | Given a is equivalent to T... |
| aistbistaandb 43140 | Given a is equivalent to T... |
| aisbnaxb 43141 | Given a is equivalent to b... |
| atbiffatnnb 43142 | If a implies b, then a imp... |
| bisaiaisb 43143 | Application of bicom1 with... |
| atbiffatnnbalt 43144 | If a implies b, then a imp... |
| abnotbtaxb 43145 | Assuming a, not b, there e... |
| abnotataxb 43146 | Assuming not a, b, there e... |
| conimpf 43147 | Assuming a, not b, and a i... |
| conimpfalt 43148 | Assuming a, not b, and a i... |
| aistbisfiaxb 43149 | Given a is equivalent to T... |
| aisfbistiaxb 43150 | Given a is equivalent to F... |
| aifftbifffaibif 43151 | Given a is equivalent to T... |
| aifftbifffaibifff 43152 | Given a is equivalent to T... |
| atnaiana 43153 | Given a, it is not the cas... |
| ainaiaandna 43154 | Given a, a implies it is n... |
| abcdta 43155 | Given (((a and b) and c) a... |
| abcdtb 43156 | Given (((a and b) and c) a... |
| abcdtc 43157 | Given (((a and b) and c) a... |
| abcdtd 43158 | Given (((a and b) and c) a... |
| abciffcbatnabciffncba 43159 | Operands in a biconditiona... |
| abciffcbatnabciffncbai 43160 | Operands in a biconditiona... |
| nabctnabc 43161 | not ( a -> ( b /\ c ) ) we... |
| jabtaib 43162 | For when pm3.4 lacks a pm3... |
| onenotinotbothi 43163 | From one negated implicati... |
| twonotinotbothi 43164 | From these two negated imp... |
| clifte 43165 | show d is the same as an i... |
| cliftet 43166 | show d is the same as an i... |
| clifteta 43167 | show d is the same as an i... |
| cliftetb 43168 | show d is the same as an i... |
| confun 43169 | Given the hypotheses there... |
| confun2 43170 | Confun simplified to two p... |
| confun3 43171 | Confun's more complex form... |
| confun4 43172 | An attempt at derivative. ... |
| confun5 43173 | An attempt at derivative. ... |
| plcofph 43174 | Given, a,b and a "definiti... |
| pldofph 43175 | Given, a,b c, d, "definiti... |
| plvcofph 43176 | Given, a,b,d, and "definit... |
| plvcofphax 43177 | Given, a,b,d, and "definit... |
| plvofpos 43178 | rh is derivable because ON... |
| mdandyv0 43179 | Given the equivalences set... |
| mdandyv1 43180 | Given the equivalences set... |
| mdandyv2 43181 | Given the equivalences set... |
| mdandyv3 43182 | Given the equivalences set... |
| mdandyv4 43183 | Given the equivalences set... |
| mdandyv5 43184 | Given the equivalences set... |
| mdandyv6 43185 | Given the equivalences set... |
| mdandyv7 43186 | Given the equivalences set... |
| mdandyv8 43187 | Given the equivalences set... |
| mdandyv9 43188 | Given the equivalences set... |
| mdandyv10 43189 | Given the equivalences set... |
| mdandyv11 43190 | Given the equivalences set... |
| mdandyv12 43191 | Given the equivalences set... |
| mdandyv13 43192 | Given the equivalences set... |
| mdandyv14 43193 | Given the equivalences set... |
| mdandyv15 43194 | Given the equivalences set... |
| mdandyvr0 43195 | Given the equivalences set... |
| mdandyvr1 43196 | Given the equivalences set... |
| mdandyvr2 43197 | Given the equivalences set... |
| mdandyvr3 43198 | Given the equivalences set... |
| mdandyvr4 43199 | Given the equivalences set... |
| mdandyvr5 43200 | Given the equivalences set... |
| mdandyvr6 43201 | Given the equivalences set... |
| mdandyvr7 43202 | Given the equivalences set... |
| mdandyvr8 43203 | Given the equivalences set... |
| mdandyvr9 43204 | Given the equivalences set... |
| mdandyvr10 43205 | Given the equivalences set... |
| mdandyvr11 43206 | Given the equivalences set... |
| mdandyvr12 43207 | Given the equivalences set... |
| mdandyvr13 43208 | Given the equivalences set... |
| mdandyvr14 43209 | Given the equivalences set... |
| mdandyvr15 43210 | Given the equivalences set... |
| mdandyvrx0 43211 | Given the exclusivities se... |
| mdandyvrx1 43212 | Given the exclusivities se... |
| mdandyvrx2 43213 | Given the exclusivities se... |
| mdandyvrx3 43214 | Given the exclusivities se... |
| mdandyvrx4 43215 | Given the exclusivities se... |
| mdandyvrx5 43216 | Given the exclusivities se... |
| mdandyvrx6 43217 | Given the exclusivities se... |
| mdandyvrx7 43218 | Given the exclusivities se... |
| mdandyvrx8 43219 | Given the exclusivities se... |
| mdandyvrx9 43220 | Given the exclusivities se... |
| mdandyvrx10 43221 | Given the exclusivities se... |
| mdandyvrx11 43222 | Given the exclusivities se... |
| mdandyvrx12 43223 | Given the exclusivities se... |
| mdandyvrx13 43224 | Given the exclusivities se... |
| mdandyvrx14 43225 | Given the exclusivities se... |
| mdandyvrx15 43226 | Given the exclusivities se... |
| H15NH16TH15IH16 43227 | Given 15 hypotheses and a ... |
| dandysum2p2e4 43228 | CONTRADICTION PRO... |
| mdandysum2p2e4 43229 | CONTRADICTION PROVED AT 1 ... |
| adh-jarrsc 43230 | Replacement of a nested an... |
| adh-minim 43231 | A single axiom for minimal... |
| adh-minim-ax1-ax2-lem1 43232 | First lemma for the deriva... |
| adh-minim-ax1-ax2-lem2 43233 | Second lemma for the deriv... |
| adh-minim-ax1-ax2-lem3 43234 | Third lemma for the deriva... |
| adh-minim-ax1-ax2-lem4 43235 | Fourth lemma for the deriv... |
| adh-minim-ax1 43236 | Derivation of ~ ax-1 from ... |
| adh-minim-ax2-lem5 43237 | Fifth lemma for the deriva... |
| adh-minim-ax2-lem6 43238 | Sixth lemma for the deriva... |
| adh-minim-ax2c 43239 | Derivation of a commuted f... |
| adh-minim-ax2 43240 | Derivation of ~ ax-2 from ... |
| adh-minim-idALT 43241 | Derivation of ~ id (reflex... |
| adh-minim-pm2.43 43242 | Derivation of ~ pm2.43 Whi... |
| adh-minimp 43243 | Another single axiom for m... |
| adh-minimp-jarr-imim1-ax2c-lem1 43244 | First lemma for the deriva... |
| adh-minimp-jarr-lem2 43245 | Second lemma for the deriv... |
| adh-minimp-jarr-ax2c-lem3 43246 | Third lemma for the deriva... |
| adh-minimp-sylsimp 43247 | Derivation of ~ jarr (also... |
| adh-minimp-ax1 43248 | Derivation of ~ ax-1 from ... |
| adh-minimp-imim1 43249 | Derivation of ~ imim1 ("le... |
| adh-minimp-ax2c 43250 | Derivation of a commuted f... |
| adh-minimp-ax2-lem4 43251 | Fourth lemma for the deriv... |
| adh-minimp-ax2 43252 | Derivation of ~ ax-2 from ... |
| adh-minimp-idALT 43253 | Derivation of ~ id (reflex... |
| adh-minimp-pm2.43 43254 | Derivation of ~ pm2.43 Whi... |
| eusnsn 43255 | There is a unique element ... |
| absnsb 43256 | If the class abstraction `... |
| euabsneu 43257 | Another way to express exi... |
| elprneb 43258 | An element of a proper uno... |
| oppr 43259 | Equality for ordered pairs... |
| opprb 43260 | Equality for unordered pai... |
| or2expropbilem1 43261 | Lemma 1 for ~ or2expropbi ... |
| or2expropbilem2 43262 | Lemma 2 for ~ or2expropbi ... |
| or2expropbi 43263 | If two classes are strictl... |
| eubrv 43264 | If there is a unique set w... |
| eubrdm 43265 | If there is a unique set w... |
| eldmressn 43266 | Element of the domain of a... |
| iota0def 43267 | Example for a defined iota... |
| iota0ndef 43268 | Example for an undefined i... |
| fveqvfvv 43269 | If a function's value at a... |
| fnresfnco 43270 | Composition of two functio... |
| funcoressn 43271 | A composition restricted t... |
| funressnfv 43272 | A restriction to a singlet... |
| funressndmfvrn 43273 | The value of a function ` ... |
| funressnvmo 43274 | A function restricted to a... |
| funressnmo 43275 | A function restricted to a... |
| funressneu 43276 | There is exactly one value... |
| aiotajust 43278 | Soundness justification th... |
| dfaiota2 43280 | Alternate definition of th... |
| reuabaiotaiota 43281 | The iota and the alternate... |
| reuaiotaiota 43282 | The iota and the alternate... |
| aiotaexb 43283 | The alternate iota over a ... |
| aiotavb 43284 | The alternate iota over a ... |
| iotan0aiotaex 43285 | If the iota over a wff ` p... |
| aiotaexaiotaiota 43286 | The alternate iota over a ... |
| aiotaval 43287 | Theorem 8.19 in [Quine] p.... |
| aiota0def 43288 | Example for a defined alte... |
| aiota0ndef 43289 | Example for an undefined a... |
| r19.32 43290 | Theorem 19.32 of [Margaris... |
| rexsb 43291 | An equivalent expression f... |
| rexrsb 43292 | An equivalent expression f... |
| 2rexsb 43293 | An equivalent expression f... |
| 2rexrsb 43294 | An equivalent expression f... |
| cbvral2 43295 | Change bound variables of ... |
| cbvrex2 43296 | Change bound variables of ... |
| 2ralbiim 43297 | Split a biconditional and ... |
| ralndv1 43298 | Example for a theorem abou... |
| ralndv2 43299 | Second example for a theor... |
| reuf1odnf 43300 | There is exactly one eleme... |
| reuf1od 43301 | There is exactly one eleme... |
| euoreqb 43302 | There is a set which is eq... |
| 2reu3 43303 | Double restricted existent... |
| 2reu7 43304 | Two equivalent expressions... |
| 2reu8 43305 | Two equivalent expressions... |
| 2reu8i 43306 | Implication of a double re... |
| 2reuimp0 43307 | Implication of a double re... |
| 2reuimp 43308 | Implication of a double re... |
| ralbinrald 43315 | Elemination of a restricte... |
| nvelim 43316 | If a class is the universa... |
| alneu 43317 | If a statement holds for a... |
| eu2ndop1stv 43318 | If there is a unique secon... |
| dfateq12d 43319 | Equality deduction for "de... |
| nfdfat 43320 | Bound-variable hypothesis ... |
| dfdfat2 43321 | Alternate definition of th... |
| fundmdfat 43322 | A function is defined at a... |
| dfatprc 43323 | A function is not defined ... |
| dfatelrn 43324 | The value of a function ` ... |
| dfafv2 43325 | Alternative definition of ... |
| afveq12d 43326 | Equality deduction for fun... |
| afveq1 43327 | Equality theorem for funct... |
| afveq2 43328 | Equality theorem for funct... |
| nfafv 43329 | Bound-variable hypothesis ... |
| csbafv12g 43330 | Move class substitution in... |
| afvfundmfveq 43331 | If a class is a function r... |
| afvnfundmuv 43332 | If a set is not in the dom... |
| ndmafv 43333 | The value of a class outsi... |
| afvvdm 43334 | If the function value of a... |
| nfunsnafv 43335 | If the restriction of a cl... |
| afvvfunressn 43336 | If the function value of a... |
| afvprc 43337 | A function's value at a pr... |
| afvvv 43338 | If a function's value at a... |
| afvpcfv0 43339 | If the value of the altern... |
| afvnufveq 43340 | The value of the alternati... |
| afvvfveq 43341 | The value of the alternati... |
| afv0fv0 43342 | If the value of the altern... |
| afvfvn0fveq 43343 | If the function's value at... |
| afv0nbfvbi 43344 | The function's value at an... |
| afvfv0bi 43345 | The function's value at an... |
| afveu 43346 | The value of a function at... |
| fnbrafvb 43347 | Equivalence of function va... |
| fnopafvb 43348 | Equivalence of function va... |
| funbrafvb 43349 | Equivalence of function va... |
| funopafvb 43350 | Equivalence of function va... |
| funbrafv 43351 | The second argument of a b... |
| funbrafv2b 43352 | Function value in terms of... |
| dfafn5a 43353 | Representation of a functi... |
| dfafn5b 43354 | Representation of a functi... |
| fnrnafv 43355 | The range of a function ex... |
| afvelrnb 43356 | A member of a function's r... |
| afvelrnb0 43357 | A member of a function's r... |
| dfaimafn 43358 | Alternate definition of th... |
| dfaimafn2 43359 | Alternate definition of th... |
| afvelima 43360 | Function value in an image... |
| afvelrn 43361 | A function's value belongs... |
| fnafvelrn 43362 | A function's value belongs... |
| fafvelrn 43363 | A function's value belongs... |
| ffnafv 43364 | A function maps to a class... |
| afvres 43365 | The value of a restricted ... |
| tz6.12-afv 43366 | Function value. Theorem 6... |
| tz6.12-1-afv 43367 | Function value (Theorem 6.... |
| dmfcoafv 43368 | Domains of a function comp... |
| afvco2 43369 | Value of a function compos... |
| rlimdmafv 43370 | Two ways to express that a... |
| aoveq123d 43371 | Equality deduction for ope... |
| nfaov 43372 | Bound-variable hypothesis ... |
| csbaovg 43373 | Move class substitution in... |
| aovfundmoveq 43374 | If a class is a function r... |
| aovnfundmuv 43375 | If an ordered pair is not ... |
| ndmaov 43376 | The value of an operation ... |
| ndmaovg 43377 | The value of an operation ... |
| aovvdm 43378 | If the operation value of ... |
| nfunsnaov 43379 | If the restriction of a cl... |
| aovvfunressn 43380 | If the operation value of ... |
| aovprc 43381 | The value of an operation ... |
| aovrcl 43382 | Reverse closure for an ope... |
| aovpcov0 43383 | If the alternative value o... |
| aovnuoveq 43384 | The alternative value of t... |
| aovvoveq 43385 | The alternative value of t... |
| aov0ov0 43386 | If the alternative value o... |
| aovovn0oveq 43387 | If the operation's value a... |
| aov0nbovbi 43388 | The operation's value on a... |
| aovov0bi 43389 | The operation's value on a... |
| rspceaov 43390 | A frequently used special ... |
| fnotaovb 43391 | Equivalence of operation v... |
| ffnaov 43392 | An operation maps to a cla... |
| faovcl 43393 | Closure law for an operati... |
| aovmpt4g 43394 | Value of a function given ... |
| aoprssdm 43395 | Domain of closure of an op... |
| ndmaovcl 43396 | The "closure" of an operat... |
| ndmaovrcl 43397 | Reverse closure law, in co... |
| ndmaovcom 43398 | Any operation is commutati... |
| ndmaovass 43399 | Any operation is associati... |
| ndmaovdistr 43400 | Any operation is distribut... |
| dfatafv2iota 43403 | If a function is defined a... |
| ndfatafv2 43404 | The alternate function val... |
| ndfatafv2undef 43405 | The alternate function val... |
| dfatafv2ex 43406 | The alternate function val... |
| afv2ex 43407 | The alternate function val... |
| afv2eq12d 43408 | Equality deduction for fun... |
| afv2eq1 43409 | Equality theorem for funct... |
| afv2eq2 43410 | Equality theorem for funct... |
| nfafv2 43411 | Bound-variable hypothesis ... |
| csbafv212g 43412 | Move class substitution in... |
| fexafv2ex 43413 | The alternate function val... |
| ndfatafv2nrn 43414 | The alternate function val... |
| ndmafv2nrn 43415 | The value of a class outsi... |
| funressndmafv2rn 43416 | The alternate function val... |
| afv2ndefb 43417 | Two ways to say that an al... |
| nfunsnafv2 43418 | If the restriction of a cl... |
| afv2prc 43419 | A function's value at a pr... |
| dfatafv2rnb 43420 | The alternate function val... |
| afv2orxorb 43421 | If a set is in the range o... |
| dmafv2rnb 43422 | The alternate function val... |
| fundmafv2rnb 43423 | The alternate function val... |
| afv2elrn 43424 | An alternate function valu... |
| afv20defat 43425 | If the alternate function ... |
| fnafv2elrn 43426 | An alternate function valu... |
| fafv2elrn 43427 | An alternate function valu... |
| fafv2elrnb 43428 | An alternate function valu... |
| frnvafv2v 43429 | If the codomain of a funct... |
| tz6.12-2-afv2 43430 | Function value when ` F ` ... |
| afv2eu 43431 | The value of a function at... |
| afv2res 43432 | The value of a restricted ... |
| tz6.12-afv2 43433 | Function value (Theorem 6.... |
| tz6.12-1-afv2 43434 | Function value (Theorem 6.... |
| tz6.12c-afv2 43435 | Corollary of Theorem 6.12(... |
| tz6.12i-afv2 43436 | Corollary of Theorem 6.12(... |
| funressnbrafv2 43437 | The second argument of a b... |
| dfatbrafv2b 43438 | Equivalence of function va... |
| dfatopafv2b 43439 | Equivalence of function va... |
| funbrafv2 43440 | The second argument of a b... |
| fnbrafv2b 43441 | Equivalence of function va... |
| fnopafv2b 43442 | Equivalence of function va... |
| funbrafv22b 43443 | Equivalence of function va... |
| funopafv2b 43444 | Equivalence of function va... |
| dfatsnafv2 43445 | Singleton of function valu... |
| dfafv23 43446 | A definition of function v... |
| dfatdmfcoafv2 43447 | Domain of a function compo... |
| dfatcolem 43448 | Lemma for ~ dfatco . (Con... |
| dfatco 43449 | The predicate "defined at"... |
| afv2co2 43450 | Value of a function compos... |
| rlimdmafv2 43451 | Two ways to express that a... |
| dfafv22 43452 | Alternate definition of ` ... |
| afv2ndeffv0 43453 | If the alternate function ... |
| dfatafv2eqfv 43454 | If a function is defined a... |
| afv2rnfveq 43455 | If the alternate function ... |
| afv20fv0 43456 | If the alternate function ... |
| afv2fvn0fveq 43457 | If the function's value at... |
| afv2fv0 43458 | If the function's value at... |
| afv2fv0b 43459 | The function's value at an... |
| afv2fv0xorb 43460 | If a set is in the range o... |
| an4com24 43461 | Rearrangement of 4 conjunc... |
| 3an4ancom24 43462 | Commutative law for a conj... |
| 4an21 43463 | Rearrangement of 4 conjunc... |
| dfnelbr2 43466 | Alternate definition of th... |
| nelbr 43467 | The binary relation of a s... |
| nelbrim 43468 | If a set is related to ano... |
| nelbrnel 43469 | A set is related to anothe... |
| nelbrnelim 43470 | If a set is related to ano... |
| ralralimp 43471 | Selecting one of two alter... |
| otiunsndisjX 43472 | The union of singletons co... |
| fvifeq 43473 | Equality of function value... |
| rnfdmpr 43474 | The range of a one-to-one ... |
| imarnf1pr 43475 | The image of the range of ... |
| funop1 43476 | A function is an ordered p... |
| fun2dmnopgexmpl 43477 | A function with a domain c... |
| opabresex0d 43478 | A collection of ordered pa... |
| opabbrfex0d 43479 | A collection of ordered pa... |
| opabresexd 43480 | A collection of ordered pa... |
| opabbrfexd 43481 | A collection of ordered pa... |
| f1oresf1orab 43482 | Build a bijection by restr... |
| f1oresf1o 43483 | Build a bijection by restr... |
| f1oresf1o2 43484 | Build a bijection by restr... |
| fvmptrab 43485 | Value of a function mappin... |
| fvmptrabdm 43486 | Value of a function mappin... |
| leltletr 43487 | Transitive law, weaker for... |
| cnambpcma 43488 | ((a-b)+c)-a = c-a holds fo... |
| cnapbmcpd 43489 | ((a+b)-c)+d = ((a+d)+b)-c ... |
| addsubeq0 43490 | The sum of two complex num... |
| leaddsuble 43491 | Addition and subtraction o... |
| 2leaddle2 43492 | If two real numbers are le... |
| ltnltne 43493 | Variant of trichotomy law ... |
| p1lep2 43494 | A real number increasd by ... |
| ltsubsubaddltsub 43495 | If the result of subtracti... |
| zm1nn 43496 | An integer minus 1 is posi... |
| readdcnnred 43497 | The sum of a real number a... |
| resubcnnred 43498 | The difference of a real n... |
| recnmulnred 43499 | The product of a real numb... |
| cndivrenred 43500 | The quotient of an imagina... |
| sqrtnegnre 43501 | The square root of a negat... |
| nn0resubcl 43502 | Closure law for subtractio... |
| zgeltp1eq 43503 | If an integer is between a... |
| 1t10e1p1e11 43504 | 11 is 1 times 10 to the po... |
| deccarry 43505 | Add 1 to a 2 digit number ... |
| eluzge0nn0 43506 | If an integer is greater t... |
| nltle2tri 43507 | Negated extended trichotom... |
| ssfz12 43508 | Subset relationship for fi... |
| elfz2z 43509 | Membership of an integer i... |
| 2elfz3nn0 43510 | If there are two elements ... |
| fz0addcom 43511 | The addition of two member... |
| 2elfz2melfz 43512 | If the sum of two integers... |
| fz0addge0 43513 | The sum of two integers in... |
| elfzlble 43514 | Membership of an integer i... |
| elfzelfzlble 43515 | Membership of an element o... |
| fzopred 43516 | Join a predecessor to the ... |
| fzopredsuc 43517 | Join a predecessor and a s... |
| 1fzopredsuc 43518 | Join 0 and a successor to ... |
| el1fzopredsuc 43519 | An element of an open inte... |
| subsubelfzo0 43520 | Subtracting a difference f... |
| fzoopth 43521 | A half-open integer range ... |
| 2ffzoeq 43522 | Two functions over a half-... |
| m1mod0mod1 43523 | An integer decreased by 1 ... |
| elmod2 43524 | An integer modulo 2 is eit... |
| smonoord 43525 | Ordering relation for a st... |
| fsummsndifre 43526 | A finite sum with one of i... |
| fsumsplitsndif 43527 | Separate out a term in a f... |
| fsummmodsndifre 43528 | A finite sum of summands m... |
| fsummmodsnunz 43529 | A finite sum of summands m... |
| setsidel 43530 | The injected slot is an el... |
| setsnidel 43531 | The injected slot is an el... |
| setsv 43532 | The value of the structure... |
| preimafvsnel 43533 | The preimage of a function... |
| preimafvn0 43534 | The preimage of a function... |
| uniimafveqt 43535 | The union of the image of ... |
| uniimaprimaeqfv 43536 | The union of the image of ... |
| setpreimafvex 43537 | The class ` P ` of all pre... |
| elsetpreimafvb 43538 | The characterization of an... |
| elsetpreimafv 43539 | An element of the class ` ... |
| elsetpreimafvssdm 43540 | An element of the class ` ... |
| fvelsetpreimafv 43541 | There is an element in a p... |
| preimafvelsetpreimafv 43542 | The preimage of a function... |
| preimafvsspwdm 43543 | The class ` P ` of all pre... |
| 0nelsetpreimafv 43544 | The empty set is not an el... |
| elsetpreimafvbi 43545 | An element of the preimage... |
| elsetpreimafveqfv 43546 | The elements of the preima... |
| eqfvelsetpreimafv 43547 | If an element of the domai... |
| elsetpreimafvrab 43548 | An element of the preimage... |
| imaelsetpreimafv 43549 | The image of an element of... |
| uniimaelsetpreimafv 43550 | The union of the image of ... |
| elsetpreimafveq 43551 | If two preimages of functi... |
| fundcmpsurinjlem1 43552 | Lemma 1 for ~ fundcmpsurin... |
| fundcmpsurinjlem2 43553 | Lemma 2 for ~ fundcmpsurin... |
| fundcmpsurinjlem3 43554 | Lemma 3 for ~ fundcmpsurin... |
| imasetpreimafvbijlemf 43555 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv 43556 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfv1 43557 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemf1 43558 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbijlemfo 43559 | Lemma for ~ imasetpreimafv... |
| imasetpreimafvbij 43560 | The mapping ` H ` is a bij... |
| fundcmpsurbijinjpreimafv 43561 | Every function ` F : A -->... |
| fundcmpsurinjpreimafv 43562 | Every function ` F : A -->... |
| fundcmpsurinj 43563 | Every function ` F : A -->... |
| fundcmpsurbijinj 43564 | Every function ` F : A -->... |
| fundcmpsurinjimaid 43565 | Every function ` F : A -->... |
| fundcmpsurinjALT 43566 | Alternate proof of ~ fundc... |
| iccpval 43569 | Partition consisting of a ... |
| iccpart 43570 | A special partition. Corr... |
| iccpartimp 43571 | Implications for a class b... |
| iccpartres 43572 | The restriction of a parti... |
| iccpartxr 43573 | If there is a partition, t... |
| iccpartgtprec 43574 | If there is a partition, t... |
| iccpartipre 43575 | If there is a partition, t... |
| iccpartiltu 43576 | If there is a partition, t... |
| iccpartigtl 43577 | If there is a partition, t... |
| iccpartlt 43578 | If there is a partition, t... |
| iccpartltu 43579 | If there is a partition, t... |
| iccpartgtl 43580 | If there is a partition, t... |
| iccpartgt 43581 | If there is a partition, t... |
| iccpartleu 43582 | If there is a partition, t... |
| iccpartgel 43583 | If there is a partition, t... |
| iccpartrn 43584 | If there is a partition, t... |
| iccpartf 43585 | The range of the partition... |
| iccpartel 43586 | If there is a partition, t... |
| iccelpart 43587 | An element of any partitio... |
| iccpartiun 43588 | A half-open interval of ex... |
| icceuelpartlem 43589 | Lemma for ~ icceuelpart . ... |
| icceuelpart 43590 | An element of a partitione... |
| iccpartdisj 43591 | The segments of a partitio... |
| iccpartnel 43592 | A point of a partition is ... |
| fargshiftfv 43593 | If a class is a function, ... |
| fargshiftf 43594 | If a class is a function, ... |
| fargshiftf1 43595 | If a function is 1-1, then... |
| fargshiftfo 43596 | If a function is onto, the... |
| fargshiftfva 43597 | The values of a shifted fu... |
| lswn0 43598 | The last symbol of a not e... |
| nfich1 43601 | The first interchangeable ... |
| nfich2 43602 | The second interchangeable... |
| ichv 43603 | Setvar variables are inter... |
| ichf 43604 | Setvar variables are inter... |
| ichid 43605 | A setvar variable is alway... |
| ichcircshi 43606 | The setvar variables are i... |
| dfich2 43607 | Alternate definition of th... |
| dfich2ai 43608 | Obsolete version of ~ dfic... |
| dfich2bi 43609 | Obsolete version of ~ dfic... |
| dfich2OLD 43610 | Obsolete version of ~ dfic... |
| ichcom 43611 | The interchangeability of ... |
| ichbi12i 43612 | Equivalence for interchang... |
| icheqid 43613 | In an equality for the sam... |
| icheq 43614 | In an equality of setvar v... |
| ichnfimlem1 43615 | Lemma for ~ ichnfimlem3 : ... |
| ichnfimlem2 43616 | Lemma for ~ ichnfimlem3 : ... |
| ichnfimlem3 43617 | Lemma for ~ ichnfim : A s... |
| ichnfim 43618 | If in an interchangeabilit... |
| ichnfb 43619 | If ` x ` and ` y ` are int... |
| ichn 43620 | Negation does not affect i... |
| ichal 43621 | Move a universal quantifie... |
| ich2al 43622 | Two setvar variables are a... |
| ich2ex 43623 | Two setvar variables are a... |
| ichan 43624 | If two setvar variables ar... |
| ichexmpl1 43625 | Example for interchangeabl... |
| ichexmpl2 43626 | Example for interchangeabl... |
| ich2exprop 43627 | If the setvar variables ar... |
| ichnreuop 43628 | If the setvar variables ar... |
| ichreuopeq 43629 | If the setvar variables ar... |
| sprid 43630 | Two identical representati... |
| elsprel 43631 | An unordered pair is an el... |
| spr0nelg 43632 | The empty set is not an el... |
| sprval 43635 | The set of all unordered p... |
| sprvalpw 43636 | The set of all unordered p... |
| sprssspr 43637 | The set of all unordered p... |
| spr0el 43638 | The empty set is not an un... |
| sprvalpwn0 43639 | The set of all unordered p... |
| sprel 43640 | An element of the set of a... |
| prssspr 43641 | An element of a subset of ... |
| prelspr 43642 | An unordered pair of eleme... |
| prsprel 43643 | The elements of a pair fro... |
| prsssprel 43644 | The elements of a pair fro... |
| sprvalpwle2 43645 | The set of all unordered p... |
| sprsymrelfvlem 43646 | Lemma for ~ sprsymrelf and... |
| sprsymrelf1lem 43647 | Lemma for ~ sprsymrelf1 . ... |
| sprsymrelfolem1 43648 | Lemma 1 for ~ sprsymrelfo ... |
| sprsymrelfolem2 43649 | Lemma 2 for ~ sprsymrelfo ... |
| sprsymrelfv 43650 | The value of the function ... |
| sprsymrelf 43651 | The mapping ` F ` is a fun... |
| sprsymrelf1 43652 | The mapping ` F ` is a one... |
| sprsymrelfo 43653 | The mapping ` F ` is a fun... |
| sprsymrelf1o 43654 | The mapping ` F ` is a bij... |
| sprbisymrel 43655 | There is a bijection betwe... |
| sprsymrelen 43656 | The class ` P ` of subsets... |
| prpair 43657 | Characterization of a prop... |
| prproropf1olem0 43658 | Lemma 0 for ~ prproropf1o ... |
| prproropf1olem1 43659 | Lemma 1 for ~ prproropf1o ... |
| prproropf1olem2 43660 | Lemma 2 for ~ prproropf1o ... |
| prproropf1olem3 43661 | Lemma 3 for ~ prproropf1o ... |
| prproropf1olem4 43662 | Lemma 4 for ~ prproropf1o ... |
| prproropf1o 43663 | There is a bijection betwe... |
| prproropen 43664 | The set of proper pairs an... |
| prproropreud 43665 | There is exactly one order... |
| pairreueq 43666 | Two equivalent representat... |
| paireqne 43667 | Two sets are not equal iff... |
| prprval 43670 | The set of all proper unor... |
| prprvalpw 43671 | The set of all proper unor... |
| prprelb 43672 | An element of the set of a... |
| prprelprb 43673 | A set is an element of the... |
| prprspr2 43674 | The set of all proper unor... |
| prprsprreu 43675 | There is a unique proper u... |
| prprreueq 43676 | There is a unique proper u... |
| sbcpr 43677 | The proper substitution of... |
| reupr 43678 | There is a unique unordere... |
| reuprpr 43679 | There is a unique proper u... |
| poprelb 43680 | Equality for unordered pai... |
| 2exopprim 43681 | The existence of an ordere... |
| reuopreuprim 43682 | There is a unique unordere... |
| fmtno 43685 | The ` N ` th Fermat number... |
| fmtnoge3 43686 | Each Fermat number is grea... |
| fmtnonn 43687 | Each Fermat number is a po... |
| fmtnom1nn 43688 | A Fermat number minus one ... |
| fmtnoodd 43689 | Each Fermat number is odd.... |
| fmtnorn 43690 | A Fermat number is a funct... |
| fmtnof1 43691 | The enumeration of the Fer... |
| fmtnoinf 43692 | The set of Fermat numbers ... |
| fmtnorec1 43693 | The first recurrence relat... |
| sqrtpwpw2p 43694 | The floor of the square ro... |
| fmtnosqrt 43695 | The floor of the square ro... |
| fmtno0 43696 | The ` 0 ` th Fermat number... |
| fmtno1 43697 | The ` 1 ` st Fermat number... |
| fmtnorec2lem 43698 | Lemma for ~ fmtnorec2 (ind... |
| fmtnorec2 43699 | The second recurrence rela... |
| fmtnodvds 43700 | Any Fermat number divides ... |
| goldbachthlem1 43701 | Lemma 1 for ~ goldbachth .... |
| goldbachthlem2 43702 | Lemma 2 for ~ goldbachth .... |
| goldbachth 43703 | Goldbach's theorem: Two d... |
| fmtnorec3 43704 | The third recurrence relat... |
| fmtnorec4 43705 | The fourth recurrence rela... |
| fmtno2 43706 | The ` 2 ` nd Fermat number... |
| fmtno3 43707 | The ` 3 ` rd Fermat number... |
| fmtno4 43708 | The ` 4 ` th Fermat number... |
| fmtno5lem1 43709 | Lemma 1 for ~ fmtno5 . (C... |
| fmtno5lem2 43710 | Lemma 2 for ~ fmtno5 . (C... |
| fmtno5lem3 43711 | Lemma 3 for ~ fmtno5 . (C... |
| fmtno5lem4 43712 | Lemma 4 for ~ fmtno5 . (C... |
| fmtno5 43713 | The ` 5 ` th Fermat number... |
| fmtno0prm 43714 | The ` 0 ` th Fermat number... |
| fmtno1prm 43715 | The ` 1 ` st Fermat number... |
| fmtno2prm 43716 | The ` 2 ` nd Fermat number... |
| 257prm 43717 | 257 is a prime number (the... |
| fmtno3prm 43718 | The ` 3 ` rd Fermat number... |
| odz2prm2pw 43719 | Any power of two is coprim... |
| fmtnoprmfac1lem 43720 | Lemma for ~ fmtnoprmfac1 :... |
| fmtnoprmfac1 43721 | Divisor of Fermat number (... |
| fmtnoprmfac2lem1 43722 | Lemma for ~ fmtnoprmfac2 .... |
| fmtnoprmfac2 43723 | Divisor of Fermat number (... |
| fmtnofac2lem 43724 | Lemma for ~ fmtnofac2 (Ind... |
| fmtnofac2 43725 | Divisor of Fermat number (... |
| fmtnofac1 43726 | Divisor of Fermat number (... |
| fmtno4sqrt 43727 | The floor of the square ro... |
| fmtno4prmfac 43728 | If P was a (prime) factor ... |
| fmtno4prmfac193 43729 | If P was a (prime) factor ... |
| fmtno4nprmfac193 43730 | 193 is not a (prime) facto... |
| fmtno4prm 43731 | The ` 4 `-th Fermat number... |
| 65537prm 43732 | 65537 is a prime number (t... |
| fmtnofz04prm 43733 | The first five Fermat numb... |
| fmtnole4prm 43734 | The first five Fermat numb... |
| fmtno5faclem1 43735 | Lemma 1 for ~ fmtno5fac . ... |
| fmtno5faclem2 43736 | Lemma 2 for ~ fmtno5fac . ... |
| fmtno5faclem3 43737 | Lemma 3 for ~ fmtno5fac . ... |
| fmtno5fac 43738 | The factorisation of the `... |
| fmtno5nprm 43739 | The ` 5 ` th Fermat number... |
| prmdvdsfmtnof1lem1 43740 | Lemma 1 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof1lem2 43741 | Lemma 2 for ~ prmdvdsfmtno... |
| prmdvdsfmtnof 43742 | The mapping of a Fermat nu... |
| prmdvdsfmtnof1 43743 | The mapping of a Fermat nu... |
| prminf2 43744 | The set of prime numbers i... |
| 2pwp1prm 43745 | For every prime number of ... |
| 2pwp1prmfmtno 43746 | Every prime number of the ... |
| m2prm 43747 | The second Mersenne number... |
| m3prm 43748 | The third Mersenne number ... |
| 2exp5 43749 | Two to the fifth power is ... |
| flsqrt 43750 | A condition equivalent to ... |
| flsqrt5 43751 | The floor of the square ro... |
| 3ndvds4 43752 | 3 does not divide 4. (Con... |
| 139prmALT 43753 | 139 is a prime number. In... |
| 31prm 43754 | 31 is a prime number. In ... |
| m5prm 43755 | The fifth Mersenne number ... |
| 2exp7 43756 | Two to the seventh power i... |
| 127prm 43757 | 127 is a prime number. (C... |
| m7prm 43758 | The seventh Mersenne numbe... |
| 2exp11 43759 | Two to the eleventh power ... |
| m11nprm 43760 | The eleventh Mersenne numb... |
| mod42tp1mod8 43761 | If a number is ` 3 ` modul... |
| sfprmdvdsmersenne 43762 | If ` Q ` is a safe prime (... |
| sgprmdvdsmersenne 43763 | If ` P ` is a Sophie Germa... |
| lighneallem1 43764 | Lemma 1 for ~ lighneal . ... |
| lighneallem2 43765 | Lemma 2 for ~ lighneal . ... |
| lighneallem3 43766 | Lemma 3 for ~ lighneal . ... |
| lighneallem4a 43767 | Lemma 1 for ~ lighneallem4... |
| lighneallem4b 43768 | Lemma 2 for ~ lighneallem4... |
| lighneallem4 43769 | Lemma 3 for ~ lighneal . ... |
| lighneal 43770 | If a power of a prime ` P ... |
| modexp2m1d 43771 | The square of an integer w... |
| proththdlem 43772 | Lemma for ~ proththd . (C... |
| proththd 43773 | Proth's theorem (1878). I... |
| 5tcu2e40 43774 | 5 times the cube of 2 is 4... |
| 3exp4mod41 43775 | 3 to the fourth power is -... |
| 41prothprmlem1 43776 | Lemma 1 for ~ 41prothprm .... |
| 41prothprmlem2 43777 | Lemma 2 for ~ 41prothprm .... |
| 41prothprm 43778 | 41 is a _Proth prime_. (C... |
| quad1 43779 | A condition for a quadrati... |
| requad01 43780 | A condition for a quadrati... |
| requad1 43781 | A condition for a quadrati... |
| requad2 43782 | A condition for a quadrati... |
| iseven 43787 | The predicate "is an even ... |
| isodd 43788 | The predicate "is an odd n... |
| evenz 43789 | An even number is an integ... |
| oddz 43790 | An odd number is an intege... |
| evendiv2z 43791 | The result of dividing an ... |
| oddp1div2z 43792 | The result of dividing an ... |
| oddm1div2z 43793 | The result of dividing an ... |
| isodd2 43794 | The predicate "is an odd n... |
| dfodd2 43795 | Alternate definition for o... |
| dfodd6 43796 | Alternate definition for o... |
| dfeven4 43797 | Alternate definition for e... |
| evenm1odd 43798 | The predecessor of an even... |
| evenp1odd 43799 | The successor of an even n... |
| oddp1eveni 43800 | The successor of an odd nu... |
| oddm1eveni 43801 | The predecessor of an odd ... |
| evennodd 43802 | An even number is not an o... |
| oddneven 43803 | An odd number is not an ev... |
| enege 43804 | The negative of an even nu... |
| onego 43805 | The negative of an odd num... |
| m1expevenALTV 43806 | Exponentiation of -1 by an... |
| m1expoddALTV 43807 | Exponentiation of -1 by an... |
| dfeven2 43808 | Alternate definition for e... |
| dfodd3 43809 | Alternate definition for o... |
| iseven2 43810 | The predicate "is an even ... |
| isodd3 43811 | The predicate "is an odd n... |
| 2dvdseven 43812 | 2 divides an even number. ... |
| m2even 43813 | A multiple of 2 is an even... |
| 2ndvdsodd 43814 | 2 does not divide an odd n... |
| 2dvdsoddp1 43815 | 2 divides an odd number in... |
| 2dvdsoddm1 43816 | 2 divides an odd number de... |
| dfeven3 43817 | Alternate definition for e... |
| dfodd4 43818 | Alternate definition for o... |
| dfodd5 43819 | Alternate definition for o... |
| zefldiv2ALTV 43820 | The floor of an even numbe... |
| zofldiv2ALTV 43821 | The floor of an odd numer ... |
| oddflALTV 43822 | Odd number representation ... |
| iseven5 43823 | The predicate "is an even ... |
| isodd7 43824 | The predicate "is an odd n... |
| dfeven5 43825 | Alternate definition for e... |
| dfodd7 43826 | Alternate definition for o... |
| gcd2odd1 43827 | The greatest common diviso... |
| zneoALTV 43828 | No even integer equals an ... |
| zeoALTV 43829 | An integer is even or odd.... |
| zeo2ALTV 43830 | An integer is even or odd ... |
| nneoALTV 43831 | A positive integer is even... |
| nneoiALTV 43832 | A positive integer is even... |
| odd2np1ALTV 43833 | An integer is odd iff it i... |
| oddm1evenALTV 43834 | An integer is odd iff its ... |
| oddp1evenALTV 43835 | An integer is odd iff its ... |
| oexpnegALTV 43836 | The exponential of the neg... |
| oexpnegnz 43837 | The exponential of the neg... |
| bits0ALTV 43838 | Value of the zeroth bit. ... |
| bits0eALTV 43839 | The zeroth bit of an even ... |
| bits0oALTV 43840 | The zeroth bit of an odd n... |
| divgcdoddALTV 43841 | Either ` A / ( A gcd B ) `... |
| opoeALTV 43842 | The sum of two odds is eve... |
| opeoALTV 43843 | The sum of an odd and an e... |
| omoeALTV 43844 | The difference of two odds... |
| omeoALTV 43845 | The difference of an odd a... |
| oddprmALTV 43846 | A prime not equal to ` 2 `... |
| 0evenALTV 43847 | 0 is an even number. (Con... |
| 0noddALTV 43848 | 0 is not an odd number. (... |
| 1oddALTV 43849 | 1 is an odd number. (Cont... |
| 1nevenALTV 43850 | 1 is not an even number. ... |
| 2evenALTV 43851 | 2 is an even number. (Con... |
| 2noddALTV 43852 | 2 is not an odd number. (... |
| nn0o1gt2ALTV 43853 | An odd nonnegative integer... |
| nnoALTV 43854 | An alternate characterizat... |
| nn0oALTV 43855 | An alternate characterizat... |
| nn0e 43856 | An alternate characterizat... |
| nneven 43857 | An alternate characterizat... |
| nn0onn0exALTV 43858 | For each odd nonnegative i... |
| nn0enn0exALTV 43859 | For each even nonnegative ... |
| nnennexALTV 43860 | For each even positive int... |
| nnpw2evenALTV 43861 | 2 to the power of a positi... |
| epoo 43862 | The sum of an even and an ... |
| emoo 43863 | The difference of an even ... |
| epee 43864 | The sum of two even number... |
| emee 43865 | The difference of two even... |
| evensumeven 43866 | If a summand is even, the ... |
| 3odd 43867 | 3 is an odd number. (Cont... |
| 4even 43868 | 4 is an even number. (Con... |
| 5odd 43869 | 5 is an odd number. (Cont... |
| 6even 43870 | 6 is an even number. (Con... |
| 7odd 43871 | 7 is an odd number. (Cont... |
| 8even 43872 | 8 is an even number. (Con... |
| evenprm2 43873 | A prime number is even iff... |
| oddprmne2 43874 | Every prime number not bei... |
| oddprmuzge3 43875 | A prime number which is od... |
| evenltle 43876 | If an even number is great... |
| odd2prm2 43877 | If an odd number is the su... |
| even3prm2 43878 | If an even number is the s... |
| mogoldbblem 43879 | Lemma for ~ mogoldbb . (C... |
| perfectALTVlem1 43880 | Lemma for ~ perfectALTV . ... |
| perfectALTVlem2 43881 | Lemma for ~ perfectALTV . ... |
| perfectALTV 43882 | The Euclid-Euler theorem, ... |
| fppr 43885 | The set of Fermat pseudopr... |
| fpprmod 43886 | The set of Fermat pseudopr... |
| fpprel 43887 | A Fermat pseudoprime to th... |
| fpprbasnn 43888 | The base of a Fermat pseud... |
| fpprnn 43889 | A Fermat pseudoprime to th... |
| fppr2odd 43890 | A Fermat pseudoprime to th... |
| 11t31e341 43891 | 341 is the product of 11 a... |
| 2exp340mod341 43892 | Eight to the eighth power ... |
| 341fppr2 43893 | 341 is the (smallest) _Pou... |
| 4fppr1 43894 | 4 is the (smallest) Fermat... |
| 8exp8mod9 43895 | Eight to the eighth power ... |
| 9fppr8 43896 | 9 is the (smallest) Fermat... |
| dfwppr 43897 | Alternate definition of a ... |
| fpprwppr 43898 | A Fermat pseudoprime to th... |
| fpprwpprb 43899 | An integer ` X ` which is ... |
| fpprel2 43900 | An alternate definition fo... |
| nfermltl8rev 43901 | Fermat's little theorem wi... |
| nfermltl2rev 43902 | Fermat's little theorem wi... |
| nfermltlrev 43903 | Fermat's little theorem re... |
| isgbe 43910 | The predicate "is an even ... |
| isgbow 43911 | The predicate "is a weak o... |
| isgbo 43912 | The predicate "is an odd G... |
| gbeeven 43913 | An even Goldbach number is... |
| gbowodd 43914 | A weak odd Goldbach number... |
| gbogbow 43915 | A (strong) odd Goldbach nu... |
| gboodd 43916 | An odd Goldbach number is ... |
| gbepos 43917 | Any even Goldbach number i... |
| gbowpos 43918 | Any weak odd Goldbach numb... |
| gbopos 43919 | Any odd Goldbach number is... |
| gbegt5 43920 | Any even Goldbach number i... |
| gbowgt5 43921 | Any weak odd Goldbach numb... |
| gbowge7 43922 | Any weak odd Goldbach numb... |
| gboge9 43923 | Any odd Goldbach number is... |
| gbege6 43924 | Any even Goldbach number i... |
| gbpart6 43925 | The Goldbach partition of ... |
| gbpart7 43926 | The (weak) Goldbach partit... |
| gbpart8 43927 | The Goldbach partition of ... |
| gbpart9 43928 | The (strong) Goldbach part... |
| gbpart11 43929 | The (strong) Goldbach part... |
| 6gbe 43930 | 6 is an even Goldbach numb... |
| 7gbow 43931 | 7 is a weak odd Goldbach n... |
| 8gbe 43932 | 8 is an even Goldbach numb... |
| 9gbo 43933 | 9 is an odd Goldbach numbe... |
| 11gbo 43934 | 11 is an odd Goldbach numb... |
| stgoldbwt 43935 | If the strong ternary Gold... |
| sbgoldbwt 43936 | If the strong binary Goldb... |
| sbgoldbst 43937 | If the strong binary Goldb... |
| sbgoldbaltlem1 43938 | Lemma 1 for ~ sbgoldbalt :... |
| sbgoldbaltlem2 43939 | Lemma 2 for ~ sbgoldbalt :... |
| sbgoldbalt 43940 | An alternate (related to t... |
| sbgoldbb 43941 | If the strong binary Goldb... |
| sgoldbeven3prm 43942 | If the binary Goldbach con... |
| sbgoldbm 43943 | If the strong binary Goldb... |
| mogoldbb 43944 | If the modern version of t... |
| sbgoldbmb 43945 | The strong binary Goldbach... |
| sbgoldbo 43946 | If the strong binary Goldb... |
| nnsum3primes4 43947 | 4 is the sum of at most 3 ... |
| nnsum4primes4 43948 | 4 is the sum of at most 4 ... |
| nnsum3primesprm 43949 | Every prime is "the sum of... |
| nnsum4primesprm 43950 | Every prime is "the sum of... |
| nnsum3primesgbe 43951 | Any even Goldbach number i... |
| nnsum4primesgbe 43952 | Any even Goldbach number i... |
| nnsum3primesle9 43953 | Every integer greater than... |
| nnsum4primesle9 43954 | Every integer greater than... |
| nnsum4primesodd 43955 | If the (weak) ternary Gold... |
| nnsum4primesoddALTV 43956 | If the (strong) ternary Go... |
| evengpop3 43957 | If the (weak) ternary Gold... |
| evengpoap3 43958 | If the (strong) ternary Go... |
| nnsum4primeseven 43959 | If the (weak) ternary Gold... |
| nnsum4primesevenALTV 43960 | If the (strong) ternary Go... |
| wtgoldbnnsum4prm 43961 | If the (weak) ternary Gold... |
| stgoldbnnsum4prm 43962 | If the (strong) ternary Go... |
| bgoldbnnsum3prm 43963 | If the binary Goldbach con... |
| bgoldbtbndlem1 43964 | Lemma 1 for ~ bgoldbtbnd :... |
| bgoldbtbndlem2 43965 | Lemma 2 for ~ bgoldbtbnd .... |
| bgoldbtbndlem3 43966 | Lemma 3 for ~ bgoldbtbnd .... |
| bgoldbtbndlem4 43967 | Lemma 4 for ~ bgoldbtbnd .... |
| bgoldbtbnd 43968 | If the binary Goldbach con... |
| tgoldbachgtALTV 43971 | Variant of Thierry Arnoux'... |
| bgoldbachlt 43972 | The binary Goldbach conjec... |
| tgblthelfgott 43974 | The ternary Goldbach conje... |
| tgoldbachlt 43975 | The ternary Goldbach conje... |
| tgoldbach 43976 | The ternary Goldbach conje... |
| isomgrrel 43981 | The isomorphy relation for... |
| isomgr 43982 | The isomorphy relation for... |
| isisomgr 43983 | Implications of two graphs... |
| isomgreqve 43984 | A set is isomorphic to a h... |
| isomushgr 43985 | The isomorphy relation for... |
| isomuspgrlem1 43986 | Lemma 1 for ~ isomuspgr . ... |
| isomuspgrlem2a 43987 | Lemma 1 for ~ isomuspgrlem... |
| isomuspgrlem2b 43988 | Lemma 2 for ~ isomuspgrlem... |
| isomuspgrlem2c 43989 | Lemma 3 for ~ isomuspgrlem... |
| isomuspgrlem2d 43990 | Lemma 4 for ~ isomuspgrlem... |
| isomuspgrlem2e 43991 | Lemma 5 for ~ isomuspgrlem... |
| isomuspgrlem2 43992 | Lemma 2 for ~ isomuspgr . ... |
| isomuspgr 43993 | The isomorphy relation for... |
| isomgrref 43994 | The isomorphy relation is ... |
| isomgrsym 43995 | The isomorphy relation is ... |
| isomgrsymb 43996 | The isomorphy relation is ... |
| isomgrtrlem 43997 | Lemma for ~ isomgrtr . (C... |
| isomgrtr 43998 | The isomorphy relation is ... |
| strisomgrop 43999 | A graph represented as an ... |
| ushrisomgr 44000 | A simple hypergraph (with ... |
| 1hegrlfgr 44001 | A graph ` G ` with one hyp... |
| upwlksfval 44004 | The set of simple walks (i... |
| isupwlk 44005 | Properties of a pair of fu... |
| isupwlkg 44006 | Generalization of ~ isupwl... |
| upwlkbprop 44007 | Basic properties of a simp... |
| upwlkwlk 44008 | A simple walk is a walk. ... |
| upgrwlkupwlk 44009 | In a pseudograph, a walk i... |
| upgrwlkupwlkb 44010 | In a pseudograph, the defi... |
| upgrisupwlkALT 44011 | Alternate proof of ~ upgri... |
| upgredgssspr 44012 | The set of edges of a pseu... |
| uspgropssxp 44013 | The set ` G ` of "simple p... |
| uspgrsprfv 44014 | The value of the function ... |
| uspgrsprf 44015 | The mapping ` F ` is a fun... |
| uspgrsprf1 44016 | The mapping ` F ` is a one... |
| uspgrsprfo 44017 | The mapping ` F ` is a fun... |
| uspgrsprf1o 44018 | The mapping ` F ` is a bij... |
| uspgrex 44019 | The class ` G ` of all "si... |
| uspgrbispr 44020 | There is a bijection betwe... |
| uspgrspren 44021 | The set ` G ` of the "simp... |
| uspgrymrelen 44022 | The set ` G ` of the "simp... |
| uspgrbisymrel 44023 | There is a bijection betwe... |
| uspgrbisymrelALT 44024 | Alternate proof of ~ uspgr... |
| ovn0dmfun 44025 | If a class operation value... |
| xpsnopab 44026 | A Cartesian product with a... |
| xpiun 44027 | A Cartesian product expres... |
| ovn0ssdmfun 44028 | If a class' operation valu... |
| fnxpdmdm 44029 | The domain of the domain o... |
| cnfldsrngbas 44030 | The base set of a subring ... |
| cnfldsrngadd 44031 | The group addition operati... |
| cnfldsrngmul 44032 | The ring multiplication op... |
| plusfreseq 44033 | If the empty set is not co... |
| mgmplusfreseq 44034 | If the empty set is not co... |
| 0mgm 44035 | A set with an empty base s... |
| mgmpropd 44036 | If two structures have the... |
| ismgmd 44037 | Deduce a magma from its pr... |
| mgmhmrcl 44042 | Reverse closure of a magma... |
| submgmrcl 44043 | Reverse closure for submag... |
| ismgmhm 44044 | Property of a magma homomo... |
| mgmhmf 44045 | A magma homomorphism is a ... |
| mgmhmpropd 44046 | Magma homomorphism depends... |
| mgmhmlin 44047 | A magma homomorphism prese... |
| mgmhmf1o 44048 | A magma homomorphism is bi... |
| idmgmhm 44049 | The identity homomorphism ... |
| issubmgm 44050 | Expand definition of a sub... |
| issubmgm2 44051 | Submagmas are subsets that... |
| rabsubmgmd 44052 | Deduction for proving that... |
| submgmss 44053 | Submagmas are subsets of t... |
| submgmid 44054 | Every magma is trivially a... |
| submgmcl 44055 | Submagmas are closed under... |
| submgmmgm 44056 | Submagmas are themselves m... |
| submgmbas 44057 | The base set of a submagma... |
| subsubmgm 44058 | A submagma of a submagma i... |
| resmgmhm 44059 | Restriction of a magma hom... |
| resmgmhm2 44060 | One direction of ~ resmgmh... |
| resmgmhm2b 44061 | Restriction of the codomai... |
| mgmhmco 44062 | The composition of magma h... |
| mgmhmima 44063 | The homomorphic image of a... |
| mgmhmeql 44064 | The equalizer of two magma... |
| submgmacs 44065 | Submagmas are an algebraic... |
| ismhm0 44066 | Property of a monoid homom... |
| mhmismgmhm 44067 | Each monoid homomorphism i... |
| opmpoismgm 44068 | A structure with a group a... |
| copissgrp 44069 | A structure with a constan... |
| copisnmnd 44070 | A structure with a constan... |
| 0nodd 44071 | 0 is not an odd integer. ... |
| 1odd 44072 | 1 is an odd integer. (Con... |
| 2nodd 44073 | 2 is not an odd integer. ... |
| oddibas 44074 | Lemma 1 for ~ oddinmgm : ... |
| oddiadd 44075 | Lemma 2 for ~ oddinmgm : ... |
| oddinmgm 44076 | The structure of all odd i... |
| nnsgrpmgm 44077 | The structure of positive ... |
| nnsgrp 44078 | The structure of positive ... |
| nnsgrpnmnd 44079 | The structure of positive ... |
| nn0mnd 44080 | The set of nonnegative int... |
| gsumsplit2f 44081 | Split a group sum into two... |
| gsumdifsndf 44082 | Extract a summand from a f... |
| gsumfsupp 44083 | A group sum of a family ca... |
| iscllaw 44090 | The predicate "is a closed... |
| iscomlaw 44091 | The predicate "is a commut... |
| clcllaw 44092 | Closure of a closed operat... |
| isasslaw 44093 | The predicate "is an assoc... |
| asslawass 44094 | Associativity of an associ... |
| mgmplusgiopALT 44095 | Slot 2 (group operation) o... |
| sgrpplusgaopALT 44096 | Slot 2 (group operation) o... |
| intopval 44103 | The internal (binary) oper... |
| intop 44104 | An internal (binary) opera... |
| clintopval 44105 | The closed (internal binar... |
| assintopval 44106 | The associative (closed in... |
| assintopmap 44107 | The associative (closed in... |
| isclintop 44108 | The predicate "is a closed... |
| clintop 44109 | A closed (internal binary)... |
| assintop 44110 | An associative (closed int... |
| isassintop 44111 | The predicate "is an assoc... |
| clintopcllaw 44112 | The closure law holds for ... |
| assintopcllaw 44113 | The closure low holds for ... |
| assintopasslaw 44114 | The associative low holds ... |
| assintopass 44115 | An associative (closed int... |
| ismgmALT 44124 | The predicate "is a magma"... |
| iscmgmALT 44125 | The predicate "is a commut... |
| issgrpALT 44126 | The predicate "is a semigr... |
| iscsgrpALT 44127 | The predicate "is a commut... |
| mgm2mgm 44128 | Equivalence of the two def... |
| sgrp2sgrp 44129 | Equivalence of the two def... |
| idfusubc0 44130 | The identity functor for a... |
| idfusubc 44131 | The identity functor for a... |
| inclfusubc 44132 | The "inclusion functor" fr... |
| lmod0rng 44133 | If the scalar ring of a mo... |
| nzrneg1ne0 44134 | The additive inverse of th... |
| 0ringdif 44135 | A zero ring is a ring whic... |
| 0ringbas 44136 | The base set of a zero rin... |
| 0ring1eq0 44137 | In a zero ring, a ring whi... |
| nrhmzr 44138 | There is no ring homomorph... |
| isrng 44141 | The predicate "is a non-un... |
| rngabl 44142 | A non-unital ring is an (a... |
| rngmgp 44143 | A non-unital ring is a sem... |
| ringrng 44144 | A unital ring is a (non-un... |
| ringssrng 44145 | The unital rings are (non-... |
| isringrng 44146 | The predicate "is a unital... |
| rngdir 44147 | Distributive law for the m... |
| rngcl 44148 | Closure of the multiplicat... |
| rnglz 44149 | The zero of a nonunital ri... |
| rnghmrcl 44154 | Reverse closure of a non-u... |
| rnghmfn 44155 | The mapping of two non-uni... |
| rnghmval 44156 | The set of the non-unital ... |
| isrnghm 44157 | A function is a non-unital... |
| isrnghmmul 44158 | A function is a non-unital... |
| rnghmmgmhm 44159 | A non-unital ring homomorp... |
| rnghmval2 44160 | The non-unital ring homomo... |
| isrngisom 44161 | An isomorphism of non-unit... |
| rngimrcl 44162 | Reverse closure for an iso... |
| rnghmghm 44163 | A non-unital ring homomorp... |
| rnghmf 44164 | A ring homomorphism is a f... |
| rnghmmul 44165 | A homomorphism of non-unit... |
| isrnghm2d 44166 | Demonstration of non-unita... |
| isrnghmd 44167 | Demonstration of non-unita... |
| rnghmf1o 44168 | A non-unital ring homomorp... |
| isrngim 44169 | An isomorphism of non-unit... |
| rngimf1o 44170 | An isomorphism of non-unit... |
| rngimrnghm 44171 | An isomorphism of non-unit... |
| rnghmco 44172 | The composition of non-uni... |
| idrnghm 44173 | The identity homomorphism ... |
| c0mgm 44174 | The constant mapping to ze... |
| c0mhm 44175 | The constant mapping to ze... |
| c0ghm 44176 | The constant mapping to ze... |
| c0rhm 44177 | The constant mapping to ze... |
| c0rnghm 44178 | The constant mapping to ze... |
| c0snmgmhm 44179 | The constant mapping to ze... |
| c0snmhm 44180 | The constant mapping to ze... |
| c0snghm 44181 | The constant mapping to ze... |
| zrrnghm 44182 | The constant mapping to ze... |
| rhmfn 44183 | The mapping of two rings t... |
| rhmval 44184 | The ring homomorphisms bet... |
| rhmisrnghm 44185 | Each unital ring homomorph... |
| lidldomn1 44186 | If a (left) ideal (which i... |
| lidlssbas 44187 | The base set of the restri... |
| lidlbas 44188 | A (left) ideal of a ring i... |
| lidlabl 44189 | A (left) ideal of a ring i... |
| lidlmmgm 44190 | The multiplicative group o... |
| lidlmsgrp 44191 | The multiplicative group o... |
| lidlrng 44192 | A (left) ideal of a ring i... |
| zlidlring 44193 | The zero (left) ideal of a... |
| uzlidlring 44194 | Only the zero (left) ideal... |
| lidldomnnring 44195 | A (left) ideal of a domain... |
| 0even 44196 | 0 is an even integer. (Co... |
| 1neven 44197 | 1 is not an even integer. ... |
| 2even 44198 | 2 is an even integer. (Co... |
| 2zlidl 44199 | The even integers are a (l... |
| 2zrng 44200 | The ring of integers restr... |
| 2zrngbas 44201 | The base set of R is the s... |
| 2zrngadd 44202 | The group addition operati... |
| 2zrng0 44203 | The additive identity of R... |
| 2zrngamgm 44204 | R is an (additive) magma. ... |
| 2zrngasgrp 44205 | R is an (additive) semigro... |
| 2zrngamnd 44206 | R is an (additive) monoid.... |
| 2zrngacmnd 44207 | R is a commutative (additi... |
| 2zrngagrp 44208 | R is an (additive) group. ... |
| 2zrngaabl 44209 | R is an (additive) abelian... |
| 2zrngmul 44210 | The ring multiplication op... |
| 2zrngmmgm 44211 | R is a (multiplicative) ma... |
| 2zrngmsgrp 44212 | R is a (multiplicative) se... |
| 2zrngALT 44213 | The ring of integers restr... |
| 2zrngnmlid 44214 | R has no multiplicative (l... |
| 2zrngnmrid 44215 | R has no multiplicative (r... |
| 2zrngnmlid2 44216 | R has no multiplicative (l... |
| 2zrngnring 44217 | R is not a unital ring. (... |
| cznrnglem 44218 | Lemma for ~ cznrng : The ... |
| cznabel 44219 | The ring constructed from ... |
| cznrng 44220 | The ring constructed from ... |
| cznnring 44221 | The ring constructed from ... |
| rngcvalALTV 44226 | Value of the category of n... |
| rngcval 44227 | Value of the category of n... |
| rnghmresfn 44228 | The class of non-unital ri... |
| rnghmresel 44229 | An element of the non-unit... |
| rngcbas 44230 | Set of objects of the cate... |
| rngchomfval 44231 | Set of arrows of the categ... |
| rngchom 44232 | Set of arrows of the categ... |
| elrngchom 44233 | A morphism of non-unital r... |
| rngchomfeqhom 44234 | The functionalized Hom-set... |
| rngccofval 44235 | Composition in the categor... |
| rngcco 44236 | Composition in the categor... |
| dfrngc2 44237 | Alternate definition of th... |
| rnghmsscmap2 44238 | The non-unital ring homomo... |
| rnghmsscmap 44239 | The non-unital ring homomo... |
| rnghmsubcsetclem1 44240 | Lemma 1 for ~ rnghmsubcset... |
| rnghmsubcsetclem2 44241 | Lemma 2 for ~ rnghmsubcset... |
| rnghmsubcsetc 44242 | The non-unital ring homomo... |
| rngccat 44243 | The category of non-unital... |
| rngcid 44244 | The identity arrow in the ... |
| rngcsect 44245 | A section in the category ... |
| rngcinv 44246 | An inverse in the category... |
| rngciso 44247 | An isomorphism in the cate... |
| rngcbasALTV 44248 | Set of objects of the cate... |
| rngchomfvalALTV 44249 | Set of arrows of the categ... |
| rngchomALTV 44250 | Set of arrows of the categ... |
| elrngchomALTV 44251 | A morphism of non-unital r... |
| rngccofvalALTV 44252 | Composition in the categor... |
| rngccoALTV 44253 | Composition in the categor... |
| rngccatidALTV 44254 | Lemma for ~ rngccatALTV . ... |
| rngccatALTV 44255 | The category of non-unital... |
| rngcidALTV 44256 | The identity arrow in the ... |
| rngcsectALTV 44257 | A section in the category ... |
| rngcinvALTV 44258 | An inverse in the category... |
| rngcisoALTV 44259 | An isomorphism in the cate... |
| rngchomffvalALTV 44260 | The value of the functiona... |
| rngchomrnghmresALTV 44261 | The value of the functiona... |
| rngcifuestrc 44262 | The "inclusion functor" fr... |
| funcrngcsetc 44263 | The "natural forgetful fun... |
| funcrngcsetcALT 44264 | Alternate proof of ~ funcr... |
| zrinitorngc 44265 | The zero ring is an initia... |
| zrtermorngc 44266 | The zero ring is a termina... |
| zrzeroorngc 44267 | The zero ring is a zero ob... |
| ringcvalALTV 44272 | Value of the category of r... |
| ringcval 44273 | Value of the category of u... |
| rhmresfn 44274 | The class of unital ring h... |
| rhmresel 44275 | An element of the unital r... |
| ringcbas 44276 | Set of objects of the cate... |
| ringchomfval 44277 | Set of arrows of the categ... |
| ringchom 44278 | Set of arrows of the categ... |
| elringchom 44279 | A morphism of unital rings... |
| ringchomfeqhom 44280 | The functionalized Hom-set... |
| ringccofval 44281 | Composition in the categor... |
| ringcco 44282 | Composition in the categor... |
| dfringc2 44283 | Alternate definition of th... |
| rhmsscmap2 44284 | The unital ring homomorphi... |
| rhmsscmap 44285 | The unital ring homomorphi... |
| rhmsubcsetclem1 44286 | Lemma 1 for ~ rhmsubcsetc ... |
| rhmsubcsetclem2 44287 | Lemma 2 for ~ rhmsubcsetc ... |
| rhmsubcsetc 44288 | The unital ring homomorphi... |
| ringccat 44289 | The category of unital rin... |
| ringcid 44290 | The identity arrow in the ... |
| rhmsscrnghm 44291 | The unital ring homomorphi... |
| rhmsubcrngclem1 44292 | Lemma 1 for ~ rhmsubcrngc ... |
| rhmsubcrngclem2 44293 | Lemma 2 for ~ rhmsubcrngc ... |
| rhmsubcrngc 44294 | The unital ring homomorphi... |
| rngcresringcat 44295 | The restriction of the cat... |
| ringcsect 44296 | A section in the category ... |
| ringcinv 44297 | An inverse in the category... |
| ringciso 44298 | An isomorphism in the cate... |
| ringcbasbas 44299 | An element of the base set... |
| funcringcsetc 44300 | The "natural forgetful fun... |
| funcringcsetcALTV2lem1 44301 | Lemma 1 for ~ funcringcset... |
| funcringcsetcALTV2lem2 44302 | Lemma 2 for ~ funcringcset... |
| funcringcsetcALTV2lem3 44303 | Lemma 3 for ~ funcringcset... |
| funcringcsetcALTV2lem4 44304 | Lemma 4 for ~ funcringcset... |
| funcringcsetcALTV2lem5 44305 | Lemma 5 for ~ funcringcset... |
| funcringcsetcALTV2lem6 44306 | Lemma 6 for ~ funcringcset... |
| funcringcsetcALTV2lem7 44307 | Lemma 7 for ~ funcringcset... |
| funcringcsetcALTV2lem8 44308 | Lemma 8 for ~ funcringcset... |
| funcringcsetcALTV2lem9 44309 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV2 44310 | The "natural forgetful fun... |
| ringcbasALTV 44311 | Set of objects of the cate... |
| ringchomfvalALTV 44312 | Set of arrows of the categ... |
| ringchomALTV 44313 | Set of arrows of the categ... |
| elringchomALTV 44314 | A morphism of rings is a f... |
| ringccofvalALTV 44315 | Composition in the categor... |
| ringccoALTV 44316 | Composition in the categor... |
| ringccatidALTV 44317 | Lemma for ~ ringccatALTV .... |
| ringccatALTV 44318 | The category of rings is a... |
| ringcidALTV 44319 | The identity arrow in the ... |
| ringcsectALTV 44320 | A section in the category ... |
| ringcinvALTV 44321 | An inverse in the category... |
| ringcisoALTV 44322 | An isomorphism in the cate... |
| ringcbasbasALTV 44323 | An element of the base set... |
| funcringcsetclem1ALTV 44324 | Lemma 1 for ~ funcringcset... |
| funcringcsetclem2ALTV 44325 | Lemma 2 for ~ funcringcset... |
| funcringcsetclem3ALTV 44326 | Lemma 3 for ~ funcringcset... |
| funcringcsetclem4ALTV 44327 | Lemma 4 for ~ funcringcset... |
| funcringcsetclem5ALTV 44328 | Lemma 5 for ~ funcringcset... |
| funcringcsetclem6ALTV 44329 | Lemma 6 for ~ funcringcset... |
| funcringcsetclem7ALTV 44330 | Lemma 7 for ~ funcringcset... |
| funcringcsetclem8ALTV 44331 | Lemma 8 for ~ funcringcset... |
| funcringcsetclem9ALTV 44332 | Lemma 9 for ~ funcringcset... |
| funcringcsetcALTV 44333 | The "natural forgetful fun... |
| irinitoringc 44334 | The ring of integers is an... |
| zrtermoringc 44335 | The zero ring is a termina... |
| zrninitoringc 44336 | The zero ring is not an in... |
| nzerooringczr 44337 | There is no zero object in... |
| srhmsubclem1 44338 | Lemma 1 for ~ srhmsubc . ... |
| srhmsubclem2 44339 | Lemma 2 for ~ srhmsubc . ... |
| srhmsubclem3 44340 | Lemma 3 for ~ srhmsubc . ... |
| srhmsubc 44341 | According to ~ df-subc , t... |
| sringcat 44342 | The restriction of the cat... |
| crhmsubc 44343 | According to ~ df-subc , t... |
| cringcat 44344 | The restriction of the cat... |
| drhmsubc 44345 | According to ~ df-subc , t... |
| drngcat 44346 | The restriction of the cat... |
| fldcat 44347 | The restriction of the cat... |
| fldc 44348 | The restriction of the cat... |
| fldhmsubc 44349 | According to ~ df-subc , t... |
| rngcrescrhm 44350 | The category of non-unital... |
| rhmsubclem1 44351 | Lemma 1 for ~ rhmsubc . (... |
| rhmsubclem2 44352 | Lemma 2 for ~ rhmsubc . (... |
| rhmsubclem3 44353 | Lemma 3 for ~ rhmsubc . (... |
| rhmsubclem4 44354 | Lemma 4 for ~ rhmsubc . (... |
| rhmsubc 44355 | According to ~ df-subc , t... |
| rhmsubccat 44356 | The restriction of the cat... |
| srhmsubcALTVlem1 44357 | Lemma 1 for ~ srhmsubcALTV... |
| srhmsubcALTVlem2 44358 | Lemma 2 for ~ srhmsubcALTV... |
| srhmsubcALTV 44359 | According to ~ df-subc , t... |
| sringcatALTV 44360 | The restriction of the cat... |
| crhmsubcALTV 44361 | According to ~ df-subc , t... |
| cringcatALTV 44362 | The restriction of the cat... |
| drhmsubcALTV 44363 | According to ~ df-subc , t... |
| drngcatALTV 44364 | The restriction of the cat... |
| fldcatALTV 44365 | The restriction of the cat... |
| fldcALTV 44366 | The restriction of the cat... |
| fldhmsubcALTV 44367 | According to ~ df-subc , t... |
| rngcrescrhmALTV 44368 | The category of non-unital... |
| rhmsubcALTVlem1 44369 | Lemma 1 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem2 44370 | Lemma 2 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem3 44371 | Lemma 3 for ~ rhmsubcALTV ... |
| rhmsubcALTVlem4 44372 | Lemma 4 for ~ rhmsubcALTV ... |
| rhmsubcALTV 44373 | According to ~ df-subc , t... |
| rhmsubcALTVcat 44374 | The restriction of the cat... |
| opeliun2xp 44375 | Membership of an ordered p... |
| eliunxp2 44376 | Membership in a union of C... |
| mpomptx2 44377 | Express a two-argument fun... |
| cbvmpox2 44378 | Rule to change the bound v... |
| dmmpossx2 44379 | The domain of a mapping is... |
| mpoexxg2 44380 | Existence of an operation ... |
| ovmpordxf 44381 | Value of an operation give... |
| ovmpordx 44382 | Value of an operation give... |
| ovmpox2 44383 | The value of an operation ... |
| fdmdifeqresdif 44384 | The restriction of a condi... |
| offvalfv 44385 | The function operation exp... |
| ofaddmndmap 44386 | The function operation app... |
| mapsnop 44387 | A singleton of an ordered ... |
| mapprop 44388 | An unordered pair containi... |
| ztprmneprm 44389 | A prime is not an integer ... |
| 2t6m3t4e0 44390 | 2 times 6 minus 3 times 4 ... |
| ssnn0ssfz 44391 | For any finite subset of `... |
| nn0sumltlt 44392 | If the sum of two nonnegat... |
| bcpascm1 44393 | Pascal's rule for the bino... |
| altgsumbc 44394 | The sum of binomial coeffi... |
| altgsumbcALT 44395 | Alternate proof of ~ altgs... |
| zlmodzxzlmod 44396 | The ` ZZ `-module ` ZZ X. ... |
| zlmodzxzel 44397 | An element of the (base se... |
| zlmodzxz0 44398 | The ` 0 ` of the ` ZZ `-mo... |
| zlmodzxzscm 44399 | The scalar multiplication ... |
| zlmodzxzadd 44400 | The addition of the ` ZZ `... |
| zlmodzxzsubm 44401 | The subtraction of the ` Z... |
| zlmodzxzsub 44402 | The subtraction of the ` Z... |
| mgpsumunsn 44403 | Extract a summand/factor f... |
| mgpsumz 44404 | If the group sum for the m... |
| mgpsumn 44405 | If the group sum for the m... |
| exple2lt6 44406 | A nonnegative integer to t... |
| pgrple2abl 44407 | Every symmetric group on a... |
| pgrpgt2nabl 44408 | Every symmetric group on a... |
| invginvrid 44409 | Identity for a multiplicat... |
| rmsupp0 44410 | The support of a mapping o... |
| domnmsuppn0 44411 | The support of a mapping o... |
| rmsuppss 44412 | The support of a mapping o... |
| mndpsuppss 44413 | The support of a mapping o... |
| scmsuppss 44414 | The support of a mapping o... |
| rmsuppfi 44415 | The support of a mapping o... |
| rmfsupp 44416 | A mapping of a multiplicat... |
| mndpsuppfi 44417 | The support of a mapping o... |
| mndpfsupp 44418 | A mapping of a scalar mult... |
| scmsuppfi 44419 | The support of a mapping o... |
| scmfsupp 44420 | A mapping of a scalar mult... |
| suppmptcfin 44421 | The support of a mapping w... |
| mptcfsupp 44422 | A mapping with value 0 exc... |
| fsuppmptdmf 44423 | A mapping with a finite do... |
| lmodvsmdi 44424 | Multiple distributive law ... |
| gsumlsscl 44425 | Closure of a group sum in ... |
| ascl1 44426 | The scalar 1 embedded into... |
| assaascl0 44427 | The scalar 0 embedded into... |
| assaascl1 44428 | The scalar 1 embedded into... |
| ply1vr1smo 44429 | The variable in a polynomi... |
| ply1ass23l 44430 | Associative identity with ... |
| ply1sclrmsm 44431 | The ring multiplication of... |
| coe1id 44432 | Coefficient vector of the ... |
| coe1sclmulval 44433 | The value of the coefficie... |
| ply1mulgsumlem1 44434 | Lemma 1 for ~ ply1mulgsum ... |
| ply1mulgsumlem2 44435 | Lemma 2 for ~ ply1mulgsum ... |
| ply1mulgsumlem3 44436 | Lemma 3 for ~ ply1mulgsum ... |
| ply1mulgsumlem4 44437 | Lemma 4 for ~ ply1mulgsum ... |
| ply1mulgsum 44438 | The product of two polynom... |
| evl1at0 44439 | Polynomial evaluation for ... |
| evl1at1 44440 | Polynomial evaluation for ... |
| linply1 44441 | A term of the form ` x - C... |
| lineval 44442 | A term of the form ` x - C... |
| zringsubgval 44443 | Subtraction in the ring of... |
| linevalexample 44444 | The polynomial ` x - 3 ` o... |
| dmatALTval 44449 | The algebra of ` N ` x ` N... |
| dmatALTbas 44450 | The base set of the algebr... |
| dmatALTbasel 44451 | An element of the base set... |
| dmatbas 44452 | The set of all ` N ` x ` N... |
| lincop 44457 | A linear combination as op... |
| lincval 44458 | The value of a linear comb... |
| dflinc2 44459 | Alternative definition of ... |
| lcoop 44460 | A linear combination as op... |
| lcoval 44461 | The value of a linear comb... |
| lincfsuppcl 44462 | A linear combination of ve... |
| linccl 44463 | A linear combination of ve... |
| lincval0 44464 | The value of an empty line... |
| lincvalsng 44465 | The linear combination ove... |
| lincvalsn 44466 | The linear combination ove... |
| lincvalpr 44467 | The linear combination ove... |
| lincval1 44468 | The linear combination ove... |
| lcosn0 44469 | Properties of a linear com... |
| lincvalsc0 44470 | The linear combination whe... |
| lcoc0 44471 | Properties of a linear com... |
| linc0scn0 44472 | If a set contains the zero... |
| lincdifsn 44473 | A vector is a linear combi... |
| linc1 44474 | A vector is a linear combi... |
| lincellss 44475 | A linear combination of a ... |
| lco0 44476 | The set of empty linear co... |
| lcoel0 44477 | The zero vector is always ... |
| lincsum 44478 | The sum of two linear comb... |
| lincscm 44479 | A linear combinations mult... |
| lincsumcl 44480 | The sum of two linear comb... |
| lincscmcl 44481 | The multiplication of a li... |
| lincsumscmcl 44482 | The sum of a linear combin... |
| lincolss 44483 | According to the statement... |
| ellcoellss 44484 | Every linear combination o... |
| lcoss 44485 | A set of vectors of a modu... |
| lspsslco 44486 | Lemma for ~ lspeqlco . (C... |
| lcosslsp 44487 | Lemma for ~ lspeqlco . (C... |
| lspeqlco 44488 | Equivalence of a _span_ of... |
| rellininds 44492 | The class defining the rel... |
| linindsv 44494 | The classes of the module ... |
| islininds 44495 | The property of being a li... |
| linindsi 44496 | The implications of being ... |
| linindslinci 44497 | The implications of being ... |
| islinindfis 44498 | The property of being a li... |
| islinindfiss 44499 | The property of being a li... |
| linindscl 44500 | A linearly independent set... |
| lindepsnlininds 44501 | A linearly dependent subse... |
| islindeps 44502 | The property of being a li... |
| lincext1 44503 | Property 1 of an extension... |
| lincext2 44504 | Property 2 of an extension... |
| lincext3 44505 | Property 3 of an extension... |
| lindslinindsimp1 44506 | Implication 1 for ~ lindsl... |
| lindslinindimp2lem1 44507 | Lemma 1 for ~ lindslininds... |
| lindslinindimp2lem2 44508 | Lemma 2 for ~ lindslininds... |
| lindslinindimp2lem3 44509 | Lemma 3 for ~ lindslininds... |
| lindslinindimp2lem4 44510 | Lemma 4 for ~ lindslininds... |
| lindslinindsimp2lem5 44511 | Lemma 5 for ~ lindslininds... |
| lindslinindsimp2 44512 | Implication 2 for ~ lindsl... |
| lindslininds 44513 | Equivalence of definitions... |
| linds0 44514 | The empty set is always a ... |
| el0ldep 44515 | A set containing the zero ... |
| el0ldepsnzr 44516 | A set containing the zero ... |
| lindsrng01 44517 | Any subset of a module is ... |
| lindszr 44518 | Any subset of a module ove... |
| snlindsntorlem 44519 | Lemma for ~ snlindsntor . ... |
| snlindsntor 44520 | A singleton is linearly in... |
| ldepsprlem 44521 | Lemma for ~ ldepspr . (Co... |
| ldepspr 44522 | If a vector is a scalar mu... |
| lincresunit3lem3 44523 | Lemma 3 for ~ lincresunit3... |
| lincresunitlem1 44524 | Lemma 1 for properties of ... |
| lincresunitlem2 44525 | Lemma for properties of a ... |
| lincresunit1 44526 | Property 1 of a specially ... |
| lincresunit2 44527 | Property 2 of a specially ... |
| lincresunit3lem1 44528 | Lemma 1 for ~ lincresunit3... |
| lincresunit3lem2 44529 | Lemma 2 for ~ lincresunit3... |
| lincresunit3 44530 | Property 3 of a specially ... |
| lincreslvec3 44531 | Property 3 of a specially ... |
| islindeps2 44532 | Conditions for being a lin... |
| islininds2 44533 | Implication of being a lin... |
| isldepslvec2 44534 | Alternative definition of ... |
| lindssnlvec 44535 | A singleton not containing... |
| lmod1lem1 44536 | Lemma 1 for ~ lmod1 . (Co... |
| lmod1lem2 44537 | Lemma 2 for ~ lmod1 . (Co... |
| lmod1lem3 44538 | Lemma 3 for ~ lmod1 . (Co... |
| lmod1lem4 44539 | Lemma 4 for ~ lmod1 . (Co... |
| lmod1lem5 44540 | Lemma 5 for ~ lmod1 . (Co... |
| lmod1 44541 | The (smallest) structure r... |
| lmod1zr 44542 | The (smallest) structure r... |
| lmod1zrnlvec 44543 | There is a (left) module (... |
| lmodn0 44544 | Left modules exist. (Cont... |
| zlmodzxzequa 44545 | Example of an equation wit... |
| zlmodzxznm 44546 | Example of a linearly depe... |
| zlmodzxzldeplem 44547 | A and B are not equal. (C... |
| zlmodzxzequap 44548 | Example of an equation wit... |
| zlmodzxzldeplem1 44549 | Lemma 1 for ~ zlmodzxzldep... |
| zlmodzxzldeplem2 44550 | Lemma 2 for ~ zlmodzxzldep... |
| zlmodzxzldeplem3 44551 | Lemma 3 for ~ zlmodzxzldep... |
| zlmodzxzldeplem4 44552 | Lemma 4 for ~ zlmodzxzldep... |
| zlmodzxzldep 44553 | { A , B } is a linearly de... |
| ldepsnlinclem1 44554 | Lemma 1 for ~ ldepsnlinc .... |
| ldepsnlinclem2 44555 | Lemma 2 for ~ ldepsnlinc .... |
| lvecpsslmod 44556 | The class of all (left) ve... |
| ldepsnlinc 44557 | The reverse implication of... |
| ldepslinc 44558 | For (left) vector spaces, ... |
| suppdm 44559 | If the range of a function... |
| eluz2cnn0n1 44560 | An integer greater than 1 ... |
| divge1b 44561 | The ratio of a real number... |
| divgt1b 44562 | The ratio of a real number... |
| ltsubaddb 44563 | Equivalence for the "less ... |
| ltsubsubb 44564 | Equivalence for the "less ... |
| ltsubadd2b 44565 | Equivalence for the "less ... |
| divsub1dir 44566 | Distribution of division o... |
| expnegico01 44567 | An integer greater than 1 ... |
| elfzolborelfzop1 44568 | An element of a half-open ... |
| pw2m1lepw2m1 44569 | 2 to the power of a positi... |
| zgtp1leeq 44570 | If an integer is between a... |
| flsubz 44571 | An integer can be moved in... |
| fldivmod 44572 | Expressing the floor of a ... |
| mod0mul 44573 | If an integer is 0 modulo ... |
| modn0mul 44574 | If an integer is not 0 mod... |
| m1modmmod 44575 | An integer decreased by 1 ... |
| difmodm1lt 44576 | The difference between an ... |
| nn0onn0ex 44577 | For each odd nonnegative i... |
| nn0enn0ex 44578 | For each even nonnegative ... |
| nnennex 44579 | For each even positive int... |
| nneop 44580 | A positive integer is even... |
| nneom 44581 | A positive integer is even... |
| nn0eo 44582 | A nonnegative integer is e... |
| nnpw2even 44583 | 2 to the power of a positi... |
| zefldiv2 44584 | The floor of an even integ... |
| zofldiv2 44585 | The floor of an odd intege... |
| nn0ofldiv2 44586 | The floor of an odd nonneg... |
| flnn0div2ge 44587 | The floor of a positive in... |
| flnn0ohalf 44588 | The floor of the half of a... |
| logcxp0 44589 | Logarithm of a complex pow... |
| regt1loggt0 44590 | The natural logarithm for ... |
| fdivval 44593 | The quotient of two functi... |
| fdivmpt 44594 | The quotient of two functi... |
| fdivmptf 44595 | The quotient of two functi... |
| refdivmptf 44596 | The quotient of two functi... |
| fdivpm 44597 | The quotient of two functi... |
| refdivpm 44598 | The quotient of two functi... |
| fdivmptfv 44599 | The function value of a qu... |
| refdivmptfv 44600 | The function value of a qu... |
| bigoval 44603 | Set of functions of order ... |
| elbigofrcl 44604 | Reverse closure of the "bi... |
| elbigo 44605 | Properties of a function o... |
| elbigo2 44606 | Properties of a function o... |
| elbigo2r 44607 | Sufficient condition for a... |
| elbigof 44608 | A function of order G(x) i... |
| elbigodm 44609 | The domain of a function o... |
| elbigoimp 44610 | The defining property of a... |
| elbigolo1 44611 | A function (into the posit... |
| rege1logbrege0 44612 | The general logarithm, wit... |
| rege1logbzge0 44613 | The general logarithm, wit... |
| fllogbd 44614 | A real number is between t... |
| relogbmulbexp 44615 | The logarithm of the produ... |
| relogbdivb 44616 | The logarithm of the quoti... |
| logbge0b 44617 | The logarithm of a number ... |
| logblt1b 44618 | The logarithm of a number ... |
| fldivexpfllog2 44619 | The floor of a positive re... |
| nnlog2ge0lt1 44620 | A positive integer is 1 if... |
| logbpw2m1 44621 | The floor of the binary lo... |
| fllog2 44622 | The floor of the binary lo... |
| blenval 44625 | The binary length of an in... |
| blen0 44626 | The binary length of 0. (... |
| blenn0 44627 | The binary length of a "nu... |
| blenre 44628 | The binary length of a pos... |
| blennn 44629 | The binary length of a pos... |
| blennnelnn 44630 | The binary length of a pos... |
| blennn0elnn 44631 | The binary length of a non... |
| blenpw2 44632 | The binary length of a pow... |
| blenpw2m1 44633 | The binary length of a pow... |
| nnpw2blen 44634 | A positive integer is betw... |
| nnpw2blenfzo 44635 | A positive integer is betw... |
| nnpw2blenfzo2 44636 | A positive integer is eith... |
| nnpw2pmod 44637 | Every positive integer can... |
| blen1 44638 | The binary length of 1. (... |
| blen2 44639 | The binary length of 2. (... |
| nnpw2p 44640 | Every positive integer can... |
| nnpw2pb 44641 | A number is a positive int... |
| blen1b 44642 | The binary length of a non... |
| blennnt2 44643 | The binary length of a pos... |
| nnolog2flm1 44644 | The floor of the binary lo... |
| blennn0em1 44645 | The binary length of the h... |
| blennngt2o2 44646 | The binary length of an od... |
| blengt1fldiv2p1 44647 | The binary length of an in... |
| blennn0e2 44648 | The binary length of an ev... |
| digfval 44651 | Operation to obtain the ` ... |
| digval 44652 | The ` K ` th digit of a no... |
| digvalnn0 44653 | The ` K ` th digit of a no... |
| nn0digval 44654 | The ` K ` th digit of a no... |
| dignn0fr 44655 | The digits of the fraction... |
| dignn0ldlem 44656 | Lemma for ~ dignnld . (Co... |
| dignnld 44657 | The leading digits of a po... |
| dig2nn0ld 44658 | The leading digits of a po... |
| dig2nn1st 44659 | The first (relevant) digit... |
| dig0 44660 | All digits of 0 are 0. (C... |
| digexp 44661 | The ` K ` th digit of a po... |
| dig1 44662 | All but one digits of 1 ar... |
| 0dig1 44663 | The ` 0 ` th digit of 1 is... |
| 0dig2pr01 44664 | The integers 0 and 1 corre... |
| dig2nn0 44665 | A digit of a nonnegative i... |
| 0dig2nn0e 44666 | The last bit of an even in... |
| 0dig2nn0o 44667 | The last bit of an odd int... |
| dig2bits 44668 | The ` K ` th digit of a no... |
| dignn0flhalflem1 44669 | Lemma 1 for ~ dignn0flhalf... |
| dignn0flhalflem2 44670 | Lemma 2 for ~ dignn0flhalf... |
| dignn0ehalf 44671 | The digits of the half of ... |
| dignn0flhalf 44672 | The digits of the rounded ... |
| nn0sumshdiglemA 44673 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglemB 44674 | Lemma for ~ nn0sumshdig (i... |
| nn0sumshdiglem1 44675 | Lemma 1 for ~ nn0sumshdig ... |
| nn0sumshdiglem2 44676 | Lemma 2 for ~ nn0sumshdig ... |
| nn0sumshdig 44677 | A nonnegative integer can ... |
| nn0mulfsum 44678 | Trivial algorithm to calcu... |
| nn0mullong 44679 | Standard algorithm (also k... |
| fv1prop 44680 | The function value of unor... |
| fv2prop 44681 | The function value of unor... |
| submuladdmuld 44682 | Transformation of a sum of... |
| affinecomb1 44683 | Combination of two real af... |
| affinecomb2 44684 | Combination of two real af... |
| affineid 44685 | Identity of an affine comb... |
| 1subrec1sub 44686 | Subtract the reciprocal of... |
| resum2sqcl 44687 | The sum of two squares of ... |
| resum2sqgt0 44688 | The sum of the square of a... |
| resum2sqrp 44689 | The sum of the square of a... |
| resum2sqorgt0 44690 | The sum of the square of t... |
| reorelicc 44691 | Membership in and outside ... |
| rrx2pxel 44692 | The x-coordinate of a poin... |
| rrx2pyel 44693 | The y-coordinate of a poin... |
| prelrrx2 44694 | An unordered pair of order... |
| prelrrx2b 44695 | An unordered pair of order... |
| rrx2pnecoorneor 44696 | If two different points ` ... |
| rrx2pnedifcoorneor 44697 | If two different points ` ... |
| rrx2pnedifcoorneorr 44698 | If two different points ` ... |
| rrx2xpref1o 44699 | There is a bijection betwe... |
| rrx2xpreen 44700 | The set of points in the t... |
| rrx2plord 44701 | The lexicographical orderi... |
| rrx2plord1 44702 | The lexicographical orderi... |
| rrx2plord2 44703 | The lexicographical orderi... |
| rrx2plordisom 44704 | The set of points in the t... |
| rrx2plordso 44705 | The lexicographical orderi... |
| ehl2eudisval0 44706 | The Euclidean distance of ... |
| ehl2eudis0lt 44707 | An upper bound of the Eucl... |
| lines 44712 | The lines passing through ... |
| line 44713 | The line passing through t... |
| rrxlines 44714 | Definition of lines passin... |
| rrxline 44715 | The line passing through t... |
| rrxlinesc 44716 | Definition of lines passin... |
| rrxlinec 44717 | The line passing through t... |
| eenglngeehlnmlem1 44718 | Lemma 1 for ~ eenglngeehln... |
| eenglngeehlnmlem2 44719 | Lemma 2 for ~ eenglngeehln... |
| eenglngeehlnm 44720 | The line definition in the... |
| rrx2line 44721 | The line passing through t... |
| rrx2vlinest 44722 | The vertical line passing ... |
| rrx2linest 44723 | The line passing through t... |
| rrx2linesl 44724 | The line passing through t... |
| rrx2linest2 44725 | The line passing through t... |
| elrrx2linest2 44726 | The line passing through t... |
| spheres 44727 | The spheres for given cent... |
| sphere 44728 | A sphere with center ` X `... |
| rrxsphere 44729 | The sphere with center ` M... |
| 2sphere 44730 | The sphere with center ` M... |
| 2sphere0 44731 | The sphere around the orig... |
| line2ylem 44732 | Lemma for ~ line2y . This... |
| line2 44733 | Example for a line ` G ` p... |
| line2xlem 44734 | Lemma for ~ line2x . This... |
| line2x 44735 | Example for a horizontal l... |
| line2y 44736 | Example for a vertical lin... |
| itsclc0lem1 44737 | Lemma for theorems about i... |
| itsclc0lem2 44738 | Lemma for theorems about i... |
| itsclc0lem3 44739 | Lemma for theorems about i... |
| itscnhlc0yqe 44740 | Lemma for ~ itsclc0 . Qua... |
| itschlc0yqe 44741 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqe 44742 | Lemma for ~ itsclc0 . Qua... |
| itsclc0yqsollem1 44743 | Lemma 1 for ~ itsclc0yqsol... |
| itsclc0yqsollem2 44744 | Lemma 2 for ~ itsclc0yqsol... |
| itsclc0yqsol 44745 | Lemma for ~ itsclc0 . Sol... |
| itscnhlc0xyqsol 44746 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol1 44747 | Lemma for ~ itsclc0 . Sol... |
| itschlc0xyqsol 44748 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsol 44749 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolr 44750 | Lemma for ~ itsclc0 . Sol... |
| itsclc0xyqsolb 44751 | Lemma for ~ itsclc0 . Sol... |
| itsclc0 44752 | The intersection points of... |
| itsclc0b 44753 | The intersection points of... |
| itsclinecirc0 44754 | The intersection points of... |
| itsclinecirc0b 44755 | The intersection points of... |
| itsclinecirc0in 44756 | The intersection points of... |
| itsclquadb 44757 | Quadratic equation for the... |
| itsclquadeu 44758 | Quadratic equation for the... |
| 2itscplem1 44759 | Lemma 1 for ~ 2itscp . (C... |
| 2itscplem2 44760 | Lemma 2 for ~ 2itscp . (C... |
| 2itscplem3 44761 | Lemma D for ~ 2itscp . (C... |
| 2itscp 44762 | A condition for a quadrati... |
| itscnhlinecirc02plem1 44763 | Lemma 1 for ~ itscnhlineci... |
| itscnhlinecirc02plem2 44764 | Lemma 2 for ~ itscnhlineci... |
| itscnhlinecirc02plem3 44765 | Lemma 3 for ~ itscnhlineci... |
| itscnhlinecirc02p 44766 | Intersection of a nonhoriz... |
| inlinecirc02plem 44767 | Lemma for ~ inlinecirc02p ... |
| inlinecirc02p 44768 | Intersection of a line wit... |
| inlinecirc02preu 44769 | Intersection of a line wit... |
| nfintd 44770 | Bound-variable hypothesis ... |
| nfiund 44771 | Bound-variable hypothesis ... |
| nfiundg 44772 | Bound-variable hypothesis ... |
| iunord 44773 | The indexed union of a col... |
| iunordi 44774 | The indexed union of a col... |
| spd 44775 | Specialization deduction, ... |
| spcdvw 44776 | A version of ~ spcdv where... |
| tfis2d 44777 | Transfinite Induction Sche... |
| bnd2d 44778 | Deduction form of ~ bnd2 .... |
| dffun3f 44779 | Alternate definition of fu... |
| setrecseq 44782 | Equality theorem for set r... |
| nfsetrecs 44783 | Bound-variable hypothesis ... |
| setrec1lem1 44784 | Lemma for ~ setrec1 . Thi... |
| setrec1lem2 44785 | Lemma for ~ setrec1 . If ... |
| setrec1lem3 44786 | Lemma for ~ setrec1 . If ... |
| setrec1lem4 44787 | Lemma for ~ setrec1 . If ... |
| setrec1 44788 | This is the first of two f... |
| setrec2fun 44789 | This is the second of two ... |
| setrec2lem1 44790 | Lemma for ~ setrec2 . The... |
| setrec2lem2 44791 | Lemma for ~ setrec2 . The... |
| setrec2 44792 | This is the second of two ... |
| setrec2v 44793 | Version of ~ setrec2 with ... |
| setis 44794 | Version of ~ setrec2 expre... |
| elsetrecslem 44795 | Lemma for ~ elsetrecs . A... |
| elsetrecs 44796 | A set ` A ` is an element ... |
| setrecsss 44797 | The ` setrecs ` operator r... |
| setrecsres 44798 | A recursively generated cl... |
| vsetrec 44799 | Construct ` _V ` using set... |
| 0setrec 44800 | If a function sends the em... |
| onsetreclem1 44801 | Lemma for ~ onsetrec . (C... |
| onsetreclem2 44802 | Lemma for ~ onsetrec . (C... |
| onsetreclem3 44803 | Lemma for ~ onsetrec . (C... |
| onsetrec 44804 | Construct ` On ` using set... |
| elpglem1 44807 | Lemma for ~ elpg . (Contr... |
| elpglem2 44808 | Lemma for ~ elpg . (Contr... |
| elpglem3 44809 | Lemma for ~ elpg . (Contr... |
| elpg 44810 | Membership in the class of... |
| sbidd 44811 | An identity theorem for su... |
| sbidd-misc 44812 | An identity theorem for su... |
| gte-lte 44817 | Simple relationship betwee... |
| gt-lt 44818 | Simple relationship betwee... |
| gte-lteh 44819 | Relationship between ` <_ ... |
| gt-lth 44820 | Relationship between ` < `... |
| ex-gt 44821 | Simple example of ` > ` , ... |
| ex-gte 44822 | Simple example of ` >_ ` ,... |
| sinhval-named 44829 | Value of the named sinh fu... |
| coshval-named 44830 | Value of the named cosh fu... |
| tanhval-named 44831 | Value of the named tanh fu... |
| sinh-conventional 44832 | Conventional definition of... |
| sinhpcosh 44833 | Prove that ` ( sinh `` A )... |
| secval 44840 | Value of the secant functi... |
| cscval 44841 | Value of the cosecant func... |
| cotval 44842 | Value of the cotangent fun... |
| seccl 44843 | The closure of the secant ... |
| csccl 44844 | The closure of the cosecan... |
| cotcl 44845 | The closure of the cotange... |
| reseccl 44846 | The closure of the secant ... |
| recsccl 44847 | The closure of the cosecan... |
| recotcl 44848 | The closure of the cotange... |
| recsec 44849 | The reciprocal of secant i... |
| reccsc 44850 | The reciprocal of cosecant... |
| reccot 44851 | The reciprocal of cotangen... |
| rectan 44852 | The reciprocal of tangent ... |
| sec0 44853 | The value of the secant fu... |
| onetansqsecsq 44854 | Prove the tangent squared ... |
| cotsqcscsq 44855 | Prove the tangent squared ... |
| ifnmfalse 44856 | If A is not a member of B,... |
| logb2aval 44857 | Define the value of the ` ... |
| comraddi 44864 | Commute RHS addition. See... |
| mvlraddi 44865 | Move LHS right addition to... |
| mvrladdi 44866 | Move RHS left addition to ... |
| assraddsubi 44867 | Associate RHS addition-sub... |
| joinlmuladdmuli 44868 | Join AB+CB into (A+C) on L... |
| joinlmulsubmuld 44869 | Join AB-CB into (A-C) on L... |
| joinlmulsubmuli 44870 | Join AB-CB into (A-C) on L... |
| mvlrmuld 44871 | Move LHS right multiplicat... |
| mvlrmuli 44872 | Move LHS right multiplicat... |
| i2linesi 44873 | Solve for the intersection... |
| i2linesd 44874 | Solve for the intersection... |
| alimp-surprise 44875 | Demonstrate that when usin... |
| alimp-no-surprise 44876 | There is no "surprise" in ... |
| empty-surprise 44877 | Demonstrate that when usin... |
| empty-surprise2 44878 | "Prove" that false is true... |
| eximp-surprise 44879 | Show what implication insi... |
| eximp-surprise2 44880 | Show that "there exists" w... |
| alsconv 44885 | There is an equivalence be... |
| alsi1d 44886 | Deduction rule: Given "al... |
| alsi2d 44887 | Deduction rule: Given "al... |
| alsc1d 44888 | Deduction rule: Given "al... |
| alsc2d 44889 | Deduction rule: Given "al... |
| alscn0d 44890 | Deduction rule: Given "al... |
| alsi-no-surprise 44891 | Demonstrate that there is ... |
| 5m4e1 44892 | Prove that 5 - 4 = 1. (Co... |
| 2p2ne5 44893 | Prove that ` 2 + 2 =/= 5 `... |
| resolution 44894 | Resolution rule. This is ... |
| testable 44895 | In classical logic all wff... |
| aacllem 44896 | Lemma for other theorems a... |
| amgmwlem 44897 | Weighted version of ~ amgm... |
| amgmlemALT 44898 | Alternate proof of ~ amgml... |
| amgmw2d 44899 | Weighted arithmetic-geomet... |
| young2d 44900 | Young's inequality for ` n... |
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